tag:blogger.com,1999:blog-40809791567002363462024-03-16T11:52:38.019-07:00Things in MotionExploring all things related to electric motors, motion control and robotics.Richard Parsonshttp://www.blogger.com/profile/08331478925576296508noreply@blogger.comBlogger13125tag:blogger.com,1999:blog-4080979156700236346.post-40812838778470157552021-02-13T01:37:00.009-08:002023-06-27T22:58:45.268-07:00Basic BLDC (PMSM) efficiency and power loss estimationsElectric motors exist to convert electrical power into mechanical power as efficiently as possible. In this post, a <i>relatively </i>simple<i> </i>analytical approach will be used to estimate the efficiency and power loss of a typical brushless <a href="https://things-in-motion.blogspot.com/2018/12/why-most-hobby-grade-bldc-out-runners.html">BLDC (PMSM)</a>. This will be achieved using easily measurable motor parameters such as the winding resistance, torque constant and stator mass.<div><div><br /></div><div>But first, a short and simplified overview of power losses in brushless electric motors. </div><div><br /></div><div>
<h3 style="text-align: left;"><span style="color: blue; font-weight: normal;">Simplified overview of power losses in brushless electric motors</span></h3>
<div>
<br /></div>There are many sources of power loss in an electric motor. However, they can generally be classified into three categories; core losses, `i^2R` losses and losses related to friction.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-12meu32-xZU/XIR8VQnUKkI/AAAAAAAAVc8/8OfV-XCtzCcbjFGngNT0odV22J1-6xJvwCLcBGAs/s1600/simple%2Bmotor%2Blosses.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="577" data-original-width="1304" height="175" src="https://1.bp.blogspot.com/-12meu32-xZU/XIR8VQnUKkI/AAAAAAAAVc8/8OfV-XCtzCcbjFGngNT0odV22J1-6xJvwCLcBGAs/s400/simple%2Bmotor%2Blosses.jpg" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><div style="text-align: left;"><ul style="text-align: left;"><li><b>Core losses</b></li></ul></div><div style="text-align: left;">Core losses typically refer to hysteresis losses and eddy current losses within a motor. <b>Hysteresis losses</b> occur anywhere that there is a changing magnetisation within ferromagnetic materials such as the stator laminations and the rotor back iron. <b>Eddy current losses</b> can occur anywhere within the motor where there is a changing magnetic field with time and something conductive. These eddy currents are mostly located in the stator laminations, which experience the largest change in magnetisation but also occur in the rotor back-iron and within the rotor magnets. Core loss typically dominates at high speeds and at relatively low torque.</div><div style="text-align: left;"><ul style="text-align: left;"><li><b>`I^2R` losses</b></li></ul></div><div style="text-align: left;"><br /></div></div>
<div>
This is the power lost in the stator windings (the armature) due to the intentional flow of current from the motor controller. This loss dominates at low speeds and high torque.</div><div><ul style="text-align: left;"><li><b>Frictional losses</b></li></ul></div><div><br /></div>
<div>Fictional losses can be used to describe the power lost in the rolling elements of the motor such as the shaft bearings. It can also be used to describe friction caused by the movement of air, also often referred to as 'windage'. This windage loss may be due to a shaft-mounted cooling fan or even just due to the movement of the rotor itself. For motors that do not include an integrated shaft-mounted cooling fan, this source of loss is relatively small. Also, with the exception of very high-speed motors, the losses within well-maintained bearings also tend to be relatively small. <b>For this reason, frictional losses will be ignored in this post.</b></div><div><b><br /></b></div><div>Please keep in mind that power loss in electric motors is a very deep topic and the above descriptions of loss is a very simplified understanding. However, this basic starting point is enough to let us explore how the power loss in a brushless electric motor can be estimated from a few simple parameters.</div><div><br /></div><div><h3><span style="color: blue;">A simplified analytical model of brushless motor power loss</span></h3></div><div>
</div>
<div><br /></div><div>To use a real-world example, let's estimate the efficiency and power loss of a <a href="http://store-en.tmotor.com/goods.php?id=325">T-motor U8 pro</a>. </div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-YFZYnDawgw4/XIR-tc6O4lI/AAAAAAAAVdI/e8EddO6_xn0ngxboXV29MsvZbC3rm3CyQCLcBGAs/s1600/t-motor_u8_pro.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="357" data-original-width="854" height="166" src="https://2.bp.blogspot.com/-YFZYnDawgw4/XIR-tc6O4lI/AAAAAAAAVdI/e8EddO6_xn0ngxboXV29MsvZbC3rm3CyQCLcBGAs/s400/t-motor_u8_pro.jpg" width="400" /></a></div>
<div>
<br /></div>
<div>This motor was selected because its efficiency and power loss has already been mapped by Ben Katz over on <a href="https://build-its.blogspot.com/p/motor-characterization.html">his blog</a>. This efficiency and power loss data will be used to evaluate the accuracy of the model described below.</div><div><br /></div><div>To model the motor we need to know a few parameters:</div>
<div>
<ul>
<li><b>Winding resistance</b> = 0.19 Ohm (line-to-line winding resistance)</li>
<li><b>Torque constant</b> = 0.075 N.m/A (uses current supplied to the motor, not the motor controller)</li>
<li><b>Pole count</b> = 42 (number of magnets on the rotor)</li>
<li><b>Approximate stator mass </b>= 0.065 kg (mass of just the iron-silicon steel laminations)</li></ul>
</div>
<div>
The stator mass was estimated by extracting the volume of the stator from <a href="https://grabcad.com/library/t-motor-u8-pro-1">this great CAD model</a> of the U8 pro and multiplying it by the density of Fe-Si steel which is around 8 `text(g/)cm^3`. If you don't know the stator mass of your own motor, you can assume it is around 1/4 to 1/3 of the total motor mass.</div>
<div><br /></div><div>Let's begin by defining the core loss and the `I^2R` loss of this motor.</div><div><div><ul style="font-size: 18.72px; font-weight: 700;"><li>Core loss estimation for a brushless motor</li></ul>For simplicity, this model makes a few key assumptions regarding core loss:</div><div><ol style="text-align: left;"><li><b>The only core losses are those located in the stator laminations</b>. All other 'core losses' (within the back iron, magnets etc.) will be ignored.</li><li><b>The magnetic polarization of the stator is due only to the magnetisation provided by the permanent magnets in the roto</b>r. That is, the core loss is independent of the current and torque output of the motor. This is a fairly safe assumption as my own testing in FEA software suggests the core loss is only slightly impacted by stator current.</li><li><b>The magnetic polarisation within the stator is constant everywhere, has a maximum of 1.5 T and a fundamental frequency equal to that of the motor's electrical frequency</b>. In a real motor, the flux density at the surface of the stator (at the teeth) tends to be larger than in the yoke. Higher-order harmonics and current ripple also lead to additional core losses. However, these extra losses are also relatively small in typical brushless motors and so can be ignored.</li></ol></div><div>All these assumptions mean that the core loss can be estimated purely from the speed at which the motor spins and the number of poles it has. The actual core loss in our motors stator can then be estimated by fitting <a href="https://www.jfe-steel.co.jp/en/products/electrical/catalog/f1e-001.pdf" target="_blank">published core loss data</a> for iron silicon steel laminations. Using this fitting, and the motor parameters, we can produce a contour plot of core loss for every speed and torque value.</div></div><div><br /></div><div>Using this model we can predict the core loss in the T-Motor U8 Pro and present it as a contour plot.</div><div><div class="separator" style="clear: both; text-align: center;"><br /></div>
<iframe frameborder="0" height="400" scrolling="no" src="//plotly.com/~rparsons01/50.embed" width="550"></iframe></div><div>As expected from our simple model, the core loss increases with motor speed and is not impacted by the torque output.</div><div><br /></div><div>Note that there is also a relatively easy way to check if this model is accurate for your motor using the steps below:</div><div><ol style="text-align: left;"><li>Measure the no-load power draw (i.e. power draw of the motor when it is not connected to anything) of your motor controller when it is spinning the motor at zero RPM, max RPM and a few points in between. </li><li>Subtract the zero RPM power draw from the other values to mostly eliminate the motor controller switching losses.</li><li>Fit the data to estimate the core loss for all points in between your measured values. </li></ol><div>You can find an example spreadsheet for this process <a href="https://docs.google.com/spreadsheets/d/1997nJuWmfzx7_oqjynEMi7Ns_LAMfhQg_VOV1f-7bA8/edit?usp=sharing">here</a>. This process works because almost all the power loss at high speed and no load is due to the losses in the core of the motor.</div></div><div>
<div class="separator" style="clear: both; text-align: center;"><br /></div></div><div class="separator" style="clear: both; text-align: center;"><ul style="text-align: left;"><li><span style="font-size: 18.72px;"><b>`I^2R` loss estimation for a brushless motor</b></span></li></ul></div><div><div>To calculate the resistive losses of the motor we must know the current required to produce a given torque. This is defined by:</div><div><br />$$ Current (Ampere) = \frac{Torque (Newton.meter)}{Torque \,constant (Newton.meter/Ampere)}$$<br /><br />Remember that the current referenced here is the line-to-line current supplied to the motor and <a href="https://things-in-motion.blogspot.com/p/blog-page.html">not the current supplied to the motor controller</a>. You can <a href="https://things-in-motion.blogspot.com/2018/12/how-to-estimate-torque-of-bldc-pmsm.html">estimate the torque output of your motor</a> from its velocity constant. With the current and winding resistance known the power loss for each torque value and velocity can also be mapped.<br /><div class="separator" style="clear: both; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><br /></div>
<iframe frameborder="0" height="400" scrolling="no" src="//plotly.com/~rparsons01/52.embed" width="550"></iframe>
<br /></div>Also as expected, the `I^2R` power loss is completely independent of the motor speed. In reality, `I^2R` loss would increase slightly with speed and torque due to <a href="https://en.wikipedia.org/wiki/Skin_effect">skin/slot</a> effects and <a href="https://en.wikipedia.org/wiki/Saturation_(magnetic)">core saturation</a> respectively.</div><div><br /></div><div>With the core loss and `I^2R` data now computed, we have all that we need to calculate the input power, output power and efficiency of our motor.</div><div><ul style="font-size: 18.72px; font-weight: 700;"><li>The combined power loss of the motor</li></ul></div><div>The core loss and the `I^2R` loss can now be combined to estimate the total power loss of the motor.</div></div><div><div class="separator" style="clear: both; text-align: center;"><br /></div>
<iframe frameborder="0" height="400" scrolling="no" src="//plotly.com/~rparsons01/46.embed" width="550"></iframe>
<div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div>It is clear that the power loss is largest at high speed and high torque. Lets now move on to the power output, power input and efficiency. </div><div><h3><ul style="font-weight: 700;"><li>Motor power output and power input mapping</li>
</ul>
</h3>
<div>The mechanical power output<span style="background-color: white;"> is estimated by the following:</span></div><div>
$$ P_{out}= Torque (Newton.meter) * Angular\, Velocity (Radian/second)$$</div><div>where $$\omega = \frac{RPM}{60} 2 \pi$$</div><div><div>Unsurprisingly, when plotted we can see that the motor produces peak power at the highest torque and speed. </div></div><div>
<iframe frameborder="0" height="400" scrolling="no" src="//plotly.com/~rparsons01/54.embed" width="550"></iframe></div><div><div>The Input power to the motor is then given by:</div></div><div>$$ P_{in}= P_{out} + P_{loss}$$</div><div>
<iframe frameborder="0" height="400" scrolling="no" src="//plotly.com/~rparsons01/48.embed" width="550"></iframe>
</div><div>From this, we can now estimate the motor efficiency and map it.</div><div><ul style="font-size: 18.72px; font-weight: 700;"><li>Motor efficiency</li></ul>The efficiency (`\eta`) of an electric motor is defined as the useful mechanical power output (`P_{ou\t}`) divided by the total power input (`P_{i\n}`). The input power is in the form of a voltage (`V`) and current* (`I`) while the useful mechanical power output is in the form of torque (`\tau`) and angular velocity (`\omega`) as defined by the following:<br />$$\eta = \frac{P_{out}}{P_{in}} = \frac{\tau \omega}{V I} $$<br />Using the power input and output we have already calculated the motor efficiency can be estimated and mapped.</div><div><div class="separator" style="clear: both; text-align: center;"><br /></div>
<iframe frameborder="0" height="400" scrolling="no" src="//plotly.com/~rparsons01/58.embed" width="550"></iframe>
<br />So how does this simple model stack up when compared to the measurement of a real motor? Below is the motor efficiency and power loss. Note that 180 rad/s is equal to approximately 1700 RPM and so the x and y-axis ranges are equivalent.</div><div><br /></div><div><div class="separator" style="clear: both; text-align: center;"><a href="https://lh3.googleusercontent.com/-hQyX1bc44aM/YB_SC_iHw3I/AAAAAAAAgqc/z0zLLZ7w-7I-Kkg7mGAunY_PtmXec_oeACNcBGAsYHQ/image.png" style="margin-left: 1em; margin-right: 1em;"><img alt="" data-original-height="905" data-original-width="1087" height="360" src="https://lh3.googleusercontent.com/-hQyX1bc44aM/YB_SC_iHw3I/AAAAAAAAgqc/z0zLLZ7w-7I-Kkg7mGAunY_PtmXec_oeACNcBGAsYHQ/w433-h360/image.png" width="433" /></a></div><br />The first thing to note is that the top right-hand corner of the measured efficiency map is missing due to limitations in the supply voltage for the motor controller. More info on why <a href="https://things-in-motion.blogspot.com/2019/05/understanding-bldc-pmsm-electric-motors.html">in this post</a>. The next important difference between the model and real data is that the numerical model predicts a far more efficient electric motor. The peak motor efficiency that was measured on the real motor was around 70% while the model predicts a peak efficiency as high as 85%. The peak efficiency region also extends to a much lower RPM and torque value than the measured efficiency map.</div><div><br /></div><div>There are two likely reasons for this overestimation in motor efficiency:</div><div><br /></div><div><b>1. The core loss is underestimated in the model</b>. The core loss used in the model is measured for flat sheets of iron silicon steel placed in a test jig. In a real motor, the sheets are stamped, compressed and then welded to form a stacked core. These processing steps can reportedly increase the core loss by a factor of 2 to 3. Therefore a 'core loss fudge factor' of 2.5 was included in the model to help compensate for this.</div>
<iframe frameborder="0" height="400" scrolling="no" src="//plotly.com/~rparsons01/61.embed" width="550"></iframe>
<div><br /></div><div>This reduced the peak motor efficiency to be more in line with that of the real motor. However, the 70% efficiency contour line extends to a much lower speed range than the real motor efficiency map. This may be due to:</div><div><br /></div><div>2. <b>The measured motor power loss also included the power loss in the motor controller. </b>This extra loss would be seen in the form of a slight increase in the I^2R loss and, perhaps more importantly for the efficiency map, the addition of a fixed power drawn by the motor controller at all times. This is the 'always on' power draw of the motor controller that is independent of the current it is supplying. It can range from less than 1 W in small motor controllers to >10 W in high-power versions. I don't know the power draw of the motor controller used in the testing and so I have just assumed a value of 8W. The resulting efficiency map is shown below.</div>
<iframe frameborder="0" height="400" scrolling="no" src="//plotly.com/~rparsons01/44.embed" width="550"></iframe>
<div> </div><div>With this change, the modelled efficiency map of the T-Motor U8 Pro now more closely resembles the measured efficiency map. This is because at low speed and torque, the motor is producing very little power. Therefore, if you add a small power loss of ~8W it can greatly reduce the system efficiency in this region. </div><div><br /></div><div><h3><span style="color: blue;">Conclusion</span></h3></div><div><span style="color: blue;"><br /></span></div><div>Some may be quick to point out that if you play with the numbers and assumptions long enough you could make an efficiency map look any way you want. However, in my opinion, the assumptions made in the model above, which are based on my experiences and observations both with real motors and simulated motors, cover the majority of the losses within a typical 'hobbyist' grade brushless motor. While this simple model is most definitely an oversimplification of a complex system it can still provide a good insight into the major sources of power loss and how they dictate the efficiency of a motor. So I encourage you to have a play with the numbers (i.e. pole count, winding resistance etc.) using the Jupyter notebook mentioned below to help improve your understanding of how aspect each impacts motor efficiency with speed and torque.</div><div><br /></div><div> <i>Equations were produced in this post with the help of <a href="http://arachnoid.com/latex/">arachnoid.com/latex/</a>. Figures were produced using <a href="https://plotly.com/">Plotyl</a> and hosted on <a href="https://chart-studio.plotly.com/">Chart Studio</a>. Online Jupyter environment using <a href="https://mybinder.org/">Binder</a>. If you have noticed any errors in the above article then please let me know.</i></div>
</div>
</div></div>Richard Parsonshttp://www.blogger.com/profile/08331478925576296508noreply@blogger.com4tag:blogger.com,1999:blog-4080979156700236346.post-34335696155457708172019-07-12T21:15:00.000-07:002020-01-02T23:24:21.764-08:00BLDC (PMSM) end turns and torque production<b>Jan 2020 update</b>: <i>I have removed the discussion of end turns in iron-core motors from this post as I have since learned that the arguments used were flawed. I will address this topic in my next post.</i><br />
<br />
End turns are not considered when calculating the torque of a hobbyist grade BLDC (<a href="https://things-in-motion.blogspot.com/2018/12/why-most-hobby-grade-bldc-out-runners.html">PMSM</a>) electric motor. This post explores why.<br />
<h2>
<span style="color: blue;">1.End turns and BLDC motors</span></h2>
End turns are the region of a BLDC motor's windings which extend out from the slots at either end of a motor.<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-LZ2rnK5KUEA/XScL5SBdHBI/AAAAAAAAXB8/ae7Ek_etkWoG0ItHp9H9ymcb1DPZKBHVwCLcBGAs/s1600/end_turns3.jpg" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="749" data-original-width="1077" height="277" src="https://1.bp.blogspot.com/-LZ2rnK5KUEA/XScL5SBdHBI/AAAAAAAAXB8/ae7Ek_etkWoG0ItHp9H9ymcb1DPZKBHVwCLcBGAs/s400/end_turns3.jpg" width="400" /></a></div>
Current supplied to a motor passes from one slot to the next via the end turns. However, end turns are not typically considered when calculating the torque produced by a BLDC motor in 2D <a href="https://en.wikipedia.org/wiki/Finite_element_method">FEA </a>software packages such as <a href="https://www.google.com/search?q=FEMM&oq=FEMM&aqs=chrome..69i57j69i60l3j69i61j69i59.1149j0j7&sourceid=chrome&ie=UTF-8">FEMM</a>.<br />
<br />
Although end turns can be ignored when it comes to calculating torque, they do contribute to the resistance of the windings. It is therefore ideal to have the end turns be as short as possible when designing a motor so as to improve motor efficiency by minimising ohmic losses in the windings.<br />
<br />
Electric motors with concentrated windings, like those found in most hobbyist outrunners, have shorter end turns than those used in motors with distributed windings. Motors that use concentrated windings are therefore more efficient in some use cases, such as in high torque density, low-speed applications. However, as <a href="https://things-in-motion.blogspot.com/2019/01/selecting-best-pole-and-slot.html">only a limited number</a> of slot and pole combinations are available for concentrated windings they are not suitable for all use cases.<br />
<h2>
<span style="color: blue;">2. Torque generation and end turns in core-less motors</span></h2>
To understand why end turns are not considered when calculating the torque output of a BLDC motor let us consider a simplified scenario. In the diagram below a single turn copper wire is placed inside a homogenous magnetic field provided by permanent magnets placed above and below the wire. The magnets are arranged so that the applied field direction is different for either side of the wire.<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-53qhIysQpuI/XScdBdkMomI/AAAAAAAAXCY/BAU3eZeuXXwtfkBiH_qTPLNkHIjCkTSiQCLcBGAs/s1600/end%2Bturn%2Boutside%2Bof%2Bfield.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="725" data-original-width="1118" height="258" src="https://1.bp.blogspot.com/-53qhIysQpuI/XScdBdkMomI/AAAAAAAAXCY/BAU3eZeuXXwtfkBiH_qTPLNkHIjCkTSiQCLcBGAs/s400/end%2Bturn%2Boutside%2Bof%2Bfield.jpg" width="400" /></a></div>
<br />
This situation is an analogous to core-less electric motors which have no ferromagnetic material in the stator. If a current flows through the wire then a <a href="https://en.wikipedia.org/wiki/Lorentz_force">Lorentz force</a> is produced on that wire as described by:<br />
<br />
<div style="text-align: center;">
<span style="font-size: large;">Force = Magnetic field `\times`</span><span style="font-size: large;">Current</span><span style="font-size: large;">`\times` </span><span style="font-size: large;">Conductor length</span></div>
<br />
where the magnetic field (B) is supplied from the magnets, the current (I) is that flowing through the wire of a given length (L). Looking at this setup from the top down we see the following.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-pa5sr_ZP2vE/XScc9Mqr7XI/AAAAAAAAXCU/PsNQBfa3nKw893crh3Gy0vRPDf-veDyTACLcBGAs/s1600/end%2Bturn%2Boutside%2Bof%2Bfield%2Btop%2Bdown.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="739" data-original-width="520" height="400" src="https://1.bp.blogspot.com/-pa5sr_ZP2vE/XScc9Mqr7XI/AAAAAAAAXCU/PsNQBfa3nKw893crh3Gy0vRPDf-veDyTACLcBGAs/s400/end%2Bturn%2Boutside%2Bof%2Bfield%2Btop%2Bdown.jpg" width="281" /></a></div>
It is clear that a Lorentz force is only generated on the wire which is located between the magnets, which is analogous to the wire located within the slots of a BLDC motor. The parts of the wire which extend outside of the magnets are not exposed to a magnetic field and so the force generated is zero. This situation is the same for end turns in a BLDC motor, and so is ignored by FEA software packages.<br />
<br />
However, for the sake of argument, let us also consider a situation where the magnets could be made long enough that they also covered the end turns. In this situation, a force is generated by the end turns. However, this force is tangential to that generated by the wires in the 'slots' and so would not contribute to the production of useful torque within a motor. This force would also be mostly cancelled out as can be seen from below.<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-5rs_9c7jbcM/XScdJXvNaXI/AAAAAAAAXCc/CrJvdnwshmENXY2wecGldz5OLv1TnOBygCLcBGAs/s1600/end%2Bturn%2Binside%2Bof%2Bfield%2Btop%2Bdown.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="928" data-original-width="757" height="400" src="https://1.bp.blogspot.com/-5rs_9c7jbcM/XScdJXvNaXI/AAAAAAAAXCc/CrJvdnwshmENXY2wecGldz5OLv1TnOBygCLcBGAs/s400/end%2Bturn%2Binside%2Bof%2Bfield%2Btop%2Bdown.jpg" width="326" /></a></div>
<br />
Another arrangement I have seen used by DIY motor makers is to take round coils and magnets and arrange them to make an axial flux motor or generator.<br />
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-jhbbY7FR-wE/XScfu097OLI/AAAAAAAAXCw/fsi_mXkWR1Q3mG2vxbIIZljX4qLago82gCLcBGAs/s1600/DIY%2Bround%2Bwindings.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="412" data-original-width="1224" height="132" src="https://1.bp.blogspot.com/-jhbbY7FR-wE/XScfu097OLI/AAAAAAAAXCw/fsi_mXkWR1Q3mG2vxbIIZljX4qLago82gCLcBGAs/s400/DIY%2Bround%2Bwindings.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
While this approach will work, it is also inefficient since the force produced on the wires is always tangential to the flow of current and so a significant fraction of the force is not available to useful work.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-lkXMWqc0zYs/XSciRXBT9uI/AAAAAAAAXC8/WRYugElIzj8UhMhaS2FXMIwkM220FCy1wCLcBGAs/s1600/end%2Bturn%2Bround%2Bmagnets%2Band%2Bcoil.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="814" data-original-width="1139" height="285" src="https://1.bp.blogspot.com/-lkXMWqc0zYs/XSciRXBT9uI/AAAAAAAAXC8/WRYugElIzj8UhMhaS2FXMIwkM220FCy1wCLcBGAs/s400/end%2Bturn%2Bround%2Bmagnets%2Band%2Bcoil.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: left;">
Note that although all of the above examples are for core-less motors, the situation is unchanged for iron cored motors.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
In my next post, the concept of flux linkage will be explored and used to understand why end turns still do not contribute to flux in conventional iron-core motors.</div>
<div>
<h2>
<span style="color: blue;">4. Conclusion</span></h2>
End turns in air-core motor designs are not considered in the calculation of torque as they are either outside of the region exposed to the permanent magnet's magnetic field or the force generated is in the wrong direction.<br />
<br />
<i>Equations were produced in this post with the help of <a href="https://arachnoid.com/latex/">arachnoid.com</a>. If you have noticed any errors in the above article then please let me know.</i></div>
Richard Parsonshttp://www.blogger.com/profile/08331478925576296508noreply@blogger.com2tag:blogger.com,1999:blog-4080979156700236346.post-38116699294809437862019-05-11T19:56:00.002-07:002019-08-04T00:06:28.218-07:00Understanding BLDC (PMSM) electric motors: Base speed, no load speed and torque vs speed<span style="background-color: white;">Search for the keywords "<b>electric motor speed vs torque</b>" and you will find many hundreds of images, with each looking different to the next.</span><br />
<span style="background-color: white;"><br /></span>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-bPABfMAhauc/XJa61oUIGgI/AAAAAAAAV3o/nUV3PGUvSCMOM6sD3GxNmyPmHlgk1LQpACLcBGAs/s1600/motor%2Bplots.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="928" data-original-width="1394" height="266" src="https://2.bp.blogspot.com/-bPABfMAhauc/XJa61oUIGgI/AAAAAAAAV3o/nUV3PGUvSCMOM6sD3GxNmyPmHlgk1LQpACLcBGAs/s400/motor%2Bplots.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">A selection of 'electric motor speed vs torque' charts</td></tr>
</tbody></table>
<span style="background-color: white;">This has, understandably, lead to considerable confusion (myself included) when discussing 'hobbyist' grade <a href="https://things-in-motion.blogspot.com/2018/12/why-most-hobby-grade-bldc-out-runners.html">BLDC (PMSM)</a> motors and the topics of:</span><br />
<ul>
<li><span style="background-color: white;"><b>no-load speed</b></span></li>
<li><span style="background-color: white;"><b>base speed</b></span></li>
<li><span style="background-color: white;"><b>torque vs speed</b></span></li>
</ul>
<span style="background-color: white;">where 'speed' is referring to how fast a motor is spinning (i.e. it's </span><span style="background-color: white;">angular velocity)</span><span style="background-color: white;">. Therefore, this post will explore these three topics. For those wanting a quick summary see the brief overview below.</span><br />
<h2>
<span style="background-color: white;"><span style="color: blue;">1.0 Brief overview</span></span></h2>
<h3>
<span style="background-color: white;">1.1 Voltage constraints</span></h3>
<h3>
<div>
<span style="font-size: small; font-weight: 400;">When a brushless motor is rotating the motion of the rotor magnets relative to the stationary stator produces a voltage in the windings. This is a brushless motor's back <a href="https://en.wikipedia.org/wiki/Electromotive_force">electromotive force</a> (back EMF). When using a hobbyist grade motor controllers the maximum speed of an unloaded motor (i.e. nothing connected to its shaft) is reached when the motor's back EMF is about equal to the supply voltage, such as from a battery. </span><span style="font-size: small; font-weight: 400;">This is the </span><u><span style="font-size: small;">no</span><span style="font-size: small;">-load speed</span></u><span style="font-size: small; font-weight: 400;"> of the motor. </span><br />
<span style="font-size: small; font-weight: 400;"><br /></span>
<span style="font-size: small; font-weight: 400;">If you have ever adjusted the speed of an unloaded brushed DC motor by adjusting its supply voltage then you have already seen the effect of supply voltage on a motors no-load speed. The situation is no different with a brushless motor, just that it is now electronically commutated. </span><br />
<span style="font-size: small; font-weight: 400;"><br /></span>
<span style="font-size: small; font-weight: 400;">At the no-load speed, the voltage difference between the motor and its power supply approaches zero and so very little current can flow through the motor. With near-zero current flowing through the motor there is near zero torque produced and so it can not spin any faster.</span><span style="font-size: small; font-weight: 400;"> </span><br />
<span style="font-size: small; font-weight: 400;"><br /></span>
<span style="font-size: small;">The speed at which you can do useful work with a hobbyist motor and motor controller is always going to be smaller than its no-load speed</span><span style="font-size: small; font-weight: 400;">. As a motors speed is reduced so too is it's back EMF. Therefore, more of the supply voltage is available to deliver current to the motor and so the capability of a motor to produce torque will increase as you move further away from its no-load speed. </span><br />
<span style="font-size: small; font-weight: 400;"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-lI2Dur4DI38/XJdHtHCJQAI/AAAAAAAAV6g/sequ3MmVpOYECEKdbTlAHt2rTpVFEYGpQCLcBGAs/s1600/brief%2Boverview%2Bvoltage%2Blimit.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="667" data-original-width="966" height="275" src="https://2.bp.blogspot.com/-lI2Dur4DI38/XJdHtHCJQAI/AAAAAAAAV6g/sequ3MmVpOYECEKdbTlAHt2rTpVFEYGpQCLcBGAs/s400/brief%2Boverview%2Bvoltage%2Blimit.jpg" width="400" /></a></div>
</h3>
<span style="background-color: white;">The amount of torque that the motor can produce continues to increase as you approach zero speed. At zero speed (i.e. shaft is stopped) the current flowing through the motor will be about equal to the supply voltage divided by the motor winding resistance. This is the 'stall current' of the motor and will typically be a very large value due to the low resistance of most brushless motors (i.e. < 0.1 Ohm) as is described below.</span><br />
<span style="background-color: white;"><br /></span>
<br />
<h3>
<span style="background-color: white;">1.2 Thermal constraints</span></h3>
<span style="background-color: white;"><br /></span><span style="background-color: white;">When in normal use, the torque output of a hobbyist brushless motor increases roughly in line with the current supplied to its windings. Therefore, if you double the current to the motor you also double the torque output. However, the resistive losses in the copper windings increase with the square of the current and so if you double the current and torque you are also producing four times as much waste heat. </span><br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-xdwKKR-eTeY/XNd6yIrROEI/AAAAAAAAWgQ/sZ9vormFhDYkcuuQjbZw3mWcmletDb6EwCLcBGAs/s1600/torque%2Band%2Bmotor%2Bcurrent.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="703" data-original-width="916" height="306" src="https://2.bp.blogspot.com/-xdwKKR-eTeY/XNd6yIrROEI/AAAAAAAAWgQ/sZ9vormFhDYkcuuQjbZw3mWcmletDb6EwCLcBGAs/s400/torque%2Band%2Bmotor%2Bcurrent.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<span style="background-color: white;">Since most brushless motors have a very low winding resistance (i.e. 0.1 Ohm) it is possible to deliver extremely large currents to a motor even with a modest supply voltage. This is of course provided you have a big enough motor controller and <a href="https://things-in-motion.blogspot.com/2018/12/how-to-select-right-power-source-for.html">power source to deliver the required current</a>. However, the use of extremely large currents will quickly overheat and destroy a motor. Therefore, most hobbyist brushless motors are not limited by how much current you can deliver to the motor, but rather by how much current you can deliver before the motor overheats. That is to say, <b>the torque output of most hobbyist brushless motors is thermally constrained</b>.</span><br />
<span style="background-color: white;"><br /></span> <span style="background-color: white;">Similarly, when a brushless motor is rotating it also produces waste heat due to core losses which are the losses generated in the iron and magnets. The faster you spin a motor, the larger these losses. Therefore your motor will overheat more quickly at higher speeds than at lower speeds for the same torque output.</span><br />
<br />
<span style="background-color: white;">This thermally constrained torque output of a motor is also often referred to as a motors </span><b><u>rated torque</u></b><span style="background-color: white;">. The maximum speed at which a motor can operate forever at its rated torque is known as a motors<b> <u>base speed</u></b>.</span><br />
<span style="background-color: white;"><br /></span>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-QvyRP7pEwcs/XJdIRQD9SnI/AAAAAAAAV6o/AahAAqwlgK0hJ9mDhEG4dmtuyAcbHmVpQCLcBGAs/s1600/brief%2Boverview%2Bthermal%2Blimit.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="702" data-original-width="997" height="281" src="https://2.bp.blogspot.com/-QvyRP7pEwcs/XJdIRQD9SnI/AAAAAAAAV6o/AahAAqwlgK0hJ9mDhEG4dmtuyAcbHmVpQCLcBGAs/s400/brief%2Boverview%2Bthermal%2Blimit.jpg" width="400" /></a></div>
<br />
It is possible to exceed a motors rated torque for short periods of time. The true maximum torque output of a motor is its <b>peak torque. </b>The peak torque output is usually defined as the maximum torque that can be sustained for some fixed period of time (i.e. 30, 60 or 180 seconds) without overheating (starting with a cold motor) or without damaging the magnets or mechanical components. The <a href="https://things-in-motion.blogspot.com/2018/12/how-to-estimate-torque-of-bldc-pmsm.html">peak torque of a motor can be estimated from its peak current</a> value. The peak current value is what is most commonly quoted by sellers of hobbyist brushless motors. Unfortunately, how the peak current is measured is usually not clear. As a rule of thumb, the rated current, and therefore torque output, of a hobbyist motor will be around one third that of the peak torque value, or around two thirds with aggressive forced air cooling.<br />
<br />
In summary:<br />
<ul>
<li><b>The no-load speed of a motor is voltage constrained.</b></li>
<li><b>Useful work can only be done below a motors no-load speed.</b></li>
<li><b>The constant torque output of a motor is thermally constrained.</b></li>
<li><b>The maximum speed which a constant torque can be produced is the base speed.</b></li>
<li><b>A motor can operate above its constant torque value for short periods of time</b></li>
</ul>
For a more in-depth discussion read on.<br />
<h2>
<span style="color: blue;">2.0 In-depth analysis </span></h2>
<h3>
2.1 Theoretical no-load speed</h3>
<span style="background-color: white;">As mentioned above, the no-load speed (</span><span style="background-color: white;">`\omega_{NL}`) </span><span style="background-color: white;">of a motor</span><span style="background-color: white;"> is the maximum rotational speed that a motor can achieve without any <b>external </b>load </span><span style="background-color: white;">(i.e. no torque output)</span><span style="background-color: white;"> placed on it. </span><span style="background-color: white;">The </span><b>theoretical </b><span style="background-color: white;">no-load speed is given by:</span><br />
<span style="background-color: white;"><br /></span> <span style="background-color: white;">$$\omega_{NL} = K_{V} \times V_{supply} $$</span><br />
<span style="background-color: white;">where `\omega_{NL}` is the theoretical no-load speed in RPM, `K_{V}` is the motor's velocity constant in RPM/V and `V_{\s\u\p\p\l\y}` is the voltage supplied at the terminals of your power source such as a battery. </span><br />
<span style="background-color: white;"><br /></span> <span style="background-color: white;">`\omega_{NL}` can be visualised as a single point on a plot of speed vs torque</span><br />
<span style="background-color: white;"><br /></span>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-oLoJxDeln34/XJa-NikEycI/AAAAAAAAV30/eZXebvN25yQeRap0rAmNknbf6shWjztPwCLcBGAs/s1600/no_load%2Bspeed%2Btheory.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="715" data-original-width="1023" height="278" src="https://2.bp.blogspot.com/-oLoJxDeln34/XJa-NikEycI/AAAAAAAAV30/eZXebvN25yQeRap0rAmNknbf6shWjztPwCLcBGAs/s400/no_load%2Bspeed%2Btheory.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<h3>
2.2 Achievable no-load speed</h3>
<span style="background-color: white;">In practice, the theoretical no-load speed can never be reached. This is due to:</span><br />
<ul>
<li><span style="background-color: white;"><b>Friction and windage</b>: All motors experience frictional and windage losses. Frictional losses originate from the bearings while windage losses originate from the relative movement of the rotor to the stationary air surrounding it. </span>These losses require torque to overcome and so the motor can never reach its true zero torque no-load speed. High rotational speed in-runner motors or out-runners with built-in<span style="background-color: white;"> cooling fans will experience larger frictional and windage losses.</span></li>
<li><span style="background-color: white;"><b>Core losses: </b>The relative movement of the rotor magnets past the stator generates hysteresis and eddy current losses within the stator, even if there is no current being delivered to the motor windings. Similarly, differences in inductance seen by the rotor magnets as they move past a slotted stator also generated eddy currents within the magnets themselves. These losses must also be overcome by the generation of torque by the motor.</span></li>
<li><span style="background-color: white;"><b>Resistive voltage drop</b>: As some small amount of torque is still required to overcome friction, windage and core losses at the no-load speed there will also be a corresponding current supplied to the motor. This current will result in a voltage drop across the motor and the motor controller, reducing the supply voltage available to overcome the motors back EMF.</span></li>
<li><span style="background-color: white;"><b>Motor controller dead time insertion</b>: Most motor controllers have some level of <a href="https://hackaday.io/project/3176-gator-quad/log/11741-pwm-control-and-dead-time-insertion">dead time when switching their MOSFETs</a> off and on so as to prevent a short between the high side and the low side of the DC bus. This dead time means that even if the motor controller is commanded to output a 100% PWM duty cycle the motor controller MOSFETs will still be off for some fraction of the time, preventing 100% utilisation of the supply voltage. It's also likely that there are other issues on the motor controller side at high electrical frequencies that may also reduce the top speed but which I don't understand at this time.</span></li>
<li><span style="background-color: white;"><b>Motor controller PWM frequency and control loop frequency.</b> If you are using a motor controller that supports <a href="https://en.wikipedia.org/wiki/Vector_control_(motor)">FOC</a> (also called vector control) then it will be producing a sine wave via PWM. Theoretically, if the frequency of this sine wave (the motors electrical frequency) surpasses approximately 1/20th the PWM frequency (the exact value will depend on the capacitance and inductance of the system) then the quality of the sine wave will <a href="https://ars.els-cdn.com/content/image/3-s2.0-B978034069143450006X-f04-35-9780340691434.jpg">start to degrade</a>. This will cause a fall-off in motor efficiency and if taken to the extreme, loss of motor control as commutation falls out of step with the rotor position. Also, if the electrical frequency of the motor approaches the control loop frequency then control will also be lost. </span></li>
</ul>
<span style="background-color: white;">Note that those losses which are internal to the motor (frictional, windage and core losses) produce no net force outside of the motor and so are not something that can be measured directly with a torque sensor. This is because the stator is both applying the torque to turn the rotor and the braking torque (losses) to stop the rotor. In general, core losses are going to be far greater than those losses due to windage and bearing losses.<br />
<span style="background-color: white;"><br /></span> <span style="background-color: white;">The no-load speed which is realistically achievable </span>(`\omega'_{NL}`) can, therefore, be approximated by:</span><span style="background-color: white;"><span style="background-color: white;"><br /></span> $$\omega'_{NL} = K_{V} \times (V_{supply} - V_{drop}) \times Modulation_{max} - \frac{P_{loss}}{\tau _{loss}}$$</span><span style="background-color: white;"><span style="background-color: white;"><br /></span> <span style="background-color: white;">where </span>`V_{drop}` is the combined resistive voltage drop in the system, `\M\o\d\u\l\a\t\i\o\n_{max}` is the maximum PWM modulation achievable by the motor controller after accounting for dead time insertion and `\frac{P_{loss}}{\tau _{loss}}` is the speed reduction brought about by the power and subsequent torque required to be produced by a motor to overcome friction, windage and core losses.<br />
<br />
</span><br />
<div class="separator" style="clear: both; text-align: center;">
<span style="background-color: white;"><a href="https://1.bp.blogspot.com/-4BHvD00aPE8/XJbEl58jnhI/AAAAAAAAV4A/69n8c03dFrEJpvJDJayz4DjG9fkdplGfQCLcBGAs/s1600/no_load%2Bspeed%2Bpractice.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="703" data-original-width="1046" height="268" src="https://1.bp.blogspot.com/-4BHvD00aPE8/XJbEl58jnhI/AAAAAAAAV4A/69n8c03dFrEJpvJDJayz4DjG9fkdplGfQCLcBGAs/s400/no_load%2Bspeed%2Bpractice.jpg" width="400" /></a></span></div>
<span style="background-color: white;"> </span><br />
<span style="background-color: white;">The size of the difference between </span><span style="background-color: white;">`\omega_{NL}` and </span><span style="background-color: white;">`\omega'_{NL}` will depend greatly on the type of motor used, its associated losses and the supply voltage. In general, it would be reasonable to expect that a typical hobbyist motor and motor controller (ESC) can reach 85% of its theoretical no-load speed. This is provided that it is operated within its 'rated' voltage. </span><br />
<span style="background-color: white;"><br /></span>
<br />
<h3>
2.3 Thermal constraints and no-load speed</h3>
<span style="background-color: white;"><br /></span> <span style="background-color: white;">Beyond voltage constraints, there are also thermal and mechanical constraints. If you try to run a '6s' (~24V motor) off a 240V motor controller then don't expect to come anywhere near its theoretical base speed before you run into thermal or mechanical (i.e. exploding rotor) limitations.</span><br />
<span style="background-color: white;"><br /></span> <span style="background-color: white;">Core losses generated within a motor at its base speed can be significant. For example, an unloaded 12n14p <a href="https://hobbyking.com/en_us/turnigy-aerodrive-sk3-5055-280kv-brushless-outrunner-motor.html?___store=en_us">HobbyKing SK3</a> 280 Kv motor produced over 100 W of losses at 10,000 RPM.</span><br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-LYJjNealeFo/XNYj2BjL0cI/AAAAAAAAWeA/lZ56J5gC4hkdOuueL0ebWLaxRWymEKSNACLcBGAs/s1600/sk3%2Bno%2Bload%2Bpower%2Bdraw%2BV2.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="715" data-original-width="966" height="295" src="https://4.bp.blogspot.com/-LYJjNealeFo/XNYj2BjL0cI/AAAAAAAAWeA/lZ56J5gC4hkdOuueL0ebWLaxRWymEKSNACLcBGAs/s400/sk3%2Bno%2Bload%2Bpower%2Bdraw%2BV2.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Power draw as measured from the power supply for an SK3 280 Kv motor</td></tr>
</tbody></table>
<span style="background-color: white;">The peak speed of 10,000 RPM only represents 75% of the maximum speed available with the 48V power source used. Despite this, the 100 W of losses within the motor at this speed is more than enough to heat the motor to the point of being damaged if operated for an extended period. Also note that the generation of core losses has nothing to do with the motor Kv and is instead determined by the pole and slot combination, with lower pole numbers, in general, producing lower losses. This is one of the reasons why a 4 pole in-runner can operate at much higher speeds than a 14 pole outrunner. </span><br />
<span style="background-color: white;"><br /></span>
<span style="background-color: white;">Due to the no-load losses within a brushless motor, the <b>maximum continuous no-load speed</b> </span><span style="background-color: white;">`\omega''_{NL}` may be far lower than the maximum achievable no-load speed. </span><br />
<span style="background-color: white;"><br /></span>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-h4Oqtj8Zhbo/XJbIUVwcFTI/AAAAAAAAV4M/LNdjw86PBUAUnydBHhbjBaEMgYA42Cy0ACLcBGAs/s1600/no_load%2Bspeed%2Bpractice%2Bno%2Boverheat.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="671" data-original-width="1042" height="257" src="https://2.bp.blogspot.com/-h4Oqtj8Zhbo/XJbIUVwcFTI/AAAAAAAAV4M/LNdjw86PBUAUnydBHhbjBaEMgYA42Cy0ACLcBGAs/s400/no_load%2Bspeed%2Bpractice%2Bno%2Boverheat.jpg" width="400" /></a></div>
<span style="background-color: white;">This could be an important consideration in some applications which require high speed but very little torque such as an engraving tool or <a href="https://en.wikipedia.org/wiki/Mirror_galvanometer">mirror galvo optical assembly</a>.</span><br />
<span style="background-color: white;"><br /></span>
<br />
<h3>
<span style="background-color: white;">2.4 Base speed</span></h3>
<span style="background-color: white;"><br /></span>
<span style="background-color: white;">Also mentioned above, the base speed (also called the rated or nominal speed) is the top speed at which a brushless motor can achieve at it's '<b>rated torque</b>'. The<b> </b>'rated torque' is the torque that a motor can output indefinitely without overheating. This speed is different to that of the achievable no-load speed because there will be a larger voltage drop across the motor and motor controller when the motor is producing its rated torque due to the requirement that a larger current be supplied to the windings.</span><br />
<span style="background-color: white;"><br /></span> <span style="background-color: white;">In all other ways the base speed of the motor (</span><span style="background-color: white;">`\omega_{BS}`) can be defined in the same way as the maximum achievable no-load speed:</span><br />
<span style="background-color: white;"><br /></span> <span style="background-color: white;">$$\omega_{BS} = K_{V} \times (V_{supply} - V_{drop}) \times Modulation_{max} - \frac{P_{loss}}{\tau _{loss}}$$</span><br />
<span style="background-color: white;">with the exception that </span><span style="background-color: white;">`V_{drop}` is now dependent upon the current required to produce the motors rated torque. Therefore, the base speed of a motor will fall off as the 'rated torque' is increased due to the additional voltage drop incurred at higher currents.</span><br />
<span style="background-color: white;"><br /></span> <span style="background-color: white;">In the figure below </span><span style="background-color: white;">`\omega_{BS}` represents the maximum speed at which a motor could produce a set level of torque. However, if less torque was required from the motor then a higher speed (following the light blue line) could be achieved.</span><br />
<span style="background-color: white;"><br /></span>
<span style="background-color: white;"><br /></span>
<span style="background-color: white;"><br /></span>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-_ahxfOkQrwo/XJbXiGN9_EI/AAAAAAAAV4k/wVby_lpygHg0UX07k_YuPeGawmMVBwpOgCLcBGAs/s1600/base%2Bspeed%2Bno%2Bcooling.jpg" style="margin-left: auto; margin-right: auto;"><span style="color: black;"><img border="0" data-original-height="508" data-original-width="753" height="268" src="https://2.bp.blogspot.com/-_ahxfOkQrwo/XJbXiGN9_EI/AAAAAAAAV4k/wVby_lpygHg0UX07k_YuPeGawmMVBwpOgCLcBGAs/s400/base%2Bspeed%2Bno%2Bcooling.jpg" width="400" /></span></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><br /></td></tr>
</tbody></table>
The gradient of the light blue line will depend on the resistance of your motor and motor controller. Higher resistance will produce a shallower gradient.<br />
<div>
<br />
<h3>
2.5 Base speed and temperature rise</h3>
<br />
So far we have assumed a fixed motor and motor controller temperature, and therefore a fixed voltage drop per amp. However, as the temperature of a motor is increased so too does its winding resistance. For example, a 50C temperature increases the winding resistance by 20% and a 135C rise by 53% [1]. Therefore, a hot motor will have a larger voltage drop across the motor and a lower base and no-load speed.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-RhQeUH-ci5o/XNv3Cx1gG-I/AAAAAAAAWl4/stt6toplphA48ntnssblXd4rIcf04Z-vgCLcBGAs/s1600/base%2Bspeed%2Bhot%2Bmotor.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="712" data-original-width="1051" height="270" src="https://2.bp.blogspot.com/-RhQeUH-ci5o/XNv3Cx1gG-I/AAAAAAAAWl4/stt6toplphA48ntnssblXd4rIcf04Z-vgCLcBGAs/s400/base%2Bspeed%2Bhot%2Bmotor.jpg" width="400" /></a></div>
Note that the above diagram is not to scale and the actual reduction in base speed for a 'hot' motor may be much smaller in reality.<br />
<br />
While it's important to keep this point in mind, for simplicity the rest of this post will assume a constant motor and motor controller temperature.<br />
<br />
<h3>
2.6 Base speed and additional cooling</h3>
<div>
<br /></div>
Adding additional cooling to a motor will allow you to increase the 'rated torque' that a motor can produce without overheating. Counterintuitively, adding additional cooling to a motor and raising its rated torque, and therefore current, will actually decrease its base speed since a larger voltage drop will produce over the motor and motor controller.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-I7Py70Rr9ek/XRf5EKzOTzI/AAAAAAAAW7s/5Ib9VcJ3t_oBLMlrGU3LUeL3-YwaYBvcQCLcBGAs/s1600/base%2Bspeed.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="736" data-original-width="1034" height="283" src="https://1.bp.blogspot.com/-I7Py70Rr9ek/XRf5EKzOTzI/AAAAAAAAW7s/5Ib9VcJ3t_oBLMlrGU3LUeL3-YwaYBvcQCLcBGAs/s400/base%2Bspeed.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
<h3>
2.7 Momentary peak torque</h3>
<br />
<div>
Momentary peak torque is important for many applications. For example, rapidly reversing the direction of an axis on a CNC machine or jumping a quadruped robot into the air. The ability to produce a large momentary peak torque is one of the largest selling points for a brushless motor over stepper motors.<br />
<br />
The peak torque output of a motor is generally limited by four factors:<br />
<br />
<ol>
<li><b>Voltage constraints</b>: As mentioned above, the maximum torque produced by a motor is reached at zero RPM (stall torque), where the current is equal to the supply voltage divided by the total resistance in the system. Therefore, a larger supply voltage will allow for a larger current to flow at zero RPM and greater peak torque. </li>
<li><b>Current constraints:</b> If using a small power source then it may not be possible to deliver the stall current demanded by the motor without the supply voltage 'sagging' under the load.</li>
<li><b>Thermal constraints: </b>Supplying a motor with a momentary large current will result in a small temperature rise. Supplying a 'momentary' large current repeatedly, or if the motor is already hot, and this temperature rise will accumulate to the point of damaging the motor. If you demand 100 A from a 10 A motor controller, then even a 'momentary' peak current and torque will likely cause it to overheat.</li>
<li><b>Material constraints</b>: Ultimately, even if all other constraints are overcome, the peak torque output of a motor is limited by the materials used in its construction as described in more detail below.</li>
</ol>
<br />
<ul style="font-weight: 700;">
<li>Material constraints: Stator saturation</li>
</ul>
<br />
The magnetically soft materials used to construct the stator, typically laminated Fe-Si steel, can only be magnetised (polarised) so far before it reaches <a href="https://en.wikipedia.org/wiki/Saturation_(magnetic)">magnetic saturation</a>. As magnetic saturation is approached there is a loss of linearity between the current supplied to the motor and the torque it produces. i.e. a 1 A increase in current may produce a 0.1 N.m increase in torque when the motor is far from saturation, but only a 0.05 N.m increase in torque close to saturation.<br />
<br />
The image below is taken from <a href="http://build-its-inprogress.blogspot.com/2019/03/hello-there-mini-cheetah.html">Ben Katz thesis</a> and nicely shows a subtle ~12% fall off in the torque constant of a motor as its current is increased due to the stator approaching saturation.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-4GvBWs8BOdw/XNYxHzijaqI/AAAAAAAAWec/AbrKgmpoM-cHMPYHJ7aFEUCj8ET17xsoACLcBGAs/s1600/torque%2Bvs%2Bcurrent.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="417" data-original-width="1061" height="155" src="https://2.bp.blogspot.com/-4GvBWs8BOdw/XNYxHzijaqI/AAAAAAAAWec/AbrKgmpoM-cHMPYHJ7aFEUCj8ET17xsoACLcBGAs/s400/torque%2Bvs%2Bcurrent.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">All credit to Ben Katz [2]. Click to enlarge</td></tr>
</tbody></table>
<br />
<br />
<ul>
<li><b>Material constraints: Magnet demagnetisation</b></li>
</ul>
Supplying a motor with a very large current produces an equally large magnetic field. If the size of this magnetic field exposed to the rotor magnets starts to approach the coercivity of the magnets then there is a risk of demagnetisation. If the rotor magnets are significantly demagnetised then the motor is permanently ruined unless the magnets can be replaced. Thankfully, modern rare earth magnets have a very large coercivity. <a href="https://things-in-motion.blogspot.com/2019/02/how-to-model-bldc-pmsm-motors-kv.html">My own simulations</a> suggest that supplying a current large enough to saturate a motor still does not produce a demagnetise risk for a typical grade N35 Nd-Fe-B magnets <u><b>at room temperature.</b></u><br />
<br />
However, the situation changes if the magnets become hot. Increase their temperature to 80C and their coercivity is about one half that at 20C. At this point, it becomes possible to demagnetise the magnets with a current approximately three times a typical brushless motor rated peak current.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-XZzImaVviX0/XNYxzd5_a7I/AAAAAAAAWek/93S_Oqsfw0QPil0ZKciNhVCu03YTog3NgCLcBGAs/s1600/magnet%2Bdemag.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="479" data-original-width="628" height="244" src="https://3.bp.blogspot.com/-XZzImaVviX0/XNYxzd5_a7I/AAAAAAAAWek/93S_Oqsfw0QPil0ZKciNhVCu03YTog3NgCLcBGAs/s320/magnet%2Bdemag.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The magnet coercivity (x-axis crossing point) falls significantly with temperature.</td></tr>
</tbody></table>
<br />
<ul>
<li><b>Material constraints: Mechanical limitations</b></li>
</ul>
Lastly, the shaft and fixings of a motor are all designed with its rated torque in mind. Exceed this value by too large a margin and you risk damaging the motor. i.e. slipping shaft where it connects to the rotor, twisting the frame etc.<br />
<br />
<h3>
2.8 Torque vs speed</h3>
<br />
From the discussion above it should now be clear why searching for a 'torque vs speed' plot yields so many results. The output torque of a brushless motor at a given speed depends on many different factors. However, these factors can be simplified by only considering a single situation which should apply to most hobbyist brushless motors and motor controllers. The assumptions are as follows:<br />
<br />
<ul>
<li>A powerful power source, such as a Lipo battery, is used which can supply current to the motor when peak torque is demanded without much in the way of voltage sag.</li>
<li>Peak torque is infrequently and only momentary required from the motor so that the motor and motor controller is maintained at a reasonable temperature of around 60C at all times.</li>
<li>The supply voltage is matched to a motor so that the motor will not overheat if run at its no-load speed indefinitely.</li>
<li>The peak torque, and therefore peak current, supplied to the motor is still well below magnetic saturation of the stator and within the safe mechanical limits of the motor.</li>
</ul>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-6OCfZO2muV4/XNY4d4aT8LI/AAAAAAAAWew/PQ_8b5Ib_XMBJfzIukyfa4Ya37qs_3GbgCLcBGAs/s1600/typical%2Bsetup.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="344" data-original-width="1275" height="107" src="https://2.bp.blogspot.com/-6OCfZO2muV4/XNY4d4aT8LI/AAAAAAAAWew/PQ_8b5Ib_XMBJfzIukyfa4Ya37qs_3GbgCLcBGAs/s400/typical%2Bsetup.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">A typical setup</td></tr>
</tbody></table>
<div>
In this scenario, the torque output of a motor below its base speed is thermally constrained while the maximum speed is voltage constrained. The peak torque that can be supplied to the motor, therefore, depends on the temperature of the motor and the length of time that torque is requied. If the motor starts at room temperature then a very large torque can be produced for a short period. Alternatively, more moderate peak torque can be supplied for a longer period. </div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-SLiEIkU_-q0/XNY7ujO4OsI/AAAAAAAAWe8/GxRjWQvHxOIB0AxjrmD9liTZtPyBTZDTwCLcBGAs/s1600/peak%2Btorque%2Bvs%2Btime.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="695" data-original-width="1025" height="270" src="https://3.bp.blogspot.com/-SLiEIkU_-q0/XNY7ujO4OsI/AAAAAAAAWe8/GxRjWQvHxOIB0AxjrmD9liTZtPyBTZDTwCLcBGAs/s400/peak%2Btorque%2Bvs%2Btime.jpg" width="400" /></a></div>
<div>
However, if the motor was already at its safe upper limit of 60C then only the motors rated torque could be supplied without heating it to the point of damage. Therefore, the available speed vs torque curve of a brushless motor also depends on its thermal history. For this reason, industrial brushless motors will typically include a temperature sensor embedded in the windings so that the motor controller can actively limit the current if the motor becomes too hot. This is also why the brushless motors supplied by <a href="http://2.8%20torque%20vs%20speed/">Odrive robotics</a> come with an embedded thermistor. </div>
</div>
<br />
Note that the power consumption of a motor producing its 'rated torque' at its base speed will be significantly higher than the same motor producing its 'rate torque' at the stall speed (zero RPM). Therefore, unless you have a powerful power supply or battery you may be unable to reach the base speed at rated torque of a motor. See <a href="https://things-in-motion.blogspot.com/2018/12/how-to-select-right-power-source-for.html">this post</a> for more details.<br />
<br />
<h2>
<span style="color: blue;">3.0 Conclusion</span></h2>
<span style="background-color: white;">In summary, the torque that a motor can produce at a given speed depends on the motor used, how the motor is cooled, the thermal history of the motor and </span><span style="background-color: white;">the power source used</span><span style="background-color: white;">. For this reason, there is no single universal 'speed vs torque' plot for any given motor type, or even for two identical motors used under different conditions.</span><br />
<span style="background-color: white;"><br /></span>
<span style="background-color: white;">However, it is a safe assumption that any given brushless motor will have a flat constant torque output vs speed due to thermal constraints. Similarly, the peak torque output will, in general, be around 2 to 3 times the constant torque output for a short period (i.e. 30 s) when starting cold. Lastly, the base speed of a motor can be approximated as roughly 0.85 times the supply voltage multiplied by the motors Kv.</span><br />
<span style="background-color: white;"><br /></span>
<span style="background-color: white;">In the next few posts, I will be using a low-cost DIY motor dynamometer to characterise these torque vs speed curves, along with motor efficiency, for serval of hobbyist grade brushless motors.</span><br />
<span style="background-color: white;"><br /></span> <span style="background-color: white;"><i>Equations were produced in this post with the help of arachnoid.com. If you have noticed any errors in the above article then please let me know.</i></span><br />
<br />
<ul>
<li><span style="background-color: white;"><i>[1] From Design of Brushless Permanent Magnet Motors by Hendershot and J. R., Miller, T. J. E.</i></span></li>
<li><span style="background-color: white;"><i>[2] Katz, Benjamin G. A low-cost modular actuator for dynamic robots. Diss. Massachusetts Institute of Technology, 2018.</i></span></li>
</ul>
<br />
<span style="background-color: white;"></span></div>
Richard Parsonshttp://www.blogger.com/profile/08331478925576296508noreply@blogger.com7tag:blogger.com,1999:blog-4080979156700236346.post-32087307444364327872019-02-09T20:49:00.001-08:002019-04-30T17:58:58.343-07:00How to model a BLDC (PMSM) motors Kv (velocity constant) and Kt (torque constant) in FEMMIn a <a href="https://things-in-motion.blogspot.com/2018/12/how-to-estimate-torque-of-bldc-pmsm.html">previous post</a>, the velocity constant (Kv) and torque constant (Kt) was estimated for a number of different 'hobbyist level' <a href="https://things-in-motion.blogspot.com/2018/12/why-most-hobby-grade-bldc-out-runners.html">BLDC (PMSM)</a> out-runner motors. In this post, I will be using <a href="http://www.femm.info/">FEMM</a> to model one of these motors as accurately as possible. In the process, you will see how to use FEMM to model such a motor and how accurately its results compare to the real thing.<br />
<h2>
<span style="color: blue;">Measuring key parameters prior to modelling</span></h2>
<div>
Before the motor can be simulated in FEMM we first need to determine its key parameters. The motor to be simulated is an Odrive Robotics N5065 motor which has previously been characterised in <a href="https://things-in-motion.blogspot.com/2018/12/how-to-estimate-torque-of-bldc-pmsm.html">this post</a>. Note that this motor is a pre-production unit that was sent to me for testing by <a href="https://github.com/madcowswe">Oskar of Odrive Robotics</a> and differs slightly (5mm shorter stator, different winding number) from <a href="https://odriverobotics.com/shopfolder/">that offered in his shop</a> today.</div>
<div>
<br /></div>
<div>
The key motor parameters are:</div>
<div>
<ul>
<li><b>The number of slots in the stator and the number of poles in the rotor</b></li>
</ul>
<div>
In order to make this, and the following steps, easier the motor was first disassembled. Disassembly was achieved by the removal of a single <a href="https://www.google.com/search?q=cir+clip&oq=cir+clip&aqs=chrome..69i57.2756j0j7&sourceid=chrome&ie=UTF-8">circlip</a> from the shaft and pulling the rotor bell away from the stator. Some force is required due to the attraction of the magnets to the stator iron. </div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-Zqbsgob73Xs/XFQfNDnCsRI/AAAAAAAAU6U/Nl7-bD8UcsMu7JJg2E4RJqN9bFb8ssv0wCLcBGAs/s1600/motor%2Bdissasembled.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="904" data-original-width="1600" height="225" src="https://2.bp.blogspot.com/-Zqbsgob73Xs/XFQfNDnCsRI/AAAAAAAAU6U/Nl7-bD8UcsMu7JJg2E4RJqN9bFb8ssv0wCLcBGAs/s400/motor%2Bdissasembled.jpg" width="400" /></a></div>
<div>
<br /></div>
<div>
With the rotor bell removed it was simply a matter of counting the number of slots/teeth on the stator and the number of magnets on the rotor. This motor has 12 slots and 14 poles (12n14p). The likely reason that the manufacturer chose this number of slots and poles is explored in detail <a href="https://things-in-motion.blogspot.com/2019/01/selecting-best-pole-and-slot.html">here</a>.</div>
<ul>
<li><b>The stator lamination dimensions and tooth profile</b></li>
</ul>
<div>
The stator lamination dimensions and tooth profile can be difficult to measure directly. If you don't mind destroying your motor then you can remove all the windings and separate a single stator lamination (or just section the entire motor with a slow speed saw) and use a flatbed scanner to capture the dimensions of the lamination without any perspective errors. Including a ruler when scanning will allow you to set a correct scale. Then you could use a software package with edge detection (such as <a href="https://inkscape.org/">Inkscape</a>, which is free) to produce a DXF file without the need for any other CAD software.</div>
<div>
<br /></div>
<div>
In my case, I didn't want to destroy or unwind the motor and so I instead opted to model it as best I could in <a href="https://www.autodesk.com/products/fusion-360/students-teachers-educators">F360</a> (free for hobby/student use) and then export the sketch as a DXF file for FEMM. By doing this in with parametric design software like F360 it also allows me to play around with motor parameters in future without the need to start from scratch.</div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-tx8YvHBoDek/XFQg-D4zqZI/AAAAAAAAU6g/9Y5kCBkR0vkDB0BIQja25QI0Murqdx8_wCLcBGAs/s1600/motor%2Bdimensions.jpg"><img border="0" data-original-height="950" data-original-width="1600" height="236" src="https://4.bp.blogspot.com/-tx8YvHBoDek/XFQg-D4zqZI/AAAAAAAAU6g/9Y5kCBkR0vkDB0BIQja25QI0Murqdx8_wCLcBGAs/s400/motor%2Bdimensions.jpg" width="400" /></a></div>
<div>
The sketch was started with the inner and outer diameters and then a single tooth and its windings were added. This tooth was then copied using the 'circular pattern tool' to make a full stator. </div>
<div>
<br /></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-2PSrftXUIX4/XFQhrnhqg1I/AAAAAAAAU6o/q65t4bElgQUdentfdI_5jOQ2aQLlGOEPACLcBGAs/s1600/F360%2Bsketch.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="834" data-original-width="869" height="383" src="https://2.bp.blogspot.com/-2PSrftXUIX4/XFQhrnhqg1I/AAAAAAAAU6o/q65t4bElgQUdentfdI_5jOQ2aQLlGOEPACLcBGAs/s400/F360%2Bsketch.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">F360 sketch of the stator</td></tr>
</tbody></table>
<div>
While this method is far from ideal it does give a good approximation as we shall see.</div>
<ul>
<li><b>The rotor magnet dimensions and grade</b></li>
</ul>
<div>
The magnets are glued to the rotor bell with epoxy and are also held at the correct separation by segments machined into the end cap and also into an aluminium retaining ring at the open end of the rotor bell.</div>
<div>
<br /></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-84FNuCCVaNM/XFQiFWLxuwI/AAAAAAAAU6w/I-xM_1BMogQv8UPljz0ysSyLp-IK_thrQCLcBGAs/s1600/magnets.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1300" data-original-width="1600" height="325" src="https://1.bp.blogspot.com/-84FNuCCVaNM/XFQiFWLxuwI/AAAAAAAAU6w/I-xM_1BMogQv8UPljz0ysSyLp-IK_thrQCLcBGAs/s400/magnets.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The silver magnets are retained in the black rotor bell with the grey epoxy and the silver coloured aluminium ring in the open end</td></tr>
</tbody></table>
<div>
This approach gives good retention of the magnets but does make it difficult to estimate the actual thickness of these magnets. I made my best approximation by taking the outer bell diameter and the inner flat-to-flat measurement of the magnets and assuming the rotor bell thickness (the black walled region in the image above) is constant. Using this data, and the assumption that the magnets are flat and not curved the rotor was also modelled in F360.</div>
<div>
<br /></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-OW0ow5FXlu0/XFQkeFJtglI/AAAAAAAAU68/eAvFpHlG1DI5no1wcUwW6yX-9YAZLsrIwCLcBGAs/s1600/rotor%2Bdimensions.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="844" data-original-width="882" height="382" src="https://1.bp.blogspot.com/-OW0ow5FXlu0/XFQkeFJtglI/AAAAAAAAU68/eAvFpHlG1DI5no1wcUwW6yX-9YAZLsrIwCLcBGAs/s400/rotor%2Bdimensions.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">F360 sketch of the rotor</td></tr>
</tbody></table>
<div>
Since the magnets are assumed to be perfectly flat and are glued to the inside of a cylinder there is a small 'air space' (actually filled with epoxy) below the magnets. When I imported this sketch as a DXF into FEMM later these small gaps were removed by FEMM. If you wish to maintain them for accuracy then you can force FEMM to recognise them by including a small line segment between the rotor bell and the magnet that crosses the thin air gap region.<br />
<br />
The magnet grade specified by the motor manufactures is N42. If you don't know the magnet grade of your motor then you can make a fairly safe guess by assuming N35 if its a cheaper model or N42 if its a higher end model. As there was in no material option in FEMM for N42, I went with N40 instead. In my case, FEMM suggests that using over N40 over N35 grade magnets leads to a ~7% increase in torque.</div>
<div>
<br /></div>
<div>
The rotor and stator sketches were exported as DXF files ready for importing into FEMM. You can find a copy of DXF files <a href="https://drive.google.com/drive/folders/1liHggQfQWpfRB10stZT3CH4fXWwsAwQJ?usp=sharing">here</a>. Note that FEMM doesn't know the difference between 'construction' geometry generated by F360 and normal sketch lines and so you may want to remove them before exporting as a DXF file.</div>
<ul>
<li><b>The diameter of the wire used, the turn number per tooth and the number of parallel strands</b></li>
</ul>
<div>
Since I didn't want to unwind the motor it was difficult to determine the number of strands and turns per tooth used. Isolating a single strand and measuring it with a calliper suggested the diameter to be around 0.35 mm and so the diameter of the wire without copper insulation was assumed to be 0.34 mm.</div>
<div>
<br /></div>
<div>
The number of parallel strands was estimated by isolating a segment of wire running from tooth to tooth and poking it with a pen until most of the individual strands were visible. </div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-wFFwWfobsZg/XFQmaGTQduI/AAAAAAAAU7I/9SGBY3I08PwfiIyOpWOyafnpGinppgvoQCLcBGAs/s1600/parallel%2Bstrands.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1600" data-original-width="1337" height="400" src="https://1.bp.blogspot.com/-wFFwWfobsZg/XFQmaGTQduI/AAAAAAAAU7I/9SGBY3I08PwfiIyOpWOyafnpGinppgvoQCLcBGAs/s400/parallel%2Bstrands.jpg" width="333" /></a></div>
<div>
<br /></div>
<div>
I count a total of 8 strands. Luckily for me, the number of turns per tooth (ten) was written right on the side of the stator stack.</div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-CQM6Q4GMs0I/XFQnDOMcRLI/AAAAAAAAU7Q/oKYywtRzk84pSyftch8h8t8hUB-a8ZgXgCLcBGAs/s1600/stator%2B10T.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="822" data-original-width="1600" height="205" src="https://1.bp.blogspot.com/-CQM6Q4GMs0I/XFQnDOMcRLI/AAAAAAAAU7Q/oKYywtRzk84pSyftch8h8t8hUB-a8ZgXgCLcBGAs/s400/stator%2B10T.jpg" width="400" /></a></div>
<div>
<br /></div>
<div>
However, it should be possible to estimate the number of turns with a reasonable margin of error from the winding resistance of the motor and the wire diameter if needed.</div>
<ul>
<li><b>The method of winding (i.e. star or delta)</b></li>
</ul>
<div>
In my previous post, I had assumed that this motor is star wound. This turns out not to be the case. As can be seen in the image above one of the leads coming from the motor has 16 parallel strands. This suggests that the motor is delta wound, with the ends of each phase winding (8 parallel strands) grouped together with the next phase for termination to give a total of 16 strands. This was further supported by the fact that that I can't see any region in the motor where a required join would be made for the natural point of a start wound motor. It is likely that all of the motors I previously look at are delta and not star wound.<br />
<br />
Other motor parameters include the lamination thickness, the lamination <a href="https://en.wikipedia.org/wiki/Magnetic_hysteresis">hysteresis loss</a>, <a href="https://en.wikipedia.org/wiki/Skin_effect">skin</a> and <a href="https://en.wikipedia.org/wiki/Proximity_effect_(electromagnetism)">proximity effect</a> of the windings, friction of the bearings etc. can be ignored here as we are only considering the motor under static conditions.</div>
</div>
<h2>
<span style="color: blue;">Modelling a motor in FEMM</span></h2>
<div>
Getting started with FEMM is fairly straight forward if you can look past its outdated interface. Before starting the motor simulation below I would highly recommend following <a href="http://www.femm.info/wiki/MagneticsTutorial">this quick tutorial</a> so that you are familiar with the interface first.<br />
<br />
When ready to model your motor, start a new magnetics problem. From the menu bar select 'problem' and check that you are using the correct units with respect to your DXF file and that have set up a planar simulation. You will also want to check that the 'depth' of your simulation matches the length of your rotor. In this case, my rotor was 30 mm long.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-bBA0ikrFFmA/XMjvRIdopJI/AAAAAAAAWUw/7ZnC9ecOfhkDjj6GUyh3NEx9E2JcrY2ggCLcBGAs/s1600/problem.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="383" data-original-width="279" height="320" src="https://3.bp.blogspot.com/-bBA0ikrFFmA/XMjvRIdopJI/AAAAAAAAWUw/7ZnC9ecOfhkDjj6GUyh3NEx9E2JcrY2ggCLcBGAs/s320/problem.jpg" width="233" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<br />
The next step is to import your DXF file(s) using the file -> import DXF option. This can take some time depending on the amount of geometry. If you have any 'construction' lines left over from your CAD model now is also time to remove those. Note that it is also important that the centre of your model be located at point (0,0) for torque estimation to be conducted later.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-IOUPx3hNuX0/XF92MqwInGI/AAAAAAAAVC4/xl7prWX26ZcfiD0q_w6XQeLVUIY7JfwbQCLcBGAs/s1600/blank%2Bmotor.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1040" data-original-width="1193" height="347" src="https://2.bp.blogspot.com/-IOUPx3hNuX0/XF92MqwInGI/AAAAAAAAVC4/xl7prWX26ZcfiD0q_w6XQeLVUIY7JfwbQCLcBGAs/s400/blank%2Bmotor.jpg" width="400" /></a></div>
<br />
<br />
With your geometry imported its now time to assign material properties. In my case, the rotor bell was assumed to be M-22 Fe-Si steel which is inaccurate. I should have used hot rolled low carbon steel or similar. However, this is unlikely to have had a large impact on the results. The shaft was selected as hot rolled low-carbon steel. The laminations were chosen as M-22 Fe-Si steel which is a common grade used for electric motors.</div>
<div>
<div style="-webkit-text-stroke-width: 0px; color: black; font-family: "Times New Roman"; font-size: medium; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
</div>
<br />
<div style="-webkit-text-stroke-width: 0px; color: black; font-family: "Times New Roman"; font-size: medium; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-decoration-color: initial; text-decoration-style: initial; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="margin: 0px;">
Magnet placement was done using the <a href="https://www.lua.org/">Lua scripting </a>language, which is natively supported by FEMM. You can find a copy of the script I used <a href="https://drive.google.com/file/d/1hGB9EqLWmYZLYx7w13FXPV5sDKvPrzBL/view?usp=sharing">here</a>. Simply enter the number of poles, the radius at which the magnets can be found and the angle (0 degrees being horizontal) that the first magnet can be found. The script will then place each material assignment at the correct orientation. If you wish to change the magnet grade just change the name of the block property in the script (i.e. "NdFeB 40 MGOe") to that of the magnet you wish to use from the list in FEMM.<br />
<br />
The windings were then assigned based on the number of turns and the number of strands. In my case, I created a new 'block property' using the property definition dialogue box. In this case, I selected plain stranded wire with a 0.34 mm diameter and 8 strands and then assigned the same properties used by the other copper conductors.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-tLyS4_yEvbU/XF93WMMvxsI/AAAAAAAAVDE/QHq0-N3G-W0BjHcmqb0VkP4poixtJTdHQCEwYBhgL/s1600/copper%2Bblock%2Bproperties.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="500" data-original-width="401" height="320" src="https://3.bp.blogspot.com/-tLyS4_yEvbU/XF93WMMvxsI/AAAAAAAAVDE/QHq0-N3G-W0BjHcmqb0VkP4poixtJTdHQCEwYBhgL/s320/copper%2Bblock%2Bproperties.jpg" width="256" /></a></div>
<br />
The actual order which the windings were placed was determined using <a href="http://www.bavaria-direct.co.za/scheme/calculator/">this handy winding layout tool</a>. By selecting 12 slots, 14 poles, two-layer winding (single winding per tooth as opposed to a single winding for every second tooth) the following is seen.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-UOHnUw7tpck/XF95roUQMPI/AAAAAAAAVDU/t8A86PffFDs6WguWQKuTh_VOkck-jjiNACLcBGAs/s1600/copper%2Bwinding%2Bdirection.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="850" data-original-width="646" height="400" src="https://4.bp.blogspot.com/-UOHnUw7tpck/XF95roUQMPI/AAAAAAAAVDU/t8A86PffFDs6WguWQKuTh_VOkck-jjiNACLcBGAs/s400/copper%2Bwinding%2Bdirection.jpg" width="303" /></a></div>
<br />
<br />
Using this as a template the winding phase and direction (either into or out of the page) was assigned. This was the most tedious process of creating the motor in FEMM. With all the materials properties assigned we see the following.<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-RaK9AzcdlKk/XFQ1HzV2qvI/AAAAAAAAU8A/SkhxQQcJTV4CsXNgVhhPJ-pks220E2diACLcBGAs/s1600/FEMM%2Bstator%2Bwith%2Bassignments.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="699" data-original-width="748" height="373" src="https://2.bp.blogspot.com/-RaK9AzcdlKk/XFQ1HzV2qvI/AAAAAAAAU8A/SkhxQQcJTV4CsXNgVhhPJ-pks220E2diACLcBGAs/s400/FEMM%2Bstator%2Bwith%2Bassignments.jpg" width="400" /></a></div>
</div>
</div>
The regions filled by air were assigned (one for the air gap region inside the rotor bell and one for outside the motor) and the open boundary builder (circle within a circle button) was used to set the boundary conditions and default settings are fine here.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-yFgkSS238OY/XF94ZRrH_0I/AAAAAAAAVDI/WFE4j_HyrnEOyVI5gLW-MJkxDXa_0a6BwCLcBGAs/s1600/finished%2Bmotor%2Bmodel.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1038" data-original-width="1043" height="318" src="https://2.bp.blogspot.com/-yFgkSS238OY/XF94ZRrH_0I/AAAAAAAAVDI/WFE4j_HyrnEOyVI5gLW-MJkxDXa_0a6BwCLcBGAs/s320/finished%2Bmotor%2Bmodel.jpg" width="320" /></a></div>
<br />
Finally, we need to set how much current is supplied to each winding. Since this motor is driven using <a href="http://scolton.blogspot.com/2010/01/3ph-duo-wrap-up-part-1-field-oriented.html">field oriented control</a> (FOC) we need only supply a current to one phase and a current opposite in sign, and twice as small, for the other two phases as per the figure below.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-Oe1fpDt345M/XF-FXMIznQI/AAAAAAAAVDg/cUFcvupF4sglI4ukYS6bPu3xd-cQkfyOgCLcBGAs/s1600/3%2Bphase.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="923" data-original-width="1227" height="300" src="https://3.bp.blogspot.com/-Oe1fpDt345M/XF-FXMIznQI/AAAAAAAAVDg/cUFcvupF4sglI4ukYS6bPu3xd-cQkfyOgCLcBGAs/s400/3%2Bphase.jpg" width="400" /></a></div>
<br />
In addition, this motor is delta wound and so the current supplied to the phase will be equal to square root(3) of the line current as described <a href="https://www.allaboutcircuits.com/textbook/alternating-current/chpt-10/three-phase-y-delta-configurations/">here</a>. In my <a href="https://things-in-motion.blogspot.com/2018/12/how-to-estimate-torque-of-bldc-pmsm.html">previous post</a> a phase current of 40 A produced a torque of 1.2 N.m. However, this value of 40 A is the line current (or phase current for a star wound motor) and so we need to divide that value by sqrt(3) to get the phase current that our delta wound motor saw. This gives a value of + 23.1 A for phase A, and - 11.55 A for phase B and C. These values are set in the circuit dialogue box with a series connection.<br />
<br />
With this, the motor is now ready for simulating. The complete FEMM file used can also be found <a href="https://drive.google.com/drive/folders/1liHggQfQWpfRB10stZT3CH4fXWwsAwQJ?usp=sharing">here</a>. Note that in order to run this script the minimum angle setting (Problem -> Min. Angle.) needs to be set to 20.<br />
<h2>
<span style="color: blue;">Torque constant estimation</span></h2>
The <a href="https://en.wikipedia.org/wiki/Motor_constants">torque constant</a> (Kt) of an electric motor can be estimated using FEMM with only a single simulation. However, you will want to make sure that your rotor is rotated so that it will produce its maximum torque per amp. In this case, peak motor torque is given when the rotors d-axis is 90 electrical degrees (360 electrical degrees is equal to 360 mechanical degrees divided by the number of pole pairs, which in this case is 7) ahead of the q-axis. A simpler 6n8p motor schematic below for illustration purposes.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-L90oBcwuwfU/XF-KfcbZ3JI/AAAAAAAAVDs/ysjB3rodqyYMTBFUl8Fr8YP4wmN_KkC1wCLcBGAs/s1600/Motor%2Bdiagram.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="906" data-original-width="1257" height="287" src="https://4.bp.blogspot.com/-L90oBcwuwfU/XF-KfcbZ3JI/AAAAAAAAVDs/ysjB3rodqyYMTBFUl8Fr8YP4wmN_KkC1wCLcBGAs/s400/Motor%2Bdiagram.jpg" width="400" /></a></div>
<br />
Simulating the 12n14p N5065 motor with the rotor 90 elec. degrees ahead of the d-axis looks like the following.<br />
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-eiuqGEt888M/XF-Vwy0nOQI/AAAAAAAAVEw/hxAQ3Z7DHfcof4Q_sULpkh4szZMY7LcqACLcBGAs/s1600/motor%2Boutput.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="875" data-original-width="1165" height="300" src="https://2.bp.blogspot.com/-eiuqGEt888M/XF-Vwy0nOQI/AAAAAAAAVEw/hxAQ3Z7DHfcof4Q_sULpkh4szZMY7LcqACLcBGAs/s400/motor%2Boutput.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<br />
To determine the torque produced the rotor needs to be selected using the area selection tool (green box inside four points) by clicking on each of the magnets and the back iron as shown below.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-RKxDEbDTbSc/XF-WA5XHGqI/AAAAAAAAVE0/idKil1osHEwfkh0SL6K8s5uhlNXN_WgWgCLcBGAs/s1600/rotor%2Bselection.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="878" data-original-width="1075" height="326" src="https://4.bp.blogspot.com/-RKxDEbDTbSc/XF-WA5XHGqI/AAAAAAAAVE0/idKil1osHEwfkh0SL6K8s5uhlNXN_WgWgCLcBGAs/s400/rotor%2Bselection.jpg" width="400" /></a></div>
<br />
With rotor selected we can find the torque around point 0.0 using the integrate option and 'Torque via Weighted Stress Tensor' selection from the drop-down menu as recommended in the FEMM manual.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-SYb1IqdDJA4/XF-SiQtrlvI/AAAAAAAAVEQ/sYa7xSaYiXEzfPf6pzlo51MGlxGsLHiEACLcBGAs/s1600/torque%2Bby%2Bweighted%2Bstress%2Btensor.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="317" data-original-width="300" src="https://3.bp.blogspot.com/-SYb1IqdDJA4/XF-SiQtrlvI/AAAAAAAAVEQ/sYa7xSaYiXEzfPf6pzlo51MGlxGsLHiEACLcBGAs/s1600/torque%2Bby%2Bweighted%2Bstress%2Btensor.jpg" /></a></div>
<br />
The result is a torque of 1.37 N.m.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-4EInuOjqY70/XF-UCOi_CAI/AAAAAAAAVEc/-uM6st632J8gsaMDl5UJgLEI-49fs45WQCLcBGAs/s1600/torque%2Bestimated.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="194" data-original-width="321" height="193" src="https://3.bp.blogspot.com/-4EInuOjqY70/XF-UCOi_CAI/AAAAAAAAVEc/-uM6st632J8gsaMDl5UJgLEI-49fs45WQCLcBGAs/s320/torque%2Bestimated.jpg" width="320" /></a></div>
<br />
For reference, the real motor measured 1.2 N.m. Therefore the FEMM value is an overestimation of about 15%. Considering the number of assumptions made with this model I was actually surprised that the FEMM estimation is as close as it is.<br />
<h2>
<span style="color: blue;">Kv, back EMF estimation</span></h2>
The back EMF constant (K<span style="background-color: white;">v) can be estimated using only the torque constant and the motor current as described in <a href="https://things-in-motion.blogspot.com/2018/12/how-to-estimate-torque-of-bldc-pmsm.html">this post</a>. In this case, our line current of 40 A and FEMM modelled torque of 1.37 N.m gives</span><br />
<br />
<span style="text-align: center;"><span style="font-size: large;">$$K_{V} \approx \frac{8.3 \times 40 }{\tau } \approx 242$$</span></span><br />
<br />
The listed Kv of this motor was 270 Kv and so once again we are only off around 15 %.<br />
<br />
In order to estimate the shape of the back EMF we need to do a series of simulations at different rotor angles and plot the torque output. This torque output can then be converted to Kv in the same way as shown in the equation above. In order to automate the rotor incrementing process, I used a Lua script which can be found <a href="https://drive.google.com/file/d/14agNqoDmL7eNf-0-SCkky7ROb2EvSwhW/view?usp=sharing">here</a>. In short, it rotates group one by x degrees around point 0,0 and solves. At the completion of solving it prints the torque around point 0,0 to the console and takes a screenshot of the resulting output. Thefeore the use of this script means that you need to assign each line segment and materials property of the rotor to group one as is shown in this <a href="https://www.youtube.com/watch?v=NZwE-aQRcsw">video</a>. Don't forget to copy the contents of the console output into another file and the completion of your simulation or otherwise modify the script to output data to a file. To use this script, place it in the same folder as your motor's .femm script so that your file is listed in line 8 "open(mydir .. "Your file name here.FEM")". You can also set how many steps you wish to make and how many degrees of rotation you desire.<br />
<br />
Note that solving for each position may take up to 30 seconds and so running hundreds of steps will take many hours to complete. Doing this for the D5065 motor model we see the sinusoidal torque output with rotor angle below.<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-rM3pDczTgW8/XFQwte7FC5I/AAAAAAAAU70/cNttYkq5E48ZCQh-wl6hXlx9EWTjRXRPQCLcBGAs/s1600/torque.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="961" data-original-width="1227" height="312" src="https://2.bp.blogspot.com/-rM3pDczTgW8/XFQwte7FC5I/AAAAAAAAU70/cNttYkq5E48ZCQh-wl6hXlx9EWTjRXRPQCLcBGAs/s400/torque.jpg" width="400" /></a></div>
<br />
This sinusoidal torque with rotor angle, and therefore back EMF as they are <a href="https://things-in-motion.blogspot.com/2018/12/how-to-estimate-torque-of-bldc-pmsm.html">the same thing</a>, agree well with the back EMF seen for this motor.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-9_C_idlOp3w/XHx9JAG6mfI/AAAAAAAAVVs/f1jkBxsUuNUgpk4dl4q9PLYGoZyjybh0ACLcBGAs/s1600/270%2BKv.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1196" data-original-width="1600" height="296" src="https://1.bp.blogspot.com/-9_C_idlOp3w/XHx9JAG6mfI/AAAAAAAAVVs/f1jkBxsUuNUgpk4dl4q9PLYGoZyjybh0ACLcBGAs/s400/270%2BKv.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<br />
<br />
Using the excellent, and free, <a href="https://obsproject.com/">OBS studio</a> in 'slide show' mode was also able to stitch all the separate solutions .bmp files that my script outputted and create this clip of the flux distribution for different rotor angles.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/S855fBqJ5xw/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/S855fBqJ5xw?feature=player_embedded" width="320"></iframe></div>
<br />
<h2>
<span style="color: blue;">Cogging torque estimation</span></h2>
The cogging torque (torque due to the attraction of the rotor magnets to the salient teeth of the stator) was estimated using the same method as the Kv estimation. The key difference being that the armature current was set to zero and a much finer step was used. The figure below shows the raw output data in black which is quite noisy and an FFT smoothed fit of the data in red.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-bLBbtyuPL-E/XFQuaooUI_I/AAAAAAAAU7o/8zKwuGFkqiMEG89mQ2_C5lBOrzTmkC6ZQCLcBGAs/s1600/cogging%2Btorque.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="978" data-original-width="1256" height="311" src="https://2.bp.blogspot.com/-bLBbtyuPL-E/XFQuaooUI_I/AAAAAAAAU7o/8zKwuGFkqiMEG89mQ2_C5lBOrzTmkC6ZQCLcBGAs/s400/cogging%2Btorque.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
As discussed in the last post, the number of cogging steps per rotation is given by the least common multiple (LCM) of the number of stator slots and rotor poles. The LCM of this 12n14p motor is therefore 84. A quick count of the peaks in the smoothed data above and extrapolated out to 1 rotation (360 mechanical degrees) and a cogging frequency of 84 is confirmed. As will be discussed in more detail in a future post, the cogging torque amplitude depends strongly on magnet placement, magnet shape, stator tooth shape etc. and so the magnitude and shape of your cogging torque estimated be FEMM will only be accurate if you have modelled your motor <u><i>exactly</i></u>.<br />
<br />
<h2>
<span style="color: blue;">Modelling motors with symmetry</span></h2>
<div>
While it is not shown here, it is important to point out that it is often not necessary to model an entire motor to determine its properties. For motors with symmetry you may only need to model a small segment of the motor and from that, you can extrapolate the results of a full-size motor. There are also a <a href="http://www.femm.info/wiki/PeriodicBoundaries">few different tools available</a> in FEMM that may be worth exploring.</div>
<h2>
<span style="color: blue;">Alternatives to FEMM</span></h2>
<div>
As can be seen from the results above FEMM is capable of approximating some properties of an electric motor. However, there are many alternatives that are also worth considering. If you would like to use FEMM on platforms other than windows then take a look at the <a href="https://sourceforge.net/p/xfemm/wiki/Home/">xFEMM project</a>. While xFEMM does away with the user interface it is also considerably faster (up to 80x in some instances) and is still a free open source.<br />
<br /></div>
<div>
If you require simulations of 3D bodies or more complicated AC loss analysis etc. then you will need to move on to an expensive commercial FEA package such as the <a href="https://www.google.com/search?q=maxwell+pacakge+ansys&oq=maxwell+pacakge+ansys&aqs=chrome..69i57.3198j0j7&sourceid=chrome&ie=UTF-8">Maxwell from Ansys</a> or <a href="https://www.comsol.com/">Comsol Multiphysics</a>. </div>
<h2>
<span style="color: blue;">Conclusion</span></h2>
</div>
<div>
Modelling an electric motor in FEMM is no easy undertaking. However, by carefully replicating all of the required components it is possible to get a reasonably accurate estimation of a motors Kt, Kv, back EMF shape and cogging torque parameters. Kt and Kv was shown to be within about 15% of that measured for a real motor and so FEMM can act as a nice starting point for verifying ideas when designing an electric motor.</div>
Richard Parsonshttp://www.blogger.com/profile/08331478925576296508noreply@blogger.com5tag:blogger.com,1999:blog-4080979156700236346.post-30082627117669469962019-01-27T15:51:00.002-08:002019-09-27T16:59:48.461-07:00Selecting the best pole and slot combination for a BLDC (PMSM) motor with concentrated windingsThe<a href="https://docs.google.com/spreadsheets/d/1AZ2w6lbniuLydnSUgLaUv4zhjWA-wICHkOnHHVaU8Mg/edit?usp=sharing"> full spreadsheet used in this post is available for download</a>. To modify the sheet just choose the 'Make a copy option' from the drop-down file menu.<br />
<h2>
Introduction </h2>
Concentrated windings have a single coil per tooth and are commonly used on 'hobbyist' style <a href="https://things-in-motion.blogspot.com/2018/12/why-most-hobby-grade-bldc-out-runners.html">BLDC (PMSM)</a> out-runner motors.<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-fmI1igpdTJU/XE4pWGVDSEI/AAAAAAAAU2s/yOoBGXp4W_o47VUILtW2ioVV7J609xGnwCLcBGAs/s1600/concentrated%2Bvs%2Bdistributed%2Bwindings.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="725" data-original-width="1394" height="207" src="https://1.bp.blogspot.com/-fmI1igpdTJU/XE4pWGVDSEI/AAAAAAAAU2s/yOoBGXp4W_o47VUILtW2ioVV7J609xGnwCLcBGAs/s400/concentrated%2Bvs%2Bdistributed%2Bwindings.jpg" width="400" /></a></div>
<br />
A key advantage of concentrated windings is that they can be quickly and cheaply wound by machine as seen in the video below.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/UtKjrGkcaLs/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/UtKjrGkcaLs?feature=player_embedded" width="320"></iframe></div>
<br />
Even motors with very thick conductors can be wound by machine.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/MuqUG_MC_DA/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/MuqUG_MC_DA?feature=player_embedded" width="320"></iframe></div>
<br />
Other advantages include short end turns, which do not contribute torque to the motor and only increase the winding resistance, and space for effective air cooling. A major drawback of concentrated windings is that without careful consideration of the stator slot number and rotor pole number the performance of the motor will be poor.<br />
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-o7-VCuPDtc8/XE4swg9Y9_I/AAAAAAAAU3A/oHY7eVal69cqovXQNuFVJgOksev75l-_ACLcBGAs/s1600/motor_schematic.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="844" data-original-width="1427" height="236" src="https://2.bp.blogspot.com/-o7-VCuPDtc8/XE4swg9Y9_I/AAAAAAAAU3A/oHY7eVal69cqovXQNuFVJgOksev75l-_ACLcBGAs/s400/motor_schematic.jpg" width="400" /></a></div>
<br />
This post will examine the advantages and disadvantages of different slot and pole combinations. The information is based off the paper '<a href="http://www.elkraft.ntnu.no/eno/Papers2006/icem-skaar-krovel-nilssen06.pdf">Distribution, coil-span and winding factors for PM machines with concentrated windings</a>' by S.E Skaar et al. and the book <a href="https://books.google.com.au/books/about/Design_of_Brushless_Permanent_magnet_Mac.html?id=n833QwAACAAJ&source=kp_book_description&redir_esc=y">'Design of Brushless Permanent-magnet Machines'</a> by J. R. Hendershot and T. Miller. See the 'recommended reading' list above for more information on this book. The<a href="https://www.emetor.com/glossary/"> emotor.com glossary page</a> is also useful for reference.<br />
<h2>
The winding factor</h2>
An electric motors 'winding factor' (not to be confused with its copper fill factor) is a number between 0 and 1 which represents the fraction of the armature current which is used to produce torque.<br />
<br />
From <a href="https://www.emetor.com/glossary/winding-factor/">emotor.com's glossary page</a> the winding factor is defined as the following:<br />
<blockquote class="tr_bq">
<em style="background-color: white; box-sizing: border-box; color: #212529; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, "Helvetica Neue", Arial, sans-serif, "Apple Color Emoji", "Segoe UI Emoji", "Segoe UI Symbol"; font-size: 16px;">The winding factor for a specific winding expresses the ratio of flux linked by that winding compared to flux that would have been linked by a single-layer full-pitch non-skewed integer-slot winding with the same number of turns and one single slot per pole per phase. The torque of an electric motor is proportional to the fundamental winding factor.</em></blockquote>
Despite being so fundamental, the calculation of the winding factor is not often discussed in textbooks. I found that the <a href="http://www.elkraft.ntnu.no/eno/Papers2006/icem-skaar-krovel-nilssen06.pdf">paper mentioned above</a> has the easiest to follow description, although I had to rely on the <a href="https://www.emetor.com/glossary/unbalanced-winding/">emotor.com description of an unbalanced winding</a> as the solution provided in the paper appears to not cover all scenarios.<br />
<br />
I will not be going into detail about the calculation of the winding factor. However, you can find the spreadsheet used to calculate the values shown in this post <a href="https://docs.google.com/spreadsheets/d/1AZ2w6lbniuLydnSUgLaUv4zhjWA-wICHkOnHHVaU8Mg/edit?usp=sharing">here</a>. This spreadsheet includes up to 108 slots and 70 poles and is easily extended further where needed.<br />
<br />
The winding factor for a 3 phase machine with a non-skewed rotor can be seen listed in the table below where the top row is the number of poles and the left-hand column is the number slots. Since this is a three-phase machine the number of slots increases by three and since a magnet has two poles (a pole pair) the pole number increases by two.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-wnhzdRY9Q9k/XE0jcqS-kCI/AAAAAAAAU0Q/i2irL-cGYmoTovA8bDvuxZW2-fHG6Ie-wCLcBGAs/s1600/no_exclusion.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="241" data-original-width="769" height="125" src="https://4.bp.blogspot.com/-wnhzdRY9Q9k/XE0jcqS-kCI/AAAAAAAAU0Q/i2irL-cGYmoTovA8bDvuxZW2-fHG6Ie-wCLcBGAs/s400/no_exclusion.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
It can be seen that some slot and pole combination have a winding factor of one while others are approaching zero, or even negative. However, not all of these slot and pole combinations can be used as is described below.<br />
<h2>
1. Exclude windings where q < 0.25</h2>
The slot to pole ratio with consideration for the number of phases is designated by the variable q. If q is less than 0.25 then the arc covered by a rotor pole is now less than half a stator tooth. This results in multiple north and south magnet poles interacting with each stator tooth and so the torque generated by the motor is reduced. Therefore, q values of less than 0.25 are generally not considered feasible and can be eliminated. The images below were created with <a href="http://www.bavaria-direct.co.za/scheme/calculator/">this winding layout tool.</a> Note that in reality, the gap between each stator tooth would be much smaller.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-3Iw5wTQb6dY/XE0ea3fSnXI/AAAAAAAAUzc/HERSd83_IaoN_6QqksFLvJtzZVvFkXrTgCLcBGAs/s1600/q_0.25_orless.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="594" data-original-width="1041" height="227" src="https://3.bp.blogspot.com/-3Iw5wTQb6dY/XE0ea3fSnXI/AAAAAAAAUzc/HERSd83_IaoN_6QqksFLvJtzZVvFkXrTgCLcBGAs/s400/q_0.25_orless.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Example winding layouts that give a q value that is not considered feasible.</td></tr>
</tbody></table>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-rwnA057RfP0/XY6hwpfDUfI/AAAAAAAAYDw/5m0_R9ivo0Qg1Ewq8PlnZt5fcvrnpIOnwCNcBGAsYHQ/s1600/q-vales.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="243" data-original-width="736" height="131" src="https://1.bp.blogspot.com/-rwnA057RfP0/XY6hwpfDUfI/AAAAAAAAYDw/5m0_R9ivo0Qg1Ewq8PlnZt5fcvrnpIOnwCNcBGAsYHQ/s400/q-vales.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The calculated q-values for different slot and plot combinations.</td></tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-PquAnXA2qd8/XE0jk4rpmxI/AAAAAAAAU0U/5l2gliF8ewIs0Ihg4W2QslQ8xw0aY5R5ACLcBGAs/s1600/q%2Bless%2Bthan%2B0.25.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="241" data-original-width="769" height="125" src="https://1.bp.blogspot.com/-PquAnXA2qd8/XE0jk4rpmxI/AAAAAAAAU0U/5l2gliF8ewIs0Ihg4W2QslQ8xw0aY5R5ACLcBGAs/s400/q%2Bless%2Bthan%2B0.25.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The winding factor of each slot and pole combination with those combinations that give q<0.25 removed.</td></tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<h2>
2. Exclude windings where q > 0.5</h2>
Alternatively, if q is greater than 0.5 then it no longer makes sense to use a concentrated winding as a single rotor pole will span over multiple teeth. Instead, a distributed windings would be used. Therefore, these combinations can also be excluded for our purposes.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-Kx2S8i-0-qA/XE0heFyMIQI/AAAAAAAAUz8/VU96_AhdM-kpZ36h7dL44N3zzl-GenylQCLcBGAs/s1600/q_greater_than_0%252C5.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="606" data-original-width="1056" height="228" src="https://3.bp.blogspot.com/-Kx2S8i-0-qA/XE0heFyMIQI/AAAAAAAAUz8/VU96_AhdM-kpZ36h7dL44N3zzl-GenylQCLcBGAs/s400/q_greater_than_0%252C5.jpg" width="400" /></a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-roeuRGYI69o/XE0jpWk51VI/AAAAAAAAU0Y/5Glyrvw2g1Q-MyouvgVVLuqGsOK-jKOiQCLcBGAs/s1600/q%2Bgreater%2Bthan%2B0.5.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="241" data-original-width="769" height="125" src="https://4.bp.blogspot.com/-roeuRGYI69o/XE0jpWk51VI/AAAAAAAAU0Y/5Glyrvw2g1Q-MyouvgVVLuqGsOK-jKOiQCLcBGAs/s400/q%2Bgreater%2Bthan%2B0.5.jpg" width="400" /></a></div>
<br />
<h2>
3. Exclude windings where Ns = Nm</h2>
If the number of slots (Ns) is equal to the number of poles (Nm) then the motor will produce large cogging torque and will no longer be self stating. This combination can therefore also be eliminated.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-o4RpkgCl0Hw/XE0kpIhK0YI/AAAAAAAAU0s/YXfq_HMwS0Yg4agj1cDNfKjd-BmabRA3ACLcBGAs/s1600/Ns%2Bequals%2BNm.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="241" data-original-width="769" height="125" src="https://3.bp.blogspot.com/-o4RpkgCl0Hw/XE0kpIhK0YI/AAAAAAAAU0s/YXfq_HMwS0Yg4agj1cDNfKjd-BmabRA3ACLcBGAs/s400/Ns%2Bequals%2BNm.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<h2>
4. Removal of unbalanced windings and motors without symmetry</h2>
<div>
A motor with balanced windings will have the same number of coils for each phase per repeating segment of the motor. A motor must have balanced windings for operation. More information <a href="https://www.emetor.com/glossary/unbalanced-winding/">here</a>. </div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-7kEw3-aSXI4/XE0yyUHnfII/AAAAAAAAU04/XeJ-ZOgVvr0UheuQIhHNCP7bIFMZjjUcwCLcBGAs/s1600/unbalanced%2Bvs%2Bbalanced.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="513" data-original-width="943" height="217" src="https://4.bp.blogspot.com/-7kEw3-aSXI4/XE0yyUHnfII/AAAAAAAAU04/XeJ-ZOgVvr0UheuQIhHNCP7bIFMZjjUcwCLcBGAs/s400/unbalanced%2Bvs%2Bbalanced.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div>
Also, it is ideal to have a motor that has a symmetry of at least 2 (i.e. the motor has two repeating sections) as this helps avoid unbalanced radial forces and noisy operation. In other words, a motor without any symmetry will produce its torque on only one side of the rotor. If a motor has a symmetry of two then its torque will be produced on opposite sides of the rotor, balancing the forces.<br />
<br /></div>
<div>
The slot and pole combinations that produced unbalanced windings and no symmetry can, therefore, be eliminated.</div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-PLvQGrtExqk/XE1EejyhlWI/AAAAAAAAU1U/WNqFSUFJr9Q4pBmQarpB2LxDhlFDNo0TgCLcBGAs/s1600/NoSym%2Band%2Bunbalanced.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="241" data-original-width="769" height="125" src="https://4.bp.blogspot.com/-PLvQGrtExqk/XE1EejyhlWI/AAAAAAAAU1U/WNqFSUFJr9Q4pBmQarpB2LxDhlFDNo0TgCLcBGAs/s400/NoSym%2Band%2Bunbalanced.jpg" width="400" /></a></div>
<br />
Based on the table above it can be seen that 12s10p and 12s14p are attractive combinations. This explains why these slot and pole combinations are so popular for 'hobbyist' out-runner electric motors.<br />
<h2>
5. Consideration of cogging torque</h2>
<div>
In addition to the winding factor, cogging torque is also an important consideration. Cogging torque creates vibration and noise during operation and acts to disturb the motor away from its desired position when used in servo applications. The cogging torque frequency is also closely correlated with the production of rotor losses both with and without current supplied to the armature.</div>
<div>
<br /></div>
<div>
The cogging torque frequency is given by the least common multiple (LCM) of the stator slots and the rotor poles. The figure below displays the possible slot and pole combinations with the winding factor replaced with the cogging torque frequency. </div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-tSBb6Ne-mCM/XE1MSGFA5NI/AAAAAAAAU1s/9zbh-o_DQYAfvxCImzjoQWRrLXeQfVvRwCLcBGAs/s1600/cogging%2Bfreq.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="241" data-original-width="769" height="125" src="https://2.bp.blogspot.com/-tSBb6Ne-mCM/XE1MSGFA5NI/AAAAAAAAU1s/9zbh-o_DQYAfvxCImzjoQWRrLXeQfVvRwCLcBGAs/s400/cogging%2Bfreq.jpg" width="400" /></a></div>
<br />
Selecting as high a cogging frequency as possible is desirable as it reduces the amplitude of the cogging torque. With this in mind, 24s22p would be an attractive option with its 264 cogging steps per rotation and 0.949 winding factor as opposed to only 84 cogging steps per rotation for 12s14p.<br />
<h2>
6. Higher order harmonics</h2>
<div>
So far we have only been considering the fundamental winding factor. However, different slot and pole combinations also affect the winding space harmonics. This topic will be covered in more detail in a future post once I have a better understanding of the topic.</div>
<h2>
7. Additional considerations</h2>
<div>
From the last two tables above its easy to conclude that a 24s22p motor is 'better' than a 12s14p motor since it has both a higher winding factor and a higher cogging frequency. However, there are other factors that also need to be considered.</div>
<h2>
7. 1 Maximum electrical frequency</h2>
<div>
The electrical frequency (current sine-wave supplied to each phase) scales linearly with the number of pole pairs (Nm/2) of a motor. Therefore, doubling the pole count of a motor will increase its core losses by a factor of four since it scales with the square of the electrical frequency. Doubling the pole count will also double the back EMF produced and so twice the required voltage will be needed to drive the motor at the same RPM. However, rewinding of the motor with a lower number of conductors per tooth can be used to offset this increase in the back EMF and <a href="https://things-in-motion.blogspot.com/2018/11/understanding-bldc-electric-motor.html">will not impact the efficiency of the motor</a>. In addition, reducing the lamination thickness will help minimise eddy current losses, which scale with the square of the lamination thickness. See <a href="https://docs.google.com/spreadsheets/d/14JMvWJfZmX69oHY8pMpoEo147WztYAYIlpZnncWnf7Y/edit?usp=sharing">this spreadsheet</a> for more details. At the extreme, the switching frequency of the motor controller used to drive the motor may also need to be increased in order to maintain a good approximation of a sine wave. This will incur additional dead time losses in the motor controller. </div>
<div>
<br /></div>
<div>
The electrical frequency required to operate the motor at 10,000 RPM is shown below.</div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-Mwv1h1i2WZc/XE46rxIpMLI/AAAAAAAAU3M/FCZqDSU0PbkPxb0PlK-YdCVUqk54slNPQCLcBGAs/s1600/electrical%2Bfreq.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="221" data-original-width="769" height="113" src="https://2.bp.blogspot.com/-Mwv1h1i2WZc/XE46rxIpMLI/AAAAAAAAU3M/FCZqDSU0PbkPxb0PlK-YdCVUqk54slNPQCLcBGAs/s400/electrical%2Bfreq.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div>
<h2>
7. 2 Mechanical winding considerations</h2>
</div>
<div>
Trying to squeeze more slots into a small motor is not always possible. Increasing the number of slots while keeping the motor diameter the same typically means that multiple smaller conductors must be used in order to bend around the tighter radii and make use of the available space. Smaller conductors have a larger fraction of their total cross section taken up by insulation and so the copper fill factor of the motor can be reduced even if the winding factor is increased. A lower copper fill factor means a higher current density in the winding and so higher I^2R losses.</div>
<div>
<h2>
7. 3 Cooling considerations</h2>
</div>
<div>
Effective cooling is critical for a high power density electric motor. Having a large number of slots and small conductors may congest the air flow path and reduce air cooling effectiveness. This, in combination with higher core losses due to an increase in the electrical frequency, can reduce the power density of a motor.</div>
<div>
<h2>
7. 4 Tooth-tip leakage</h2>
</div>
<div>
Squeezing more slots into a motor may also require that the gaps between the teeth be reduced. Reducing the size of this gap can allow some flux to jump (or zig-zag) from one tooth to the next, reducing the effective torque produced by the motor.</div>
<div>
<h2>
7. 5 Rotor skewing</h2>
</div>
<div>
If you plan on skewing your motor in an effort to reduce its cogging torque then it is important that the required skew does not lower the winding factor too much. The reduction in the winding factor is given by the skew factor discussed <a href="https://www.emetor.com/glossary/winding-factor/">here</a>. This topic will be covered in more detail in a future post.</div>
<div>
<span style="color: red;"></span><br />
<h2>
7. 6 Reduced rotor and stator back-iron (yoke) mass</h2>
</div>
<div>
Increasing the pole and slot count of an electric motor has the advantage of reducing the flux density in the back-iron. This is because the higher number of stator slots and rotor poles means that flux does not need to travel as far around the stator or rotor to make a closed magnetic path. For every doubling of the pole or slot number, the flux density is halved and so the thickness, and therefore mass, of the back-iron can also be halved. This acts to increase the power density of the motor. </div>
<div>
<br /></div>
<div>
For example, Siemens appears to have increased the gravimetric torque density (torque per unit mass) by 50% for their 260 kW light aircraft electric motor by increasing its pole and slot count from an already high 36s30p to 72s60p. Note that this assumption is based solely on the images from their <a href="https://www.engineering.com/DesignSoftware/DesignSoftwareArticles/ArticleID/17821/Siemens-Electric-Aircraft-Propulsion-Unit-Inside-the-Digital-Twin-Design-Strategy.aspx">promotional material</a>.</div>
<h2>
Conclusion</h2>
<div>
The tables above provide guidance when selecting the number of slots and poles for a BLDC (PMSM) motor with concentrated windings. If the diameter of your motor is less than approximately 60 mm and if your maximum RPM is less than 10,000 then 10s14p or 12s14p may be a good choice. For larger diameter and/or motors that rotate more slowly a higher slot and pole count can be used, such as 24s26p with a reduced number of turns per tooth and thinner Fe-Si laminations. Higher pole count motors will have a higher electrical frequency, increasing losses, but will also have a lower mass since a reduced back-iron thickness can be used while also having a smaller cogging torque amplitude.</div>
Richard Parsonshttp://www.blogger.com/profile/08331478925576296508noreply@blogger.com34tag:blogger.com,1999:blog-4080979156700236346.post-68588552022352024572018-12-27T13:49:00.000-08:002018-12-28T01:48:06.874-08:00The advantages and disadvantages of using a Halbach array with a BLDC (PMSM) motorIn the <a href="https://things-in-motion.blogspot.com/2018/12/understanding-bldc-pmsm-electric-motor_27.html" target="_blank">last post</a> it was shown that the length of the rotor magnets has an impact on the specific torque density of an electric motor. Longer magnets will, in general, produce a larger flux density at the poles but this comes at the expense of a larger flux gap. Therefore, the optimum magnet length for our motor model was around 2-4 mm.<br />
<br />
However, there is an alternative arrangement of magnets in the rotor that, according to many a forum post all over the internet, will significantly increase the torque density of any motor.<br />
<br />
<h2>
<span style="color: blue;">The Halbach array</span></h2>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-sMJi4MZXz4M/XBToDHI8h6I/AAAAAAAAUGE/0ToZCwap1WwoNqkDQw0PMzLdclD95VzPwCLcBGAs/s1600/Halbach_array_field.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="774" data-original-width="1119" height="276" src="https://4.bp.blogspot.com/-sMJi4MZXz4M/XBToDHI8h6I/AAAAAAAAUGE/0ToZCwap1WwoNqkDQw0PMzLdclD95VzPwCLcBGAs/s400/Halbach_array_field.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Credit: Wikipedia</td></tr>
</tbody></table>
If you are not familiar with the concept of a Halbach array then its <a href="https://en.wikipedia.org/wiki/Halbach_array" target="_blank">wiki page</a> has all the relevant information. In this post we will be testing the impact of adding a simple Halbach array to our model motor in <a href="http://www.femm.info/wiki/HomePage" target="_blank">FEMM</a>.<br />
<div>
<h2>
<span style="color: blue;">Four different scenarios for a simple motor model</span></h2>
While more details regarding this motor can be found <a href="https://things-in-motion.blogspot.com/2018/12/understanding-bldc-pmsm-electric-motor.html" target="_blank">in this post </a>a brief description is as follows: It is a 6 slot, 8 pole motor with three phases wound as concentrated windings in a pattern of ABCABC. All current in the windings is on the q-axis and in this case there is a 4.5 mm flux gap with 4 mm long magnets. FEMM simulation results shown below.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-gEAhDmHrJHw/XAdRt6F0KmI/AAAAAAAAT5w/GJQ_09MoMQswmuLn5h2kRuFcrkSLqjTcACLcBGAs/s1600/4mm%2Bwith%2Bbackiron%2Band%2Bno%2Bhalbach%2B0.714%2BN.m.jpg" imageanchor="1"><img border="0" data-original-height="547" data-original-width="567" height="308" src="https://4.bp.blogspot.com/-gEAhDmHrJHw/XAdRt6F0KmI/AAAAAAAAT5w/GJQ_09MoMQswmuLn5h2kRuFcrkSLqjTcACLcBGAs/s320/4mm%2Bwith%2Bbackiron%2Band%2Bno%2Bhalbach%2B0.714%2BN.m.jpg" width="320" /></a></div>
<div style="font-size: 12.8px; text-align: center;">
<span style="font-size: 12.8px;"><br /></span></div>
<div style="text-align: center;">
<span style="color: blue; font-size: large;">Back iron only</span></div>
<div style="font-size: 12.8px; text-align: center;">
<b><span style="font-size: large;">Torque output: 0.714 N.m </span></b></div>
<div style="font-size: 12.8px; text-align: center;">
<br /></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-tdfAlesXWlY/XAdRuYJjUVI/AAAAAAAAT54/VzYH_h6mYsMHqOM8ylZt2fCjL6Vs91QHgCLcBGAs/s1600/4mm%2Bwith%2Bbackiron%2Band%2Bno%2Bhalbach%2B0.717%2BN.m.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="556" data-original-width="551" height="320" src="https://2.bp.blogspot.com/-tdfAlesXWlY/XAdRuYJjUVI/AAAAAAAAT54/VzYH_h6mYsMHqOM8ylZt2fCjL6Vs91QHgCLcBGAs/s320/4mm%2Bwith%2Bbackiron%2Band%2Bno%2Bhalbach%2B0.717%2BN.m.jpg" width="317" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div style="text-align: center;">
<span style="color: blue; font-size: large;">Back iron and Halbach</span></div>
<div style="font-size: 12.8px; text-align: center;">
<b><span style="font-size: large;">Torque output: 0.714 N.m </span></b></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-38rZonesS98/XAdRuBlgWgI/AAAAAAAAT50/doe4j4Og5f48bUsguQenhmJHKm29wxX_QCLcBGAs/s1600/4mm%2Bwithout%2Bbackiron%2Band%2Bwith%2Bhalbach%2B0.683%2BN.m.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="585" data-original-width="578" height="320" src="https://3.bp.blogspot.com/-38rZonesS98/XAdRuBlgWgI/AAAAAAAAT50/doe4j4Og5f48bUsguQenhmJHKm29wxX_QCLcBGAs/s320/4mm%2Bwithout%2Bbackiron%2Band%2Bwith%2Bhalbach%2B0.683%2BN.m.jpg" width="316" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div style="text-align: center;">
<span style="color: blue; font-size: large;">Halbach Only</span></div>
<div style="font-size: 12.8px; text-align: center;">
<b><span style="font-size: large;">Torque output: 0.683 N.m </span></b></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-jhZYKp-zIP0/XAdRu8Ix3II/AAAAAAAAT58/pEx5LOtVekEVNTGzNlu6M6rtxq3JzW2ewCLcBGAs/s1600/4mm%2Bwithout%2Bbackiron%2Band%2Bwith%2Bno%2Bhalbach%2B0.435%2BN.m.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="572" data-original-width="574" height="318" src="https://4.bp.blogspot.com/-jhZYKp-zIP0/XAdRu8Ix3II/AAAAAAAAT58/pEx5LOtVekEVNTGzNlu6M6rtxq3JzW2ewCLcBGAs/s320/4mm%2Bwithout%2Bbackiron%2Band%2Bwith%2Bno%2Bhalbach%2B0.435%2BN.m.jpg" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div style="text-align: center;">
<span style="color: blue; font-size: large;">Neither back iron or Halbach</span></div>
<div style="text-align: center;">
<div style="font-size: 12.8px;">
<b><span style="font-size: large;">Torque output: 0.435 N.m </span></b></div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
It's clear that adding a Halbach style arrangement of magnets to the rotor has no impact on the torque produced when back iron is also used. However, it makes a considerable difference when the rotor back iron is removed, giving roughly 50% more torque than the non-Halbach arrangement. </div>
<div style="text-align: left;">
<br /></div>
<h2 style="text-align: left;">
<span style="color: blue;">A simplified example</span></h2>
<div style="text-align: left;">
The reason for this is clear when you look at a simplified arrangement of magnets.</div>
<div style="text-align: left;">
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-zaKyNc8hEU4/XBXQEXJS6wI/AAAAAAAAUHM/J-EFCB89z4AdLH1MdLyuwsSHd4e5Ggx-ACLcBGAs/s1600/no%2Bhallbach%2Bwith%2Bbackiron.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="360" data-original-width="683" height="168" src="https://3.bp.blogspot.com/-zaKyNc8hEU4/XBXQEXJS6wI/AAAAAAAAUHM/J-EFCB89z4AdLH1MdLyuwsSHd4e5Ggx-ACLcBGAs/s320/no%2Bhallbach%2Bwith%2Bbackiron.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="color: blue; font-size: large;">Back iron only</span></td></tr>
</tbody></table>
<br /></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-ic5Dvzc8D1U/XBXQDpfXuLI/AAAAAAAAUHI/QQAYijqhJOY6E_k65hO9gH6KnJiyeCOzwCLcBGAs/s1600/halbach%2Bwith%2Bback%2Biron.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="405" data-original-width="689" height="188" src="https://2.bp.blogspot.com/-ic5Dvzc8D1U/XBXQDpfXuLI/AAAAAAAAUHI/QQAYijqhJOY6E_k65hO9gH6KnJiyeCOzwCLcBGAs/s320/halbach%2Bwith%2Bback%2Biron.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="color: blue; font-size: large;">Back iron and Halbach</span></td></tr>
</tbody></table>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-wN5xjtX3TFg/XBXQDgtvMuI/AAAAAAAAUHE/cBSz0pYQ3UgzlbX9d3wbKTgCBZSqXY0hwCLcBGAs/s1600/halbach%2Bwithout%2Bback%2Biron.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="376" data-original-width="698" height="172" src="https://3.bp.blogspot.com/-wN5xjtX3TFg/XBXQDgtvMuI/AAAAAAAAUHE/cBSz0pYQ3UgzlbX9d3wbKTgCBZSqXY0hwCLcBGAs/s320/halbach%2Bwithout%2Bback%2Biron.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="color: blue; font-size: large;">Halbach Only</span></td></tr>
</tbody></table>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-C5zM4BCz9mA/XBXQDLz2fnI/AAAAAAAAUHA/7r3mXxVlv20w2frKz4gB2af8mZXQhueMwCLcBGAs/s1600/no%2Bhalbach%2Bwith%2Bno%2Bback%2Biron.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="365" data-original-width="686" height="170" src="https://1.bp.blogspot.com/-C5zM4BCz9mA/XBXQDLz2fnI/AAAAAAAAUHA/7r3mXxVlv20w2frKz4gB2af8mZXQhueMwCLcBGAs/s320/no%2Bhalbach%2Bwith%2Bno%2Bback%2Biron.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="color: blue; font-size: large;">Neither back iron or Halbach</span></td></tr>
</tbody></table>
<br />
<div style="text-align: left;">
The back iron produces a high magnetic permeability path for the permanent magnet 'flux' to pass through. The use of a Halbach array eliminates the need for back iron and so the use of both a Halbach array and back iron will only increase the cost and weight of a motor, with no improvement in performance.<br />
<br />
If we draw a downwards line from the surface of the exposed magnets in the images above and plot the flux density at each point we see the following.<br />
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-lL2BMUbsJEA/XCV3CJYcs8I/AAAAAAAAUa0/dH-h1bl2qQ8VunNdZhSRmxbOdKagqknbACLcBGAs/s1600/halbach.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="889" data-original-width="1187" height="298" src="https://1.bp.blogspot.com/-lL2BMUbsJEA/XCV3CJYcs8I/AAAAAAAAUa0/dH-h1bl2qQ8VunNdZhSRmxbOdKagqknbACLcBGAs/s400/halbach.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<br />
It is clear that the flux density is the same with or without the Halbach configuration provided that back iron is used. Therefore, a Halbach array does not act to redirect the flux from one side of a magnet, concentrating it on the other, as is sometimes stated. Therefore, it only makes sense to use a Halbach array when designing a motor which has no 'back iron' in the rotor. However, having no back iron in the rotor is essentially the same as having an infinite flux gap. As was shown in the last post, the highest torque density was achieved for our model motor when the flux gap was kept small using thin magnets. So not only would a Halbach configuration for this motor be considerably more expensive to manufacture, it would also have a lower torque density.<br />
<br /></div>
<div style="text-align: left;">
There are of course exceptions to this example. Completely core-less electric motors (no stator or rotor iron) <a href="http://build-its-inprogress.blogspot.com/2015/02/coreless-axial-flux-motors.html" target="_blank">such as this example</a> often use Halbach arrays as a means to increase their otherwise terrible torque density. The benefit to this design is that the lack of iron core losses means that these motors can be quite efficient provided eddy currents in the windings and magnets is adequately controlled. They also produce no cogging torque. As far as I can tell, core-less motors are also popular among hobbyist because they eliminate the need to cut your own Fe-Si steel lamination. If you are looking to make a one off custom motor as a hobbyist my advice would be to re-use an off the shelf stator (either new or from a donor motor) and modify it to your own needs. This approach will always give you a higher torque density than a core-less motor and will often be cheaper since you don't need to purchase as many, or as large, expensive magnets or <a href="https://en.wikipedia.org/wiki/Litz_wire" target="_blank">litz wire</a> for the windings. </div>
</div>
</div>
Richard Parsonshttp://www.blogger.com/profile/08331478925576296508noreply@blogger.com7tag:blogger.com,1999:blog-4080979156700236346.post-26389524358913502712018-12-27T02:35:00.001-08:002018-12-28T01:56:05.370-08:00Understanding BLDC (PMSM) electric motor constants - Optimal magnet length for high torque densityIn the last post it was shown that the torque density of a motor can be improved by making the flux gap as small as possible. It was also seen that the rotor magnets are considered part of the flux gap. Therefore, it would appear that an ideal motor will always have rotor magnets that are as thin as possible.<br />
<br />
However, it is also well known that the longer you make a permanent magnet, the larger the flux density at its surface. This raises an important question: <b>Are long magnets and a large flux gap better than short magnets and a small flux gap when it comes to producing the most torque?</b><br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-U5MgQG-GoCA/XBSbZqnEuXI/AAAAAAAAUEc/rWRenUprhZ8ObQnFPlzYoLHx0A2W4IueACLcBGAs/s1600/Magnet%2Blength.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="748" data-original-width="1179" height="253" src="https://3.bp.blogspot.com/-U5MgQG-GoCA/XBSbZqnEuXI/AAAAAAAAUEc/rWRenUprhZ8ObQnFPlzYoLHx0A2W4IueACLcBGAs/s400/Magnet%2Blength.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Small flux gap and magnets on the left, large magnets and flux gap on the right.</td></tr>
</tbody></table>
<h2>
<span style="color: blue;">The short answer: </span></h2>
For a 'hobby grade' out-runner electric motor the optimal magnet length will depend on your exact motor design, but in general, it will likely be around 1 to 4mm in length. Shorter magnets see a rapid fall off in their flux contribution to the airgap while longer magnets increase the magnetic reluctance of the magnetic circuit, reducing the stator contribution. Very long magnets will also cause the stator and rotor back iron to saturate which reduces performance.<br />
<br />
Read on if you would like a more detailed understanding.<br />
<h2>
<span style="color: blue;">Permanent magnet self demagnetisation</span></h2>
<div>
Below are four magnets modelled in <a href="http://www.femm.info/wiki/HomePage" target="_blank">FEMM</a> with a length of 1, 2, 4 and 8 mm </div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-Kf4hps_PFV0/W_5-GWwvNxI/AAAAAAAATtU/EhgWjxA5bXwTiZwxY-F8eJ6ZaqQ4gjI7wCLcBGAs/s1600/2mm%2Bmagnet.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="545" data-original-width="558" height="195" src="https://1.bp.blogspot.com/-Kf4hps_PFV0/W_5-GWwvNxI/AAAAAAAATtU/EhgWjxA5bXwTiZwxY-F8eJ6ZaqQ4gjI7wCLcBGAs/s200/2mm%2Bmagnet.jpg" width="200" /></a><a href="https://4.bp.blogspot.com/-_qbEdZB-i_Q/W_5-GUlAkNI/AAAAAAAATtQ/uCoXZzLA_R8houi6b14RI1KzmNwGvZZgACLcBGAs/s1600/1mm%2Bmagnet.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="538" data-original-width="547" height="196" src="https://4.bp.blogspot.com/-_qbEdZB-i_Q/W_5-GUlAkNI/AAAAAAAATtQ/uCoXZzLA_R8houi6b14RI1KzmNwGvZZgACLcBGAs/s200/1mm%2Bmagnet.jpg" width="200" /></a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-PuaLwcDpCVw/W_5-GsXWxqI/AAAAAAAATtY/W1-MSfDLGuczVRussOZMQGFATL9RXaZYACLcBGAs/s1600/4mm%2Bmagnet.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="533" data-original-width="540" height="196" src="https://1.bp.blogspot.com/-PuaLwcDpCVw/W_5-GsXWxqI/AAAAAAAATtY/W1-MSfDLGuczVRussOZMQGFATL9RXaZYACLcBGAs/s200/4mm%2Bmagnet.jpg" width="200" /></a><a href="https://1.bp.blogspot.com/-DdpYCZ7R7FQ/W_5-G8ebYrI/AAAAAAAATtc/2AuYYQX-HEEvpSnsgduqhHXL9McqSD_0ACLcBGAs/s1600/8mm%2Bmagnet.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="545" data-original-width="539" height="200" src="https://1.bp.blogspot.com/-DdpYCZ7R7FQ/W_5-G8ebYrI/AAAAAAAATtc/2AuYYQX-HEEvpSnsgduqhHXL9McqSD_0ACLcBGAs/s200/8mm%2Bmagnet.jpg" width="197" /></a></div>
<br />
Despite each magnet being made of the same material there is a clear difference in the flux density present at the surface poles. The reason for this effect is that shorter magnets have a higher <a href="https://en.wikipedia.org/wiki/Demagnetizing_field" target="_blank">demagnetisation factor</a> in that direction. The demagnetisation factor reduces the B field inside the magnet and is dependent upon the magnet geometry. The concept of a demagnetisation factor also applies to soft magnetic materials, not just permanent magnets. Long magnets will have a lower demagnetisation factor than shorter magnets. Unfortunately, there is no simple equation that can be used to describe the demagnetisation factor for something as basic as a cube. However, <a href="http://www.shlucky.com/en/index.php?case=archive&act=show&aid=67" target="_blank">there is a simple relationship for an ellipsoid</a>. Note that if you have a magnetic circuit that makes a closed circuit then the demagnetisation factor is zero but there are also no magnetic poles.<br />
<br />
If we draw a line projecting out from the surface pole of each magnet and measure the flux density at each point we get the following plot.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-bOgZuphY6VA/W_6Bvn_LByI/AAAAAAAATt8/22aApqMMmVQNRtbnOSv1TmUhr2C2QbiqwCLcBGAs/s1600/Magnet%2Blength2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="845" data-original-width="1134" height="297" src="https://2.bp.blogspot.com/-bOgZuphY6VA/W_6Bvn_LByI/AAAAAAAATt8/22aApqMMmVQNRtbnOSv1TmUhr2C2QbiqwCLcBGAs/s400/Magnet%2Blength2.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
Here we can see that the flux density in air for the 8 x 8 mm magnet at a distance of ~ 10 mm is the same as the surface (0 mm distance) for the 1 x 8 mm. This trend of increasing flux density with magnet length does not continue forever. The flux density in air at a distance of 0.5 mm from the surface of the magnets is plotted vs magnet length below.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-pQzn94aEdwk/XAHTmeo75bI/AAAAAAAATyw/V9swUNxBe1kim-2uR6dTEpSaSJ7StoIAgCLcBGAs/s1600/field%2Bstrength%2Bwith%2Bmagnet%2Blength.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="897" data-original-width="1221" height="293" src="https://1.bp.blogspot.com/-pQzn94aEdwk/XAHTmeo75bI/AAAAAAAATyw/V9swUNxBe1kim-2uR6dTEpSaSJ7StoIAgCLcBGAs/s400/field%2Bstrength%2Bwith%2Bmagnet%2Blength.jpg" width="400" /></a></div>
<br />
<div class="separator" style="clear: both; text-align: left;">
As the magnet is made longer, and its demagnetisation factor in that direction decreases, the field produced at the surface poles would eventually approach that of the magnets <a href="https://en.wikipedia.org/wiki/Remanence" target="_blank">remanent magnetisation</a>. The plot above will look quite different if the same magnet was instead placed into the rotor of a motor since you then also have high magnetic permeability material in the stator and rotor helping guide the flux from the magnet, reducing its demagnetisation factor. However the overall concept remains the same.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<h2 style="text-align: left;">
<span style="color: blue;">Effect of magnet length on the torque produced by a simple motor</span></h2>
<div class="separator" style="clear: both; text-align: left;">
As in the <a href="https://things-in-motion.blogspot.com/2018/12/understanding-bldc-pmsm-electric-motor.html" target="_blank">previous post</a>, we can use a simple model of a motor to test how different magnet lengths impact the torque produced for a fixed winding current. Below are four different scenarios. Each motor has a gap between the stator and the magnets of 0.5 mm. Therefore, the total flux gap is given by the magnet length + 0.5 mm. </div>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-tqYBZTl4ObI/XAUlOaX3gkI/AAAAAAAAT3U/gdduWQUossgxz7a_nkD-Wv7O-5h4jWM-wCLcBGAs/s1600/1mm%2Bmagnet%2Bflux.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="701" data-original-width="750" height="299" src="https://1.bp.blogspot.com/-tqYBZTl4ObI/XAUlOaX3gkI/AAAAAAAAT3U/gdduWQUossgxz7a_nkD-Wv7O-5h4jWM-wCLcBGAs/s320/1mm%2Bmagnet%2Bflux.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">1 mm rotor magnet</td></tr>
</tbody></table>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-RXz8BoSwQyc/XAUlOocM6EI/AAAAAAAAT3Y/khoTHmdLUxMGhm1JHHZYq4LEYQeQGDKdACLcBGAs/s1600/2mm%2Bmagnet%2Bflux.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="754" data-original-width="698" height="320" src="https://3.bp.blogspot.com/-RXz8BoSwQyc/XAUlOocM6EI/AAAAAAAAT3Y/khoTHmdLUxMGhm1JHHZYq4LEYQeQGDKdACLcBGAs/s320/2mm%2Bmagnet%2Bflux.jpg" width="296" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">2 mm rotor magnet</td></tr>
</tbody></table>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-4LCSXA0Hrjs/XAUlOzP7GaI/AAAAAAAAT3c/z6IJlUJFzsYy36ovLGBLGIkY48iRAMYKwCLcBGAs/s1600/4mm%2Bmagnet%2Bflux.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="732" data-original-width="725" height="320" src="https://2.bp.blogspot.com/-4LCSXA0Hrjs/XAUlOzP7GaI/AAAAAAAAT3c/z6IJlUJFzsYy36ovLGBLGIkY48iRAMYKwCLcBGAs/s320/4mm%2Bmagnet%2Bflux.jpg" width="316" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">4 mm rotor magnet</td></tr>
</tbody></table>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-Qzf6XcBiAVc/XAUlPb5SvjI/AAAAAAAAT3g/RTVbiB2NJSIuNuJFmXRw1HdgsxJOsqavQCLcBGAs/s1600/8mm%2Bmagnets%2Bflux.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="738" data-original-width="701" height="320" src="https://1.bp.blogspot.com/-Qzf6XcBiAVc/XAUlPb5SvjI/AAAAAAAAT3g/RTVbiB2NJSIuNuJFmXRw1HdgsxJOsqavQCLcBGAs/s320/8mm%2Bmagnets%2Bflux.jpg" width="303" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">8 mm rotor magnet</td></tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: left;">
We can see a few things right away. First, the flux density in the rotor 'back iron' increases considerably as you make the magnets longer to the point that the back iron begins to saturate. This can also be seen in the stator teeth. Secondly, we can see that more flux escapes the rotor and fringes into the surrounding air. This is due to the saturation of the back iron. </div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
The flux density in the flux gap is plotted below with different length magnets. The flux contribution from only the stator windings was estimated by removing the magnets and simulating the motor. The same was done for the flux density contribution for the magnets, this time with the stator winding current set to zero.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-SNJIq3KRP2Y/XBW8oik2SeI/AAAAAAAAUG0/psDPaNDKLvMW8s8peRK5Q-ri3_D7B21VACLcBGAs/s1600/Flux%2Bdensity%2Bfrom%2Bmagnets%2Band%2Bstator.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="903" data-original-width="1218" height="295" src="https://1.bp.blogspot.com/-SNJIq3KRP2Y/XBW8oik2SeI/AAAAAAAAUG0/psDPaNDKLvMW8s8peRK5Q-ri3_D7B21VACLcBGAs/s400/Flux%2Bdensity%2Bfrom%2Bmagnets%2Band%2Bstator.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
It is clear that as you make the magnets longer the flux contribution from the stator becomes smaller due to the increase in the flux gap size. On the other hand, the flux density contribution from the magnets increases as they are made longer. Based on this plot we would expect that the torque will continue to rise as the magnets are increased in length. However, when the rotor torque is plotted with respect to the magnet length we can see that maximum torque is reached for magnets that are about 4 mm long. Further increasing the magnet length sees the torque slowly fall off.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-XggEDmJC6q4/XBSujt0ycyI/AAAAAAAAUE8/0rfyWDHUEOMp8XBNnTOw60VHojB8Fc-2gCLcBGAs/s1600/total%2Btorque%2Bvs%2Bmagnet%2Blength.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="640" data-original-width="896" height="285" src="https://3.bp.blogspot.com/-XggEDmJC6q4/XBSujt0ycyI/AAAAAAAAUE8/0rfyWDHUEOMp8XBNnTOw60VHojB8Fc-2gCLcBGAs/s400/total%2Btorque%2Bvs%2Bmagnet%2Blength.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<span style="text-align: start;"><br /></span></div>
<div class="separator" style="clear: both; text-align: left;">
This fall off in torque for magnets longer than about 4 mm is likely due to the stator core and rotor back iron beginning to saturate. Perhaps more interestingly is if we plot the specific torque density (torque per unit volume) and gravimetric torque density (torque per unit mass). </div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-Q8hkN-H44mI/XBSZ0MXCCgI/AAAAAAAAUEQ/Ul4A9la4mncGd6jb-eELpvUQzLx9YFjKwCLcBGAs/s1600/Magnet%2Blength%2Bcomparsion.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="510" data-original-width="752" height="271" src="https://1.bp.blogspot.com/-Q8hkN-H44mI/XBSZ0MXCCgI/AAAAAAAAUEQ/Ul4A9la4mncGd6jb-eELpvUQzLx9YFjKwCLcBGAs/s400/Magnet%2Blength%2Bcomparsion.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
When the magnets are made longer they are adding mass and volume to the entire motor while the torque gradually decreases. Therefore, there is a sharp fall off in the specific torque density for magnets longer than about 2 mm. Note that in the above example only the magnet length was changed. If more than one parameter was refined for, such as the thickness of the back iron, then the results will differ from those above.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
In addition to just the torque output there are many other factors which need to be considered when you change the length of the magnets contained in a motor. A few that come to mind may include:</div>
<div class="separator" style="clear: both; text-align: left;">
</div>
<ul>
<li>Core losses are likely to increase when magnet length is increased as the stator and rotor iron is operating closer to saturation. Stray flux from the magnets may also cause more eddy current losses in the windings.</li>
<li>Magnets are easily the most expensive part of a hobby grade electric motor. Therefore, motor cost will increase considerably if you were to use longer magnets, even if a redesigned motor did see a slight increase in torque with magnet length.</li>
<li>Increasing the magnet length will add more mass to the rotor which increases its <a href="https://en.wikipedia.org/wiki/Moment_of_inertia" target="_blank">moment of inertia</a>, reducing the dynamic response time of the motor.</li>
<li>Cogging torque generated by the salient poles of the stator will likely be much worse with more powerful magnets</li>
<li>There are many different grades of magnetic materiel. Using more or less powerful magnets would likely change the optimum magnet length.</li>
</ul>
<h2 style="clear: both; text-align: left;">
<span style="color: blue;">Conclusion</span></h2>
<div>
Increasing the length of the permanent magnets used in a <a href="https://things-in-motion.blogspot.com/2018/12/why-most-hobby-grade-bldc-out-runners.html" target="_blank">'BLDC' (PMSM)</a> motor will increase the torque produced only up to a point. The optimum magnet length will depend on many factors, but as a general rule of thumb, this length will be between 1 and 4 mm for 'hobby grade' out-runner electric motors constructed with back iron in the rotor. Further increasing the magnet length will only reduce the motor performance and increase the motors production costs.</div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
<i>If you have noticed any errors in the above article then please let me know. If you would like to play around with any of the models shown in this post in FEMM you can find the files hosted <a href="https://drive.google.com/drive/folders/1cFgdnIDvC_3F-CNXVnokyaIVGl2Y5z72?usp=sharing" target="_blank">here</a>. <a href="http://www.femm.info/wiki/MagneticsTutorial" target="_blank">This tutorial</a> gives you enough information to get started if you have never used FEMM before.</i></div>
Richard Parsonshttp://www.blogger.com/profile/08331478925576296508noreply@blogger.com2tag:blogger.com,1999:blog-4080979156700236346.post-65088926219813697722018-12-26T18:30:00.001-08:002019-03-12T16:06:13.312-07:00Understanding BLDC (PMSM) electric motor constants - Optimal flux gap for high torque densityFor many weight sensitive applications in robotics, it is desirable to have high torque density actuators. It is also often desirable to have relatively low gear ratios as this improves un-sensored output torque accuracy and helps to minimise rotational inertia, which improves angular acceleration. This is the basis of <a href="http://build-its-inprogress.blogspot.com/2018/" target="_blank">Ben Katz</a> low-cost <a href="https://dspace.mit.edu/handle/1721.1/105580" target="_blank">modular actuator design</a>. There is, therefore, a need for electric motors with as high a gravimetric torque density (torque per unit mass) as possible.<br />
<br />
Despite this, I have so far been unable to find much information online in the 'hobbyist community' (i.e. non-academic) about which aspects of an electric motor are important for torque density and how they can be optimised. Therefore, for the next series of posts I will be using <a href="http://www.femm.info/wiki/HomePage">FEMM </a>and a simple motor model to try and develop a working understanding of how different motor parameters (e.g. flux gap size, magnet length, stator tooth shape, slot and pole number etc.) impact the torque density of a brushless permanent magnet synchronous motor.<br />
<br />
Let's get started.<br />
<h2 style="text-align: left;">
<span style="color: blue;">The flux gap</span></h2>
<div style="text-align: left;">
The flux gap is the distance between the high magnetic permeability material in the stator (stator 'iron') and the corresponding high magnetic permeability material in the rotor (rotor 'back iron'). This material is normally made of thin laminations of Fe-Si steel.</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
It is well known that, in general, the flux gap should be as small as possible.</div>
<div style="text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-yo3Mg1lNjQk/W_5vHyjBw2I/AAAAAAAATsc/o8-Cuko2vJ4ZiQ1Ji2k4ktQcVLRYrf2bQCLcBGAs/s1600/flux%2Bgap%2Bcomparsion%2B2.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="674" data-original-width="1021" height="263" src="https://1.bp.blogspot.com/-yo3Mg1lNjQk/W_5vHyjBw2I/AAAAAAAATsc/o8-Cuko2vJ4ZiQ1Ji2k4ktQcVLRYrf2bQCLcBGAs/s400/flux%2Bgap%2Bcomparsion%2B2.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div style="text-align: left;">
It is important to note here that <b><u>the flux gap includes the magnets</u></b>. Rare earth magnets (magnetised or un-magnetised) have a magnetic permeability essentially the same as air. Therefore, from the stators perspective, a magnet is no different than air and should be included as part of the flux gap. </div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
Let's start by considering two simple magnetic circuits simulated in <a href="http://www.femm.info/wiki/HomePage">FEMM</a>; one with a 1 mm flux gap and one with a 4 mm flux gap.</div>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-0uWuoWpb1gQ/XACOH7Tv_rI/AAAAAAAATwY/-rxrCst0D08PrBx9YDxO_KLGUHoFNb_SgCLcBGAs/s1600/1mm_flux_gap_round.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="573" data-original-width="561" height="400" src="https://1.bp.blogspot.com/-0uWuoWpb1gQ/XACOH7Tv_rI/AAAAAAAATwY/-rxrCst0D08PrBx9YDxO_KLGUHoFNb_SgCLcBGAs/s400/1mm_flux_gap_round.jpg" width="391" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">1 mm flux gap core</td></tr>
</tbody></table>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-NgUL69uu52Y/XACPK8CUslI/AAAAAAAATwo/77_zMXL576wKT8s7GYyuT0f2H4V_jDM7ACLcBGAs/s1600/4mm_flux_gap_round.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="585" data-original-width="513" height="400" src="https://2.bp.blogspot.com/-NgUL69uu52Y/XACPK8CUslI/AAAAAAAATwo/77_zMXL576wKT8s7GYyuT0f2H4V_jDM7ACLcBGAs/s400/4mm_flux_gap_round.jpg" width="350" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">4 mm flux gap</td></tr>
</tbody></table>
The magnetic circuit consists of a ring of soft iron (literally pure annealed iron), copper windings represented by the green rectangles (2A, 250 turns) and an air gap in the ring. The flux density (unit of Tesla) is represented by how close the lines of flux are together and by the colour, with <span style="color: red;">red </span>being the highest density and <span style="color: blue;">blue</span> being the lowest. The flux density in the ring with the small flux gap is clearly the largest. This ring also has the least amount of flux 'leaking' out into the surrounding air.<br />
<br />
The reason for this difference is that an air gap increases the <a href="https://en.wikipedia.org/wiki/Magnetic_reluctance">magnetic reluctance</a> of the circuit. Magnetic reluctance is to flux in a magnetic circuit what resistance is to a current in an electric circuit. Therefore, the magnetic flux in the circuit is dependent upon the total magnetic reluctance and the applied magneto-motive force (turns times current) just as an electrical current is dependent upon the total resistance in an electric circuit and the applied voltage. For a nice overview of the concept check out Ben Krasnow's <a href="https://www.youtube.com/watch?v=4UFKl9fULkA&t=51s">video on the topic</a>.<br />
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div style="text-align: left;">
<div>
Let's look more closely at how the flux changes over the flux gap itself. We can do this by drawing a line over the flux gap and measuring the flux density at each point on the line.</div>
<div>
<br /></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-nylqdRkD2DY/W_50oWR1CxI/AAAAAAAATs8/ibQCFhg2URceb8OSlFi_5mLeI11I0055ACLcBGAs/s1600/flux_meas_region.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="727" data-original-width="988" height="293" src="https://1.bp.blogspot.com/-nylqdRkD2DY/W_50oWR1CxI/AAAAAAAATs8/ibQCFhg2URceb8OSlFi_5mLeI11I0055ACLcBGAs/s400/flux_meas_region.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The 4 mm flux gap and line which we will measure flux density</td></tr>
</tbody></table>
<div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-LDXRuSngAhI/XACRgIsIUaI/AAAAAAAATw0/-QIzsviOSyUp-R1NRhm6ZO5Qvx6oYBY8wCLcBGAs/s1600/flux%2Bdensity%2Bin%2Bthe%2Bgap.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="911" data-original-width="1162" height="311" src="https://1.bp.blogspot.com/-LDXRuSngAhI/XACRgIsIUaI/AAAAAAAATw0/-QIzsviOSyUp-R1NRhm6ZO5Qvx6oYBY8wCLcBGAs/s400/flux%2Bdensity%2Bin%2Bthe%2Bgap.jpg" width="400" /></a></div>
<br /></div>
<div>
Doing this for both the 1 mm and 4 mm flux gap it is clear that the flux in the middle of each gap remains constant. It can also be seen that the flux is four times smaller in the 4 mm flux gap than the 1 mm flux gap. Therefore, in order to produce the same flux density in the 4 mm gap, we would need to either add four times as many windings at the same current or alternatively, keep the same number of windings and add four times as much current. <b>This concept can also be applied to electric motors </b>and explains why engineers generally do everything they can to keep the flux gap as small as possible.<br />
<br />
The flux density in the flux gap can be approximated using the following equation:<br />
<br />
<div style="text-align: center;">
<span style="font-size: large;">`B=\frac{\mu_{0}NI}{g}`</span></div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
where B is the flux density (Tesla), <span style="text-align: center;">`mu_{0}` </span><span style="text-align: center;">is the magnetic permeability of free space </span><span style="text-align: center;">`(4\pi\times10^{-7})`, N is the number of turns of wire, I is the current (Ampere) and g is the flux gap (meters). Plotting B vs g we see the following:</span></div>
<div style="text-align: left;">
<span style="text-align: center;"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-Ww8b016SVtI/XACW7smcbDI/AAAAAAAATxA/0haGOBHk4rkkzIBXkLSxPD_QMdytG9T4ACLcBGAs/s1600/flux_density_vs_flux_gap.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="889" data-original-width="1177" height="301" src="https://3.bp.blogspot.com/-Ww8b016SVtI/XACW7smcbDI/AAAAAAAATxA/0haGOBHk4rkkzIBXkLSxPD_QMdytG9T4ACLcBGAs/s400/flux_density_vs_flux_gap.jpg" width="400" /></a></div>
<div style="text-align: left;">
<span style="text-align: center;"><br /></span></div>
This equation assumes that the reluctance of the iron core is negligible. This is a safe assumption in this example as reluctance of the Fe core is around <span style="text-align: center;">`10^{-7}` </span>times smaller than the reluctance of the flux gap and so can be disregarded. However, if your core is close to saturation, as would be the case if you reduce the flux gap to zero, then this will not always be the case. Also, this equation can only be used for a constant cross-section like that of the 'racetrack' core shown above, but does provide a good starting point for cores of other shapes. For best results a <a href="https://en.wikipedia.org/wiki/Finite_element_method">FEA </a>package like FEMM (its free!) will give the best approximation.<br />
<br />
Unlike the simple magnetic circuits shown above, the flux gap problem is made more complicated for BLDC motors for a few reasons:<br />
<ol>
<li>There are multiple flux gaps. The stator flux must travel across to the rotor and back again and can do so at multiple points.</li>
<li>The magnetic permeability, and hence the magnetic reluctance, of the ferromagnetic stator and rotor back iron, is not constant but instead depends on the total amount of flux in that region.</li>
</ol>
<div>
Note that simply embedding the magnets in the rotor back iron does not eliminate the flux gap, it only moves it further back into the rotor. Embedded rotor magnets do have their own advantages (improved field weakening performance, control over reluctance torque) but they are topics for another day.<br />
<h2 style="text-align: left;">
<span style="color: blue;">Effect of flux gap size on torque for a simple motor</span></h2>
</div>
<div>
In order to explore the impact of the flux gap size on something more closely resembling a real motor I have simulated a 6 slot, 8 pole 'out runner'. The motor was sketched in <a href="https://www.autodesk.com/products/fusion-360/overview">F360</a> and exported as a dxf file for use in FEMM. It has a stator diameter of 57 mm and a rotor length (into screen) of 10 mm. This motor has three phases which are wound with <a href="https://www.emetor.com/glossary/concentrated-winding/" target="_blank">concentrated windings</a> (<a href="https://www.emetor.com/glossary/number-of-winding-layers/" target="_blank">double layered</a>) as ABCABC. A current of 50A is supplied on phase A, and -25A is supplied on phase B and C so that all of the flux is directed on the Q-axis where it will produce the most torque.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-nMpgIhT3U6g/XBDRTCKm1dI/AAAAAAAAUCg/qINvaBJC6i4pb4mB2IL_xvb33gPtIsSIgCLcBGAs/s1600/3%2Bphase.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="923" data-original-width="1227" height="300" src="https://3.bp.blogspot.com/-nMpgIhT3U6g/XBDRTCKm1dI/AAAAAAAAUCg/qINvaBJC6i4pb4mB2IL_xvb33gPtIsSIgCLcBGAs/s400/3%2Bphase.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The motor is simulated with a fixed current at 90 degree</td></tr>
</tbody></table>
<a href="http://scolton.blogspot.com/2010/01/3ph-duo-wrap-up-part-1-field-oriented.html">Shane Colton's blog post on field orientation</a> has a good rundown on the q-axis and d-axis argument. In short, the phase with the most current on it (phase A) is 90 electrical degrees ahead (q-axis) of the direct flux of the magnets (d-axis) where it will produce the most torque per amp. If the rotor was rotating then so too would the magnetic field generated in the stator so that the torque remains constant and proportional to the current.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-ENI_xJ8YEFQ/XBDWFjzAumI/AAAAAAAAUDA/ZXoYVjY2qTgzQgnCMgAPrEUyj2XkBz4vACLcBGAs/s1600/FEMM%2Bmotor%2Bdiagram.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="866" data-original-width="1286" height="268" src="https://4.bp.blogspot.com/-ENI_xJ8YEFQ/XBDWFjzAumI/AAAAAAAAUDA/ZXoYVjY2qTgzQgnCMgAPrEUyj2XkBz4vACLcBGAs/s400/FEMM%2Bmotor%2Bdiagram.jpg" width="400" /></a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-vZK116eTzLs/XBW0h7apOtI/AAAAAAAAUGo/vqmnQNfgSz0ya53TYcWD4JrToxJqmJEKQCLcBGAs/s1600/flux%2Bdue%2Bto%2Bstator%2Bonly2.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="564" data-original-width="566" height="318" src="https://4.bp.blogspot.com/-vZK116eTzLs/XBW0h7apOtI/AAAAAAAAUGo/vqmnQNfgSz0ya53TYcWD4JrToxJqmJEKQCLcBGAs/s320/flux%2Bdue%2Bto%2Bstator%2Bonly2.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Flux density due to the stator windings only (magnets removed)</td></tr>
</tbody></table>
However, in this simulation, both the current and rotor position are fixed and we are instead only solving for the flux density generated in the air gap by the stator windings and the magnets.<br />
In the simulation seen below the flux is seen to be concentrated in the high magnetic permeability stator teeth and in the back iron of the rotor. This flux almost exclusively crosses at the flux gap.<br />
<br /></div>
</div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-ypS-Zxj8d2g/XBDURVehQmI/AAAAAAAAUCs/iQXSkdNICXM-o_TKSP2mQOriVij4ATsawCLcBGAs/s1600/1mm%2Bmagnet%2Bflux%2Bwith%2B1mm%2Bflux%2Bgap.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="651" data-original-width="649" height="320" src="https://3.bp.blogspot.com/-ypS-Zxj8d2g/XBDURVehQmI/AAAAAAAAUCs/iQXSkdNICXM-o_TKSP2mQOriVij4ATsawCLcBGAs/s320/1mm%2Bmagnet%2Bflux%2Bwith%2B1mm%2Bflux%2Bgap.jpg" width="319" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">1 mm flux gap</td></tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: left;">
For comparison, a motor with a 3mm flux gap is shown below. Here we can see that the total flux density in the stator teeth and back iron is greatly reduced due to the increased magnetic reluctance in the magnetic circuit. This decrease is also seen in the flux gap where the magnets are located.</div>
</div>
<div style="text-align: left;">
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-Pcr7vIgaLck/XBDUY5PBmaI/AAAAAAAAUCw/tz5OamK2X2kxndW3OwpMzerC-YHec_qVgCLcBGAs/s1600/1mm%2Bmagnet%2Bflux%2Bwith%2B4mm%2Bflux%2Bgap2.jpg" style="margin-left: auto; margin-right: auto; text-align: center;"><img border="0" data-original-height="619" data-original-width="679" height="291" src="https://1.bp.blogspot.com/-Pcr7vIgaLck/XBDUY5PBmaI/AAAAAAAAUCw/tz5OamK2X2kxndW3OwpMzerC-YHec_qVgCLcBGAs/s320/1mm%2Bmagnet%2Bflux%2Bwith%2B4mm%2Bflux%2Bgap2.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">3 mm flux gap</td></tr>
</tbody></table>
If you would like to play around with these models in FEMM you can find the files hosted <a href="https://drive.google.com/drive/folders/1cFgdnIDvC_3F-CNXVnokyaIVGl2Y5z72?usp=sharing" target="_blank">here</a>. <a href="http://www.femm.info/wiki/MagneticsTutorial" target="_blank">This tutorial</a> gives you enough information to get started if you have never used FEMM before.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<div style="text-align: left;">
Using FEMM it is also possible to estimate the static torque that this 10 mm long rotor would produce. The torque and flux density within the flux gap with respect to the flux gap size is shown below.</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-t8DIMWVGKq0/XBSM5FPVfpI/AAAAAAAAUEE/MViPOBME1iQt4aGnHFvZClPDjeP6-XyvwCLcBGAs/s1600/Torque%2Band%2BFlux%2Bvs%2Bflux%2Bgap3.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="936" data-original-width="1325" height="282" src="https://4.bp.blogspot.com/-t8DIMWVGKq0/XBSM5FPVfpI/AAAAAAAAUEE/MViPOBME1iQt4aGnHFvZClPDjeP6-XyvwCLcBGAs/s400/Torque%2Band%2BFlux%2Bvs%2Bflux%2Bgap3.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<h2>
</h2>
We can see two things quite clearly: i) the torque produced by the motor is dependent upon the flux density in the air gap and ii) the torque falls off asymptotically as the flux gap size is increased. Note that in this example we are increasing the diameter of the rotor to increase the flux gap. Since motor torque increases with the square of the <a href="https://www.emetor.com/glossary/airgap-diameter/" target="_blank">flux gap diameter</a> the fall off in torque would be much steeper if we were to instead decrease the size of the stator to increase the flux gap.<br />
<br />
From the above discussion it is clear that, in general, we want to have as small a flux gap as physically possible so as to increase the motors torque output and, therefore, it's motor constant. However, aside from needing to consider manufacturing tolerances, we also need to consider the thickness of the magnets. In general, if you make the rotor magnets longer then the flux density at their poles is also increased. This will act to increase the torque output of your motor.<br />
<br />
The impact of magnet length to motor torque will, therefore, be the topic of the next post.<br />
<h2>
<span style="color: blue;">Conclusion</span></h2>
<div>
Increasing the size of the flux gap for a motor will increase the magnetic reluctance in the magnetic circuit which reduces the flux density in the air gap. The torque generated by a motor is proportional to the flux in the air gap. Therefore, increasing the size of the flux gap will reduce the torque generated by a motor for a fixed winding current, which reduces the overall motor constant. </div>
<br />
<div>
<i>Equations were produced in this post with the help of <a href="https://arachnoid.com/latex/">arachnoid.com</a> and are based on those found in the book <a href="https://books.google.com.au/books?id=Cf78BAAAQBAJ">Electric Motors and Drives: Fundamentals, types and applications</a> by Austin Hughes. If you have noticed any errors in the above article then please let me know.</i></div>
</div>
Richard Parsonshttp://www.blogger.com/profile/08331478925576296508noreply@blogger.com0tag:blogger.com,1999:blog-4080979156700236346.post-57188564413335948432018-12-25T17:13:00.004-08:002020-01-06T15:00:25.607-08:00How to estimate the torque of a BLDC (PMSM) electric motor using only its Kv and current drawThere exists a fundamental relationship between an electric motors velocity constant <b style="text-align: center;">`(K_{V}),` </b><span style="text-align: center;">armature* current <b style="text-align: center;">`(I_{A})`</b> and torque </span><b style="text-align: center;">`(\tau)`</b><span style="text-align: center;">. </span>This relationship is as follows:<br />
<br />
<div style="text-align: center;">
<span style="font-size: large;">`\tau \approx \frac{8.3 \times I_{A} }{K_{V}}`</span></div>
<br />
were <b style="text-align: center;">`\tau` </b><span style="text-align: center;">is in N.m, </span><b style="text-align: center;">`I_{A}` </b><span style="text-align: center;">is in A and </span> <b style="text-align: center;">`K_{V}` </b><span style="text-align: center;">is in RPM/V. This relationship is extremely useful since most 'hobby grade' <a href="https://things-in-motion.blogspot.com/2018/12/why-most-hobby-grade-bldc-out-runners.html" target="_blank">BLDC/PMSM</a> manufacturers do not publish the usual<a href="https://en.wikipedia.org/wiki/Motor_constants" target="_blank"> motor constants</a> you would expect from an industrial product while </span><b style="text-align: center;">`K_{V}` </b><span style="text-align: center;">and </span><span style="text-align: center;">peak </span><b style="text-align: center;">`I_{A}` </b><span style="text-align: center;">is normally specified on sites like <a href="https://hobbyking.com/" target="_blank">hobbyking.com</a>.</span><br />
<br />
The above relationship works for two reasons:<br />
<ol>
<li> A motors <b style="text-align: center;">`K_{V}`</b><span style="text-align: center;"> and its torque constant </span><b style="text-align: center;">`(K_{\tau})` </b><span style="text-align: center;">are fundamentally the same thing.</span></li>
<li> The torque generated by a motor for a given current is governed by <b style="text-align: center;">`K_{\tau}`</b>.</li>
</ol>
Of course, there are limitations to this approach. If a motor is close to saturation or if the current waveform supplied by a motor controller does not exactly match a motors back EMF waveform (i.e. the current is not exactly on the <a href="http://scolton.blogspot.com/2010/01/3ph-duo-wrap-up-part-1-field-oriented.html" target="_blank">q-axis </a>at all times) then your torque will be less than that suggested above. However, my own testing shows that this approximation is quite accurate when a 'hobby grade' PMSM motor is driven using field oriented control (FOC). Even when not using FOC or a PMSM this approach should still provide a good starting point.<br />
<br />
Read on if you are interested in a more detailed understanding of why this relationship works and the limitations of this approach.<br />
<h2>
<span style="color: blue;">Fundamental torque production</span></h2>
Electric motors do useful work by producing torque and rotation. The amount of <b>steady state</b> torque produced by a <b>well optimised</b> and <b>non-salient machine</b>** with a <b>specific volume</b> (specific torque density) is ultimately dependent upon three factors:<br />
<ul>
<li><b>The average flux density acting upon the armature</b>. For a <a href="https://things-in-motion.blogspot.com/2018/12/why-most-hobby-grade-bldc-out-runners.html" style="text-align: center;" target="_blank">BLDC/PMSM</a> motor this 'flux' is provided by the rotor's permanent magnets.</li>
<li><b>The average current that can be maintained by the armature without overheating</b>**. For a <a href="https://things-in-motion.blogspot.com/2018/12/why-most-hobby-grade-bldc-out-runners.html" style="text-align: center;" target="_blank">BLDC/PMSM</a> motor the armature consists of the copper windings in the stator.</li>
<li><b>The total length of the armature windings which has 'flux' acting upon it</b>. For a <a href="https://things-in-motion.blogspot.com/2018/12/why-most-hobby-grade-bldc-out-runners.html" style="text-align: center;" target="_blank">BLDC/PMSM</a> motor this length represents the number of turns of wire in the stator armature which interact with the 'flux' provided by the rotor.</li>
</ul>
<div>
In other words, all the complexity of an electric motor ultimately boils down to the "BIL" <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfor.html#c2">Lorentz force</a> law:</div>
<div>
<br />
<div style="text-align: center;">
<span style="font-size: large;">Force = Magnetic field `\times`</span><span style="font-size: large;">Current</span><span style="font-size: large;">`\times` </span><span style="font-size: large;">Conductor length</span></div>
</div>
<div>
<div style="text-align: center;">
</div>
</div>
<div style="text-align: center;">
<br /></div>
<div style="text-align: left;">
Increase one of the above terms without negatively impacting another and you have increased the torque output of your system. When discussing motor constants I find it useful to keep this simple relationship in mind.<br />
<h2>
<span style="color: blue;">Velocity constant = Torque constant</span></h2>
<div>
In the 'hobby' community most people seem comfortable with the concept of a motors velocity constant <b style="text-align: center;">`(K_{V})`</b><span style="text-align: center;">.</span><b style="text-align: center;"> </b><b style="text-align: center;">`K_{V}` </b><span style="text-align: center;">is <a href="https://fishpepper.de/2017/10/17/tutorial-how-to-measure-the-kv-of-a-brushless-motor/" target="_blank">easily measured</a> and is given by half the peak to peak back EMF generated line to line (across any two motor leads) for a given mechanical frequency with a unit RPM/V. </span><br />
<span style="text-align: center;"><br /></span> <span style="text-align: center;">What seems less well understood is that an electric motors</span> <a href="https://en.wikipedia.org/wiki/Motor_constants" target="_blank">torque constant</a> <b style="text-align: center;">`(K_{\tau})`</b> <b>is equal to</b> <b style="text-align: center;">`K_{V}`</b> so that<br />
<div style="text-align: center;">
<span style="font-size: large;">`K_{\tau}^* = K_{V}^*`</span><br />
<div style="text-align: left;">
<span style="text-align: center;"><br /></span></div>
<div style="text-align: left;">
<span style="text-align: center;">where here </span><b style="text-align: center;">`K_{\tau}^*` </b><span style="text-align: center;">is a motors <b>per-phase</b> torque constant and </span><b style="text-align: center;">`K_{V}^*`</b><span style="text-align: center;"> is a motors <b>per-phase</b> velocity constant </span><span style="text-align: center;">given by half the peak to peak back EMF generated <b>line to </b></span><span style="text-align: center;"><b>neutral</b> </span><span style="text-align: center;">(from the centre tap point on a star wound motor) for a given electrical frequency </span>(Elec. Freq. = Mech. Freq. / No. Pole Pairs) with the units <b style="text-align: center;">`\frac{V \cdot S}{rad}`</b><span style="text-align: center;">.</span></div>
<div style="text-align: left;">
<span style="text-align: center;"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-fTJkKDcHwZQ/XCHVrIFGZNI/AAAAAAAAUYM/P4CuLRtDIM0ccCPB84nHp17CTs6lBZ8jwCLcBGAs/s1600/Kv%2Bvs%2BKv2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="715" data-original-width="810" height="352" src="https://3.bp.blogspot.com/-fTJkKDcHwZQ/XCHVrIFGZNI/AAAAAAAAUYM/P4CuLRtDIM0ccCPB84nHp17CTs6lBZ8jwCLcBGAs/s400/Kv%2Bvs%2BKv2.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
When first learning about electric motors it took me a while to accept that <b style="text-align: center;">`K_{\tau}^*` </b><span style="text-align: center;">does indeed </span><span style="text-align: center;">equal to </span><b style="text-align: center;">`K_{V}^*`</b><span style="text-align: center;">. The mathematics is clear and their </span><span style="text-align: center;">SI units are equivalent but something about it just didn't feel right. To overcome this I find it helps to keep in mind that all aspects of an electric motor which impact its </span><b style="text-align: center;">`K_{V}^*` </b><span style="text-align: center;">(e.g. flux gap size, magnet strength, winding turn number, rotor length etc.) also impact the amount of torque a motor can produce for a given current.</span></div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
<span style="text-align: center;"> </span><b style="text-align: center;">`K_{V}^*` </b>is not especially useful since we are normally interested in a motors 'total torque constant' produced by all three phases and not that of a single phase. The 'total torque constant' of a 3 phase PMSM<b style="text-align: center;"> </b><span style="text-align: center;">as estimated using its line-line `K_{V}` (RPM/V) is instead given by</span></div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
<span style="text-align: center;"></span></div>
<div style="text-align: center;">
<span style="font-size: large;">`K_{\tau} = \frac{3}{2} \times \frac{1}{\sqrt{3}}\times \frac{60}{2\pi} \times \frac{1}{K_{V}}`</span></div>
<div style="text-align: left;">
<span style="text-align: center;"><span style="text-align: center;"><br /></span></span>
<span style="text-align: center;"><span style="text-align: center;">As is discussed by <a href="https://github.com/madcowswe">Oskar Weigl</a> <a href="https://discourse.odriverobotics.com/t/where-does-the-formula-for-calculating-torque-come-from/1169/2?u=madcowswe">here</a>, the factor </span><b style="text-align: center;">`\frac{3}{2}` </b><span style="text-align: center;">is derived from eq. 3.2 in <a href="https://pdfs.semanticscholar.org/e2be/948549aaf744c73b36d303cf7165041222f4.pdf">this paper,</a> the </span></span><b style="text-align: center;">`\frac{1}{\sqrt{3}}` </b><span style="text-align: center;">is for converting the line-line voltage (which is what is commonly used by hobby motor manufacturers in the determination of the `K_{V}`) to phase voltage (example <a href="https://electronics.stackexchange.com/questions/92678/three-phase-power-supply-what-is-line-to-line-voltage/92826#92826">here</a>) and </span><b style="text-align: center;">`\frac{60}{2\p}` </b><span style="text-align: center;">is to convert from rpm to rad/s, which is needed to estimate the torque constant from the voltage constant, due to the voltage constant being specified in units of RPM/V. </span></div>
<h2 style="text-align: left;">
<span style="color: blue;">Torque constant determines torque output for a given current</span></h2>
<div style="text-align: left;">
<b style="text-align: center;">`K_{\tau}` </b><span style="text-align: center;">is also defined as</span></div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: center;">
<span style="font-size: large;">`K_{\tau} = \frac{\tau}{I_{A}}`</span></div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
where <b style="text-align: center;">`I_{A}` </b><span style="text-align: center;">was the armature current. Therefore </span>the 'overall torque' output of a 3 phase PMSM is given by<br />
<br /></div>
<div style="text-align: center;">
<span style="font-size: large;">`\tau = \frac{3}{2} \times \frac{1}{\sqrt{3}}\times \frac{60}{2\pi} \times \frac{1}{K_{V}}\times I_{A}`</span></div>
<div style="text-align: center;">
<br /></div>
<div style="text-align: left;">
<span style="text-align: center;"><br /></span>
<span style="text-align: center;">This </span>finally leads us to our 'conversion constant' of approximately 8.3 mentioned in the introduction</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: center;">
<span style="font-size: large;">`\tau \approx 8.3 \times \frac{1}{K_{V}} \times I_{A} \approx \frac{8.3 \times I_{A} }{K_{V}}`</span></div>
<div style="text-align: center;">
<br /></div>
</div>
</div>
<div style="text-align: left;">
and so the 'conversion constant' of 8.3 is just a simplified approximation for converting <b style="text-align: center;">`K_{V}` </b><span style="text-align: center;">to </span><b style="text-align: center;">`K_{V}^*`.</b><br />
<br />
It is important to point out here that this relationship is true for any three PMSM. Star, delta, b<span style="text-align: center;">ig, small, h</span>igh <b style="text-align: center;">`K_{V}`, </b> low <b style="text-align: center;">`K_{V}`</b><span style="text-align: center;">, </span><span style="text-align: center;">in-runner, out-runner, cored or core-less, 'weak' or 'strong' magnets, it doesn't make any difference. Equally important, the torque referenced above is the '<i><b>electromagnetic</b></i>' torque produced by the motor. This torque represents a 100% efficient conversion from electrical </span><span style="text-align: center;">energy</span><span style="text-align: center;"> </span><span style="text-align: center;">to mechanical </span><span style="text-align: center;">energy</span><span style="text-align: center;">. The '<b><i>shaft</i></b>' torque, which is the usable output torque of the motor, will always be less than the </span><span style="text-align: center;"><b style="font-style: italic;">electromagnetic</b> torque due losses in the system.</span><span style="text-align: center;"> </span><span style="text-align: center;">If you would like to go deeper into all the topics described above than this then I highly recommend<a href="http://krex.k-state.edu/dspace/bitstream/2097/1507/1/JamesMevey2009.pdf"> </a></span><a href="http://krex.k-state.edu/dspace/bitstream/2097/1507/1/JamesMevey2009.pdf" style="text-align: left;">James Mevey's Master's thesis</a><span style="text-align: left;"> from Kansas State which which was recommended and summarised by </span><a href="http://scolton.blogspot.com/2009/11/everything-you-ever-wanted-to-know.html" style="text-align: left;">Shane Colton in his blog</a><span style="text-align: left;">.</span><br />
<br />
<span style="text-align: center;">All this theory is great, but lets see if it actually works in practice.</span></div>
<h2>
<span style="color: blue; font-weight: normal;">Measuring the motor constants of <span style="text-align: center;">some real motors</span></span></h2>
<div style="text-align: left;">
In order to put all this theory to the test I have measured the motor constants for the following collection of 'hobby grade' out-runner motors.</div>
<div style="text-align: left;">
<span style="text-align: center;"><br /></span></div>
<div style="text-align: left;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-icvRWmHuGJ0/XDVwDd4Uu-I/AAAAAAAAUoE/PbPTkRt7EvA7qXoLc_FdPuuEShlo0GEFgCLcBGAs/s1600/motor%2Blables.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="578" data-original-width="1087" height="212" src="https://4.bp.blogspot.com/-icvRWmHuGJ0/XDVwDd4Uu-I/AAAAAAAAUoE/PbPTkRt7EvA7qXoLc_FdPuuEShlo0GEFgCLcBGAs/s400/motor%2Blables.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<br /></div>
<div style="text-align: left;">
<b style="text-align: center;">`K_{V}` </b><span style="text-align: center;">was first measured </span><span style="text-align: center;">using the method described </span><a href="https://fishpepper.de/2017/10/17/tutorial-how-to-measure-the-kv-of-a-brushless-motor/" style="text-align: center;">here</a><span style="text-align: center;">. I took a photo of my oscilloscope at around 500 RPM for each motor as shown below.</span></div>
<div style="text-align: left;">
<div class="separator" style="clear: both; text-align: center;">
</div>
<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-2Kx2lB3IlJ8/XB19QrUW0II/AAAAAAAAUNM/-o8kVNyLgMM25S6Bju-DRct89-r3vzJbQCLcBGAs/s1600/1000%2BkV.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1210" data-original-width="1600" height="240" src="https://3.bp.blogspot.com/-2Kx2lB3IlJ8/XB19QrUW0II/AAAAAAAAUNM/-o8kVNyLgMM25S6Bju-DRct89-r3vzJbQCLcBGAs/s320/1000%2BkV.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">1000 Kv Racerstar BR2212</td></tr>
</tbody></table>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-tMoUxgGbIQo/XB19RdiSl2I/AAAAAAAAUNU/9B3nWe7uuwYELqv3PqX_uqwSxzSjZBKrQCLcBGAs/s1600/190%2BKv.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1208" data-original-width="1600" height="240" src="https://4.bp.blogspot.com/-tMoUxgGbIQo/XB19RdiSl2I/AAAAAAAAUNU/9B3nWe7uuwYELqv3PqX_uqwSxzSjZBKrQCLcBGAs/s320/190%2BKv.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">190 Kv Keda 6364</td></tr>
</tbody></table>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-9sEbauaQI3M/XB19RcPoQxI/AAAAAAAAUNQ/41UOgW16L3ABUqRCu1b0VQdAN4lCEXlQwCLcBGAs/s1600/150%2BKv.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1229" data-original-width="1600" height="244" src="https://3.bp.blogspot.com/-9sEbauaQI3M/XB19RcPoQxI/AAAAAAAAUNQ/41UOgW16L3ABUqRCu1b0VQdAN4lCEXlQwCLcBGAs/s320/150%2BKv.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">150 Kv Odrive 6374</td></tr>
</tbody></table>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-6dvra6GTD6g/XB19ZHHOvbI/AAAAAAAAUNc/gZgj2hNqi5YAAkpUcB17kcztg9FTc6UNwCLcBGAs/s1600/280%2BKv.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1211" data-original-width="1600" height="241" src="https://1.bp.blogspot.com/-6dvra6GTD6g/XB19ZHHOvbI/AAAAAAAAUNc/gZgj2hNqi5YAAkpUcB17kcztg9FTc6UNwCLcBGAs/s320/280%2BKv.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">280 kV Turnigy SK3 5055</td></tr>
</tbody></table>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-Wm4FDPa7Re8/XB19XXvRkNI/AAAAAAAAUNY/k0U0ukAVcgIesE21t4DX3_qR0dVwNvb4wCLcBGAs/s1600/270%2BKv.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1197" data-original-width="1600" height="238" src="https://3.bp.blogspot.com/-Wm4FDPa7Re8/XB19XXvRkNI/AAAAAAAAUNY/k0U0ukAVcgIesE21t4DX3_qR0dVwNvb4wCLcBGAs/s320/270%2BKv.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">270 Kv Odrive N5065</td></tr>
</tbody></table>
<span style="text-align: center;"><br /></span><span style="text-align: center;">All of these motors are 12N14P and of a similar construction. I</span><span style="text-align: center;">ts clear that the shape of the back EMF is sinusoidal and not</span> trapezoidal. <b><a href="https://things-in-motion.blogspot.com/2018/12/why-most-hobby-grade-bldc-out-runners.html" target="_blank">Therefore, these motors are technically permanent magnet synchronous motors (PMSM) and not brushless DC motors</a></b>. <span style="text-align: center;">Overall, the measured </span><b style="text-align: center;">`K_{V}` </b><span style="text-align: center;">of each motor was quite close to their labelled values as will be shown in a table later.</span><br />
<span style="text-align: center;"><br /></span> To estimate <span style="text-align: center;">`K_{\tau}` the output torque of each motor was measured with respect to the armature current. This was done by winding a string around the rotor of each motor, attaching that string to a lever arm that then pulled down onto a laboratory balance. This balance then measured the weight, and therefore force, produced by the motor. </span><span style="text-align: center;">This method is only possible thanks to the use of a high resolution encoder (8192 counts per rotation) and an</span><span style="text-align: center;"> </span><a href="https://odriverobotics.com/" style="text-align: center;">Odrive motor controller</a><span style="text-align: center;"> which uses field oriented control (FOC) to place all current on the motors q-axis for maximum torque, even when stationary. </span><br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-FaH1oXQDRO8/XCCtmPeX-nI/AAAAAAAAUTs/sw3yO3TlAtceLuNrkW5FOonlToJ51LFrwCLcBGAs/s1600/torque%2Bmeasurement.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="760" data-original-width="1048" height="290" src="https://1.bp.blogspot.com/-FaH1oXQDRO8/XCCtmPeX-nI/AAAAAAAAUTs/sw3yO3TlAtceLuNrkW5FOonlToJ51LFrwCLcBGAs/s400/torque%2Bmeasurement.jpg" width="400" /></a></div>
<br /></div>
<div style="text-align: left;">
<span style="text-align: center;">The setup is crude and would have been much cleaner if I had just attach an arm to each motors rotor. However, by using a string I didn't need to make a new arm adaptor for each motor and could instead just consider the diameter of the rotor in the final calculations. This was enough to get the job done.</span><br />
<span style="text-align: center;"><br /></span> <span style="text-align: center;">Using this setup and a simple script I slowly stepped up the commanded current for each motor while manually recording the weight on the scale. After some back calculations the torque output for each motor with commanded current was found.</span><br />
<span style="text-align: center;"><br /></span>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-cImBhtUCj6Y/XCHBj2jud1I/AAAAAAAAUXw/JDxSV6SHztgyjL34hlOc3j-RESDvJvvJACLcBGAs/s1600/motor%2Btorque%2Bvs%2Bcurrent%2B2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="905" data-original-width="1189" height="303" src="https://1.bp.blogspot.com/-cImBhtUCj6Y/XCHBj2jud1I/AAAAAAAAUXw/JDxSV6SHztgyjL34hlOc3j-RESDvJvvJACLcBGAs/s400/motor%2Btorque%2Bvs%2Bcurrent%2B2.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<span style="text-align: center;">No surprises so far with the bigger motors producing more torque per amp and the torque increasing linearly with current. </span><br />
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
A summary of the motor parameters can be seen below and the raw data can be found <a href="https://docs.google.com/spreadsheets/d/1fR4AZi9J2d8eAKPYbgm9BNy-5hfkOhjNoTNe5huGxxs/edit?usp=sharing" target="_blank">here</a>. </div>
<style type="text/css">
.tg {border-collapse:collapse;border-spacing:0;}
.tg td{font-family:Arial, sans-serif;font-size:14px;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:black;}
.tg th{font-family:Arial, sans-serif;font-size:14px;font-weight:normal;padding:10px 5px;border-style:solid;border-width:1px;overflow:hidden;word-break:normal;border-color:black;}
.tg .tg-yj5y{background-color:#efefef;border-color:inherit;text-align:center;vertical-align:top}
.tg .tg-v0hj{font-weight:bold;background-color:#efefef;border-color:inherit;text-align:center;vertical-align:top}
.tg .tg-c3ow{border-color:inherit;text-align:center;vertical-align:top}
.tg .tg-7btt{font-weight:bold;border-color:inherit;text-align:center;vertical-align:top}
</style>
<br />
<table class="tg">
<tbody>
<tr>
<th class="tg-yj5y"></th>
<th class="tg-yj5y"></th>
<th class="tg-v0hj">Racerstar BR2212</th>
<th class="tg-v0hj">Turnigy SK3 5055</th>
<th class="tg-v0hj">Odrive D5065</th>
<th class="tg-v0hj">Keda 6364</th>
<th class="tg-v0hj">Odrive D6374</th>
</tr>
<tr>
<td class="tg-7btt">Rated kV</td>
<td class="tg-c3ow">rpm/V</td>
<td class="tg-c3ow">1000</td>
<td class="tg-c3ow">280</td>
<td class="tg-c3ow">270</td>
<td class="tg-c3ow">190</td>
<td class="tg-c3ow">150</td>
</tr>
<tr>
<td class="tg-7btt">Measured kV</td>
<td class="tg-c3ow">rpm/V</td>
<td class="tg-c3ow">1058</td>
<td class="tg-c3ow">276</td>
<td class="tg-c3ow">259</td>
<td class="tg-c3ow">182</td>
<td class="tg-c3ow">151</td>
</tr>
<tr>
<td class="tg-7btt">Phase Resistance</td>
<td class="tg-c3ow">Ohm</td>
<td class="tg-c3ow">0.128</td>
<td class="tg-c3ow">0.032</td>
<td class="tg-c3ow">0.039</td>
<td class="tg-c3ow">0.039</td>
<td class="tg-c3ow">0.039</td>
</tr>
<tr>
<td class="tg-7btt">Phase Inductance</td>
<td class="tg-c3ow">H</td>
<td class="tg-c3ow">1.84E-05</td>
<td class="tg-c3ow">1.33E-05</td>
<td class="tg-c3ow">2.02E-05</td>
<td class="tg-c3ow">2.13E-05</td>
<td class="tg-c3ow">2.81E-05</td>
</tr>
<tr>
<td class="tg-7btt">Weight</td>
<td class="tg-c3ow">kg</td>
<td class="tg-c3ow">0.045</td>
<td class="tg-c3ow">0.389</td>
<td class="tg-c3ow">0.411</td>
<td class="tg-c3ow">0.647</td>
<td class="tg-c3ow">0.885</td>
</tr>
<tr>
<td class="tg-7btt">Price</td>
<td class="tg-c3ow">$ USD</td>
<td class="tg-c3ow">6</td>
<td class="tg-c3ow">52</td>
<td class="tg-c3ow">69</td>
<td class="tg-c3ow">47</td>
<td class="tg-c3ow">99</td>
</tr>
<tr>
<td class="tg-7btt">Torque constant (Kt)</td>
<td class="tg-c3ow">N·m/A</td>
<td class="tg-c3ow">0.008</td>
<td class="tg-c3ow">0.029</td>
<td class="tg-c3ow">0.030</td>
<td class="tg-c3ow">0.042</td>
<td class="tg-c3ow">0.053</td>
</tr>
<tr>
<td class="tg-7btt">Kt/kg</td>
<td class="tg-c3ow">N·m/A/kg</td>
<td class="tg-c3ow">0.172</td>
<td class="tg-c3ow">0.073</td>
<td class="tg-c3ow">0.073</td>
<td class="tg-c3ow">0.065</td>
<td class="tg-c3ow">0.060</td>
</tr>
<tr>
<td class="tg-7btt">Kt/$ USD</td>
<td class="tg-c3ow">N·m/A/$ USD</td>
<td class="tg-c3ow">0.00129</td>
<td class="tg-c3ow">0.00055</td>
<td class="tg-c3ow">0.00043</td>
<td class="tg-c3ow">0.00090</td>
<td class="tg-c3ow">0.00054</td>
</tr>
<tr>
<td class="tg-7btt">Motor constant (Km)</td>
<td class="tg-c3ow">Nâ‹…m/sqrt(W)</td>
<td class="tg-c3ow">0.02</td>
<td class="tg-c3ow">0.16</td>
<td class="tg-c3ow">0.15</td>
<td class="tg-c3ow">0.21</td>
<td class="tg-c3ow">0.27</td>
</tr>
<tr>
<td class="tg-7btt">Km/kg</td>
<td class="tg-c3ow">Nâ‹…m/sqrt(W)/kg</td>
<td class="tg-c3ow">0.485</td>
<td class="tg-c3ow">0.417</td>
<td class="tg-c3ow">0.369</td>
<td class="tg-c3ow">0.329</td>
<td class="tg-c3ow">0.308</td>
</tr>
<tr>
<td class="tg-7btt">Conversion constant</td>
<td class="tg-c3ow"></td>
<td class="tg-7btt">8.1</td>
<td class="tg-7btt">8.2</td>
<td class="tg-7btt">8.2</td>
<td class="tg-7btt">8.2</td>
<td class="tg-7btt">8.2</td>
</tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<br /></div>
<div style="text-align: left;">
I have calculated the 'conversion constant' based on an average of a few different current vs torque measurements. The calculated 'conversion constant' is actually very close to the theoretical value, with measured values between 8.1 and 8.2. Any error is likely due to the friction in the system and my less than ideal measurement setup. Also note that these values are based on the manufacture rated <b style="text-align: center;">`K_{V}`</b> of the motors and they are little lower if my own <b style="text-align: center;">`K_{V}` </b>is used instead. I'm not sure why this is the case but it may have something to do with the 'fudge factor' each manufacture assumes when estimating <b style="text-align: center;">`K_{V}`.</b><br />
<br />
Also listed is <b style="text-align: center;">`K_{\tau}`</b> with respect to motor weight and motor cost. Surprisingly, the smallest motor (1000 <b style="text-align: center;">`K_{V}` </b><span style="text-align: center;">BR2212) </span>came out on top by a considerable margin in both cases, with a <b style="text-align: center;">`K_{\tau}` </b><span style="text-align: center;">more than double that of the other motors. This suggests that the electrical loading (current density in the copper windings) is much higher in the smallest motor when compared to the others. This same result could be achieved for the other motors by re-winding them to have a lower </span><b style="text-align: center;">`K_{V}`</b><span style="text-align: center;">. However, s</span><span style="text-align: center;">ince the 'torque efficiency' (torque produced per Watt) is the same <a href="https://things-in-motion.blogspot.com/2018/11/understanding-bldc-electric-motor.html" target="_blank">no matter how a motor is wound</a> provided the amount of copper remains the same, and that the mass fraction of copper is likely to be about the same for these motors, the smallest motor will produce no more torque per unit weight than the rest when thermally limited. </span><span style="text-align: center;">This assumption is backed up by the fact that the <a href="https://en.wikipedia.org/wiki/Motor_constants#Motor_constant" target="_blank">motor constant</a> </span><b style="text-align: center;">`(K_{M})` </b><span style="text-align: center;">per weight is about the same for all motors tested. Its likely that the motor manufacture decided to wind the smallest motor this way so that its base speed matched that required by an appropriately sized prop and the </span><span style="text-align: center;">typical battery voltage used on model aircraft and drones.</span></div>
<div style="text-align: left;">
<h2>
<span style="color: blue;">Limitations of this approach</span></h2>
<div style="text-align: left;">
Using <b style="text-align: center;">`K_{V}` </b><span style="text-align: center;">and a motors current draw to estimate its torque output only works provided that: </span><br />
<ul>
<li><span style="text-align: center;">A motor does not produce any useful </span><a href="https://en.wikipedia.org/wiki/Reluctance_motor" style="text-align: center;" target="_blank">reluctance</a> torque (i.e. its a <i> non-salient machine**</i>). This is true for essentially all <span style="text-align: center;">'hobby grade' electric motors. </span></li>
<li><span style="text-align: center;">A motors torque increase linearly with current. </span>This is not true if your motor is close to saturation. However, most 'hobby grade' electric motors are designed to operate with a current limit well below that needed to saturate them and so this is generally a safe assumption for stead state use.</li>
<li>A motor is operated below its base speed. Operating a motor above its base speed by field weakening effectively lowers a motors <b style="text-align: center;">`K_{V}` </b><span style="text-align: center;">in that region and so unless you know by how much </span><b style="text-align: center;">`K_{V}` </b><span style="text-align: center;">is reduced you can not calculate the torque output. However, since no 'hobby grade' motor controllers (ESC's) that I know of utilise field weakening this is also not an issue.</span></li>
<li>The current waveform supplied by a motor controller exactly matches a motors back EMF waveform (i.e. the current is not exactly on the <a href="http://scolton.blogspot.com/2010/01/3ph-duo-wrap-up-part-1-field-oriented.html" target="_blank">q-axis </a>at all times). This is generally true when using field oriented control (FOC) <a href="https://things-in-motion.blogspot.com/2018/12/why-most-hobby-grade-bldc-out-runners.html" target="_blank">on a PMSM motor</a> that has inbuilt position sensors (hall effect, encoder etc.). Operating a motor with 'six step 120 degree' commutation at low speed without a position sensor will result in less torque being produced than that predicted using the 'conversion constant' while high speed operation should be pretty close.</li>
</ul>
</div>
</div>
<h2>
<span style="color: blue;">Conclusion</span></h2>
<div>
The torque produced by brushless permanent magnet synchronous motor can be easily estimated so long as its <b style="text-align: center;">`K_{V}`</b><span style="text-align: center;"> and armature current is known. This relationship works because a motors </span><b style="text-align: center;">`K_{V}` </b><span style="text-align: center;">is fundamentally the same thing as its motor torque constant provided the right units are used, which is where a 'conversion constant' of ~8.3 is required.</span></div>
<div>
<span style="color: blue;"><br /></span></div>
<div>
<i>* The armature is considered the winding in which a rotational 'back emf' would be generated if the motor were used as a generator. In some motor designs the armature is on the rotor (e.g. brushed DC motor) or in the stator (e.g. brushless DC motor).</i><br />
<i><br /></i> <i>** A non-salient machine in this context is any motor which does not derive useful torque from <a href="https://en.wikipedia.org/wiki/Reluctance_motor">reluctance torque</a>. A motor can have <a href="https://en.wikipedia.org/wiki/Rotor_(electric)#Salient_pole_rotor">salient poles</a> on the rotor or stator and still be considered a non-salient machine with this definition.</i></div>
<div>
<br /></div>
<div>
<i>Equations were produced in this post with the help of <a href="https://arachnoid.com/latex/">arachnoid.com</a>. If you have noticed any errors in the above article then please let me know.</i><br />
<br /></div>
</div>
Richard Parsonshttp://www.blogger.com/profile/08331478925576296508noreply@blogger.com13tag:blogger.com,1999:blog-4080979156700236346.post-5013952631626726092018-12-23T19:39:00.000-08:002019-02-08T16:50:31.927-08:00Why most hobby grade BLDC out runners are actually permanent magnet synchronous motors (PMSM)Look anywhere online for a brushless direct current (BLDC) electric motor and you will likely find something that looks like the out-runner shown below:<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-W0rCwViiRus/XB7xxbEs6TI/AAAAAAAAUQc/uDFhZpdM9VUQIf2-6nySTBQ9OdYqCaVogCLcBGAs/s1600/stator%2Blabel.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="921" data-original-width="1030" height="357" src="https://2.bp.blogspot.com/-W0rCwViiRus/XB7xxbEs6TI/AAAAAAAAUQc/uDFhZpdM9VUQIf2-6nySTBQ9OdYqCaVogCLcBGAs/s400/stator%2Blabel.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">A hobby grade 'BLDC' motor</td></tr>
</tbody></table>
However, it turns out <b>this is technically not a <a href="https://en.wikipedia.org/wiki/Brushless_DC_electric_motor" target="_blank">BLDC</a> motor</b>. By most definitions, it is actually a permanent magnet synchronous motor (<a href="https://en.wikipedia.org/wiki/Synchronous_motor#Permanent_magnet_motors" target="_blank">PMSM</a>). If you own a similar looking out-runner style hobby motor then you too may have a PMSM.<br />
<br />
This post will briefly cover the difference between a BLDC motor and a PMSM and when this difference matters.<br />
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div>
<h2>
<span style="color: blue;">BLDC vs PMSM</span></h2>
<div>
BLDC and PMSM have a lot in common. Neither have brushes, both have permanent magnets on the rotor and have an armature on the stator. Where they differ is in how the magnetic field produced by the rotor magnets interacts with the windings of the armature.<br />
<br />
Let's consider an 'out-runner' motor of the style shown below.</div>
<div style="text-align: left;">
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td><a href="https://4.bp.blogspot.com/-2H9e15x0hMI/XCAU69iN1XI/AAAAAAAAUSo/S89kOtCLKhw9NLh14rUIl2Ljr6n02zo7ACLcBGAs/s1600/BPMS%2Bmotors3.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="865" data-original-width="1288" height="267" src="https://4.bp.blogspot.com/-2H9e15x0hMI/XCAU69iN1XI/AAAAAAAAUSo/S89kOtCLKhw9NLh14rUIl2Ljr6n02zo7ACLcBGAs/s400/BPMS%2Bmotors3.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 12.8px;">A cross-sectional view of a 6 slot, 8 pole (6N8P) motor</td></tr>
</tbody></table>
The movement of the rotor magnets past the stator teeth induces a back EMF (<a href="https://en.wikipedia.org/wiki/Counter-electromotive_force" target="_blank">bEMF</a>) into the windings of the armature. When the phase-phase bEMF is plotted with electrical angle (360 electrical degrees = 360 mechanical degrees / no. pole pairs) it may look sinusoidal, trapezoidal, or somewhere in between.<br />
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-gH2HSfOi60s/XB8VG1BI6QI/AAAAAAAAUSE/Qh-UqWlzKhEmG-OhgJVqxgHcVEQdq8zWwCLcBGAs/s1600/back%2Bemf.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="493" data-original-width="1069" height="183" src="https://3.bp.blogspot.com/-gH2HSfOi60s/XB8VG1BI6QI/AAAAAAAAUSE/Qh-UqWlzKhEmG-OhgJVqxgHcVEQdq8zWwCLcBGAs/s400/back%2Bemf.jpg" width="400" /></a></div>
<div>
<br /></div>
<div style="text-align: center;">
<div style="text-align: left;">
If a motor produces a trapezoidal bEMF then it is considered BLDC motor since this shape is similar to what would be seen from a conventional brushed DC motor. If your motor produces a sinusoidal bEMF then it is generally considered to be a PMSM. </div>
</div>
<div>
<h2>
<span style="color: blue;">How to tell if your motor is a BLDC or PMSM? Measure its bEMF</span></h2>
</div>
<div>
If you have access to an oscilloscope then determining if your motor is a PMSM or a BLDC motor is as simple as measuring across any two phases and spinning the rotor to observe the bEMF shape. This can be done by hand for low kV motors or with the help of a power drill for high kV motors. After doing this for a 12 slot, 14 pole (12N14P) out-runner the following shape was seen:</div>
<div>
<br /></div>
<div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td><a href="https://2.bp.blogspot.com/-oW1R9bQFGAI/XB73BQI7qKI/AAAAAAAAUQ0/axb7LZTHFoAPRtivZzPNTDyzq1DRUQP6ACLcBGAs/s1600/270%2BKv.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1197" data-original-width="1600" height="298" src="https://2.bp.blogspot.com/-oW1R9bQFGAI/XB73BQI7qKI/AAAAAAAAUQ0/axb7LZTHFoAPRtivZzPNTDyzq1DRUQP6ACLcBGAs/s400/270%2BKv.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 12.8px;">Back EMF produced by a 12N14P out-runner</td></tr>
</tbody></table>
</div>
<div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td><img alt="IMG_20180423_171938@0,5x.jpg" height="300" src="https://static1.squarespace.com/static/58aff26de4fcb53b5efd2f02/58d831ee03596e5dedde2772/5ade91e61ae6cf4ebd6296d1/1530597241757/IMG_20180423_171938%400%2C5x.jpg?format=750w" style="margin-left: auto; margin-right: auto;" width="400" /></td></tr>
<tr><td class="tr-caption"><span style="font-size: 12.8px;">D5065 270 kV 12N14P brushless out-runner from </span><a href="https://odriverobotics.com/shop/odrive-custom-motor-d5065" style="font-size: 12.8px;" target="_blank">Odrive Robotics</a></td></tr>
</tbody></table>
This motor clearly has a sinusoidal bEMF and so would be considered a PMSM. Repeating this same measurement for a couple of other motors from different manufacturers also yielded the same results.<br />
<br />
A sinusoidal bEMF typically means a motor has been wound with <a href="https://www.chegg.com/homework-help/definitions/windings-4" target="_blank">distributed windings</a>, where the windings are distributed over many slots, and is more common for large electric motors. Distributed windings are easily spotted by the overlap at the end of the motor.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-jscknixxGmA/XCBMOSfc4lI/AAAAAAAAUS0/hQNbCdNpxeMJbdU1zJYmf2YZi1ZyVxH8gCLcBGAs/s1600/concentrated%2Bvs%2Bdistributed%2Bwindings.jpg"><img border="0" data-original-height="725" data-original-width="1394" height="207" src="https://4.bp.blogspot.com/-jscknixxGmA/XCBMOSfc4lI/AAAAAAAAUS0/hQNbCdNpxeMJbdU1zJYmf2YZi1ZyVxH8gCLcBGAs/s400/concentrated%2Bvs%2Bdistributed%2Bwindings.jpg" width="400" /></a></div>
<br />
However, hobby grade brushless motors are almost universally constructed with concentrated windings. It is, therefore, a little surprising that they can produce a sinusoidal bEMF. The reason for a sinusoidal bEMF in the out-runner tested above is apparently related to its 12N14P configuration in combination with its doubly wound concentrated windings. I will explore this in more detail at some point in the future. Note that so far I have only measured 12N14P out-runner style motors and so you may see different results for motors with different slot/pole ratios or for in-runners.</div>
<div>
<h2>
<span style="color: blue;">BLDC or PMSM - Does it matter?</span></h2>
</div>
<div>
Useful torque is produced by an electric motor when you feed in a current waveform to each phase that perfectly opposes the a generated bEMF. Importantly, the shape of the waveform does not matter, so long as it exactly matches the bEMF waveform. For example, consider the torque produced by a PMSM and BLDC motor as seen by the figure below which were taken from <a href="http://scolton.blogspot.com/2009/11/everything-you-ever-wanted-to-know.html" target="_blank">James Mavey's</a> excellent masters thesis.</div>
<div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-nHQcHjNLA7U/XB72yAErWkI/AAAAAAAAUQw/sYv6RP2LeMQbeI2D9xCfOX6n6PHHSwiiACLcBGAs/s1600/BPMS%2Bmotors%2Btorque%2Boutput.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="580" data-original-width="757" height="306" src="https://2.bp.blogspot.com/-nHQcHjNLA7U/XB72yAErWkI/AAAAAAAAUQw/sYv6RP2LeMQbeI2D9xCfOX6n6PHHSwiiACLcBGAs/s400/BPMS%2Bmotors%2Btorque%2Boutput.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Torque output vs current input for a PMSM (left) and a BLDC (right) motor</td></tr>
</tbody></table>
<div>
When you sum up the current contributions from each phase for the sinusoidal waveform (PMSM) and for the trapezoidal waveform (BLDC) you see the same result; a perfect constant output current, and therefore a constant output torque. Also, in theory at least, both motors should be equally efficient. In reality, even if you could perfectly match the current to the bEMF, the rapid change in flux density seen by the stator in a BLDC motor due to the use of a trapezoidal waveform is likely to induce larger eddy current losses than a comparable sinusoidal PMSM.</div>
<div>
<br /></div>
<div>
The problems begin when you use a motor controller that outputs a current waveform which does not exactly match the bEMF of your motor. Most low-cost hobby grade motor controllers (ESC's) only output a 'six-step 120 degree' current waveform like that shown for the BLDC motor above. Therefore, if you use a PMSM with one of these ESC's it's torque output will be choppy, which creates audible noise, vibration, and will be quite inefficient. Instead, you would ideally use a motor controller which supports <a href="http://scolton.blogspot.com/2010/01/3ph-duo-wrap-up-part-1-field-oriented.html" target="_blank">field oriented controlled</a> (FOC) and that outputs a sinusoidal current waveform that more closely matches that of your motor.<br />
<br />
Furthermore, the trapezoidal bEMF produced by a BLDC motor can vary quite a lot from motor to motor. This means that the current waveform produced by an ESC will never perfectly match the bEMF of a BLDC motor. This means that even if you match a BLDC motor with an ESC you will still have some amount of motor noise, vibration, and decreased efficiency. A PMSM has no such problem since ideally, ever motor produces the same sinusoidal bEMF.<br />
<br />
For most hobby applications (e.g. small model planes, boats, and cars) using a PMSM with a conventional six step ESC won't cause any noticeable problems. However, for high-performance applications (e.g. multi-rotors used for cinematography, robotics and EV applications) the reduced noise, vibration and increased efficiency that comes from using a FOC motor controller with a PMSM may mean it's worth the extra investment.</div>
<div>
<h2>
<span style="color: blue;">Conclusion</span></h2>
</div>
<div>
If a motor produces a sinusoidal bEMF then its a PMSM and not a BLDC motor. A PMSM is best driven by a sinusoidal current as this reduces noise, vibration and improves efficiency.<br />
<br />
In order to avoid confusion, going forward I will be simply referring to both BLDC and PMSM as brushless permanent magnet synchronous motors (PMSM).</div>
<div>
<i><br /></i></div>
<div>
<i>If you have noticed any errors in the above article then please let me know.</i></div>
</div>
</div>
Richard Parsonshttp://www.blogger.com/profile/08331478925576296508noreply@blogger.com8tag:blogger.com,1999:blog-4080979156700236346.post-5963943799951505962018-12-08T23:26:00.000-08:002018-12-11T22:44:14.619-08:00How to select the right power source for a hobby BLDC (PMSM) motorIf you have a '90 A, 2000 W, 6 to 10S' rated BLDC* motor such as <a href="https://hobbyking.com/en_us/kd-53-30-high-voltage-brushless-outrunner-190kv.html">this</a>, do you then need a 90 A, 2000 W, 22 to 37 V power source?<br />
<br />
<h3>
<b><span style="color: blue;">The short(er) answer:</span></b></h3>
<div>
<b><span style="color: blue;"><br /></span></b></div>
In most cases, no, not even close. Select your power source (battery or mains power supply) based on your own specific requirements using the following three steps:<br />
<br />
<b>1. Required Current:</b> The current supplied to a motor controller <b style="font-style: italic;">does not </b>equal the current supplied to a BLDC motor. Motor controllers ('ESC') take a relatively high voltage and low current power source and, though <a href="https://learn.sparkfun.com/tutorials/pulse-width-modulation/all">pulse width modulation</a> (PWM), converts it into a low voltage and a high current for use with a motor. <a href="http://tinyurl.com/y99benpx">This single phase simulation</a> (thanks to <a href="https://github.com/madcowswe">Oskar Weigl</a>) demonstrates how a controlled current is supplied to a motor and has typical values for the resistance, inductance and capacitance seen in each part of the circuit. In short, most motors have a very low resistance (i.e. < 0.1 Ohm) and so a high current can be supplied with relatively little power. Therefore, you only need to consider the voltage and total power output of your power source and your motor controller will take care of the rest.<br />
<br />
<div>
<b>2. Required Voltage: </b>The voltage required from your power source will depend on your motors velocity constant (`K_{V}`) and the the top speed you require. For example, if you required 3000 RPM from a 190 `K_{V}` motor then the power source voltage needed (`V_{PSU}`) is give by<br />
<ul>
</ul>
<div style="text-align: center;">
<span style="font-size: large;">`V_{PSU} = \frac{RPM_{max}}{K_{V}} \times 1.25 = 19.7 V`</span></div>
<div style="font-weight: bold;">
<b><br /></b></div>
<div>
The value of 1.25 is a safety margin since `K_{V}` is always measured with no load. Most fixed voltage mains power supplies come in voltage steps of 12, 15, 24, 36 and 48V. Therefore, provided it did not exceed the voltage limit of your motor controller, you would select a 24 V power supply.</div>
<div style="font-weight: bold;">
<b><br /></b></div>
<b>3. Required Power: </b>As a rough rule of thumb the peak power (`P_{max}`) required from your power source will depend on the maximum motor current (`I_{max}`) needed at your maximum RPM as given by:</div>
<div>
<br /></div>
<div style="text-align: center;">
<span style="font-size: large;">`P_{max} = I_{max} \times (\frac{RPM_{max}}{K_{V}}) \times 1.25`</span></div>
<div>
<div>
<br />
were the value of 1.25 is again a safety margin is to account for inefficiencies in the motor and motor controller. Once you know the peak power required you can then select a power supply which meets your needs. At the end of this post I recommend a few different mains powered fixed voltage power supplies.<br />
<br />
The remainder of this post will develop a more detailed understanding of when and why BLDC motors draw power.</div>
<div style="text-align: left;">
<h2>
<span style="color: blue;">Estimating power draw</span></h2>
The mechanical power produced by a motor is given by<br />
<br />
<div style="text-align: center;">
<span style="font-size: large;">`P = \tau \times \omega = \tau \times \pi \times \frac{RPM}{30}`</span></div>
<div style="text-align: center;">
<br /></div>
where `\tau` is the motor torque in N.m and `\omega` is its rotational velocity in radians per second. If we assuming for the moment that a motor is 100% efficient then its power consumption can be mapped as follows<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-ILgLuiIShUQ/XAyb6tUs5-I/AAAAAAAAT_Y/jldxetyaws0IPTva3r7eq08I-Ns0-q1mgCLcBGAs/s1600/Motor%2Bpower%2Bmap.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="919" data-original-width="1010" height="363" src="https://2.bp.blogspot.com/-ILgLuiIShUQ/XAyb6tUs5-I/AAAAAAAAT_Y/jldxetyaws0IPTva3r7eq08I-Ns0-q1mgCLcBGAs/s400/Motor%2Bpower%2Bmap.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: left;">
where the values labelling each contour line are the motors power draw. Of course, real BLDC hobby motors <a href="https://build-its.blogspot.com/p/motor-characterization.html">have an efficiency far lower than 100%</a>. An electric motor is least efficient at low speeds and at high torque where the winding loss is largest with respect to the mechanical power output. We can estimate the winding losses for a motor using the following equation:</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<span style="font-size: large;">`P_{loss} = I^{2}R`</span></div>
<br />
Using the 190Kv motor mentioned above as an example, the current required to produce a given torque can be estimated using the motors <a href="https://discourse.odriverobotics.com/t/motor-torque-and-temperature-measurements-for-keda-6364-motors-and-mosfet-temperature-rise/356">known torque constant</a> and its winding resistance which I have measured to be 0.0447 Ohm. Combining this power loss with the power output of the motor we can produce a slightly more realistic power map.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-V6m_IdmRqEc/XAye09MUmqI/AAAAAAAAT_k/QVqFrE_LaoI-_k1H87P9kjKYPdLsSj-YgCLcBGAs/s1600/Motor%2Bpower%2Bmap%2Bwith%2Bwinding%2Blosses.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="930" data-original-width="992" height="375" src="https://1.bp.blogspot.com/-V6m_IdmRqEc/XAye09MUmqI/AAAAAAAAT_k/QVqFrE_LaoI-_k1H87P9kjKYPdLsSj-YgCLcBGAs/s400/Motor%2Bpower%2Bmap%2Bwith%2Bwinding%2Blosses.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: left;">
All of the contour lines are now moved closer to the left at high torque levels. Its clear that even a relatively small 450 W PSU can produce full torque up to ~1000 RPM or reduced torque up to 5000 RPM. Note that this ignores any power loss in the motor controller and core losses, which will dominate at higher speeds. However, at these speed ranges and currents both of these losses will be fairly small relative to winding losses and so can be ignored.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
If this motor is powered with a 24 V power supply voltage then its not possible to reach all regions of the power map due to the <a href="https://en.wikipedia.org/wiki/Counter-electromotive_force">back EMF</a> created by the motor as it spins. The motor can no longer reach a required torque when the back EMF plus the voltage drop across the motor exceeds the supply voltage. Using this cutoff the achievable torque and speed is shown below.</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-4DZVctnj2As/XAyhrp_N_xI/AAAAAAAAUAA/xYmLbqFDLJQp7-OeESF2eMickikr7YzMgCLcBGAs/s1600/Motor%2Bpower%2Bmap%2Bwith%2Bwinding%2Blosses%2Bwith%2BV%2Blimit3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="926" data-original-width="990" height="373" src="https://1.bp.blogspot.com/-4DZVctnj2As/XAyhrp_N_xI/AAAAAAAAUAA/xYmLbqFDLJQp7-OeESF2eMickikr7YzMgCLcBGAs/s400/Motor%2Bpower%2Bmap%2Bwith%2Bwinding%2Blosses%2Bwith%2BV%2Blimit3.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: left;">
In reality the drop off in torque will be a smooth curve but here it has a 'saw tooth' pattern due to the speed only being plotted at 100 RPM intervals.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
It is important to note here that the peak torque in the above power map would require a current of more than 70 A and a loss in the motor windings of more than 200 W. <a href="https://discourse.odriverobotics.com/t/motor-torque-and-temperature-measurements-for-keda-6364-motors-and-mosfet-temperature-rise/356">Testing has shown</a> that this is enough to permanently damage the motor windings in about 30 seconds. The steady state torque output of the motor will depend on the cooling provided but for this motor a safe assumption is about 30 A, or one third of its 'peak power' value. This is equal to about 1.3 N.m and is shown by the dashed line in the power map above. The reason that this line is fixed at a constant torque and not at a constant power is that almost all the heat is generated by the previously mentioned winding losses. These losses are dependent upon the winding current which in turn determines the torque.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
It can also be seen from the power contour lines that a 450 W power supply is able to provide enough power for any speed and torque up to 4000 RPM at 1 N.m. Rather than simply limiting the motor current for all speeds, and therefore the motor torque, motor controllers with active power management could instead actively limit the current depending on the motor speed, keeping the power draw always below a set limit. This is a planned feature for <a href="https://odriverobotics.com/">Odrive Robotics motor controllers</a>.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
In order to operate this motor at higher speeds we need a higher supply voltage. The power map for a 48 V supply voltage is shown below.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-LcGelwgaIps/XBCtDL5mZSI/AAAAAAAAUB4/Uc3opW6a2UgDZKWNqOUdWBfpmzMNjg-FQCLcBGAs/s1600/RPM%2Bvs%2Btorque%2B48V.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="817" data-original-width="869" height="375" src="https://4.bp.blogspot.com/-LcGelwgaIps/XBCtDL5mZSI/AAAAAAAAUB4/Uc3opW6a2UgDZKWNqOUdWBfpmzMNjg-FQCLcBGAs/s400/RPM%2Bvs%2Btorque%2B48V.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
With a 48 V power supply you would now need to limit your peak torque to less than 0.6 N.m to in order to not exceed the 450 W power rating previously used. Therefore, if you are planning on operating at higher speeds and voltages you will generally need a more powerful power supply.</div>
<div>
<div>
<div style="text-align: center;">
<div style="text-align: left;">
<h2 style="text-align: left;">
<span style="color: blue;">A few suggestions for power supplies</span></h2>
<div>
If you are uncomfortable working with mains voltages then a high current, low voltage mains laboratory bench power supply from a reputable supplier (<a href="http://www.manson.com.hk/product/sps-9602/">Manson is a good choice</a>) is the way to go.<br />
<br />
If you are comfortable with having semi-exposed mains wiring and in adding your own plug, then any fixed voltage power supply will be much cheaper. I recommend a well known brand, such as the <a href="https://www.amazon.com/MEAN-WELL-SE-450-24-Supply-Single/dp/B005T6NNKO/ref=sr_1_1?ie=UTF8&qid=1544325154&sr=8-1&keywords=SE-450-24">Mean Well SE series</a>. You can find cheaper copies but they tend have lower real world outputs, have noisy always-on fans, poorly implemented protection and lower quality components that may fail sooner or even be a fire risk.<br />
<br /></div>
<div>
If you are more adventurous and want the absolute highest kW/$ then consider picking up a new-old-stock or used server power supply off ebay. For example, I picked up this 6.5kW 42V server power supply for less than $100 USD delivered. </div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-Rfjz5cUdVrk/XAsvrqOgQfI/AAAAAAAAT8M/neEP6BNCrWoxbKSChbCmq-FCNsIigZl1ACLcBGAs/s1600/IMG_20180831_183931.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1200" data-original-width="1600" height="240" src="https://2.bp.blogspot.com/-Rfjz5cUdVrk/XAsvrqOgQfI/AAAAAAAAT8M/neEP6BNCrWoxbKSChbCmq-FCNsIigZl1ACLcBGAs/s320/IMG_20180831_183931.jpg" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-GHVldPYAugs/XAsxX1IVdGI/AAAAAAAAT8k/99T5EP9zEAQLTdyopQuUzlg5PZXFg3s5QCLcBGAs/s1600/IMG_20180901_094012.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1200" data-original-width="1600" height="240" src="https://3.bp.blogspot.com/-GHVldPYAugs/XAsxX1IVdGI/AAAAAAAAT8k/99T5EP9zEAQLTdyopQuUzlg5PZXFg3s5QCLcBGAs/s320/IMG_20180901_094012.jpg" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-l2G2Q1GMz90/XAswavHlp2I/AAAAAAAAT8Y/AnWmhvbYQdkQbv8QK0CmX2rmi4NyKHdyACLcBGAs/s1600/IMG_20180901_094206.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1200" data-original-width="1600" height="240" src="https://2.bp.blogspot.com/-l2G2Q1GMz90/XAswavHlp2I/AAAAAAAAT8Y/AnWmhvbYQdkQbv8QK0CmX2rmi4NyKHdyACLcBGAs/s320/IMG_20180901_094206.jpg" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
These power supplies are commonly used in the RC community to charge batteries but will also work fine with a motor controller in most situations. See <a href="https://www.rcgroups.com/forums/showthread.php?1292514-How-to-convert-Server-Power-Supplies">this thread</a> for more details. These power supplies can be modified to output slightly different voltage ranges if needed (~ 35 to 50V). The downside of using these power supplies is that the server fan can be quite noisy, and so may need replacing, and that its not always easy to get access to the output. An approach suggested by Macaba on the <a href="https://discord.gg/rKSFWg">Odrive Robotics discord server</a> was to drill holes in the case and attach cables to the internal binding posts. </div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-USdUZd1GUKM/XAsyEbxaruI/AAAAAAAAT8s/8EF_W8wQI5oQmWCdp5fzqA8W9UMtTtk6QCLcBGAs/s1600/IMG_7869.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1200" data-original-width="1600" height="239" src="https://1.bp.blogspot.com/-USdUZd1GUKM/XAsyEbxaruI/AAAAAAAAT8s/8EF_W8wQI5oQmWCdp5fzqA8W9UMtTtk6QCLcBGAs/s320/IMG_7869.JPG" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Image credit Macaba</td></tr>
</tbody></table>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-XptQZWWI0Vg/XAsyRwMo8zI/AAAAAAAAT8w/rHSpJkgXMsYwX0sR_EAKk2cFnch7FVV4QCLcBGAs/s1600/IMG_20180901_105341.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1200" data-original-width="1600" height="240" src="https://4.bp.blogspot.com/-XptQZWWI0Vg/XAsyRwMo8zI/AAAAAAAAT8w/rHSpJkgXMsYwX0sR_EAKk2cFnch7FVV4QCLcBGAs/s320/IMG_20180901_105341.jpg" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
Also, its worth remember that these power supplies can easily draw enough power to trip your mains breaker and so make sure you have the capacity to run them before buying. </div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
Provided you have the mains capacity, one of these power supplies should be able to meet even the highest peak power draws of any hobby BLDC motor.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
* I'm told that most hobby motors produce a back EMF wave form that is closer to a pure sine wave than that of a traditional brushed motors trapezoidal wave form. Therefore most hobby 'BLDC' motors should more correctly be called permanent magnet synchronous motors (PMSM). This topic will receive more attention in the future.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
<i>The spreadsheet used to create the figures in this post can be found <a href="https://docs.google.com/spreadsheets/d/1Nwk89I9T-gcxIV0Q85pvfX2R_qOgznXD37wqrkEBT9U/edit?usp=sharing">here</a>. The contour figures were produced in <a href="https://www.originlab.com/Origin">Origin</a>. If you have noticed any errors in the above article then please let me know.</i></div>
</div>
</div>
</div>
<br /></div>
</div>
Richard Parsonshttp://www.blogger.com/profile/08331478925576296508noreply@blogger.com2tag:blogger.com,1999:blog-4080979156700236346.post-86132324788047880112018-11-23T18:07:00.000-08:002019-02-09T16:22:07.674-08:00Why you don't see tape wound foil conductors used on electric motors2019 update: Induced eddy currents also contribute significant losses to tape wound motors. This argument has now also been added.<br />
<br />
--------------------------------------------------------------------------------------------------------------------<br />
<br />
Most electric motors are wound using circular conductors. Some motors are wound using square or rectangular shaped conductors. However, I have never seen a motor with conductors that have a flat, tape-like cross section made from thin foils which are then wound onto a tooth like a roll of sticky tape.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-cZz0-H0ccgQ/W_VELfLdWEI/AAAAAAAATkI/GREaA1ExyNgJg42n3Ik_rMbCC3rOG2oXwCLcBGAs/s1600/conductor%2Bround%2Band%2Bsquare.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="583" data-original-width="1275" height="181" src="https://3.bp.blogspot.com/-cZz0-H0ccgQ/W_VELfLdWEI/AAAAAAAATkI/GREaA1ExyNgJg42n3Ik_rMbCC3rOG2oXwCLcBGAs/s400/conductor%2Bround%2Band%2Bsquare.jpg" width="400" /></a></div>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-XIZJhHr7ASM/W_VELcer6QI/AAAAAAAATkE/GKww4DdNHy0Bd1Gg0as1sMeJM664kUxjQCLcBGAs/s1600/tape%2Bwound%2Bconductors.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="609" data-original-width="821" height="295" src="https://1.bp.blogspot.com/-XIZJhHr7ASM/W_VELcer6QI/AAAAAAAATkE/GKww4DdNHy0Bd1Gg0as1sMeJM664kUxjQCLcBGAs/s400/tape%2Bwound%2Bconductors.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">A comparison of the different winding topologies on a stator (black)</td></tr>
</tbody></table>
That was until <a href="http://scolton.blogspot.com/2009/11/epic-axial-motor-epic-axial-update.html">this blog post</a> by Shane Colton come to my attention.<br />
<br />
This raised a question: <b>Why is it that this style of winding is not used for electric motors?</b><br />
<b><br /></b> <b>The short answer:</b> If you wind a thick (i.e. mm) tape wound conductor then you will likely see very large induced eddy current losses in your conductors due to the changing flux they are exposed to. If you use very thin tape wound conductors (i.e. < 0.1 mm thickness) then your copper fill factor will be very poor and you would be better off using a more typical stranded conductor.<br />
<br />
Read on to see why.<br />
<h3>
<span style="color: blue;">Potential benefits for using a tape wound foil conductor</span></h3>
First, why use this style of conductors? The benefits may include:<br />
<ol>
<li>A high <a href="https://en.wikipedia.org/wiki/Coil_winding_technology#Calculation_of_the_fill_factor">copper fill factor</a>. Perfectly stacked round conductors will have a maximum fill factor of ~0.9. Tape wound conductors may improve on this. (This turns out to not be true in most cases)</li>
<li>Ease of winding. A single wide tape may be easier to stack than many small conductors.</li>
<li>High conductor cross section while maintaining a small <a href="https://en.wikipedia.org/wiki/Bend_radius">minimum bend radius</a>. For an equal conductor area, a flat foil will have a much smaller minimum bend radius than a round conductor.</li>
<li>Improved conductor utilisation at high frequency. Due to the <a href="https://en.wikipedia.org/wiki/Skin_effect">skin effect</a>, the centre of a round conductor will carry little current when operating at high electrical frequencies. </li>
<li>The possibility of improved cooling capacity. The large uniform area of a tape wound conductor may prevent hot spot formation and facilitate better heat conduction to the surface of a winding stack when compared to round wires.</li>
</ol>
<h3>
<span style="color: blue;">Potential negative consequences for using a tape wound foil conductor</span></h3>
<div>
Potential drawbacks to this style of conductors include:</div>
<div>
<ol>
<li>Difficulty in winding a tape wound foil conductor due to the surrounding stator teeth getting in the way.</li>
<li>Stray flux crossing the conductor at right angles (normal to the flat surface of the foil) may significantly increase eddy current losses in a motor.</li>
<li>A high aspect ratio may mean that the electrical insulation occupies a significant fraction of the available area, reducing the copper fill factor.</li>
<li>Due to the <a href="https://en.wikipedia.org/wiki/Proximity_effect_(electromagnetism)">proximity effect</a> and the difference in inductance seen by different parts of a tape wound conductor embedded in the slot of a motor, a larger AC current will tend to flow in the region of a conductor closer to the top of a stator slot than the bottom. This will cause a larger than expected AC resistance. See chapter 3.6.7.4 of <a href="https://books.google.com.au/books/about/Design_of_Brushless_Permanent_magnet_Mot.html?id=HhXBQgAACAAJ&source=kp_cover&redir_esc=y">this book</a> for further details.</li>
</ol>
<div>
Let's look at each of these drawbacks one by one:</div>
</div>
<div>
<br /></div>
<h3>
1. Difficulty in winding around stator teeth</h3>
<div>
For small radial flux motors, it may be difficult to automate the winding process. However, segmented radial flux motors or<a href="http://www.mojaladja.com/upload/elmotor/Analysis%20of%20the%20Yokeless%20and%20Segmented%20Armature%20machine.pdf"> yokeless and segmented armature</a> (YASA) axial flux motors, such as those made by <a href="https://www.magnax.com/">Magnax</a>, <a href="http://emrax.com/">Emrax</a> and <a href="https://www.yasa.com/">Yasa</a>, and used by Shane in his blog post, are becoming increasingly common.</div>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-xSPqdxwgw58/W_VImjFJnlI/AAAAAAAATkY/zDqt1WsbyJsmtAzCuWbcM-MWEYvIl9JuwCLcBGAs/s1600/yasa%2Bmotor.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="688" data-original-width="870" height="253" src="https://1.bp.blogspot.com/-xSPqdxwgw58/W_VImjFJnlI/AAAAAAAATkY/zDqt1WsbyJsmtAzCuWbcM-MWEYvIl9JuwCLcBGAs/s320/yasa%2Bmotor.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">A YASA motor with segmented stator teeth.</td></tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: left;">
A YASA axial flux motor has a stator constructed from individual core sections which can each be wound separately before final assembly of the motor. These motors could, therefore, be easily wound with a tape-style conductor.</div>
<div>
<br /></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><img alt="Magnax axial flux windings and cores 2.jpg" height="282" src="https://www.magnax.com/hs-fs/hubfs/Magnax_Feb2017/Images/Magnex%20Product/Magnax%20axial%20flux%20windings%20and%20cores%202.jpg?t=1542731676757&width=800&name=Magnax%20axial%20flux%20windings%20and%20cores%202.jpg" style="margin-left: auto; margin-right: auto;" width="400" /></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stator segments wound and ready for assembly of a <a href="https://www.magnax.com/product">Magnax </a>motor</td></tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: left;">
Therefore for segmented core motors winding a tape wound foil should pose no problem.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<h3 style="clear: both;">
2. Induced eddy current losses</h3>
<div>
Let's consider the worst-case scenario. Below is a 2D <a href="http://www.femm.info/wiki/HomePage">FEMM </a>simulation of an open-ended 30 mm diameter, 80 mm long mild steel rod with 10 turns of conductors added to one end with a current of 6A. <i>Side note: Drawing lines in FEMM is a pain. I recommend using a dedicated CAD program like <a href="https://www.autodesk.com/products/fusion-360/students-teachers-educators">F360</a> and then just exporting your model as a DXF and importing that into FEMM </i></div>
<div>
<i><br /></i></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-HVb0fAFymdc/W_ipkiKn8DI/AAAAAAAATnU/a-pphEMgfUELT6LFhZLzsEtSqD4OU0XXQCLcBGAs/s1600/rod%2Bsim.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="643" data-original-width="739" height="278" src="https://1.bp.blogspot.com/-HVb0fAFymdc/W_ipkiKn8DI/AAAAAAAATnU/a-pphEMgfUELT6LFhZLzsEtSqD4OU0XXQCLcBGAs/s320/rod%2Bsim.jpg" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: left;">
It's quite clear that a large portion of the flux is crossing the windings placed at one end and so it would be expected that a foil conductor would have large induced eddy currents. To test if this is really a problem when operating at electrical typical electrical frequencies you would find in an electric motor I wound two rods with the same dimensions as mentioned above. One with round conductors and one with a foil conductor. The foil was cur to a width such that its CSA was the same as the round conductor and both had ten turns applied.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-_R5-y9fxyr8/W_iqptQAvjI/AAAAAAAATng/yXRiCEo1bZca3zXxxxeGVCvb4qt4K7wKACLcBGAs/s1600/IMG_20181121_173238.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="800" data-original-width="1600" height="160" src="https://3.bp.blogspot.com/-_R5-y9fxyr8/W_iqptQAvjI/AAAAAAAATng/yXRiCEo1bZca3zXxxxeGVCvb4qt4K7wKACLcBGAs/s320/IMG_20181121_173238.jpg" width="320" /></a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-5z1YXz4saKg/W_iqy1ZmlAI/AAAAAAAATno/DM9YIDQU5VMEZvzbJPwt3yFQ20EqEyVNwCLcBGAs/s1600/IMG_20181121_175841.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="800" data-original-width="1600" height="160" src="https://4.bp.blogspot.com/-5z1YXz4saKg/W_iqy1ZmlAI/AAAAAAAATno/DM9YIDQU5VMEZvzbJPwt3yFQ20EqEyVNwCLcBGAs/s320/IMG_20181121_175841.jpg" width="320" /></a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-1draSFE0CtI/W_iqm8WRvTI/AAAAAAAATnc/G6Kut5Y3cScVI0Um6EVasAJ5b3G1AlSRgCLcBGAs/s1600/IMG_20181121_180118.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="800" data-original-width="1600" height="160" src="https://3.bp.blogspot.com/-1draSFE0CtI/W_iqm8WRvTI/AAAAAAAATnc/G6Kut5Y3cScVI0Um6EVasAJ5b3G1AlSRgCLcBGAs/s320/IMG_20181121_180118.jpg" width="320" /></a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-9t71AlxFeNk/W_iq2DgDaVI/AAAAAAAATns/iaGBg6GdU8IikcpZ3N90j0HLA8-EsxB_ACLcBGAs/s1600/IMG_20181121_180625.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="800" data-original-width="1600" height="160" src="https://1.bp.blogspot.com/-9t71AlxFeNk/W_iq2DgDaVI/AAAAAAAATns/iaGBg6GdU8IikcpZ3N90j0HLA8-EsxB_ACLcBGAs/s320/IMG_20181121_180625.jpg" width="320" /></a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-5eTmogH-CqI/W_iqxLO3P0I/AAAAAAAATnk/TR1uYHj1ZAI8wzpoJphoIKwyBL2zpOZtQCLcBGAs/s1600/IMG_20181121_181235.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="800" data-original-width="1600" height="160" src="https://1.bp.blogspot.com/-5eTmogH-CqI/W_iqxLO3P0I/AAAAAAAATnk/TR1uYHj1ZAI8wzpoJphoIKwyBL2zpOZtQCLcBGAs/s320/IMG_20181121_181235.jpg" width="320" /></a></div>
<div class="image-gallery">
</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
Using an AC BH tracer I thin applied a 1 kHz and 10 kHz sinusoidal current to the windings. The result:</div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-Iz_x0IozAMg/W_itYf5LyVI/AAAAAAAAToA/3S44qa53d4o76EqCsauh8EKJiFUpnif2QCLcBGAs/s1600/AC%2BBH.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="636" data-original-width="1134" height="223" src="https://3.bp.blogspot.com/-Iz_x0IozAMg/W_itYf5LyVI/AAAAAAAAToA/3S44qa53d4o76EqCsauh8EKJiFUpnif2QCLcBGAs/s400/AC%2BBH.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">AC BH loops</td></tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: left;">
It's clear that at 1 kHz there is little difference in the shape of the BH loop while at 10 kHz they are quite distinct. This indicates that even under this worst-case scenario a 10 kHz frequency, which is well above the typical electrical frequency used by electric motors, is required to produce significant eddy currents in the foil conductors. </div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
This suggests that for an electric motor, which will likely operate at a lower frequency and has less stray flux than this example, will not suffer from significant eddy currents in the conductors.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
For a real BLDC electric mot<span style="background-color: white;">or, the flux crossing the conductors due to the </span>relative movement of the magnets on the rotor must also be considered. This flux can be considerably larger than that induced by the stator windings alone. You can use<a href="https://docs.google.com/spreadsheets/d/14JMvWJfZmX69oHY8pMpoEo147WztYAYIlpZnncWnf7Y/edit?usp=sharing"> this sheet</a> to estimate the expected eddy current losses in a tape wound conductor. To edit the sheet just make a copy using the 'file -> make a copy' option.</div>
<h3 style="clear: both; text-align: left;">
3. Copper fill factor</h3>
<div>
It is desirable to have as high a copper fill factor as possible when constructing an electric motor so as to reduce the current density in the conductors so as to minimise `I^2R` losses. The copper fill factor is impacted by:</div>
<div>
<ul>
<li>The stacking factor ( `P_{d}`): How efficiently a conductor is arranged in the slot.</li>
<li>The copper to insulation area ratio: A thick layer of insulation gives a low area ratio and a low fill factor.</li>
</ul>
<div>
As previously mentioned, round conductors have a fundamental stacking factor limit of ~0.9. Square and tape conductors can have a theoretical 100% stacking factor.</div>
</div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://4.bp.blogspot.com/-hgQVoguw4DU/W_ixOPJIToI/AAAAAAAAToU/SmCQy4frstUi5OrsG5M2feB1DhFo2M-CACLcBGAs/s1600/conductor%2Bstacking%2B2.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="656" data-original-width="1101" height="237" src="https://4.bp.blogspot.com/-hgQVoguw4DU/W_ixOPJIToI/AAAAAAAAToU/SmCQy4frstUi5OrsG5M2feB1DhFo2M-CACLcBGAs/s400/conductor%2Bstacking%2B2.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
I imagine that it is this desire for a high stacking factor without the need to purchase expensive square conductors that convinced Shane to use tape wound conductors. </div>
<div>
<br /></div>
<div>
Even though square and tape wound conductors can have a theoretical 100% fill factor, the copper fill factor (ratio of copper to total area) can never be 100% due to the need for an insulation layer in-between each winding. The thicker this insulation layer, the lower the copper (Cu) fill factor. Therefore the copper stacking factor is given by:</div>
<div>
<br /></div>
<div style="text-align: center;">
<span style="text-align: start;"><span style="font-size: large;">Cu fill factor` =\frac{CSA_{Cu}}{(CSA_{Cu}+CSA_{Ins})}\times `Stacking factor </span></span></div>
<div>
<br /></div>
<div>
According to <a href="https://temcoindustrial.com/product-guides/wire-cable-and-accessories/magnet-wire/magnet-wire-faq">this site</a>, a typical 24 gauge magnet wire has an insulation layer thickness of ~0.025 mm. For Shane's motor, he used a single <a href="https://en.wikipedia.org/wiki/Kapton">polyimide </a>(Kapton) tape, which has a typical thickness of 0.03mm, to insulate just one side of his copper foil.<br />
<br />
Using this information I have estimated the copper fill factor when winding a conductor with a cross sectional area of 13.1 `mm^2` which is 2.54 mm wide, like that used by Shane's design. The spreadsheet can be found <a href="https://docs.google.com/spreadsheets/d/1cs33IUC9fdb8YB1kTnIzR3BUnfpVOkYCMEdhbVN7RIc/edit?usp=sharing">here</a>.</div>
<div>
<br /></div>
<div>
Here are the results:</div>
<div>
<ul>
<li>Round conductor:<b> 88.5 %</b> Copper fill factor</li>
<li>Square conductor: <b>97.3 %</b> Copper fill factor </li>
<li>Tape conductor: <b>94.5 %</b> Copper fill factor </li>
</ul>
<div>
This is quite a respectable copper fill factor and would only be superseded by the use of square conductors.<b> Shane is therefore justified in his choice of a tape wound conductor.</b></div>
</div>
<div>
<br /></div>
<div>
However, one thing to note here is that the currents used by Shane's motor are unusually high, and his winding turn number is quite low. You will more typically encounter motors that have dozens to hundreds of turns and that operate at well less than 50 A. In this situation, a tape wound conductor may make less sense. </div>
<div>
<br /></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><img height="360" src="https://cdn.shopify.com/s/files/1/0496/8205/products/KDE2315XF_V_3c418db0-98f9-4523-bdc5-69d435976bac_2048x2048.JPG?v=1526409284" style="margin-left: auto; margin-right: auto;" width="400" /></td></tr>
<tr><td class="tr-caption" style="text-align: center;">A more typical motor that operates at < 50A with many turns per tooth</td></tr>
</tbody></table>
<div>
<br /></div>
<div>
For example, if you wound a small motor with a tape conductor that has a cross-section equal to a 24 gauge conductor (0.5 mm diameter, 0.205`mm^2`) onto a 10 mm stator tooth your fill factor is as follows:<br />
<ul>
<li>Round conductor:<b> 75 %</b> Copper fill factor</li>
<li>Square conductor: <b>80.8 %</b> Copper fill factor </li>
<li>Tape conductor:<span style="color: #38761d;"> </span><b><span style="color: red;">40.6 %</span></b> Copper fill factor </li>
</ul>
<div>
All of the fill factors listed above are relatively low since the insulation layer is fixed while the motor size is reduced. The tape fill factor is also unacceptably small since the area per turn of the insulating Kapton layer at ~0.1 `mm^2` is now comparable to that of the copper foil itself at 0.2 `mm^2`.</div>
<div>
<br /></div>
<div>
To really drive this point home lets consider the copper fill factor of Shane's motor if he were to use lower currents and a higher number of turns, and therefore smaller cross section conductors.</div>
</div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-mmM-hk7f3OI/W_jAv5iAYUI/AAAAAAAATog/5eNUYo_dryMdOl669qzo-hDlUIpQg7TYgCLcBGAs/s1600/copper%2Bcsa%2Bvs%2Bcopper%2Bfill%2Bfactor.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="952" data-original-width="1259" height="301" src="https://3.bp.blogspot.com/-mmM-hk7f3OI/W_jAv5iAYUI/AAAAAAAATog/5eNUYo_dryMdOl669qzo-hDlUIpQg7TYgCLcBGAs/s400/copper%2Bcsa%2Bvs%2Bcopper%2Bfill%2Bfactor.jpg" width="400" /></a></div>
<div>
<br /></div>
<div>
If you were to halve the peak current of Shane's motor while maintaining the same current density (i.e. also halve the copper cross section) then it no longer makes sense to use a tape wound conductor. At this point, you are probably better off <a href="https://www.alibaba.com/trade/search?fsb=y&IndexArea=product_en&CatId=&SearchText=square+magnet+wire">purchasing flat magnet</a> wire or just sticking with round wire.<br />
<h2>
<span style="color: blue;">Additional points to consider:</span></h2>
User Kindiana over on the <a href="https://discord.gg/kPHsE3G">Odrive Robotics discord channel</a> also pointed out that motor manufacturers are even less likely to be interested in tape wound conductors for the following reasons:<br />
<ul>
<li>The extra weight and cost of copper is likely not worth the efficiency improvements at low torque for RC applications and is likely also why they don't stack windings closer together.</li>
<li>The tooling for machine winding foil may be far more complicated for a one piece stator compared to round wires or bundles of round wires.</li>
<li>There may be no readily available source of pre-insulated copper foils with custom dimensions, so manufacturers may need to do their own insulation.</li>
</ul>
<h2>
<span style="color: blue;">Conclusion</span></h2>
</div>
<div>
At first glance, a tape wound conductor may appear as an attractive means to increase your electric motors copper fill factor without the need for expensive square conductors. However, when you look at the numbers more closely it becomes apparent that the high aspect ratio of a tape wound conductor will likely result in a worse fill factor for most applications. Furthermore, when a tape wound conductor is made quite thick then you will likely encounter considerable induced eddy currents in the conductor itself. Therefore careful consideration is required before you decide on a conductor geometry for your electric motor.<br />
<br /></div>
<div>
<i>Equations were produced in this post with the help of <a href="https://arachnoid.com/latex/">arachnoid.com</a>. If you have noticed any errors in the above article then please let me know.</i></div>
Richard Parsonshttp://www.blogger.com/profile/08331478925576296508noreply@blogger.com4tag:blogger.com,1999:blog-4080979156700236346.post-33303545942476392152018-11-17T16:56:00.001-08:002021-01-21T01:16:47.250-08:00Understanding BLDC electric motor constants - The Kv torque fallacyIt is a common misconception that if you have two otherwise identical electric motors, one with a low <a href="https://en.wikipedia.org/wiki/Motor_constants#Motor_velocity_constant,_back_EMF_constant">Kv</a> and one with a high Kv, the lower Kv motor will be capable of producing more torque with less waste heat.<br />
<br />
<div style="text-align: left;">
<b>This assumption is incorrect.</b></div>
<br />
The specific torque density of an electric motor (torque per unit volume) is independent of its Kv. Similarly, the heat generated by an electric motor while producing a given torque value is also independent of Kv. Read on to see why.<br />
<br />
<h2>
<span style="color: blue;">Torque produced by a BLDC motor for a fixed current density</span></h2>
<div>
The torque capability of a BLDC motor is determined by the average magnetic field strength produced by the stator which acts on the rotor, the average magnetic field strength produced by the rotor magnets which act on the stator and the dimensions of the rotor itself.</div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-6Zt-o6J9DrM/W_CgQevwi-I/AAAAAAAATeI/QQoiQF-5blkma8b-mv1xobe04bEF5VW2wCLcBGAs/s1600/stator%2Blabel.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="921" data-original-width="1030" height="286" src="https://3.bp.blogspot.com/-6Zt-o6J9DrM/W_CgQevwi-I/AAAAAAAATeI/QQoiQF-5blkma8b-mv1xobe04bEF5VW2wCLcBGAs/s320/stator%2Blabel.jpg" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
If we have two otherwise identical motors, one with a low Kv and one with a high Kv, then we can assume that the average magnetic field produced by the rotor magnets and the dimensions of the rotor itself (i.e. its radius and length) are the same. This leaves only the average magnetic field strength produced by the stator as a possible difference.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
The average magnitude of flux provided by the stator which acts over the entire surface of the rotor is determined by many factors (flux gap size, stator core material, the geometry of the motor etc.) but we can once again assume that these are all the same between our high and low Kv motors. Therefore,<b> the only possible difference between our two motors can come from the average current density in the stator windings.</b></div>
<div class="separator" style="clear: both; text-align: left;">
<b><br /></b></div>
<div class="separator" style="clear: both; text-align: left;">
Looking at a cross-sectional view of the stator we can see that there is only a fixed area available to place the copper windings.</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://2.bp.blogspot.com/-qRMR914zCHc/W_CnOwe54EI/AAAAAAAATeg/yMIeu7NPnNEKOPYx76FhYZNN0NgizvkEwCLcBGAs/s1600/stator%2Bfill%2Bfactor3.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="856" data-original-width="736" height="320" src="https://2.bp.blogspot.com/-qRMR914zCHc/W_CnOwe54EI/AAAAAAAATeg/yMIeu7NPnNEKOPYx76FhYZNN0NgizvkEwCLcBGAs/s320/stator%2Bfill%2Bfactor3.jpg" width="275" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
Let's look at just a single stator tooth and the impact that a different turn number will have on the applied magnetic field strength when placed in the available winding area.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://3.bp.blogspot.com/-0K7eGGAQPIU/W_DLbN-nI0I/AAAAAAAATf0/y412eVnbZakuUVSsYnlqdRvqjifpgO9DQCLcBGAs/s1600/low%2Bvs%2Bhigh%2BKV%2Bwith%2Bdifferent%2Bturn%2Bnumber2.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="500" data-original-width="956" height="208" src="https://3.bp.blogspot.com/-0K7eGGAQPIU/W_DLbN-nI0I/AAAAAAAATf0/y412eVnbZakuUVSsYnlqdRvqjifpgO9DQCLcBGAs/s400/low%2Bvs%2Bhigh%2BKV%2Bwith%2Bdifferent%2Bturn%2Bnumber2.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
The motor with fewer turns of wire will have a lower induced voltage produced by the rotor magnets as they pass by the tooth, giving it its high Kv rating when compared to the motor with more turns.<br />
<br />
The high Kv motor has 4 turns of wire each at 10 A for a combined total of 40 A/tooth. The low Kv motor has 10 turns of wire each at 4 A, for <b>the same total</b> of 40A/tooth. <b>Therefore these two motors will provide the same magnetic field strength and have the same torque output. </b><br />
<b><br /></b>
Yes, you could increase the current in the low Kv motor to be the same as the high Kv motor at 10A and produce more torque. However, this is fundamentally no different than increasing the current in the low Kv motor with the same end result. Therefore, rewinding a motor to increase its Kv only makes sense when you wish to match the motor current draw to the current limit of your existing motor controller (ESC). You could just as easily achieve a higher torque output by purchasing a new motor controller with a higher current limit and keeping your existing motor unchanged. Alternatively, if you have a motor with a very poor copper fill factor (area in the stator slot filled with copper vs empty air) then it may also make sense to rewind your motor.<br />
<br />
Note that for the purposes of this argument we are ignoring the production of any useful reluctance torque (like that used by a <a href="https://en.wikipedia.org/wiki/Reluctance_motor">reluctance motor</a>) which will be true for almost all motor you encounter as a hobbyist.<br />
<br />
Now let's consider waste heat generation for our high and low Kv motors.<br />
<h2>
<span style="color: blue;">Waste heat produced by a BLDC motor for a fixed torque</span></h2>
The power dissipated by a motor winding is given by:<br />
<br />
<div style="text-align: center;">
<span style="font-size: large;">`P=I^2R` </span></div>
<br />
where I is the current in the windings and R is the resistance of the windings. As the power dissipation in the motor scales with the square of the stator current, it feels only natural to assume that the low Kv motor, with its 4A current draw, will produce less heat than our high Kv motor with its 10A current draw. However, this assumption fails to take into consideration that the total area of the copper windings is fixed and therefore the current density remains the same.<br />
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
The DC resistance of a wire is given by:<br />
<br />
<div>
<div style="text-align: center;">
<span style="font-size: large;">`R_{DC}=\frac{l\rho}{A}`</span></div>
<br />
<div style="text-align: left;">
where l is the wire length, <span style="text-align: center;">`\rho` is the conductivity of the conductor and A is the conductor area. In order to simplify this argument lets assume we are using square cross-section conductors.</span></div>
<div style="text-align: left;">
<span style="text-align: center;"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-cUWgj0Al8Y4/W_DLIUqSy4I/AAAAAAAATfs/Qw12IjtU1aUrOaF-YcZorATYsekkOPINwCLcBGAs/s1600/low%2Bvs%2Bhigh%2BKV%2Bwith%2Bdifferent%2Bturn%2Bnumber%2Barea%2Bsquare3.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="720" data-original-width="960" height="300" src="https://1.bp.blogspot.com/-cUWgj0Al8Y4/W_DLIUqSy4I/AAAAAAAATfs/Qw12IjtU1aUrOaF-YcZorATYsekkOPINwCLcBGAs/s400/low%2Bvs%2Bhigh%2BKV%2Bwith%2Bdifferent%2Bturn%2Bnumber%2Barea%2Bsquare3.jpg" width="400" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
<span style="text-align: center;">In order to fit more turns into the same area, we had to reduce the cross-section of each individual conductor, which reduced its area and therefore increased its resistance. </span>If we assume that each turn of wire has a length of 1 and that the total conductor cross-sectional area is also 1, then enter the current and turn numbers listed above we find:</div>
<div style="text-align: center;">
<div style="text-align: left;">
<br /></div>
<span style="font-size: large;">`P=I^2R=I^2\frac{l\rho}{A}=10^2\frac{4\rho}{1/4} = 4^2\frac{10\rho}{1/10} = 1600\rho`</span><br />
<div>
<br /></div>
</div>
<div style="text-align: left;">
<b style="text-align: center;">Therefore, for a given torque (fixed current density), copper fill factor and copper winding area, the power dissipation is not changed by altering the motor Kv. </b></div>
<div style="text-align: center;">
<br /></div>
<div style="text-align: left;">
If you do wish to increase the specific torque density of an existing BLDC motor then you have a few options:</div>
<div style="text-align: left;">
</div>
<ol>
<li>Rewinding the motor to increase the copper fill factor by more efficiently packing the conductors.</li>
<li>Replace the permanent magnets in the rotor with higher energy density magnets.</li>
<li>Reduce the flux gap distance between the rotor and the stator.</li>
</ol>
Increasing your peak current output of your motor controller so that the motor windings run with a higher current density will of course also increase your peak torque output. However, this will require additional cooling to handle the extra waste heat and you run the risk of <a href="https://en.wikipedia.org/wiki/Saturation_(magnetic)">saturating the core</a> material</div>
<div>
<br />
<div>
</div>
<div>
Note that in the above example we assumed a DC resistance. In reality, a motor will operate with an AC current. At very high frequencies it may make sense to rewind a motor to use <a href="https://en.wikipedia.org/wiki/Litz_wire">many parallel small conductors</a> rather than singular thick conductors in order to minimise the <a href="https://en.wikipedia.org/wiki/Skin_effect">skin effect</a>.</div>
<br />
<div>
<h2>
<span style="color: blue;">Conclusion</span></h2>
<div>
You will not improve the specific torque density or lower the power dissipation for a given torque output by rewinding a motor to have a lower Kv. However, it can make sense to rewind a motor so that its peak current draw will be better matched with an existing motor controller. </div>
<div style="text-align: center;">
<br /></div>
<i>Equations were produced in this post with the help of <a href="https://arachnoid.com/latex/">arachnoid.com</a> and are based on those found in the book <a href="https://books.google.com.au/books?id=Cf78BAAAQBAJ">Electric Motors and Drives: Fundamentals, types and applications</a> by Austin Hughes. If you have noticed any errors in the above article then please let me know.</i></div><div><i><br /></i></div><div><i><br /></i></div><div><i><b><u>January 2021 addendum</u></b></i></div><div><i><b><u><br /></u></b></i></div><div>An anonymous commenter has pointed out that the above argument does not consider the impact that changing the wire diameter has on the lengths of wire between each wound tooth or to the ESC. See the comment below for more details. In short, if you were to decrease the Kv of a motor by doubling the number of turns and halving the conductor area you may think that the total length of wire from one phase terminal to the next is also doubled. However, this turns out not to be the case because the length of wire from the ESC to the motor and from one wound tooth to the next does not actually change. Therefore, the total increase in the length of wire is slightly less than double, making the lower Kv motor technically more efficient at producing the same amount of torque. However, this effect is likely to be very small in most scenarios. </div><div><br /></div><div>Thanks, Anonymous!</div><div><i><b><u><br /></u></b></i></div><div><i><b><u><br /></u></b></i></div>
</div>
Richard Parsonshttp://www.blogger.com/profile/08331478925576296508noreply@blogger.com5