Jun 08, 2014, 03:06 PM Registered User Spain, Community of Madrid, Madrid Joined Jun 2014 9 Posts Help! Brushless motor model Hello all, Hope this is the right place to put this long question, …, maybe problem… A year ago after a debate in a Spanish forum I started to develop a model to calculate brushless motors performances. I recognize that at the beginning I thought it was going to be easy. I found and interesting thread from you Ron (hope you read this post), and from there I took some ideas: one of it was to use FEMM as open code for magnetic calculations. I started to create a program with Excel that allows me to generate in a simply (relatively) manner the geometries of the motors. There are parameters to set the geometry of the stator, number of poles, winding, strands, wire turns, materials, material properties, etc. For the rotor is similar, but simpler as the geometry is also simpler. Here below, I attach an animation of an invented geometry used to test the program. I also made a Lua program that writes in a file the flux linkage and I post process it with an Excel spreadsheet to get the back emf shape. Up to here has been fine. A lot of work in visualbasic and Lua, but I get something. Arriving here has already not being easy. My problem comes now when I try to link supply voltage and current with the motor magnetics. I have been looking in internet and I found hundreds of technical reports, which all are fine. They help me to understand very well how this motors work and the mathematics behind, but I don’t see how to solve the basic equation that link all the parts (I am not an electronic engineer, but mechanical…) The coupled circuit equations of the stator windings in terms of motor electrical constants are (copied from one report to help me in this post): Now comes my question, can you help me? I am pretty sure that any idea will help me. Thanks. Xavier
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 Jun 08, 2014, 06:06 PM Registered User Joined Nov 2007 913 Posts Since the three phases are all the same you only need to consider one phase. The equation for one phase would be: V= Ri + L di/dt + e Now comes the hard part, trying to solve the equation. V, I and e are all sinusoidal and have different phases. The motor can run with both a leading and lagging power factor, e can be larger or smaller then V. These motors can be tortured to run in many ways. With a model airplane ESC the motor is forced to run with a constant timing which puts some limits on both the power factor and phase of e. You have to take into account the ESC and how it limits the motor, otherwise the solution for the motor equation gets very complex. Also you cannot solve the equation for trapezoidal drive. It would be much easier to calculate the KV of a motor. KV is the open circuit voltage generated at a given RPM. Since it is open circuit voltage it does not depend on R, L or I. The generated voltage is the number of lines of flux cut per second times the number of windings. This would only depend on the physical dimensions of the motor and magnet strength. (There is a fudge factor needed to convert generator KV to model airplane KV) Assuming a model airplane ESC all the motor equations would apply. Last edited by mjsas; Jun 08, 2014 at 09:27 PM.
Jun 10, 2014, 05:22 PM
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Hello mjsas

Thanks for your answer. I know that the problem is not easy to be solved, maybe that’s the reason it pushes me to work on it…

The KV problem is already solved. The figures, graphics and tables are the calculation for one example motor.
The first figure is an animated gif with the test motor.
The motor definition is:
• 5 turns of one wire. Wire radius 0.75mm.
• Housing: Al6061
• Material for the stator: 1020 steel. Depth: 19.7mm. laminated thickness: 0.35mm
• Magnet: NdFeB 52 MGOe
• Shaft: Std steel
The stator shape is optimized to avoid material saturation.

The graphics below are the flux linkage per phase, Back-EMF, voltage per radian and cogging torque

For this particular design the KV value I got is 4958,28 rpm/V

Coming back to the problem I am trying to solve. My next step is suppose a speed control in the easiest way, something like what is shown in the scheme below:

Let’s suppose that we have electronics capable to switch the gates S1 to S6, every 60 motor electrical degrees in the right order. For the time being, also as initial simplification, we don’t consider the power control, ie, no PWM. Electrical scheme like:

We can also consider that the timing is 0. With these assumptions, how can I solve the problem equations?

The final goal solving this problem is to have the capability to predict motor torque function of timing. As you know the torque curve of a motor is something like the one shown in the graphic below.

If we are capable to develop this model we will be able to predict torque capability vs timing, and it will be something like the following graphic

With the knowledge of the effects of timing we will know the current required, then motor temperature, battery autonomy, etc. Finally we can also increase the efficiency of the motor by setting the adequate timing curve function of rpms in the ESC.
Xavier

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 Jun 11, 2014, 03:22 PM Registered User Joined Nov 2007 913 Posts Given that the simple equation for one phase is: V= Ri + L di/dt + e What makes this so difficult is V, I and e all have different phases. In this case both V and e are known so V - e becomes V'sin(wt). V'sin(wt) = Ri + Ldi/dt This a more simple equation to solve for i. If i is known you should be able to calculate torque. With a brushed motor the torque is a function of Kv and current. For these motors Kv is sort of a phony number and can only be used to estimate torque. It seems to me that torque should be a function of generator Kv, I, the phase angle of I and the cogging torque. With sinusoidal drive it is possible to do something, with trapezoidal drive it would be impossible. The power between the battery and ESC is just V * I. However, between the motor and ESC the voltage and current are out of phase so the power factor has to be added in. That means the motor - ESC current is always larger then the battery - ESC current. The larger the phase angle between V and I is the lower the efficiency is as I becomes larger. Adjusting the timing would reduce the phase angle and improve efficiency. To get maximum efficiency at a given operating point you have to adjust the timing to minimize the phase angle between V and I which also minimizes I.
 Jun 12, 2014, 04:19 PM Registered User Spain, Community of Madrid, Madrid Joined Jun 2014 9 Posts Hello mjsas, This exchange of posts is helping me to understand the electrics behind. I will try to be more precise on the exercise that I will have to solve finally if I understood it correctly. For the specific case of a motor Y winded, for every 60 electrical degrees, I will have to solve an equation like the following one, based on the electrical scheme. The back-emf, e is a function not sinusoidal, as seen in the graphic above, that´s the reason I considered it as f(wt). Two poles will be connected in series, but their back-emf de-phased 60 degrees. R and L should be the same. V= Ri + L di/dt + f(wt) + Ri + L di/dt + f(wt + π/3) V= 2Ri + 2L di/dt + f(wt) + f(wt + π/3) V - f(wt) - f(wt + π/3) = 2Ri + 2L di/dt If f’(wt) = f(wt) + f(wt + π/3) then V – f’(wt) = 2Ri + 2L di/dt To solve this equation for i, should I suppose that i will be V - f’(wt-90)? as de current is 90 degrees de-phased backwards from voltage. It seems to me that torque should be a function of generator Kv, I, the phase angle of I and the cogging torque. Electromagnetic Torque equation is: T = i * (f(wt) + f(wt + π/3)) / w With sinusoidal drive it is possible to do something, with trapezoidal drive it would be impossible. I don’t understand this statement… The power between the battery and ESC is just V * I. However, between the motor and ESC the voltage and current are out of phase so the power factor has to be added in. That means the motor - ESC current is always larger then the battery - ESC current. The larger the phase angle between V and I is the lower the efficiency is as I becomes larger. Adjusting the timing would reduce the phase angle and improve efficiency. To get maximum efficiency at a given operating point you have to adjust the timing to minimize the phase angle between V and I which also minimizes I. Correct. Timing can be used in two ways: 1.- improve motor efficiency 2.- increase motor speed at the cost of efficiency Xavier
 Jun 13, 2014, 01:53 PM Registered User Joined Nov 2007 913 Posts It has been a long time since I graduated from collage so give me a small break. One problem with any computer model it that they print out the answer with too much accuracy. You noted that the KV was calculated to be: (For this particular design the KV value I got is 4958,28 rpm/V). That would have six decimal places of accuracy. The real answer should have been 5000 +-1000 or some such. Each time that two uncertain numbers are multiplied or divided, the uncertainty of the two numbers is added. For example if 10+-1 is multiplied by 20+-2 the answer is 200 +-40. It went from a 10% uncertainty to a 20% uncertainty. If the program does many complex math operations in a row the output becomes, basically nothing useful. Given the equation: V – f’(wt) = 2Ri + 2L di/dt If V is a square wave and the BEMF, f'(wt) is some weird waveform, the equation gets to be impossible for normal humans to solve. Nature can solve any equation so there is a solution. I assume your goal is to design an ESC that dynamically changes the timing to maximize efficiency. All autos have a control system to control the mixture. The computer uses many sensors and tables to get near the optimum mixture and than fine tunes the mixture using the O2 sensors. The computer constantly sweeps the mixture from lean to rich and monitors the O2 sensor to determine the peak value. The only way to find the peak value is to sweep across the peak. For your application the ESC would use a table to get near the optimum timing and then sweep the timing to find the peak or minimum current. The main point that I am trying to make is that the model is very useful for understanding, but in the end may not be able to give answers to enough accuracy to design an ESC without some sort of feedback. The auto industry has very good models of their engines but in the end the engine computer has to find the optimum operating point by itself. Last edited by mjsas; Jun 14, 2014 at 04:50 PM.
 Jun 14, 2014, 07:58 AM Registered User Spain, Community of Madrid, Madrid Joined Jun 2014 9 Posts “Given the equation: V – f’(wt) = 2Ri + 2L di/dt If V is a square wave and the BEMF, f'(wt) is some weird waveform, the equation gets to be impossible for normal humans to solve. Nature can solve any equation so there is a solution.” I know about those difficulties, and that’s the reason for the finite elements methods, but it is essential to minimize the errors to start with the right equations and boundary conditions. I believe that this equation is the right one, although up to now we still have not taken into account all the factors, for example L, in reality is not constant, it varies with rotor angle and current (if current is so high that the stator material falls into saturation), but it is known once the geometry is frozen and in case of material saturation it has to be calculated. Also the back-emf has to be calculated as it depends of flux linkage which changes with the current. This makes the process to set up this simulation more complex and the computing time increase a lot. For every angle and current step, L and back-emf has to be calculated… almost nothing… To simplify the computation let’s consider R constant, which is a wrong assumption because it changes with the temperature, but let’s consider a constant temperature simulation :-). Also it is important to know the physics behind. I remember from me collage time (30 +/- 1 years :-) ) that in an electrical circuit with a coil the current go 90 degrees behind the voltage with the same curve shape but different value (resistance and impedance are in the relation). I understand that the motor windings are coils in series, and then a similar approach could be applied. If this is the case, once I know the shape of the voltage, I know the shape of the current (90 degrees de-phased) and the relation between them is the resistance + impedance effect. Are these assumptions correct? “The main point that I am trying to make is that the model is very useful for understanding” This is exactly my objective developing this simulation: understand how a brushless motor works and how the different variables affect torque, current, speed. Xavier
 Jun 14, 2014, 08:16 AM Registered User Spain, Community of Madrid, Madrid Joined Jun 2014 9 Posts I just realize that may be I had to mention that part of the complexity of the problem that we are trying to solve comes from the fact that I am using a static tool, FEMM, to help me to solve a dynamic problem. But, I choose to use FEMM and save me to develop a solver for Maxwell, Gauss, etc equations in dynamic mode. That would be my second thesis!!!
 Jun 14, 2014, 12:40 PM homo ludens modellisticus The Netherlands, GE, Nijmegen Joined Feb 2001 10,924 Posts See also the attached www.microchip.com files (not the pictures) in this post www.wattflyer.com/forums/showthread.php?p=950237#post950237 Vriendelijke groeten Ron
Jun 14, 2014, 05:26 PM
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I am going to cheat and just copy some pages from my textbook. This book claims that R is very small compared to the impedance of L so R can be ignored. Note that V are I in phase when V+e leads V by 90 degrees, assuming R=0. This gives the highest efficiency. The ESC needs to control the timing of I, not the BEMF.

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 Jun 15, 2014, 02:26 AM Registered User Spain, Community of Madrid, Madrid Joined Jun 2014 9 Posts Then, in simple words and for anybody who is following this threat right now… (poor crazy guys ) : the current I lags 90 degrees V – f’(wt), supposing R=0. For R different than 0, It will lag V – f’(wt) an angle lower than 90 deg. Any clue on how can I estimate that angle?
Jun 15, 2014, 02:58 AM
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Hello Ron,
First of all: congratulations 1-5 is a sweet revenge from last world cup.
You can also have a look at the following report.

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 Jun 29, 2014, 09:01 PM Registered User Joined Mar 2013 350 Posts Em ..... What happened !!! I was following this post and I waited and waited and nothing happened Not that I understand all that was written, but it was very interesting anyway. I am trying to rewind a bigger Brushless motor for sometime now, and I read what ever I get my hands on to, but I have to admit I really regret all thoes Electronic hours which I went after my Girlfriend insted Good Summer Everyone and by the way is there a video or something of Ron wining ?
 Jul 01, 2014, 02:46 PM Registered User Spain, Community of Madrid, Madrid Joined Jun 2014 9 Posts Be patient. I am preparing another long post...
Jul 01, 2014, 03:08 PM
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Joined Mar 2013
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Quote:
 Originally Posted by PDN2000 Be patient. I am preparing another long post...
Hi
I am rewinding a Car. Alt. ( Oh I cant remember the name ) ) You know the one which Produce el. in the car, anyway. I have Clean a couple of them, A very hard job I have to Admit, specially when they are so Old, and dearty.
Tomarrow I will take them to a shop near us, so that I can Polish them and clean them such so I can put the 2 together, without any Cm. Difference.

Believe me or not I use to help others with English, and now I fell I am getting dumer and dumer some how!!? Maybe because I am out of Work, and sitting home for a couple oc mounth
anyway, I hope that you Guys can help me with rewinding them, Can I put Pictures in here, or do I Just start a new post do you think ?

Thank you

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