Example Data for Electric Drives Experiment 6 Analysis and Control of a Permanent Magnet AC (PMAC) Motor The intent of this document is to provide example data for instructors and TAs, to help them prepare for the electric drives laboratory activities. This document is informal and does not represent a laboratory report. Checklist of items requested in these lab procedures: Record the DC armature voltage applied to the drive motor while measuring the PMAC motor s induced EMF. Also record the PMAC motor s speed in this configuration. Va = -3.0, Wm = 29 rad/sec Save an image of the ControlDesk plotter showing a few cycles of the PMAC motor s encoder and induced EMF waveforms. Given the number of AC cycles observed in the EMF waveform for one complete rotation of the motor, determine how many magnets (poles) are mounted on the rotor of the PMAC motor. 4 sinusoidal cycles per rotation => 8 poles (magnets), or 4 pole-pairs
Save an image of the ControlDesk plotter showing the time difference between the rising EMF zero crossing and the encoder s zero position. Also record the time offset. The time offset, as shown at the top of the plotter window, is 0.0440693 seconds. Calculate the value of index for your lab station s PMAC motor. Theta_index = 0.0441seconds * 4 pole-pairs * 29 rad/sec = 5.12 radians. This value is different for each motor. Calculate the angular offsets between the encoder s zero position and the first maximum and minimum magnetic flux locations on the PMAC s rotor. Theta_da+ = Theta_index 7/6 pi = 1.45 radians Theta_da- = Theta_index pi/6 = 4.59 radians Use a cursor to measure the maximum value (i.e., amplitude) of the EMF waveform displayed on the ControlDesk plotter. Save an image of this cursor measurement, and also explicitly state the maximum EMF value in your report. The peak EMF is listed as 2.9248 V. A precision of 2.9 V is more appropriate.
Calculate the PMAC motor s voltage constant, k E. eˆab ke = 2.9/29 = 0.1 V-s/radian m Record the voltage constant listed on the PMAC motor s nameplate. Convert that value to units of V-sec/radian and compare it to the value you determined experimentally. Nameplate Ke = 9.5 V/Krpm x 60/(2*pi*1000) = 0.091 Fairly close. Calculate the proportional and integral constants for the PMAC motor s current and speed controllers: k ii, k pi, k iω, and k pω. Show the relevant equations and your work in calculating these values. Given: R a L a 0.625 Ω 0.45 mh J eq 5.0 x 10-4 kg m 2
Also know that K_pwm = 40 ω ci = 2πf ci = 2*pi*400 = 2513 τ e represents the electrical time constant, L a /R a = 0.00045/.625 =.00072 ci Ra From lab 4, kii =2513*.625/40 = 39.3, close to stated value of 40 k k pi e ii PWM k = 0.028, close to the stated 0.03 ω cω = 2πf cω = 2*pi*10 = 62.8 k i k T J 2 2 eq c eq c 2 2kT 1 tan( 120 ) J =9.86, close to stated value of 10 k p kitan( 120 ) ki 3 =0.27, close to stated value of 0.3 c c Record your calculation for Theta_da_Initial. (Different for each motor.) /2 Theta _ da _ Initial da = 1.45 + pi/2 = 3.02 radians pole _ pairs Record the limits and center of the range of Theta_da_Initial values which produced positive rotation for your PMAC motor. How does this measured value compared to the Theta_da_Initial value that you calculated earlier? Range is 2.6 to 4.6 radians, center value is 3.6 radians. (Different for each motor.) Record the motor s average speed at the center Theta_da_Initial value. Speed approx. 73 rad/sec. Save an image showing a few periods of the I a, I b, and I c waveforms. Does phase a lead phase b by 120, as expected, and does phase b lead phase c by 120? Yes about the 120 offsets (although the waveforms are not ideal sinusoids!).
Reduce the desired quadrature current to 1.3 Amps, and record the motor s average speed. Does the change in the PMAC motor s speed after a decrease in I q make sense? Explain. Motor speed decreased to about 65 radians/second after decreasing the current. Yes, a lower current results in a smaller magnetic flux which will result in less torque applied to the rotors magnets. Record the limits and middle of the range of Theta_da_Initial values where the PMAC motor did not spin. Stopped between 4.8 and 5.4, center value of 5.1 radians. (Different for each motor.) How far, in radians, is this center value from the Theta_da_Initial value which produced the maximum speed? 5.1-3.6 = 1.5 radians Theoretically, how far of a phase shift from the quadrature alignment would cause the rotor s magnets to be perfectly aligned with the magnetic field generated by the stator, resulting in no motion? 90 degrees, or pi/2 = 1.57 radians Is the theoretical expectation close to your measurement? 1.5 vs. 1.57, yes, close
Record the limits and center of the range of Theta_da_Initial values that produced a speed for the PMAC motor did not spin. Negative rotation for theta values from -0.6 to 1.0 => center value of 0.2 How does this center measurement compare to the value of Theta_da_Initial which is predicted to produce the maximum negative rotation (there may be a phase difference of 2π)? Predicted Theta _ da _ Initial / 2 = 4.59 + 1.57 = 6.16 6.16 2pi = -0.12, a decent match to the range shown above da How does the measured center angle compare that which resulted in the maximum positive speed, and does that difference match the theoretical expectation? You may want to refer to Figure 6.6 in your explanation. The center of the negative speed range should be pi away from the center of the positive speed range. 0.2 + 3.14 = 3.34, reasonable comparison to the measured value of 3.6 Save an image of the plotter showing the desired and actual speeds when the speed controller was added to the model, and analyze the quality of the PMAC speed controller. Good response time and steady state error, although the speed varies quite a bit around the desired value. Note that each lab motor is different, but I made the students use the same motor parameters in creating their controllers. Therefore, the controllers performance will vary widely among the lab groups. Some students observed a surprisingly sluggish response time when the controllers were applied to their particular motors.
I created and then deleted the following content from this lab, as it seemed to just confuse the students too many concepts in the lab at that point. Others may find the material useful, however, so saving it here. In order to use the current controller design process that was developed for the PMDC motor and followed above, the currents for the a, b, and c phases need to be combined into one equivalent current. Engineers have found that it works well to combine the three current phases into a stator-current space vector, i s (t), which points in the direction of the total magnetic field produced by the three phase currents. Ideally, the stator s magnetic field vector would be located at quadrature or 90 degrees from a peak magnetic flux on the rotor (in terms of the rotor s magnetic field waveform, rather than a mechanical angle) to produce the maximum torque. If the three phase currents create a magnetic field component that aligns directly with the maximum magnetic flux on the rotor rather than at quadrature, which is undesirable but can happen, that component is called the direct component. The direct magnetic field component created by the stator works to hold the rotor magnets in place, rather than pull or push the rotor s magnets in a rotational direction. So, i s(t) would ideally equal i q (t) but it may have both i q (t) and i d (t) components. The following transformation is often used to convert from a-b-c to d-q current components: 2 4 cosda cos da cos da Ia Id 2 3 3 I b I q 3 2 4 sin da sin da sin da I c 3 3 This relationship assumes that the I a waveform leads I b by 120, and that I b leads I c by 120. In addition, the coils for I a, I b, and I c should be positioned 120 apart on the stator, with 120 representing the phase difference of the rotor s magnetic field rather than a mechanical angle. The d-q current components are converted back to a-b-c components as follows: cosda sin da Ia 2 2 2 Id I b cosda sin da 3 3 3 I q I c 4 4 cosda sin da 3 3 Note that these transformations are used before and after the current controller in the MATLAB Simulink model, so that the simple PMDC version of the current controller can be used. In your lab report, calculate the phase a, b, and c currents that would produce a direct current of 0 Amps and a quadrature current of 2.0 Amps when da 90 / 2. Show your work. For example,
2 I cos sin 2 0 0 1 2 2 a Id Iq 2 1.63 3 2 2 Amps 3 3 Double-check your calculations against the knowledge that for three-phase systems, Ia Ib Ic 0 at each point in time. Note, as shown in the MATLAB Simulink model, that da is equal to the encoder s radial position plus the da offset that you determined earlier.