Simulation of Permanent-magnet Motor Drive

Li Rui

Abstract

This project mainly focuses on analysis of the ripple component of the battery voltage and cur-rent of an ideal permanent-magnet synchronous machine (PMSM) control system. First, detaileddescriptions will be given for model setups of the battery, filter, inverter, Sine-Triangle modulator,and the motor. Then, calculation will be conducted to determine V r

qs and V rds needed to generate

designated torque at required engine speed. After that, a Simulink based simulation of this systemwill be developed and results shown to verify the calculation results. Finally, the ripple componentin the battery voltage and current, as well as other variables of this system will be plotted, andanalysis in detail will be given for each plot.

Model Setup

Battery Model

The battery model used in this project is basically derived from Project 2. However, as weassume the battery SOC to be constant here, and thus Voc, all the relevant parameters within thebattery model are in turn fixed. Also, as we are controlling the motor by voltage, the currentoutput of the battery model is not necessary any more. Therefore, slight changes are needed uponour previous battery model. Figure 1 shows the battery model used in this project.

Please note that the input port iin stands for the current measured by the Filter block shownby Figure 2 on Page 2. It is actually flowing out of the battery model, rather than into the battery,as it seems. This current signal is used for calculating the terminal voltage of the battery model.

Filter Model

Using KCL and KVL, we can assume iL = ibatt, and vC = vS during each short time interval.Thus, the voltage across the inductor, and the current through the capacitor can be calculated by

vL = vbatt − vS

iC = ibatt − iS

iL = ibatt =1

L

∫vLdt

vC = vS =1

C

∫iCdt

(1)

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Figure 1: Block Diagram of The Battery Model

The input of this block will be vbatt from the battery model, and the iS measured by the Inverterblock shown in Figure 3 on Page 3 . The Simulink Model of the filter is shown in Figure 2.

Figure 2: Block Diagram of The Filter Model

Inverter Model

The inverter model is based on the suggested one given from the instruction. It is built by basicswitching components in Simulink for simplicity. The Inverter is shown in Figure 3. The currentinputs, ias, ibs, and ics here are measured inside the Motor model shown in Figure 4 on Page 4 tosimplify the model setup.

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Figure 3: Block Diagram of The Inverter Model

Sine-Triangle Modulator Model

The Sine-Triangle Modulator used in this project is based on the provided one from the instruc-tion. Because the ideal motor of this project is operating at constant speed and load, and thusconstant Te, the inputs vr∗qs and vr∗ds can be calculated in advance by hand, and stay always constantduring operation. For simplicity, no voltage or current feedback is involved here, which will makethis control system unstable. This instability can be seen in the plot of torque output Te, when themotor is starting.

Motor Model

The motor model used in this project is based on Parks equivalent circuit of PMSM, and thiscan simplify the model build-up by a large scale. As we assuming ideal control over the motor,we can minimize ids to be zero, and this furthermore simplify the model and the calculation of Te.The calculation of Te is based on vqs, vds, and engine speed ωrm. In calculation, iqs will first beencalculated, and put out to Inverter block. Additional Park transformation and retransformationblocks will be needed at the expense of simplicity of motor model. The motor model is shown inFigure 4.

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Figure 4: Block Diagram of The Motor Model

Parks Transformation Model

The Parks transformation and retransformation blocks are based on Parks transformation ma-trix.

Overall Block Diagram

The overall block diagram of this control system is shown in Figure 5. In this model, voltagesignals always serve as control signal, while current signals as reference signal for parameter cal-culation. Please note that in this control system, battery cells are connected by six in parallel toform one set and 100 sets are connected in series to form the whole battery pack. Thus, the batteryvoltage and current need to be multiplied by a certain factor.

Calculation and Simulation

Calculation of V rqs and V r

ds

The fundamental control parameters of this control system is V rqs and V r

ds. To calculate them,we first assume that the control system is ideal, and thus Irds can be zero. Then, as the motor hasconstant torque output and speed, we can perform our calculation in steady state. Furthermore,we assume the engine is ideal, and thus its mechanical torque output Tm equals to its electrical

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Figure 5: Block Diagram of The Overall System

torque Te. The simplified equations for calculation are shown in Equations 2.

Te =3

2

P

2λmI

rqs

V rqs = rsI

rqs + ωrλm

V rds = −ωrLqI

rqs

(2)

Substituting Te with 250 N·m, and ωrm with 3200 rpm and thus 335.1 rad/s, we get

V rqs = 111.995101V

V rds = −157.769876V

(3)

Simulation Results

Average Torque

By putting the two control parameters calculated above into the control system, specifically theSine-Triangle modulator, we can now perform the simulation, and verify the output of torque aswell as calculate the ripple component of battery voltage and current. Figure 6 shows the simulationresult of torque from 0 to 0.01s, which is unstable. And 7 shows the result from 0.19 to 0.2s, whichis relatively stable.

The maximum value of Figure 6 is 483 Nm, while the minimum value is 0 Nm. The maximumvalue of Figure 7 is 254 Nm, while the minimum value is 246 Nm. We can see from here the averagetorque value in steady state is exactly 250 Nm, with a peak-peak value of 8 Nm. Please note that

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Figure 6: Average Torque at 0.01 s

Figure 7: Average Torque at 0.19 s

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the ranges of Y axes of these two plots are different, and this is why the first plot seems to be morestable.

Battery Voltage Ripple

Figure 8 shows the battery voltage at steady state (from 0.19s to 0.20s). From the plot we cansee the maximum value for vbatt is 340.3 V, while the minimum value is 330.8 V, and thus thepeak-peak value is 9.5 V.

Figure 8: Battery Voltage Ripple at 0.19 s

Battery Current Ripple

Figure 9 shows the battery current at steady state (from 0.19s to 0.20s). From the plot wecan see the maximum value for ibatt is 297.9 A, while the minimum value is 228.4 A, and thus thepeak-peak value is 69.5 A.

Analysis of Simulation Results

vas: Voltage of Stator of Phase A

The plot of variable vas at steady state (from 0.19 to 0.20s) is shown in Figure 10. The waveformis formed by adding a series of square waves.

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Figure 9: Battery Current Ripple at 0.19 s

Figure 10: Voltage of Stator of Phase A at 0.19 s

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As shown in Figure 10, the magnitude of vas is 225 V, with an equivalent frequency of 158.25Hz. And this is close to ideal frequency of vas, which can be calculated by ωr/2π = 160 Hz. Thedifference between the ideal one and real one can be because of the distortion of the waveform.Calculating from this plot, the line to line voltage generated by the inverter is 389.7 V.

ias: Current of Stator of Phase A

The plot of variable ias at steady state (from 0.185 to 0.20s) is shown in Figure 11.

Figure 11: Current of Stator of Phase A at 0.185 s

As shown in the plot, the magnitude of ias is 525.5 A, with an equivalent frequency of 157.89Hz. And this is close to ideal frequency 160 Hz. The serrate areas at the each concave and convexis because of the distortion of voltage waveform. As we can see, the current waveform here is closeto desired sinusoidal wave, despite the fact that vas is distorted severely.

vbatt: Terminal Voltage of Battery

The plot of variable vbatt at steady state (from 0.175 to 0.20s) is shown in Figure 12.From the plot we can calculate the frequency of the primary component of vbatt is about 158Hz.

The current ripple has a fundamental component of a frequency of 10kHz, which is the switchingfrequency. As we discussed above, the average value of vbatt is 335.55 V with the peak-peak valueof its ripple component being 9.5 V. The ripple takes up 2.83% of the average value.

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Figure 12: Terminal Voltage of Battery at 0.175 s

ibatt: Terminal Current of Battery

The plot of variable ibatt at steady state (from 0.185 to 0.20s) is shown in Figure 13.From the plot we can calculate the frequency of the primary component of ibatt is about 158Hz.

The current ripple has a fundamental component of a frequency of 10kHz, which is the switchingfrequency. As we discussed above, the average value of ibatt is 263.15 A with the peak-peak valueof its ripple component being 69.5 A. The ripple takes up 26.41% of the average value.

vS: Input Voltage of Inverter

The plot of variable vS at steady state (from 0.185 to 0.20s) is shown in Figure 14.The average value of vS is 335.85 V, with its ripple being 11.5 V. The waveform is a sum of a

DC component and an AC ripple with a frequency of 10kHz.

vS: Input Current of Inverter

The plot of variable vS at steady state (from 0.195 to 0.20s) is shown in Figure 15.As shown is the plot, the input current of the inverter is distorted severely. The frequency of

fundamental component within current waveform is approximately 1 kHz, and this is determinedmainly by the time constant of the Filter. The switching frequency is approximately 10 kHz, as wedesigned. As we can see, there are some areas where iS is less than zero. However, when we lookat the current waveform at the battery terminal, it is always greater than zero. These areas standfor current flow back into the inverter, which generates reactive power.

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Figure 13: Terminal Current of Battery at 0.185 s

Figure 14: Input Voltage of Inverter at 0.185 s

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Figure 15: Input Current of Inverter at 0.195 s

Te: Electrical Torque of Motor

The plot of Te in steady state is shown in Figure 7 on Page 6 . As we discussed above, themaximum value is 254 Nm, while the minimum value is 246 Nm. The average torque value in steadystate is exactly 250 Nm, with a peak-peak value of 8 Nm. The ripple component takes up 3.2% ofits total torque. Also, as we can see from Figure 6, the torque value hits its maximum at starting,which is approximately three times its rated value. This is usually not desirable.

Conclusion

This project gives detailed analysis upon the steady-state operating parameters of an idealPMSM. As we discussed above, when we use fixed value of V r

qs and V rds, the average torque generated

by the motor will be 250 Nm at a speed of 3200 rpm. The ripple component of both battery voltageand current largely depends on the switching frequency of the inverter. Although we find thatthe waveform of vas is a sum of a series of square waves, and distorted severely from sinusoidalwave, the waveform of ias is close to desired sinusoidal wave. During operation, iS may change itsdirection frequently, while ibatt will not, and this may be one of the major functions of the Filter.In all the plots, we can see sharp rising edges at the starting time. To analyze this, additionalknowledge about transient analysis upon motors, and more advanced models will be needed, whichis beyond our scope. However, as our model is set up by steady-state models, we should ignoretransient performance and focus mainly on steady-state performance of the system. In this case,the simulation results in this project are acceptable.