Analysis of Voltage Source InverterInduction Machine Drive

Nathan P. Petersennpetersen2@wisc.edu

Dept. of Electrical and Computer Engineering1415 Engineering Drive

Madison, WI 53706

Abstract—The induction machine is widely used for industrialapplications due to its simplicity and ease-of-use. For many years,constant frequency drives (e.g. the grid) were used to power theinduction motor. With the introduction of the controllable powerelectronic switch, induction motor drives have transitioned awayfrom the grid and moved to the voltage source inverter (VSI).The VSI typically uses high frequency modulation techniquesto accurately deliver electrical power to the motor. This paperinvestigates the inherently dynamic nature of the induction motordriven by the VSI. Several different modulation techniques arestudied and simulated for the VSI, including both standard sine-triangle PWM and space vector PWM with enhancements forreduced harmonics. Key performance metrics are summarizedfor each modulation technique.

I. BACKGROUND

The induction machine is one of the oldest AC electro-mechanical power conversion devices [1]. In the late 19thcentury, engineers pioneered the first single-phase inductionmotor, followed soon by the three-phase variant. Comparedwith other popular motor types (e.g. DC or synchronous),the induction motor is typically lower cost and has higherreliability. This has pushed the technology to become thedominant player in industrial applications. See Fig. 1a for anexample induction motor.

To fully realize the potential of the induction motor atvarious torque / speed operating points, it must be driven froma variable frequency power source. In the mid to late 20thcentury, a breakthrough in power electronics was made withthe introduction of the transistor. This device enabled efficient,robust, high-frequency power converters to be created. Thevoltage source inverter (VSI) became the most prevalent foun-dation for today’s variable frequency drives used to controlmodern induction motors.

A. Voltage Source Inverter (VSI)

The voltage source inverter (VSI) is used to convert betweenDC and AC power. Fig. 1b shows a VSI connected to aninduction machine. For normal motoring operation, power isdrawn from the DC input power supply (Vdc) and is modulatedto AC power on the output phases (vas, vbs, and vcs).

Internally, the VSI is composed of several components. Toprovide a stiff voltage source, a DC link capacitor is connectedacross the DC input. Three half-bridges are connected across

(a) Cutaway view of an induction motor showing the statorwindings, rotor iron, and shaft bearings.

Vdc

+

−VSI

vas

vbs

vcsIM

(b) Voltage source inverter (VSI) driving induction motor (IM).

(c) Schematic diagram of VSI showing DC link capacitor andthree half-bridges. The mid-point of the half-bridges connectsto the phases of the induction motor.

Fig. 1: Typical induction motor set-up including three-phasevoltage source inverter.

the DC link capacitor. Each half-bridge is composed of twopower electronic switches. The mid-point of each half-bridgebecomes the phase connection for the induction motor. SeeFig. 1c for a schematic representation.

B. Recent Technology Advancements

The typical power electronic switch that is used in the VSIis either the insulated gate bipolar transistor (IGBT) or themetal-oxide semiconductor field effect transistor (MOSFET).MOSFETs are usually used for high frequency applicationswith lower voltage requirements, while IGBTs tend to be usedfor lower frequency, higher power applications. In recent years,two new promising switch technologies have been introduced:gallium-nitride (GaN) and silicon-carbide (SiC) [2]. Thesedevices enable power converters to switch at higher frequen-cies while maintaining lower switching losses. When the VSIdrives the induction motor using higher switching frequencies,the output voltage better approximates the reference and leadsto lower current ripple due to the machine inductance actingas a filter. This reduces the losses in the induction motor andreduces torque ripple.

II. MOTIVATION

To create steady-state torque which rotates an inductionmotor, it can be shown that a three-phase balanced set ofsinusoidal voltages must be applied to the motor phases, asshown in Fig. 2. These voltages can be expressed as (1), whereVo is the amplitude of the voltage and ωe is the electricalfrequency. vasvbs

vcs

= Vo

cos(ωet+ 0π3 )

cos(ωet+ 2π3 )

cos(ωet+ 4π3 )

(1)

Alternatively, these voltages can be represented in thecomplex dq plane (vsqd = vsq − jvsd) as

vsqd = Voejωet (2)

0 60 120 180 240 300 360

Electrical Angle [deg]

-100

-50

0

50

100

Volt

age

[V]

Fig. 2: Ideal sinusoidal voltage excitation for three-phaseinduction machine.

qs

ds

1

65

4

3 2

0 7

(a)

0 60 120 180 240 300 360

Electrical Angle [deg]

-100

-50

0

50

100

Volt

age

[V]

(b)

Fig. 3: (a) Complex dq plane representation of the eightdiscrete VSI voltage vectors. (b) Six-step excitation phasevoltages and resulting fundamental component.

III. MODULATION TECHNIQUES

The goal of the VSI is to reproduce the waveforms shownin Fig. 2 as accurately as possible. However, because theinverter uses switched-mode operation, only eight discretevoltage states are available at the phase terminals. Typically,duty cycle techniques are employed to create a time averagingeffect. On average, the VSI replicates the desired referencewaveforms, but at any instance in time, the waveforms arecomprised of the eight discrete voltage states.

A. Six-Step Excitation

The most simple technique for driving an induction motorusing the VSI involves direct use of the six active voltagevectors (1-6) to approximate the reference waveforms. Fig. 3ashows all possible VSI voltage vectors on the complex dqplane. The six-step modulation technique applies each of theactive voltage vectors for a portion of time. If each is appliedfor a 60 segment, the fundamental component of the voltagewaveform will equal the desired reference waveforms. Fig. 3bshows both the actual phase voltages and the fundamentalcomponent for six-step excitation. While simple, six-step op-eration creates large harmonics in the induction motor current,thus creating torque ripple. Alternative modulation techniquesare preferred due to better harmonic properties.

0 60 120 180 240 300 360

-100

0

100V

olt

age

[V]

0 60 120 180 240 300 360

-1

0

1

Du

ty R

atio

0 60 120 180 240 300 360

Electrical Angle [deg]

0

0.5

1

Gat

e S

ign

al

Fig. 4: The sine-triangle modulation waveforms with voltagereference v(t) = 100 cos(ωt) and Vdc = 240V .

B. Sine-Triangle Modulation

The sine-triangle modulation approach is shown in Fig. 4for a single desired reference voltage (e.g. vas). To implementthis modulation, a high frequency triangle wave, referred toas the carrier wave, is required. The desired voltage referenceis scaled by the DC bus voltage and compared to the trianglecarrier. The output of this comparison is used for the gatingsignals to each half-bridge in the VSI.

The advantages of the sine-triangle modulation techniqueinclude the ability to generate arbitrary reference waveformsusing pulse-width modulation (PWM) and operating with afixed switching frequency, but limit the induction motor phasevoltage magnitude to 1

2Vdc. This limitation occurs because thecommon-mode voltage on all three phases is fixed at half ofVdc.

C. Space Vector Modulation

The space vector modulation (SVM) algorithm can berelated to both the six-step and sine-triangle modulation tech-niques. As extension to six-step operation, SVM adds theconcept of duty cycle averaging to achieve average phasevoltages which can arbitrarily lie on the complex dq planewithin the hexagon (see Fig. 3a), as opposed to the six fixedactive voltage vectors [3]. SVM achieves this by mixing partsof the active voltage vectors (1-6) along with the zero states(0,7) within each switching interval.

Alternatively, SVM can be thought of as an extension tothe standard sine-triangle modulation [4]. For this, a zero-sequence portion is added to each voltage command whichdepends on the instantaneous voltage references. To controlthe ratio of time spent in the two zero states (0,7), a parameter0 ≤ ko ≤ 1 is defined. The final voltage reference is definedas v∗∗abc = v∗abc + v∗zs where

v∗zs = −[(1− 2ko) + kov∗max + (1− ko)v∗min] (3)

0 60 120 180 240 300 360

-100

0

100

V*

[V

]

0 60 120 180 240 300 360-50

0

50

Vzs

[V

]

0 60 120 180 240 300 360

Electrical Angle [deg]

-100

0

100

V*

* [

V]

Fig. 5: Space vector modulation implemented as zero-sequenceinjection per Eq. (3). Plot shown for ko = 0.5.

Fig. 5 shows the voltage references for the VSI when usingSVM and generating the same voltage references as in Fig. 2.The zero-sequence component is shown in black. By addingthis component to each phase voltage, the common-modevoltage does not appear on the induction motor, but acts toextend the usable DC bus voltage to about 15% more thanwith no zero-sequence injection.

IV. SIMULATION

The induction machine can be modeled as a coupled set ofnon-linear differential equations. Typically, complex variablesare used to represent the electrical quantities (encapsulatingvector magnitude and phase information). When the mechan-ical system is included, the induction motor becomes a non-linear fifth order system.

The electrical quantities can be viewed from various refer-ence frames, with common choices representing the stationary,rotor, or synchronous reference frames. Eq. (4) and Eq. (5)model the stator and the rotor of the induction machine inthe arbitrary reference frame, respectively. Eq. (6) models thedynamics of the mechanical system. The coupling between theelectrical and mechanical systems occurs due to the developedtorque, which can be represented as the right hand side ofEq. (6). Together, these three equations model the entireinduction machine dynamics.

vqds = rsiqds + Ls(p+ jω)iqds + Lm(p+ jω)iqdr (4)

vqdr = rriqdr + Lr[p+ j(ω − ωr)]iqdr+ Lm[p+ j(ω − ωr)]iqds (5)

Jdωmdt

+ TL =3

2

P

2Lm Imisqdsi

s∗qdr (6)

v∗abc

fsw ko

Vdc −+

SVMVSI

vas

vbs

vcsIM

LOAD

Fig. 6: Open-loop simulation diagram of the induction motordriven by the space vector modulated VSI, connected to amechanical load.

VR 460 Vrms,l−l

PR 20 hpfR 60 HzP 4 polesZB 14.2 ΩVB 376 Vpk

IB 26.5 Apk

TB 79.16 N-mrs = rr 0.355 Ωxls = xlr 1.42 Ωxm 34.1 ΩM 1.4 secRated slip 0.03135

TABLE I: Induction motor parameters used for simulation.

In the above equations, rs and rr are the stator and rotorresistances, Ls = Lm+Lls is the total stator inductance, Lr =Lm +Llr is the total rotor inductance, Lm is the magnetizinginductance, p = d

dt denotes the differentiation operator, ω isthe arbitrary reference frame, ωr is the rotor electrical speed,ωm is the rotor mechanical speed, P is the number of machinepoles, and J is the mechanical rotary inertia.

The choice of ω determines the reference frame; in thispaper, the synchronous frame is used such that ω = ωe. Forthe simulation results that will be presented and discussedbelow, Table I summarizes the induction motor parameters.Fig. 6 shows the simulation block diagram that will be used.The inverter switching frequency fsw and zero state allocationscheme ko with be investigated.

A. Baseline: Steady-State Operation

The first condition of interest is the baseline steady-stateoperation, where the induction machine falls into a stableoperating point depending on the values of the resulting speedand torque. A constant load of 50% rated torque is applied. Forthis simulation, the inverter is operating at a fixed switchingfrequency of fsw = 3kHz at a DC voltage of 650V. Spacevector modulation is used with the modulation index Mi = 0.9and ko = 0.5. Note that the ratio of the switching frequencyto the fundamental is 50.

Fig. 7 shows the resulting waveforms from the simulation,with Fig. 7a showing the time domain waveforms and Fig. 7bshowing the frequency content of the waveforms. In the timedomain, the PWM nature of the applied phase voltage is

0.484 0.486 0.488 0.49 0.492 0.494 0.496 0.498 0.5

-500

0

500

Vas [

V]

0.484 0.486 0.488 0.49 0.492 0.494 0.496 0.498 0.5

-20

0

20

I as [

A]

0.484 0.486 0.488 0.49 0.492 0.494 0.496 0.498 0.535

40

45

Te [

Nm

]

0.484 0.486 0.488 0.49 0.492 0.494 0.496 0.498 0.5

Time [sec]

1700

1750

1800

m [

RP

M]

(a)

100

101

102

103

100

Vas [

V]

100

101

102

103

100

I as [

A]

100

101

102

103

Harmonic #

100

Te [

Nm

]

(b)

Fig. 7: (a) Simulated steady-state time domain waveforms forfsw = 3kHz, ko = 0.5. (b) Frequency content of waveforms.

obvious, but the resulting phase current is very smooth andsinusoidal due to the filtering effects of the motor inductance.The resulting torque has a DC component, but nearly 10%ripple. The resulting speed is approximately constant at 1755RPM. To verify the steady-state simulation results, we cananalyze the induction machine as its Thevenin equivalentcircuit, where the equivalent complex impedance is given by

Zth = rs + xls +xm(xlr + rr/s)

xm + xlr + rr/s(7)

The critical or break frequency of an RL circuit is givenby fc = R/(2πL). Using Zth for the simulated inductionmachine, this results in fc = 101Hz. Because the phase voltageexcitation PWM is at 3kHz, this is nearly 30x greater than thebreak frequency of the machine, so the nearly smooth currentis reasonable.

-500

0

500

Vas [

V]

100

Vas [

V]

-20

0

20

I as [

A]

100

I as [

A]

0.484 0.486 0.488 0.49 0.492 0.494 0.496 0.498 0.5

Time [sec]

35

40

45

Te [

Nm

]

100

101

102

103

Harmonic #

100

Te [

Nm

]

(a)

-500

0

500

Vas [

V]

100

Vas [

V]

-20

0

20

I as [

A]

100

I as [

A]

0.484 0.486 0.488 0.49 0.492 0.494 0.496 0.498 0.5

Time [sec]

35

40

45

Te [

Nm

]

100

101

102

103

Harmonic #

100

Te [

Nm

](b)

Fig. 8: (a) Simulated VSI operation at various fsw. (a) fsw = 2kHz (b) fsw = 10kHz

B. Effects of VSI Switching Frequency

The switching frequency of the VSI plays a major role in theresulting ripple on the current and torque waveforms. Fig. 8shows the simulated waveforms for different VSI switchingfrequencies. Fig. 8a and Fig. 8b show the resulting waveformswhen fsw = 2kHz (20x machine break frequency) and fsw =10kHz (100x machine break frequency), respectively. Noticethat at large fsw, the harmonic content of the waveforms isshifted to higher frequencies such that the signal energy nearthe fundamental is reduced. This results in cleaner current andtorque time domain waveforms with less ripple. By increasingfsw, harmonic losses in the induction motor are reduced,while inverter switching losses are increased. Full systemapplications must make an engineering trade-off between themachine and inverter losses.

To characterize the improvement from increasing theswitching frequency, various metrics can be analyzed. In thispaper, the phase current ripple is analyzed. The expectedphase current is ideally sinusoidal, but due to the PWM VSI,current ripple increases as fsw decreases. The total harmonicdistortion (THD) of a sinusoidal waveform can be calculatedas Eq. (8) where x1 denotes the magnitude of the fundamentalcomponent, x2 is the magnitude of the second harmonic, etc.This provides a metric for the amount of energy present in thewaveform not at the fundamental frequency.

THD =

√x22 + x23 + x24 + ...

x1(8)

65

70

75

80

85

Vas

TH

D [

%]

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Switching Frequency [Hz]

0

5

10

15

I as T

HD

[%

]

Fig. 9: Simulated voltage and current total harmonic distortion(first 800 harmonics) for various VSI switching frequencies.

To relate the VSI switching frequency to the THD of thephase voltage and current, multiple simulations were run forvarious fsw. Fig. 9 shows the simulation results: when fsw =1kHz (10x break frequency) and fsw = 10kHz (100x breakfrequency), the THD of the current waveform is 13% and1.2%, respectively, showing an inverse logarithmic relationshipbetween fsw and current THD. However, the voltage kept itshigh THD across fsw.

-500

0

500

Vas [

V]

100

Vas [

V]

-20

0

20

I as [

A]

100

I as [

A]

0.484 0.486 0.488 0.49 0.492 0.494 0.496 0.498 0.5

Time [sec]

30

40

50

Te [

Nm

]

100

101

102

103

Harmonic #

100

Te [

Nm

]

(a)

-500

0

500

Vas [

V]

100

Vas [

V]

-20

0

20

I as [

A]

100

I as [

A]

0.484 0.486 0.488 0.49 0.492 0.494 0.496 0.498 0.5

Time [sec]

30

40

50

Te [

Nm

]

100

101

102

103

Harmonic #

100

Te [

Nm

](b)

Fig. 10: Simulated VSI operation at fsw = 2kHz and various ko. (a) ko = 0.2 (a) ko = 0.35

75

80

85

90

95

100

Vas T

HD

[%

]

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Ko [-]

0

5

10

15

20

I as T

HD

[%

]

Fig. 11: Simulated voltage and current total harmonic distor-tion (first 800 harmonics) for various VSI zero state ratios.

C. Steady-State Operation With Various Constant ko

The previous simulation results used a fixed value ofko = 0.5. This partitions the zero voltage vector statesevenly between vector 0 and 7 (see Fig. 3a for voltage vectordefinitions). By changing ko in Eq. (3), the ratio of the zerostates is adjusted. This section of the paper investigates theeffects of setting ko to a constant value not equal to 0.5. A

range of 0.2 to 0.8 is investigated. For example, when ko =0.2, voltage vector 7 is used for 80% of the zero state, and theremaining 20% is spent in voltage vector 0. The theoreticaltime-averaged voltage in a single PWM cycle is the same, butit is achieved using different VSI switch combinations.

Fig. 10 shows simulation results for two values of ko = 0.2and 0.35 (fsw = 2kHz). The current waveforms roughly matchFig. 8a, the equivalent scenario but with ko = 0.5. However, thetorque waveform has much more ripple and harmonics thanpreviously. When ko = 0.35, the torque waveform reduces itsharmonics, but adds other high frequency content. The effectson the torque waveform occur due to the interactions betweenthe three phase currents in the synchronous reference frame,as noted in Eq. (6).

Furthermore, by setting ko 6= 0.5, the frequency contentof the waveforms is spread and shifted. In Fig. 10a, thefrequencies between the fundamental and PWM carrier hassignificantly more energy than in Fig. 8a. By setting ko = 0.5,this spread of energy is reduced.

To further investigate the effects of ko on the inductionmotor drive, the full range of ko = 0.2 to 0.8 is simulated. Thetotal harmonic distortion (THD) of the resulting voltage andcurrent waveforms is calculated and presented in Fig. 11. Atko = 0.5, both the phase voltage and current have a minimumin THD. As ko is brought away from its nominal value of 0.5,the THD of both the voltage and current increases. This leadsto the conclusion that, for fixed ko, the least THD in voltageand current is achieved at ko = 0.5.

-500

0

500V

as*

[V

]

0

0.5

1

Ko [

-]

-200

0

200

Vzs

[V]

-500

0

500

Vas*

* [

V]

0.484 0.486 0.488 0.49 0.492 0.494 0.496 0.498 0.5

Time [sec]

-500

0

500

Vas

[V]

Fig. 12: Simulated waveforms when ko jumps between 0.2and 0.8 per PWM cycle.

D. Steady-State Operation With Changing ko

Per the previous section, the optimal constant ko value is0.5; this reduces THD in both voltage and current waveformsto the minimum value. However, by dynamically changing ko,a further reduction in THD is possible.

Fig. 12 presents the flow of signals inside the SVM VSI.The input voltage reference v∗as is summed with the zero-sequence voltage v∗zs (based off of the present ko value) toform the final voltage command v∗∗as . This voltage referenceis then modulated with the PWM triangle carrier to form thevoltage applied to the induction motor vas. Notice that thewaveform for ko is not constant; each PWM cycle, it jumpsbetween a minimum and maximum value for 50% of thetime. If the minimum and maximum values for ko are both0.5, this approach simplifies to the results from the previoussection. However, by changing the min and max ko values,the harmonic performance is changed.

Fig. 13 shows the results of 169 simulations, one for eachbox in the map. In each simulation, fsw is held constant at3kHz, but the min and max values for ko are changed. TheTHD is calculated for both the phase voltage and current, andexpressed as a percent corresponding to a color. The previoussection results for ko = 0.5 is found in the lower right-handbox, resulting in a current THD of about 7%. Following aline diagonal from the bottom right to the top left results inthe THD values for when the min and max ko is symmetricalaround the nominal 0.5 value, while off-diagonal boxes areasymmetrical. By alternating between symmetrical values ofko, the THD of both the voltage and current waveforms isreduced. At ko = 0.2 and 0.8, the current THD is 61% of thenominal baseline (ko = 0.5).

Fig. 13: THD map of both phase voltage and current (fsw= 3kHz) when ko is dynamically changed half-way betweenPWM cycles. ko values are swept across a range of 0.5 ± 0.3.

V. SUMMARY & CONCLUSIONS

This paper analyzed the induction motor when driven bythe voltage source inverter. Several modulation techniqueswere presented and related to each other. The space vectormodulation technique was shown to have superior performanceby extending the phase voltage range to 15% more than thesine-triangle modulation technique.

The induction motor was simulated using several configura-tions of the SVM VSI. Analysis of PWM switching frequencyand zero voltage vector placement was performed. Resultswere compared in the time and frequency domain. Totalharmonic distortion was used as the metric for evaluating theperformance of the inverter configuration. It was found thathigher switching frequencies result in less THD, along withdynamically changing zero voltage vector placements.

Future work may investigate changing the load of theinduction motor and the dynamic effects of the modulationtechniques. For example, using real-world drive sequences canhelp the application engineer determine optimal values for fswand ko depending on system level requirements.

REFERENCES

[1] B. G. Lamme, “The story of the induction motor,” Journal of theAmerican Institute of Electrical Engineers, vol. 40, no. 3, pp. 203–223,March 1921.

[2] J. Millan, P. Godignon, X. Perpina, A. Perez-Tomas, and J. Rebollo, “Asurvey of wide bandgap power semiconductor devices,” IEEE Transac-tions on Power Electronics, vol. 29, no. 5, pp. 2155–2163, 2014.

[3] H. W. van der Broeck, H. . Skudelny, and G. V. Stanke, “Analysis andrealization of a pulsewidth modulator based on voltage space vectors,”IEEE Transactions on Industry Applications, vol. 24, no. 1, pp. 142–150,1988.

[4] V. Blasko, “A hybrid pwm strategy combining modified space vectorand triangle comparison methods,” in PESC Record. 27th Annual IEEEPower Electronics Specialists Conference, vol. 2, June 1996, pp. 1872–1878 vol.2.