45

Chapter 4

SPACE VECTOR MODULATED THREE

PHASE VOLTAGE SOURCE INVERTER

4.1. Introduction

The Pulse Width Modulation (PWM) technique is applied in the

inverter (DC/AC converter) to obtain an AC waveform with variable voltage

and variable frequency for variable speed motor drives. The basic idea is to

modulate the duration of the pulses or duty ratio in order to achieve

controlled voltage/current/power and frequency, satisfying the criteria of

equal area. The implementations of the complex PWM algorithms have been

made easier with the advent of fast digital signal processors,

microcontrollers and FPGAs.

At present, the control strategies are implemented in digital systems,

which demands for digital modulating techniques such as SVM technique.

SVM is one of the most popular PWM techniques used due to higher DC bus

utilization compared to the classical SPWM. In SVM, the gating time of each

power switch is directly calculated from the analytical time equations while

the high frequency carrier-wave is compared with the sinusoidal modulating

signals to generate the appropriate gate signals in SPWM. The main

objective of PWM is to generate PWM load line voltages that are on average

equal to given load line voltages, with lowest Total Harmonic Distortion

(THD) in the output voltages.

In this chapter, the operational principle of SVM is illustrated and

simulation model developed for FOC induction motor drive. In the

46

simulation results, the performance of the developed FOC induction motor

drive model with SVM inverter is compared with that of SPWM.

4.2. Principle of SVM

The concept of the SVM relies on the representation of the inverter

output as space vectors or space phasors. Space vector representation of

output voltages of the inverter is realized for the implementation of SVM.

Space vector represents three phase quantities as a single rotating vector

where the three phases are assumed as only one quantity.

4.2.1. Space Vector Transformation

Any three phase set of variables that add up to zero in the stationary

a-b-c frame can be represented in a complex plane by a complex vector that

contains a real � and imaginary β component and Fig. 4.1 shows space

vector transformation.

Fig. 4.1 Space vector transformation

A three phase system defined by Va(t), Vb(t) and Vc(t)) can be

represented uniquely by a rotating vector is given as:

( )2 /3 2 /32( ) ( ) ( )

3

j ja b cV V t V t e V t e

π π−= + +

(4.1)

where, 2

3 is the

( ) sina mV t V t

( ) sinb mV t V t

( ) sinc mV t V t

4.2.2. Operational Principle of SVM

In a three phase

and frequency are always controllable. The standard three phase VSI

topology is shown in Fig.

(i.e., +0.5VDC or -0.5V

001, 010, 011,100, 101, 110, 111).

shown in Fig. 4.3. Here,

represents the upper switch is ‘ON’.

correspond to short circuit on the output producing zero AC line voltage.

The other six states can be considered to form stationary vectors in the

complex plane. Only one switch in an inverter leg

time. It cannot be switched ‘ON’

a short circuit

Fig. 4.2 Three phase VSI configuration

47

is the normalization factor,

( )( ) sina mV t V tω= 2

( ) sin3

b mV t V tπ

ω = −

2( ) sin

3c mV t V t

πω = +

Operational Principle of SVM

phase Voltage Source Inverter (VSI), the amplitude, phase

and frequency are always controllable. The standard three phase VSI

topology is shown in Fig. 4.2. Since the inverter can attain

VDC, VDC or 0), the total possible outputs are 2

001, 010, 011,100, 101, 110, 111). The basic inverter s

Here, ‘0’ indicates that the upper switch is ‘OFF’ and

represents the upper switch is ‘ON’. Two of these states,

correspond to short circuit on the output producing zero AC line voltage.

The other six states can be considered to form stationary vectors in the

Only one switch in an inverter leg can be

switched ‘ON’ simultaneously because this would

Fig. 4.2 Three phase VSI configuration

Voltage Source Inverter (VSI), the amplitude, phase

and frequency are always controllable. The standard three phase VSI

Since the inverter can attain only two states

or 0), the total possible outputs are 23 (000,

inverter switch states are

the upper switch is ‘OFF’ and ‘1’

Two of these states, V0 and V7

correspond to short circuit on the output producing zero AC line voltage.

The other six states can be considered to form stationary vectors in the α-β

can be turned ‘ON’ at a

this would result in

48

Fig. 4.3 Eight possible phase leg switch combinations for a VSI

across the DC link voltage supply. Similarly, in order to avoid undefined

states in the VSI, and thus undefined AC output line voltages, the switches

of any leg of the inverter cannot be switched ‘OFF’ simultaneously as this

will result in voltages that will depend upon the respective line current

polarity.

49

The possible space vectors are computed using (4.1) and listed in table

4.1. The tip of the space vectors, when joined together form a hexagon,

which is shown graphically in Fig.4.4. The hexagon consists of six distinct

sectors (I to VI) spinning over 360 degrees (one sinusoidal wave cycle

corresponds to one rotation of the hexagon) with each sector of 60 degrees.

Fig. 4.4 Location of eight possible voltage space vectors for a VSI

Space vectors 1 to 6 are called active state vectors and 7 and 8 are called

zero state vectors. The magnitude of each of the six active vectors is equal to

(2/3)VDC. The zero state vectors are redundant vectors but they are used to

minimize the switching frequency.

The space vectors are stationary while reference vector ‘Vref’, is

rotating at speed of the fundamental frequency of the inverter output

voltage. It circles once for one cycle of the fundamental frequency, ‘ω’. For

generating a given voltage waveform, the inverter moves from one state to

another and it circles for one cycle of the fundamental frequency. Each

stationary vector corresponds to a particular fundamental angular position

as shown in Fig. 4.5. The reference voltage follows a circular trajectory in a

linear modulation range and the output is sinusoidal.

50

Table 4.1 Vector definition

Space

vector

Switching

state

Vector

definition

V0 000 0

V1 100 2/3VDC���

V2 110 2/3VDC���/�

V3 010 2/3VDC����/�

V4 011 2/3VDC����/�

V5 001 2/3VDC����/�

V6 101 2/3VDC���/�

V7 111 0

Fig. 4.5 Inverter phasor angular position in fundamental cycle

4.3. SVM Compared to SPWM

In the linear operating region, the maximum line-to-line voltage

amplitude can be achieved when Vref is rotated along the largest inscribed

circle in the space vector hexagon. In Fig. 4.6, the different reference vector

51

loci are presented, in which, OQ = (2/3)VDC, OP = (1/√3)VDC and OR =

(1/2)VDC. The maximum possible output voltage using SPWM is (1/2)VDC

whereas for SVM, it is (1/√3)VDC. Hence the increase in the output voltage,

when using SVM, is (1/√3VDC)/(1/2VDC) = 1.154. Thus it is possible to get

line-to-line voltage amplitude as high as VDC using the SVM in the linear

operating range. Due to higher line-to-line voltage amplitude, the torque

generated by the motor is higher. In the linear operating range, modulation

index range is 0.0 to 1.0 in SPWM, whereas in SVM, it is 0 to 0.866. Line-to-

line voltage is 15% more in the SVM compared to SPWM. Hence, SVM has

the better usage of the modulation index depth.

The symmetry in the output waveforms are mainly responsible for

having lower THD in SVM compared to SPWM in linear operating region. The

phase-to-centre voltage containing the triple order harmonics that are

generated by SVM compared to SPWM reference voltage as shown in Fig.

4.6. But the triple order harmonics are not appeared in the phase-to-phase

voltage as well. This leads to the higher modulation index compared to

SPWM.

Fig. 4.6 Locus comparison of SPWM and SVM

52

4.4. Implementation of SVM Algorithm

For implementing the SVM, the reference voltage is synthesized by

using the nearest two neighboring active vectors and zero vectors. The

choice of the active vectors depends upon the sector number in which the

reference is located. Once the reference voltage is located, the vectors to be

used for the SVM implementation to be identified. The next task is to find

the time of application of each vector, called the ‘dwell time’. The output

voltage frequency of the inverter is the same as that of the speed of the

reference voltage and the output voltage magnitude is the same as the

reference voltage.

For each switching period Ts, the reference vector as a geometric

summation of two nearest space vectors is shown in Fig.4.7 and is

expressed mathematically by applying volt-second balancing,

Fig. 4.7 Reference vector as a geometric summation of space vectors

01 21 2 0ref ref

s s s

TT TV V V V V

T T Tθ= ∠ = + +

uuuv

(4.2)

where, T1 is the time for which space vector V1 is applied and T2 is the

respective time for which the basic space vector V2 should be applied within

53

the time period Ts and T0 is the course of time for which the null vectors V0

and V7 are applied. The block diagram for generating SVM pulses is shown

in Fig. 4.8 and SVM algorithm is implemented through the following steps:

1. Computation of reference voltage and angle ‘θ’,

2. Identification of sector number,

3. Computation of space vector duty cycle,

4. Computation of modulating function,

5. Generation of SVM pulses.

Fig. 4.8 Block diagram for SVM pulse generation

4.4.1. Computation of Reference Voltage and Angle ‘θ’

The transition of reference vector moving from one sector in the space

vector diagram to the next requires minimum number of switching in order

to reduce the switching losses. The space vector, Vref is normally represented

in complex plane and the magnitude as,

2 2refV V Vα β= +

(4.3)

1tan

Vt

V

β

α

ω θ − ⋅ = =

(4.4)

4.4.2. Identification of Sector Number

The six active vectors are of equal magnitude and are mutually phase

displaced by π/3. These vectors divide the complex plane into the six sectors

I to VI. Any desired reference voltage vector within the hexagon can be

54

synthesized by decomposing it into components which lie along the active

voltage vectors. The general expression is represented by,

( 1) /32.

3

j nn DCV V e

π−=

(4.5)

where, n = 1, 2,….., 6

Determination of the sector in which the reference vector needed is

done by considering the expression of the vector in the α-β coordinate. To

calculate the projections Va, Vb, and Vc of the reference voltage vector in the

a-b-c plane and these projections are compared with zero. From the Clark’s

transformation, the projections Rf1, Rf2 and Rf3 as shown in table 4.2 and

are obtained as,

1Rf Vβ=

(4.6)

23Rf V Vα β= −

(4.7)

33Rf V Vα β= − −

(4.8)

Table 4.2 Sector identification

Sector

Reference voltages

Rf1 Rf2 Rf3

1 > 0 > 0 ≤ 0

2 > 0 ≤ 0 ≤ 0

3 > 0 ≤ 0 > 0

4 ≤ 0 ≤ 0 > 0

5 ≤ 0 > 0 > 0

6 ≤ 0 > 0 ≤ 0

4.4.3. Computation of Space Vector Duty Cycle

The duty cycle computation is done for each triangular sector formed

by two state vectors. The reference vector could be synthesized by the

adjacent switching state vectors V1 and V2, the duty cycle of each being d1

55

and d2 respectively and d0 as the zero vector duty cycle. The individual duty

cycles for each sector boundary state vector and the zero state vectors are

given by,

1 2

1 2

1 2 0

0 0

s ST TT T

ref

T T

V dt V dt V dt V dt= + +∫ ∫ ∫ ∫

(4.9)

3 sin( / 3 ) 3 sin( / 3 ).

2 sin( / 3) 2 sin( / 3)

ref

DC

Vd m

Vα

π θ π θπ π

− −= =

(4.10)

3 sin 3 sin.

2 sin( / 3) 2 sin( / 3)

ref

DC

Vd m

Vβ

θ θπ π

= =

(4.11)

01d d dα β= − −

(4.12)

where,

m is the modulation index, 0 0.866m≤ ≤

01 2

0, ,

s s s

TT Td d d

T T Tα β= = =

(4.13)

The dwell times for the seven segment add up to the sampling period Ts. The

switching time (dwell time) duration at sector-I is calculated as follows:

1

3 sin( / 3 ) 3 sin( / 3 ).

2 sin( / 3) 2 sin( / 3)

ref

s s

DC

VT T m T

V

π θ π θπ π

− −= =

(4.14)

2

3 sin 3 sin.

2 sin( / 3) 2 sin( / 3)

ref

s s

DC

VT T m T

V

θ θπ π

= =

(4.15)

0 1 2sT T T T= − −

(4.16)

where, 1

s

s

Tf

=

General expression for switching time during any sector can be represented

by,

( ) ( )1 1sin( / 3 ) sin( / 3 )

3 33 3.2 sin( / 3) 2 sin( / 3)

ref

n s s

DC

n nV

T T m TV

π θ π π θ π

π π

− −− + − +

= =

(4.17)

56

( ) ( )

1

1 1sin sin

3 33 3.

2 sin( / 3) 2 sin( / 3)

ref

n s s

DC

n n

VT T m T

V

θ π θ π

π π+

− −− −

= =

(4.18)

0 1s n nT T T T += − −

(4.19)

where, n = 1, 2, ….6 and 0 ≤ θ ≤ 60°

This gives switching times T0, T1 and T2 for each inverter state for a

total switching period, Ts. Applying both active and zero vectors for the time

periods ensures that average voltage has the same magnitude as desired.

The determination of time T1 and T2 of each adjacent vector is given by

simple projections and with the α-β component values of the vectors, dwell

time is calculated in terms of α-β component as,

( )

( )

1

3 sin ( / 3 )

2 sin ( / 3)

sin ( / 3) cos( ) cos( / 3). sin ( )3

2 sin ( / 3)

3 1cos( ) sin ( )

2 3

3 13 3

2 23

s

ref

D C

s s

D C D C

T m T

m

V

V

T TV V V V

V Vα β α β

π θπ

π θ π θ

π

θ θ

−=

−=

= −

= − = −

(4.20)

2

3 sin.

2 sin ( / 3)

3 sin

2 sin ( / 3)

3 2. . 3

2 3

re f

s

D C

s

s s

D C D C

VT T

V

m T

T TV V

V Vβ β

θπ

θπ

=

=

= = (4.21)

Commutation duration can be calculated for every sector and the time

of vector application is related to the following variables as:

3.

2

s

D C

TU V

Vβ=

(4.22)

57

( )13 3 .

2 2

s

D C

TV V V

Vα β= +

(4.23)

( )13 3 .

2 2

s

D C

TW V V

Vα β= − +

(4.24)

Sector number belonging to the related reference voltage vector can be

easily calculated from the (4.22) to (4.24). Table 4.3 shows the expression of

Tn and Tn+1 for different sectors in terms of U, V and W.

Table 4.3 Operation time of fundamental vector for different sectors

Sector 1 2 3 4 5 6

Tn -W W U -U -V V

Tn+1 U V -V W -W -U

4.4.4. Computation of Modulating Function

Switching time for each sector is shown in Fig. 4.9. The four

modulating functions, m0, m1, m2 and m3, in terms of the duty cycle for the

space vector modulation scheme is expressed as,

Fig. 4.9 Switching time for each sector

58

00

2

dm =

(4.25)

1 0m m dα= +

(4.26)

2 1m m d β= + (4.27)

3 0m m d β= + (4.28)

4.4.5. Generation of SVM Pulses

The required pulses can be generated by comparing the modulating

functions with the triangular waveform. A symmetric seven segment

technique is used to alternate the null vector in each cycle and to reverse

the sequence after each null vector. The switching pulse pattern for the

three phases in the six sectors can be generated. The redundant switching

states are utilized to reduce the number of switching per sampling period.

The switching state ‘111’ is selected for the T0 segment in the centre of the

sampling period, where as the state 000 is used for the segment T0/2 on

both sides. A typical seven segment switching sequence for generating

reference vector in sector-I is shown in Fig. 4.10. Table 4.4 shows

calculation of switching time at each sector.

Average variation of the voltage space vector will move along a circle with

uniform velocity. Since T0 period equally divided, that is equal duration for

average variation. So, T0 will not contribute to average variation. The mean

values of the inverter pole voltages averaged over one switching cycle are,

Table 4.4 Switching time calculation

Sector Upper switches

S1 S3 S5

I T1+T2+T0/2 T2+T0/2 T0/2

II T1+T0/2 T1+T2+T0/2 T0/2

III T0/2 T1+T2+T0/2 T2+T0/2

IV T0/2 T1+ T0/2 T1+T2+T0/2

V T2+T0/2 T0/2 T1+T2+T0/2

VI T1+T2+T0/2 T0/2 T1+ T0/2

59

Fig. 4.10 Switching logic signals for Sector-I

[ ]0 01 2 1 2

2 2.

2 2

DC DC

ao

s s

V V

T TV T T T T

T T

= − + + + = +

(4.29)

[ ]0 01 2 1 2

2 2. .

2 2

DC DC

bo

s s

V V

T TV T T T T

T T

= − − + + = − +

(4.30)

60

[ ]0 01 2 1 2 0

2 2. .

2 2

DC DC

co A

s s

V V

T TV T T T T V

T T

= − − − + = − − = −

(4.31)

For the first sector 00 60θ≤ ≤

( )3sin / 3

2ao refV V θ π= +

(4.32)

( )3sin / 6

2bo refV V θ π= −

(4.33)

( )3sin / 3

2co ref AOV V Vθ π= − + = −

(4.34)

Thus the resulting AC output line voltages consist of discrete values of

voltages that are +VDC/2, 0 and -VDC/2 for the topology shown in Fig. 4.11.

Fig. 4.11 Pole voltage for space vector modulation

4.5 Simulation of FOC Drive with SVM and SPWM Inverters

The Block diagram of FOC induction motor drive with SVM and SPWM

inverters are presented in Chapter 3 (Fig. 3.17) and validated using

MATLAB/Simulink. The simulation model is developed using the

mathematical model of SVM and SPWM inverters. The DC link voltage is

assumed as 400 V, the fundamental output frequency is chosen as 50 Hz,

the switching frequency is kept at 5 kHz, and the modulation index as 0.8.

The induction motor parameters used for the simulation are given in

Appendix - I.

61

4.6 Simulation Results and Discussion

Sector corresponds to the location of voltage in the circular locus

traced by the rotating reference vector of SVM inverter and is divided into six

sectors of 60° each as shown in Fig. 4.12. A careful observation shows that

the order of sectors is the same as in Fig. 4.10, where the vector rotates in

counter clockwise direction.

Fig. 4.12 Sector selection of voltage vector

Switching reference function represents the duty ratio of each inverter

leg or the conduction time normalized to the sampling period for a given

switch and it is a mathematical function with variation between 0 and 1

centered around 0.5. The reference function for the regular space vector is

shown in Fig. 4.13.

Projection vectors of the reference voltage vector on a-b-c plane are

presented in Fig. 4.14 with time domain. The six non-zero switch

combinations seems to be stationary snap shots of a three phase set of time

varying sinusoids with a phase voltage magnitude as shown in Fig. 4.14.

0

1

2

3

4

5

6

7

0.005 0.015 0.025 0.035 0.045Time (sec)

Sec

tor

Nu

mb

er

62

Fig. 4.13 Switching Reference Function

Fig. 4.14 Projection vectors of the reference voltage on a-b-c plane

The output voltage vector in the form of hexagon is shown in Fig. 4.15,

which shows the space vector representation of all the possible switching

states. In figure the entire space is distinctively divided into six equal sized

sectors of 60°, where each sector is bounded by two active vectors.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.01 0.02 0.03 0.04 0.05

Time (sec)

Ref

eren

ce F

un

ctio

n

-150

-100

-50

0

50

100

150

0 0.01 0.02 0.03 0.04 0.05Time (sec)

Ref

eren

ce v

olt

age

am

pli

tud

e (v

olt

)

63

Fig. 4.15 Space Vector hexagon

SVM sampled signal (reference voltage) can be observed in Fig. 4.16.

Switching pulses generated are presented in Fig. 4.17. Line-to-neutral

voltage in the form of frequent pulses and Line-to-Line voltage is shown in

Figs. 4.18 and 4.19.

Fig. 4.16 SVM output with the signal sample

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Vαααα (volt)

Vββ ββ

(volt

)

-400

-200

0

200

400

0 0.01 0.02 0.03 0.04 0.05

Time (sec)

Vββ ββ

(volt

)

-400

-200

0

200

400

0 0.01 0.02 0.03 0.04 0.05Time (sec)

Vαα αα

(volt

)

64

Fig. 4.17 Switching Pulses

0.0

0.2

0.4

0.6

0.8

1.0S

5

0.0

0.2

0.4

0.6

0.8

1.0

S3

0.0

0.2

0.4

0.6

0.8

1.0

0 0.0005 0.001 0.0015 0.002

Time (sec)

S1

65

Fig. 4.18 Line-to-neutral voltage output of SVM

-400

-200

0

200

400V

cN (volt

)

-400

-200

0

200

400

Vb

N (v

olt

)

-400

-200

0

200

400

0 0.01 0.02 0.03 0.04 0.05

Va

N (volt

)

Time (sec)

66

Fig. 4.19 Line-to-Line voltage output of SVM

For comparing the simulation results of FOC model with sensor using

SPWM and SVM inverters, the motor starts from a standstill state with

reference speed 104 rad/sec and application of a load torque, TL = 5 Nm at

time, t = 1 sec. Figures 4.20 and 4.21 show the response of rotor speed and

electromagnetic torque with time for SPWM and SVM inverters respectively.

The motor torque has a high initial value in the speed acceleration zone,

then the value decreases to zero and increases to the applied load torque

and performed well in both cases.

-500

-250

0

250

500

Vca

(volt

)

-500

-250

0

250

500

Vb

c (v

olt

)

-500

-250

0

250

500

0 0.01 0.02 0.03 0.04 0.05

Va

b (volt

)

Time (sec)

67

a) Rotor speed vs. time b) Torque vs. time

Fig. 4.20 Torque and speed responses of FOC with SPWM

a) Rotor speed vs. time b) Torque vs. time

Fig. 4.21 Torque and speed responses of FOC with SVM

4.7 Summary

This chapter contains complete review of the SVM modulation

techniques with significance advantages, which can be implemented in

special application of IM drives. FOC models with SVM inverter and SPWM

inverter are simulated using MATLAB/Simulink and the models are

validated by analyzing the results.

4.8. Publications Related to this Chapter

International Conference: 1. G. K. Nisha, S. Ushakumari and Z. V. Lakaparampil “CFT Based Optimal PWM

Strategy for Three Phase Inverter”, IEEE International conference on Power, Control

68

and Embedded Systems (ICPCES’12), Allahabad, India, pp. 1-6, 17-19 December

2012.

2. G. K. Nisha, S. Ushakumari and Z. V. Lakaparampil “Harmonic Elimination of Space

Vector Modulated Three Phase Inverter”, Lecture Notes in Engineering and Computer

Science: Proceedings of the International Multi-conference of Engineers and Computer

Scientists 2012, (IMECS 2012), Hong Kong, pp. 1109-1115, 14-16 March 2012.

3. G. K. Nisha, S. Ushakumari and Z. V. Lakaparampil “Method to Eliminate

Harmonics in PWM: A Study for Single Phase and Three Phase”, International

conference on Emerging Technology, Trends on Advanced Engineering Research,

Kollam, India, pp. 598-604, 20-21 February 2012.

International Journal: 1. G. K. Nisha, S. Ushakumari and Z. V. Lakaparampil “Online Harmonic Elimination

of SVPWM for Three Phase Inverter and a Systematic Method for Practical

Implementation”, IAENG International Journal of Computer Science, vol. 39, no. 2, pp.

220-230, May 2012.