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CHAPTER 4

SPACE VECTOR PULSE WIDTH MODULATION

4.1 INTRODUCTION

The main objectives of space vector pulse width modulation

generated gate pulse are the following.

Wide linear modulation range

Less switching loss

Less total harmonic distortion in the spectrum of switching

waveform

Easy implementation and less computational calculations

With the emerging technology in microprocessor the SVPWM has

been playing a pivotal and viable role in power conversion (Jenni and Wueest

1993). It uses a space vector concept to calculate the duty cycle of the switch

which is imperative implementation of digital control theory of PWM

modulators.

Before getting into the space vector theory it is necessary to know

about the harmonic analysis of power converters. With the application of

Fourier analysis the harmonic content of any waveform can be determined.

A brief description of such analysis is presented here. This study is with a

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view to measure total harmonic distortion which will indicate the probable

losses in the output.

4.2 HARMONIC ANALYSIS OF INVERTER OUTPUT

Any periodic function can be represented by fundamental sine and

cosine waves and their harmonics as illustrated in Equation (4.1).

F(x)= (4.1)

where ao through an and b1 through bn are constants, which can be determined

as illustrated in Equations (4.2) and (4.3).

na nxdxcos)x(f/1 (n=0, 1, 2 …) (4.2)

nxdxsin)x(f/1bn (n=1, 2, 3…) (4.3)

When this analysis is applied to a voltage waveform such as e ( )t ,

Equation (4.1) becomes,

e (ωt) = ω ω ω ω ω ω (4.4)

(or)

1n

nn0)tnsinbtncosa()2/a()t(e (4.5)

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The constants are the magnitudes of the nth harmonics except a0

where a0 is the DC component of the voltage waveform. These magnitudes

are determined from Equations (4.6) and (4.7).

)t(d)`t(ncos)t(e/1an ( ,....)3,2,1,0n (4.6)

nb )t(d)t(nsin)t(e/1 ( ,.....)3,2,1n (4.7)

The output voltage of an inverter is a square wave as shown in

Figure 4.1. This square wave is taken as an example to explain about

harmonics.

Figure 4.1 Typical Inverter Output Voltage

With )t(e as a square wave , it is advantageous of selecting t=0 at

a particular point. If t=0 is chosen as the starting of the positive half cycle of

)t(e , then Equations (4.6) and (4.7) become

Equations (4.8) and (4.9).

an = 0 (4.8)

e (ωt)

0

Em

-Em

л

Л

2л 3Л

50

л∫ ω л л ω ω (n=0, 1, 2…) (4.9)

The voltage function for the square wave of Figure 4.1 is given by

Equations (4.10) and (4.11).

e (ωt) = Em, for 0 ≤ e (ωt) ≤ л (4.10)

e (ωt) = -Em, for л ≤ e(ωt) ≤ 2 л (4.11)

Substituting these relationships into Equation (4.8), the coefficients

are found as given in Equation (4.12).

bn = π , (n=1,3,5…..) (4.12)

Substituting Equations (4.8) and (4.12) in Equation (4.5),

e (ωt)=

ω (4.13)

From Equation (4.13), it is known that the output voltage contains

odd harmonics. To eliminate the third harmonic and its multiples present in

the inverter output, third harmonic injection technique is followed which can

be done using space vector pulse width modulation. Different types of

harmonics are illustrated in Figure 4.2.

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Figure 4.2 Theoretical Harmonic Identification of Inverter Output

0

0

0

0

0

0

1

1

1

1

1

1

2

+1

-1

+2/3

-2/3

+2/3

ωt

ωt

ωt

ωt

ωt

ωt

Fundamental is Integral Product over 2 half

cycle

Output of the Inverter

Second Harmonics: Area is ½ of the

Fundamental half Cycle. Net Integral of

fundamental half cycle is zero.

Third Harmonics: Area is 2/3 of the

Fundamental half cycle. Net integral product

is 2/3

Fourth Harmonics: Net integral product over

fundamental half cycle is zero

Fifth Harmonics: Net integral product is 2/3

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Equation (4.14) is used to find the number of harmonic components

in the output voltage. Output signal harmonics are equal to Mf ±1. When

switching frequency increases than the fundamental frequency the effect of

output harmonics will decrease. Increase in switching frequency leads to high

switching losses and decrease in output voltage.

Mf = (fm / fc) (4.14)

where Mf = Modulation ratio,

fc = Carrier frequency,

fm = Fundamental frequency

In Equation (4.15), Vc increases with an increase of M. It is called

over modulation. Space vector pulse width modulation scheme is a method

directly implemented using digital computer. The following theory gives

different types of modulation schemes and space vector theory.

M = ( Vc/Vt) (4.15)

where

M = Modulation index

Vc = Control signal value

Vt = Carrier signal value

4.3 DIFFERENT TYPES OF MODULATION SCHEMES

Different types of modulation schemes are analyzed. Venturini has

developed first modulation scheme for matrix converter. Maximum voltage

transfer ratio 50% is possible in Venturini algorithm. Implementation of

Venturini algorithm involves difficult calculation. An improvement in the

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achievable voltage ratio to 87% is possible by adding common mode voltage

to the target output ( Kaura and Blasko 1996). In this analysis maximum

voltage transformation ratio is determined for the different types of

modulation scheme as explained below. The relationship between the space

vector pulse width modulation duty cycle and output voltage is described.

4.3.1 Venturini Modulation Method (Venturini First Method)

It is a type of modulation scheme used to operate matrix converter.

However calculating the switching timings directly from the modulation

solutions is difficult from practical point of view. The relationship between

output voltage and duty cycle is shown in Equation (4.16). It is more

conveniently expressed in terms of the input voltages and the target output

voltages assuming unity displacement factor. The formal statement of the

algorithm, including displacement factor control (Alesina and Venturini 1988)

is rather complex and appears unsuited for real time implementation. Figure

4.3 illustrates maximum voltage transformation ratio is limited to 50%. It

shows relationship between input voltage envelope and output target voltage.

Figure 4.3 Wave form Illustrating 50% Voltage Transformation Ratio

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Assume a converter having „j‟ input lines and „k‟ output lines.

Then modulation function of switch connecting jth

input with kth

output is

illustrated in Equation (4.16).

]v

vv21[

3

1

T

tm

im2

kj

seq

kj

kj (4.16)

For 3 phase input/3phase output converter, the input terminals of

the matrix converter are j=A, B, C and the output terminals are k=U, V, W.

mkj = Modulation function of switch connecting jth

input with

kth

output

vj = Input voltage vector

vk = Output voltage vector

vim = Maximum input voltage

tkj = Switching time connecting jth

input with kth

output

Tseq = Time taken over the switching sequence

4.3.2 Venturini Optimum Method (Venturini Second Method)

It is also known as displacement factor control. Displacement factor

control can be introduced by inserting a phase shift between the measured

input voltages (vj) and inserted voltage (vk) as shown in the Equation (4.17). It

employs common mode addition that helps to achieve the maximum

transformation ratio of 87%. The relationship between output voltage and

duty cycle is illustrated in Equation (4.17).

)]t3sin()tsin(33

q4

v

vv21[

3

1m

jkjim

2

jk

kj (4.17)

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For j=A, B, C and k=U, V, W

k = 0, 2Π/3, 4Π/3 for k = U, V, W respectively

where Vim = Maximum input voltage

k = Output amplitude of harmonic component

q = Voltage ratio

ωi = Harmonic component of input

4.3.3 Scalar Modulation Method

In this method of modulation the switch actuation signals are

calculated directly from measurement of input voltages. This method yields

virtually identical switching timings to the optimum Venturini method. The

relationship between output voltage and duty cycle is shown in Equation

(4.18). The voltage transformation ratio of the scalar modulation method is

87%.

)]t3sin()tsin(3

2

v

vv21[

3

1m

jkjim

2

jk

kj (4.18)

where ωj = harmonic component of input

k

= output amplitude of harmonic components

4.3.4 Indirect Modulation Method

This method aims to increase the maximum voltage ratio above

86.6% limit of other methods. The voltage output is greater than the previous

method. For the values q>0.866, as shown in the Equation (4.19) the mean

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output voltage, V0 no longer equals the target output voltage in each switching

interval. This inevitably leads to low frequency distortion in the output

voltage and /or the input current compared to other methods with q<0.866.

For q<0.866, the indirect method yields results similar to the direct method.

]

45π5π/tcos(ω2ππ/3tcos(ωt)cos(ω

2

VK3K)BAv(V

i

i

i

imBA

io (4.19)

The voltage ratio is q = 3 KA K B /2. Clearly A and B modulation

steps are not continuous in time as shown above. The KA and K B are output

modulation steps.

KA = 2 √3 Vim/Π and K B = 2 / Π

Then, q = 6 sqrt (3) / Π2 = 105.3%

4.3.5 Space Vector Pulse Width Modulation Control Algorithm

Space vector pulse width modulation is applied to output voltage

and input current control. This method is an advantage because of increased

flexibility in the choice of switching vector for both input current and output

voltage control. It can yield useful advantage under unbalanced conditions.

The three phase variables are expressed in space vectors. For a sufficiently

small time interval, the reference voltage vector can be approximated by a set

of stationary vectors generated by a matrix converter.

If this time interval is the sample time for converter control, then at

the next sampling instant when the reference voltage vector rotates to a new

angular position, it may correspond to a new set of stationary voltage vectors

(Casadei et al 1993). Carrying this process onwards by sampling the entire

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waveform of the desired voltage vector being synthesized in sequence, the

average output voltage would closely emulate the reference voltage.

Meanwhile, the selected stationary vectors can also give the desirable phase

shift between input voltage and current. The modulation process thus required

consists of two main parts: selection of the switching vectors and computation

of the vector time intervals.

The above methods give the theoretical maximum voltage gain of

0.866, though they use different approaches. This is realized in Venturini

method.

Modulation of the line to line voltage naturally gives an extended

output voltage capability. The computational procedure required by SVPWM

method is less complex than that for Venturini method because of the reduced

number of sine function computations (Kolar et al 1991). The number of

switch commutations per switching cycle for SVPWM method is 20% less

than that of Venturini method.

Roots of vectorial representation of three-phase systems are

presented in the research contributions of Park and Kron, but the decisive step

on systematically using the Space Vectors was done by Kovacs and Racz

(Park 1933). They provided both mathematical treatment and a physical

description and understanding of the drive transients even in the cases when

machines are fed through electronic converters (Maamoun et al 2010).

SVPWM refers to a special switching sequence of the upper three

power transistors of a three-phase power inverter. It has been shown to

generate less harmonic distortion in the output voltages and or currents

applied to the phases of an AC motor and to provide more efficient use of

supply voltage. There are two possible vectors called zero vector and Active

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vector. The objective of space vector PWM technique is to approximate the

reference voltage vector Vref using the eight switching patterns. One simple

method of approximation is to generate the average output of the inverter in a

small period, T to be the same as that of Vref in the same period. Therefore,

space vector PWM can be implemented by the following steps:

Step 1 : Determine Vd, Vq, Vref, and angle ( )

Step 2 : Determine time duration T1, T2, T0

Step 3 : Determine the switching time of each transistor

(S1 to S6)

All sectors in SVPWM are shown in Figure 4.4. It uses a set of vectors

that are defined as instantaneous space vectors of the voltages and currents at

the input and output of the inverter. These vectors are created by various

switching states that the inverter is capable of generating.

Figure 4.4 Space Vector Diagram with Sectors

d Axis

1

2

3

5

6

4

√

( )

√ √

√

q Axis

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Figure 4.5 shows the maximum control voltages obtained using

sine wave pulse widh modulation which is (1/2)Vdc and space vector pulse

width modulation scheme which is (1/√3)Vdc.

P]o

Figure 4.5 Maximum Voltage Transformation Ratio

To implement the space vector PWM, the voltage equations in the

ABC reference frame can be transformed into the stationary dq reference

frame. Relating the three phase voltages and currents in terms of „ωt‟ is

difficult to handle directly. It can be transformed into two reference frames by

using Park‟s transform (Bernard Adkins and Harley 1975) and their

C

Space Vector Pulse Width Modulation

B

q

A

d

√

Sine PWM

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relationships are shown in Equation (4.20). That consists of the horizontal (d)

and vertical (q) axes as shown in Figure 4.6.

Figure 4.6 dq and ABC Reference Frame

fdqo = Ks fabc (4.20)

[ √ ⁄ √ ⁄ ]

where „f‟ is a voltage or current

In dq reference frame, there are six sectors. Each sector is divided

equally by sixty degrees. Basic Vectors are V1, V2, V3, V4, V5 and V6. These

vectors are shown in Figure 4.4.

4.3.5.1 Calculation of time period for Sector I

At sector I, V1 and V2 are voltage vectors. Assume Vref makes „ ‟

phase angle difference with V1. This Vref can be calculated using vector

d Axis

q Axis B

C

A

Vref

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calculus by referring Figure 4.7. „Tz„is switching time interval at which output

voltage of inverter is constant. T1 and T2 are switching time duration of

voltage space vectors V1 and V2.

Figure 4.7 Reference Vector with respect to Sector I

∫ = ∫ ∫ ∫ =

| | = [ π π ] (4.21)

From Equation (4.21),

| | ] = ] π (4.22)

| | ] = π (4.23)

From Equations (4.22) and (4.23) it is obtained

0 V1

V2

Vref

(T2/TZ)V2

(T1/TZ)V1

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= π π (4.24)

= π (4.25)

= | | (4.26)

4.3.5.2 Switching Time at Any Duration (T1, T2, T0)

Switching time at any instant can be illustrated in Equation (4.27)

to (4.29). For „n‟ number of samples T1, T2 and T0 are,

T1 = √ | | π π

= √ | | π

= √ | | π π (4.27)

T2 = √ | | π

= √ | | π π (4.28)

T0 = (4.29)

where, n=1 through 6 (that is sector 1 to 6), 0 ≤ ≤ 60 4.3.5.3 Determination of switching time

Figures 4.8 to 4.13 show the switching time each transistor of an

inverter system.

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Figure 4.8 Swtching Time in Sector I

Figure 4.9 Switching Time in Sector II

64

Figure 4.10 Swtching Time in Sector III

Figure 4.11 Swtching Time in Sector IV

65

Figure 4.12 Swtching Time in Sector V

Figure 4.13 Swtching Time in Sector VI

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Table 4.1 shows the 6 sectors and the time calculation of each

switch. This can be easily calculated using above switching states.

Table 4.1 Switching Time Calculation of Each Section switch (VSI)

Sector Upper switch Lower switch

1 S1=T1+T2+T0/2

S3= T2+T0/2

S5= T0/2

S4=T0/2

S6= T1+T0/2

S2= T1+T2+T0/2

2 S1= T1+T0/2

S3= T1+T2+T0/2

S5= T0/2

S4= T2+T0/2

S6= T0/2

S2= T1+T2+T0/2

3 S1= T0/2

S3=T1+T2+T0/2

S5= T2+T0/2

S4= T1+T2+T0/2

S6= T0/2

S2= T1+T0/2

4 S1= T0/2

S3= T1+T0/2

S5= T1+T2+T0/2

S4= T1+T2+T0/2

S6= T2+T0/2

S2=T0/2

5 S1= T2+T0/2

S3= T0/2

S5=T1+T2+T0/2

S4= T1+T0/2

S6= T1+T2+T0/2

S2=T0/2

6 S1=T1+T2+T0/2

S3= T0/2

S1= T1+T0/2

S4=T0/2

S6= T1+T2+T0/2

S2= T2+T0/2

4.4 SVPWM BASED DUTY CYCLE CALCULATION FOR

RECTIFIER

The rectifier gate drive duty cycle based on voltage space vector is

illustrated here. For speed control applications rectifier fed inverter system is

employed. This system converts fixed AC to variable AC voltage using two

conversion stages. The matrix converter is a direct conversion system. To get

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variable AC, switches in rectifier as well as the inverter must be switched on

at the same instant. Switch on time of both the systems is calculated. This is

used to find out duty cycle of the matrix converter. This section describes

duty cycle calculation of rectifier for the inverter.

Let

For standalone current controlled rectifier, adjacent switching

vectors are and as shown in Figure 4.14.

Figure 4.14 Reference Vector with Respect to Current

Let are duty cycles corresponding to adjacent switching

vectors i1 and i2. Rectifier for the inverter switching time interval during

constant output current is . This tz is equal to Tz shown in Equation (4.29).

From Figure 4.14 i1* can be witten as follows.

⁄ π

0 i1

i2

i1*

(t2/tz) i2

(t1 /tz) i1

c

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⁄

⁄ where Angle of the reference current vector

To find current modulation index power balance condition can be

used. With balanced output load current condition such as,

(

(

√ (

where

[ ] . ( [ ] [ ]) [ ] (4.30)

[ ] (4.31)

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Equations (4.30) and (4.31) describe DC voltage and current in

terms of duty cycle. This is used to find mathematical relationship between

duty cycle and output voltage with respect to space vector pulse width

modulation.

4.5 CONCLUSION

In this chapter space vector pulse width modulation is discussed.

The basic principle of harmonic identification is explained. Graphical

representation of various harmonics is also shown. Identification of different

types of modulation schemes is analyzed. Space vector algorithm based

switching time is calculated for inverter. Mathematical modeling of SVPWM

based duty cycle is described for current source rectifier. This duty cycle is

used to find duty cycle of matrix converter described in chapter 8.