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).

a n 1 / f ( x ) cos nxdx (n=0, 1, 2 …)

(4.2)

b n 1 / f ( x ) sin nxdx (n=1, 2, 3…)

(4.3)

When this analysis is applied to a voltage waveform such as e ( t ) , Equation (4.1) becomes, ω

e (ωt) = ω

ω ω

ω ω

(4.4)

(or)

e(t ) (a 0 / 2) (a n cos nt b n sin nt )

n 1

(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).

a n 1 / e(t ) cos n (t )`d(t )

bn 1 / e(t ) sin n (t )d(t )

( n 0,1,2,3,....)

(4.6)

( n 1,2,3,.....)

(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. e (ωt)

Em л

3Л

2л

0 Л -Em

Figure 4.1 Typical Inverter Output Voltage With e(t ) 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

e(t ) ,

then

Equations

(4.6)

and

(4.7)

become

Equations (4.8) and (4.9). an =

0

(4.8)

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л л

∫ л

ω

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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Output of the Inverter

1

ωt

0

1

Fundamental is Integral Product over 2 half cycle

2 0

ωt Second Harmonics: Area is ½ of the Fundamental half Cycle. Net Integral of fundamental half cycle is zero.

1 +1

ωt

0 -1

Third Harmonics: Area is 2/3 of the Fundamental half cycle. Net integral product is 2/3

1 -2/3

ωt

0 +2/3

1

0

1

0

+2/3 Fourth Harmonics: Net integral product over fundamental half cycle is zero

ωt

Fifth Harmonics: Net integral product is 2/3 ωt

Figure 4.2 Theoretical Harmonic Identification of Inverter Output

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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.

where

Mf

=

(fm / fc)

Mf

=

Modulation ratio,

fc

=

Carrier frequency,

fm

=

Fundamental frequency

(4.14)

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)

M =

Modulation index

Vc =

Control signal value

Vt =

Carrier signal value

(4.15)

where

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).

m kj

t kj Tseq

2v j v k 1 [1 ] 3 v 2 im

(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 = 4.3.2

Time taken over the switching sequence

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).

2v k v j 4q 1 sin( j t k ) sin(3 j t )] m kj [1 2 v im 3 3 3

(4.17)

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For j=A, B, C and k=U, V, W = 0, 2Π/3, 4Π/3 for k = U, V, W respectively

k

where

4.3.3

Vim

k

=

Maximum input voltage

=

Output amplitude of harmonic component

q

=

Voltage ratio

ωi

=

Harmonic component of input

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%.

2v k v j 2 1 m kj [1 2 sin( j t k ) sin(3 j t )] 3 v im 3 where

4.3.4

(4.18)

ωj = harmonic component of input

k = output amplitude of harmonic components

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