47
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
48
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 nt  b n sin nt )

n 1
(4.5)
49
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)
50
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.
51
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
52
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
53
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
54
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)
55
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
56
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.
cos(ωi t)


3K A K B Vim 
cos(ωi t  2ππ/3]
Vo  (Avi )B 

2
cos(ωi t  45π5π/ 
The voltage ratio is q = 3 KA K
B
(4.19)
/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
Then,
q
6 sqrt (3) / Π2 = 105.3%
4.3.5
Space Vector Pulse Width Modulation Control Algorithm
=
KB =2/Π
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
57
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
58
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.
q Axis
√
√
2
(
)
3
1
d Axis
4
6
5
√
√
Figure 4.4 Space Vector Diagram with Sectors
59
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.
q
B
Space Vector Pulse Width Modulation
√
P]o
d
A
Sine PWM
C
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
60
relationships are shown in Equation (4.20). That consists of the horizontal (d)
and vertical (q) axes as shown in Figure 4.6.
q Axis
B
Vref
A
d Axis
C
Figure 4.6 dq and ABC Reference Frame
fdqo = Ks fabc
[
where
(4.20)
√ ⁄
√ ⁄ ]
„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
61
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.
V2
Vref
(T2/TZ)V2
V1
0
(T1/TZ)V1
Figure 4.7 Reference Vector with respect to Sector I
= ∫
∫
=
∫
∫
π
|
|
[
=
π
]
(4.21)
From Equation (4.21),
|
|
]=
|
|
]=
]
π
π
From Equations (4.22) and (4.23) it is obtained
(4.22)
(4.23)
62
π
=
(4.24)
π
=
(4.25)
π
|
=
4.3.5.2
|
(4.26)
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
=
√
|
|
=
√
|
|
=
T2
T0
=
√
|
|
=
√
|
|
π
π
|
|
√
π
π
π
(4.27)
π
π
=
π
(4.28)
(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.
63
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
66
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
1
2
3
4
5
6
4.4
Lower switch
S1=T1+T2+T0/2
S4=T0/2
S3= T2+T0/2
S6= T1+T0/2
S5= T0/2
S2= T1+T2+T0/2
S1= T1+T0/2
S3= T1+T2+T0/2
S4= T2+T0/2
S6= T0/2
S5= T0/2
S2= T1+T2+T0/2
S1= T0/2
S4= T1+T2+T0/2
S3=T1+T2+T0/2
S5= T2+T0/2
S6= T0/2
S2= T1+T0/2
S1= T0/2
S4= T1+T2+T0/2
S3= T1+T0/2
S5= T1+T2+T0/2
S6= T2+T0/2
S2=T0/2
S1= T2+T0/2
S4= T1+T0/2
S3= T0/2
S5=T1+T2+T0/2
S6= T1+T2+T0/2
S2=T0/2
S1=T1+T2+T0/2
S3= T0/2
S4=T0/2
S6= T1+T2+T0/2
S1= T1+T0/2
S2= T2+T0/2
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
67
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.
i2
i1*
(t2/tz) i2
c
i1
0
(t1 /tz) i1
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
68
⁄
⁄
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)
69
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.