31
CHAPTER 3
TRADITIONAL CONTROL TECHNIQUE
3.1
INTRODUCTION
The distribution of shoot-through states in the switching waveforms
of a traditional PWM concept is the key factor to control Z-source inverter. In
the carrier based implementation such as simple boost control, maximum
boost control and constant boost control are studied in detail in the literature;
the three phase sinusoidal modulating signals with 120 degree phase shift are
compared with the high frequency carrier triangular signal. In space vector
modulation technique, the shoot-through state is inserted before or after the
active states by keeping the time period of the active states constant (Loh et al
2005). Shoot-through states again boost the DC link capacitor voltages and
could partially supplement the null states within a fixed switching cycle
without altering the normalized volt–sec average, since both states similarly
short-circuit the inverter three phase output terminals, producing zero voltage
across the AC load. Shoot-through states could therefore be inserted to the
existing PWM state patterns of a conventional voltage source inverter to
derive different modulation strategies for controlling a three phase Z-source
inverter.
3.2
SPACE VECTOR MODULATION
Space vector pulse width modulation (SVPWM) method often
referred as space vector modulation (SVM), is an advanced, computation
32
intensive PWM method and is possibly the best among all the PWM
techniques for variable speed drive applications (Rashid 2003). Because of its
superior performance characteristics, it has been finding widespread
applications in recent years. All the other PWM methods have only
considered implementation on a half bridge of a three phase bridge inverter. If
the load neutral is connected to the center tap of the DC supply, all three
bridges operate independently, giving satisfactory PWM performance. With
the machine load, load neutral is normally isolated, which causes interaction
among the phases. This interaction was not considered in other PWM
techniques (Lee et al 1998). SVM considers this interaction of the phases and
optimizes the harmonic content of the three phase isolated neutral load. There
are eight traditional states in the three phase voltage source three leg inverter,
in which six are active states producing output voltage either positive or
negative magnitude and two are traditional zero vectors producing zero output
voltage. Basically, this zero vector time period is the difference between the
total switching time period and sum of the time periods of the adjacent active
vectors. This zero state vector is distributed uniformly and inserted in the
switching waveform of the inverter either before or after the active vector
periods without changing the active vector time period (Broeck et al 1988).
The space vector based modulating technique is a digital technique
in which the objective is to generate PWM line voltages that are on average
equal to given load line voltages. This is done in each sampling period (Ts) by
properly selecting the switch states from the valid ones of the voltage source
inverter and by proper calculation of the period of times they are used. The
selection and calculation time are based on the space transformation (Bowes
et al 1997, Blasko et al 1997).
33
3.2.1
Space Transformation
Any three phase set of variables that add up to zero in the stationary
a-b-c frame could be represented in a complex plane by a complex vector that
contains a real ( ) and an imaginary ( ) component (Bose 2005). For instance,
the three phase line modulating signals could be represented as follows
va (t)
Vm cos t
v b (t)
Vm cos
t
2
3
vc (t)
Vm cos
t
2
3
(3.1)
The above three functions of time satisfy
va (t) v b (t) vc (t) 0
(3.2)
This could be represented in a two dimensional space by space
transformation
V(t)
2
va
3
v be j(2/3)
v ce
j(2/3)
(3.3)
where 2/3 is a scaling factor.
Equation (3.3) could be written as a complex function as
V(t)
vx
jv y
(3.4)
The co-ordinate transformation could be obtained from the a-b-c
axis to the x-y axis by using the equations (3.3) and (3.4).
34
vx
vy
1
2
2
3
3
0
2
1
2
1
va
3
2
(3.5)
vb
vc
Equation (3.5) could also be written as follows
vx
vy
2
va 0.5(v b
3
3
(v b vc )
3
vc )
(3.6)
The stationary frame is transformed to a rotating frame by the
transformation from x-y axis to
axis with an angular velocity of
. The
rotating frame could be obtained by rotating the x-y axis with t as given by,
v
v
cos( t) cos(
sin( t) sin(
2
2
t)
t)
vx
vy
cos( t) sin( t)
sin( t) cos( t)
vx
vy
(3.7)
From equation (3.7), the space vector representation is obtained as
V(t)
Ve j
Ve j t
(3.8)
This is the vector of magnitude Vm rotating at a constant speed
in
rad/sec. e j may be interpreted as vector rotational operator that converts
rotating frame variables into stationary frame variables (Bose 2005). The
direction of rotation depends on the phase sequence of the voltages. With the
sinusoid three phase command voltages, the composite PWM fabrication at
35
the inverter output should be such that the voltages follow these command
voltages with a minimum amount of harmonic distortion (Holtz 1992).
3.2.2
Inverter Switching States
A three phase voltage source inverter has 23 = 8 permissible
switching states. For example in state-I, switches S4, S3 and S5 are open and
S1, S6 and S2 are closed. In this sate, phase a is connected with positive DC
terminal of the battery and phases b and c are connected with negative
terminal of the battery (Holtz et al 1993). The phase-neutral voltage is
represented by
van
v bn
vcn
2
Vdc
3
1
Vdc
3
1
V
3 dc
(3.9)
The inverter has six active states, where the voltage is impressed
across the load and two zero states when the machine terminals are shorted
through, either upper or lower devices of the inverter bridge. Figure 3.1 shows
the trajectory of voltage space vectors for the traditional three phase PWM
inverters. The output voltage of the inverter is determined by the different
voltages between each inverter arm and the time duration in which the voltage
is
maintained.
Eight
voltage
vectors
V0 , V1 , V2 , ... V7 are
defined
corresponding to the switching states S0 = [0 0 0], S1 = [1 0 0] ..................
S7 = [1 1 1] respectively. V1 , V2 , ... V6 are called active vectors V0 and V7
are called traditional zero vectors.
36
Figure 3.1 Voltage space vector for voltage source inverter
Length of the active vectors is unity and length of the zero vectors
is zero. In one sampling interval TS, the output voltage vector of the
traditional inverter V is split into the two nearest adjacent voltage vectors.
These two nearest active vectors and the traditional zero vectors are used to
synthesize the output voltage vector. Two nearest vectors Vn and Vn
1
(where n = 0……6) are applied at times T1 and T2 respectively, and zero
vectors are applied at TZ time period (Bose 2005). For example in sector-I
0
3
, output voltage vector V could be synthesized as
V
where 2n
T1
T2
V1
V2
Ts
Ts
wt
2n
Tz
V 0orV7
Ts
3
(3.10)
37
The time periods for the first and second active vectors, T1 and T2
could be calculated by the following equations.
T1
2
V cos(
3
T1 Ts ma sin(
T2
T2
3
2
V cos(
3
Ts ma sin( )
6
)Ts
(3.11)
)
3
)Ts
2
(3.12)
From the above equations (3.11) and (3.12), time period for the
traditional zero vector, TZ could be calculated as follows
Tz
Ts (T1 T2 ) Ts T1 T2
1
2
V cos
3
3
Ts (3.13)
The trajectory of voltage vector V should be a circular (as shown
in Figure 3.1) while maintaining pure sinusoidal output line–to-line voltages.
Time duration for the active vectors are kept constant throughout the
operation and the zero vector time is conveniently placed depending upon the
angle of the space vector (TZ is decreased when the length of the output
voltage vector is increased). Maximum output line-to-line voltage is obtained
when the voltage vector trajectory becomes a inscribed circle of the hexagon
and V becomes
3 / 2 . This limitation of the length of the active vector
affects the smooth operation of loads like motor drives where overdrive is
desired (Holtz et al 1993). Figure 3.2 shows the construction of the
symmetrical pulse pattern for a sampling period (Ts). Here, Ts=1/fs
(fs = switching frequency) is the sampling time. The null time has been
conveniently distributed in the start and end of the switching cycle to describe
38
the symmetrical pulse widths. This type of symmetrical pulse pattern gives
minimal output harmonics.
Figure 3.2 Switching pattern of VSI for sector-I
In overmodulation, the reference vector follows a circular trajectory
that extends the bounds of the hexagon (Hava et al 1998). Hence the boundary
of the linear modulation and over modulation is a hexagon. Portions of the
circle inside the hexagon utilize the same equations (3.11)-(3.13) for
determining state times T1, T2 and TZ. However, the portions of the circle
outside of the hexagon are limited by the boundaries of the hexagon (Zhou et
al 2002), and the corresponding time states could be found from,
39
T1 Ts
3 cos( ) sin( )
3 cos( ) sin( )
T2
Ts
2sin( )
3 cos( ) sin( )
Tz
Ts T1 T2
(3.14)
0
The maximum modulation index (ma) for SVM is
For 0 ma
1 , the inverter operates in a normal SVM, and for ma
3/2 .
3/2 ,
the inverter operates completely in the six-step output mode. Six step
operations, switches the inverter only to six active vectors, there by
minimizing the number of switching at one time. For 1 m a
2 / 3 , the
inverter operates in overmodulation, which is normally used as a transition
step from the SVM techniques in to six-step operation. Although
overmodulation allows more utilization of the DC input voltage than the
linear mode, it results non sinusoidal voltages with a high degree of distortion,
especially at a low output frequency (Boost et al 1988, Radomski 2009). SVM
with significant modifications allows the Z-source inverters to be operated at
over modulation region and is discussed in the next section.
3.3
MODULATION FOR Z-SOURCE INVERTER
All the traditional PWM schemes could be used to control the
Z-source inverter with proper modifications and their theoretical input–output
relationships still hold. When the DC voltage is high enough to generate
desired AC voltage, the conventional PWM inverter is used. In every
switching cycle, two non shoot-through zero states are used along with two
adjacent active states to synthesize the desired voltage (Trzynadlowski et al
1994). While the DC voltage is not enough to directly generate a desired
40
output voltage, a modified PWM with shoot-through zero states would be
used to boost the DC link voltage.
For a three phase voltage source inverter, both continuous
switching (e.g., centered SVM) and discontinuous switching (e.g., 60
discontinuous PWM) are possible with each having its own unique null
placement at the start and end of a switching cycle and an optimum harmonic
spectrum (Tran et al 2009).
Table 3.1 Switching table of three phase Z-source inverter
State
Switching
Output
voltage
S1
S3
S5
S2
S4
S6
Active
100
Non-zero
1
0
0
0
1
1
Active
110
Non-zero
1
1
0
0
0
1
Active
010
Non-zero
0
1
0
1
0
1
Active
011
Non-zero
0
1
1
1
0
0
Active
001
Non-zero
0
0
1
1
1
0
Active
101
Non-zero
1
0
1
0
1
0
Null
000
Zero
0
0
0
1
1
1
Null
111
Zero
1
1
1
0
0
0
Shoot-through
E1
Zero
1
1
S3
0
S5
0
Shoot-through
E2
Zero
S1
!S1
1
1
S5
!S5
Shoot-through
E3
Zero
S1
!S1
S3
!S3
1
1
Shoot-through
E4
Zero
1
1
1
1
S5
!S5
Shoot-through
E5
Zero
1
1
S3
!S3
1
1
Shoot-through
E6
Zero
S1
!S1
1
1
1
1
Zero
1
1
1
1
1
1
Shoot-through
E7
(!Sx is complement of Sx)
This section now extends the analysis presented in the previous
section, to derive the continuous PWM strategy for a three phase Z-source
inverter with each having the same characteristic spectrum as its conventional
counterpart. The above Table 3.1 lists the fifteen switching states of a three
41
phase Z-source inverter. In addition to the six active and two null states
associated with a conventional voltage source inverters, the Z-source inverter
has seven shoot-through states representing the short circuiting of a phase leg
(shoot-through states E1 to E3), two phase-legs (shoot-through states E4 to
E6) or all three phase-legs (shoot-through state E7). With three state
transitions, three equal interval shoot-through states (T0/3) could be added
immediately adjacent to the active states per switching cycle. This preferred
state sequence and placement of shoot-through states are shown in the
Figure 3.3, where the middle shoot-through state is symmetrically placed
about the original switching instant.
Figure 3.3 Switching pattern for three phase Z-source inverter (sector I)
42
The active states {100} and {110} are left/right shifted accordingly
by T0 / 6
with their time intervals kept constant, and the remaining two
shoot-through states are lastly inserted within the null intervals, immediately
adjacent to the left of the first state transition and to the right of the second
transition. This way of sequencing inverter states also ensures a single device
switching at all transitions, and allows the use of only shoot-through states
E1, E2, and E3. The other shoot-through states could not be used since they
require the switching of at least two phase-legs at every transition. The state
sequence and placement of shoot-through states in Figure 3.3 could similarly
be generated through carrier based implementation. Starting with a set of
three phase sinusoidal signals (Va, Vb and Vc) and noting that the first VSI
transition
with Vmax
is
triggered
by
the
intersection
of
the
falling carrier
max(Va , Vb , Vc ) , the modified references for inserting the first
shoot-through state E1. Vmax(sp) is for inserting shoot-through state E1 by
turning ON the upper (odd-numbered) switch of the relevant phase-lag at
prior to T0 / 6 , while Vmax(sn) is for ending the shoot-through by turning
OFF the lower (even numbered) switch at T0 / 6 earlier. In the same way,
second and third VSI transitions are triggered by Vmid
and Vmin
mid(Va , Vb , Vc )
min(Va , Vb ,Vc ) respectively (Loh et al 2005). The traditional
reference voltages could be represented by the following expressions.
Vmax(sp)
Vmax
Voff
T
Vmax(sn)
Vmax
Voff
T
3
Vmid(sp)
Vmid
Vmid(sn)
Vmid
Voff
Voff
T
3
T
3
(3.15)
(3.16)
43
Vmin(sp)
Vmin
Voff
Vmin(sn)
Vmin
Voff
T
3
T
(3.17)
{sp,sn} {1, 4},{3, 6},{5, 2}
where T
T0 / TS
shoot-through duty ratio and sp and sn are switches
connected in positive DC rail and negative DC rail respectively and Voff
represents the triplen offset needed for implementing centered SVM.
The active state {100} is left shifted by (T0/6) and the other active
state {110} is right shifted by (T0/6) to insert the middle shoot-through for the
duration (T0/3). While shifting the active states towards the traditional zero
states, the length of the each traditional zero states {000} and {111} are
reduced by (T0/2). Remaining two shoot-through states are inserted before and
after the active states {100} and {110} respectively. Since the shoot-through
time is acquired from the traditional zero time periods, the length of the
traditional zero state is reduced to (TZ/2-T0/2). It should be noted that each
phase leg still switches ON and OFF once per switching cycle. Without
changing the total active state time interval, shoot-through zero states are
evenly allocated into each phase. That is, the active states are unchanged.
However, the equivalent DC link voltage to the inverter is boosted because of
the shoot-through states. It is noticeable here that, the equivalent switching
frequency viewed from the Z-source network is six times the switching
frequency of the main inverter, which greatly reduces the required inductance
size of the Z-source network. For shoot-through duty ratio = 0.2, the space
vector modulated pulses for all the switches in all the six sectors of the space
vector trajectory is shown in Figure 3.4.
44
Figure 3.4
Sector 1
Sector 2
Sector 3
Sector 4
Sector 5
Sector 6
Placement of shoot-through states in all sectors by
traditional SVM
45
Expressions for calculating different performance parameters of the
Z-source inverter are summarized as follows (Shen et al 2006)
Shoot-through duty ratio D0
Boost factor B
4
9 3m a
Voltage gain G
vˆ i
Vdc / 2
Voltage stress Vs
3.4
3 2
4
3 3m a
2
2
4 ma
9 3ma 2
9 3G 4
Vdc
2
(3.18)
(3.19)
(3.20)
(3.21)
TRADITIONAL CONTROLLER ARRANGEMENT
To achieve good performance for both the DC boost and AC line
voltage control in the Z-source inverter fed ASD, motor terminal voltage and
Z-source capacitor voltage are sensed and compared with the pre-defined
reference values. The implementation block diagram of traditional control
system is depicted in Figure 3.5. Depending up on the shoot-through time
period, boost factor of the capacitor voltage is determined and the output
voltage is obtained. Boost factor and the modulation index are the control
variables to control the Z-source capacitor voltage and motor terminal voltage
respectively (Gajanayake et al 2007). Control modes of Z-source inverter fed
induction motor drives are classified depending on the boost or buck
operation. During voltage sag, the Z-source capacitor voltage is increased in
relation to the DC input voltage by inserting the shoot-through states
periodically. The capacitor voltage is regulated depending upon the supply
voltage sag by the capacitor voltage feed back control loop. In this mode,
46
voltage across the Z-source capacitor is measured and compared with the
pre-defined reference values, and then the error is processed to adjust the time
period of the shoot-through. When the terminal voltage needs to be adjusted,
the AC output voltage of the Z-source inverter is controlled by the modulation
index. Modulation index is adjusted in accordance to the error of the terminal
voltage feedback control loop. While controlling the Z-source inverter,
modulation index could take values from zero to one and boost factor could
take values from one to infinity. So their multiplication gives all levels of
desired voltages at the output. However, there are some practical limitations
for both parameters. Reduction in the modulation index increases the voltage
stress on the switches, whereas controlling the voltage level when
shoot-through duty cycle is near 0.5 (or boosting factor is near infinity)
becomes hard because of the sharp increase of the voltage gain. It needs two
independent closed loop controllers to be designed separately and the
presence of more number of controllers leads instability to overall system
(Botteron et al 2001).
AC Supply
CVref
-+
Rectifier
PI
Controller
CV
Three phase
Inverter
Driver circuit for
IGBTs
Z-Source
S1 6
DSP Processor
(MSVM)
S0
abcTransformation
Induction
Motor drive
V
Vabc
Vref
+-
PI
Controller
e
Figure 3.5 Implementation block diagram of the traditional system
47
There are two parameters to be changed in order to get the desired
output AC voltage in the Z-source inverter: (i) the modulation index, which
also exists in traditional voltage source inverters (ii) the boost factor, which
depends on the shoot-through time period inserted in the traditional switching
waveform (Tran et al 2007). Theoretically, the modulation index could take
values from zero to one, while the boost factor could take values from one to
infinity. Small signal model of the Z-source inverter shows that there is a
non-minimum phase behavior in the control-to-capacitor voltage transfer
function, which limits the control range of the system (Liu et al 2007). In
chapter 2, a detailed explanation of Z-source inverter topology is given.
3.5
SUMMARY
The basics of SVM concept for three phase inverter system have
been analyzed. The traditional space vector modulation concept to switch a
Z-source inverter while retaining all the features has been discussed in this
chapter. A traditional way of inserting the shoot-through time period in a
switching waveform is depicted with its features. Then the traditional multi
controller system for Z-source inverter fed induction motor has been outlined.