15

CHAPTER 2

Z-SOURCE INVERTER FED INDUCTION MOTOR

2.1 INTRODUCTION

The traditional inverter based induction motor drive system consists

of a front end three phase diode rectifier, DC link LC filter, and three phaseinverter bridge as shown in Figure 2.1. It has some common limitations and

problems such as the obtainable output voltage is limited quite below theinput line voltage. Momentary supply voltage sags can interrupt the

performance of the overall drive system and shut down critical loads and

processes. Hence the performance and reliability are compromised bytraditional inverter structure (Hava et al 1999, Alan et al 2000).

Figure 2.1 Traditional VSI based ASD system

Two inverter topologies are used with the existing induction motordrives to supply required power to the motor terminals: (i) a conventionalthree phase PWM based voltage source inverter and (ii) a three phase PWM

16

inverter with a DC-DC boost converter, which is also very popular in otherapplications such as photovoltaic and fuel cell. For the wide voltage range and

limited voltage level, the conventional PWM inverter topology imposes highstresses to the switching devices and motor, consequently limits the motors

constant power-speed ratio (Mohan et al 2004). The DC-DC boost PWMinverter topology can alleviate the stresses and limitations, however it suffers

the problem of high cost and complexity associated with the two-stage powerconversion. A Z-source inverter could elevate most of the problemsassociated with traditional voltage source and current source inverters. Unique

features of the Z-source inverter provide a cheaper, simpler, and single stagepower conversion structure for induction motor drives. Presence of Z-network

highly enhances the reliability of the inverter since the shoot-through can nolonger destroy the inverter (Peng et al 2003).

Table 2.1 Operation conditions of motor drive at different power

Power rating of the motor 10kW 20kW 30kW 40kW 50kWDC supply voltage (V) 340 325 305 280 250

Motor phasevoltage (V)

Conventional PWMinverter 120.2 117.9 107.8 99 88.4

DC-DC boost+PWM inverter 148.5 148.5 148.5 148.5 148.5

Z-source inverter 155.2 152.1 148 142.9 136.8

Motorcurrent (A)

Conventional PWMinverter 39.7 77.3 115.9 158.5 209.4

DC-DC boost+PWM inverter 32.1 59.9 84.2 105.6 124.7

Z-source inverter 29.6 56.2 81.1 105.3 129.5

Table 2.1 illustrates the advantages of the Z-source inverter based

system over traditional power conversion topologies for induction motordrives (Olszewski 2005). A conventional PWM inverter is always operating atmodulation index of 1; the DC-DC boost PWM converter boosts the DC

17

voltage to a desired level since the inverter always operates with modulationindex of 1; the Z-source inverter outputs a maximum obtainable voltage while

keeping the device voltage under a given value. With these assumptions, anobtainable motor phase voltage and current for various loads are shown in

Table 2.1. It can be seen that, for low and medium power applications,Z-source inverter based operation provides better motor phase voltages and it

ensures the wide range of operation of the drive for the same input DCvoltage. Depending upon the time period of the shoot-through in oneswitching cycle, the DC link voltage of the inverter could be increased or

reduced as required by the application (Chen et al 2007).

2.2 MODES OF OPERATION

Figure 2.2 Z-source inverter fed induction motor arrangement

Figure 2.2 shows the power circuit configuration of the Z-source

inverter fed induction motor drive system. Similar to that of a traditionalinverter fed induction motor system, the Z-source inverter fed induction

motor drive systems power circuit consists of four major parts: a front enddiode rectifier, Z-network, an inverter bridge and a three phase inductionmotor load. The differences are that a DC link circuit is implemented by the

Z-source network (C1, C2, L1 and L2). Small range of input capacitors (Ca, Cband Cc) is connected to the front end diode rectifier. These input capacitorsalso serves as a DC source feeding the Z-source network and are used to

18

suppress voltage surge that may occur due to the line inductance during diodecommutation and shoot-through mode of the inverter, thus requiring a small

value of capacitance. These changes can easily be realized and implementedfrom the traditional inverter fed induction motor drive systems. Since the

Z-source inverter bridge could boost the DC link capacitor (C1 and C2) voltageto any desired value that is above the average DC value of the rectifier, a

desired output voltage is always obtainable regardless of the line voltage. Fora 230V adjustable speed drive system, the Z-source capacitor voltage could beboosted to 350V or greater in order to produce 230V rms AC outputregardless of the line voltage. Theoretically, the Z-source capacitor voltagecan be boosted to any value above the inherent average DC voltage

(310325V for a 230V line) of the rectifier rating in practical use (Peng et al2005b).

From the symmetry of the Z-source and equivalent circuit one has,

C1 C2 C L1 L2 LV V V ; v v v? ? ? ? (2.1)Based on the switching states, the operation of the Z-source inverter

could be classified in to three operating modes, namely traditional active (nonshoot-through) mode, shoot-through mode and traditional zero mode.

2.2.1 Traditional Active State

Figure 2.3 shows an equivalent circuit of the active mode, duringone of the six traditional active states, the inverter bridge acts as a traditional

voltage source inverter, thus acting as a current source viewed from theZ-source circuit. The positive group diodes (Dpa) and the negative groupdiodes (Dnb) conduct and carry currents. In traditional inverter based inductormotor drive system, the diode bridge may conduct depending on the DC linkcapacitor voltage level.

19

Figure 2.3 Equivalent circuit under active state

However, the Z-source circuit always forces diodes to conduct and

carry a current difference between the inductor current L(i ) and current through the inverter out(i ) . Because of the symmetrical configuration of the circuit, both the equal inductors would have identical current value. The diode in the equivalent circuit would be forward biased in this case. From the

symmetry of Z-source network and equivalent circuit one could have,

L dc C d dc

i C L C dc

v V V ; v Vv V v 2V V

= =

= = (2.2)

where, iv is the peak DC link voltage, dcV is DC supply voltage, vd is the

voltage before the Z-source.

2.2.2 Shoot-through State

Figure 2.4 shows the equivalent circuit representation of

shoot-through mode of operation. In this mode, the inverter bridge is under one of the shoot-through states for an interval (T0), over a sampling period

1L

2L

2C1CpaD

nbD

aC

sai

1Lv

2Lv

+

C2V+

C1Vdc dV v=

outi

L out2i i

Li

20

(Ts). The diode in the equivalent circuit would be reverse biased in this case. During the shoot-through state, both diodes are off, separating the DC link from the AC line and line current flows to the capacitor (Ca). This shoot-through period (T0) is applied in every switching cycle (Ts) and acquired from the traditional zero time (TZ) period generated by the PWM control.

Figure 2.4 Equivalent circuit under shoot-through state

Depending on a voltage boost needed, the shoot-through time interval or its duty cycle is determined. It could be seen that the shoot-through interval is only a fraction of the switching cycle; therefore it requires a

relatively small capacitor to suppress the voltage.

The voltage across the impedance elements could be related as

L C d C iv V ; v 2V ; v 0= = = (2.3)

where iv is the DC link voltage of the inverter.

Li

paD

nbD

aC

sai

1Lv

2Lv

+

C2V+

C1VdcV

Li

paD

nbD

aC

sai

1Lv

2Lv

+

C2V+

C1VdcV

21

2.2.3 Traditional Zero State

During one of the two traditional zero states (TZ), (i.e. shorting through either the upper or lower three switches) the inverter bridge is acting as an open circuit viewed from the Z-source circuit and the diodes conduct.

The equivalent circuit of Z-source inverter fed drive system during traditional

zero state is shown in Figure 2.5. Since the inverter bridge is in any one of the traditional zero states (000 or 111), the instantaneous output voltage of the inverter is zero.

Figure 2.5 Equivalent circuit under traditional zero state

The peak DC link voltage across the inverter bridge is expressed in

equation (2.2) and it could be written as,

i C L C dc dc dc0

1v V v 2V V V BV

1 2D= = = =

(2.4)

where B is boost factor and D0 is shoot-through duty ratio ( )0 0 SD T / T= . The voltage stress across the switch is equal to the peak DC link voltage i dcv BV= ,

therefore, to minimize the voltage stress for any given voltage gain (G=Bma),

1L

2L

2C1C

Li

paD

nbD

aC

sai

1Lv

2Lv

+

C2V+

C1VdcV

+

iv

22

one need to minimize B and maximize ma, with the restriction of that theirproduct is the desired value. On the other hand, one should maximize B for

any given modulation index ma, to achieve the maximum voltage gain.

The voltage across the Z-source capacitors could be written as

0

S 0c1 c2 c dc dc dc

00

S

T1T 1 D B 1V V V V V V

1 2D 2T1 2T

? ?? ? ? ? ?? ?? ?? ?? ? ? ? ?? ? ? ??? ? ? ?? ?? ? ?? ?(2.5)

where,0

1B1 2D

? ? .Peak value of the AC output voltage could be expressed as

dcac a

Vv m B

2? (2.6)

where

dc LL LL3 2V V 1.35V?? ? (2.7)

Vdc is the inherent DC voltage of the rectifier fed from the line witha line to line rms value of VLL, assuming that voltage drop on the lineimpedance is negligible. In addition to the above equations, it should be notedthat the equivalent DC voltage across the inverter bridge Vd, is different fromthe DC capacitor voltage (Vc1 and Vc2), or when the boost factor is greaterthan 1, DC link voltage is expressed as

dc link i dc C2BV v BV V

B 1? ? ?? ? ?? ??? ? (2.8)

23

For a traditional control strategy having utility supply of 230V,50 Hz, AC, the maximum DC link voltage is restricted to 310V and DC linkvoltage of the inverter is again increased due to the presence of the Z-source.

2.3 DYNAMIC MODELING

2.3.1 Small Signal Modeling

Power inverters require feedback control in order to keep thedesired output AC voltage/current values. This could be achieved through acontroller circuit that can vary the inverter control input such that the outputvoltage/current is regulated around a pre-defined value. The feedback controlsystem composed of a power inverter and a control circuit should be stableand the converter transient parameters such as percentage overshoot, settlingtime and steady state error should meet specifications. The design of such acontrol circuit requires an accurate dynamic model of the inverter. Thisdynamic model should give an idea of how the output voltage of the converteris affected by the changes in the input voltage, the load current, and dutycycle. Modeling is the representation of a physical system using mathematicaltools. In general, engineering models include significant system behaviorswith neglected second order effects. This system model gives a physicalinsight into the system dynamics and helps in designing a proper controller.Accuracy and complexity of the model are inversely related and depend onthe level of the assumptions made. So far, different modeling techniques aswell as various models with different levels of accuracy and complexity areused for power electronic circuits. The use of a dynamic model depends onthe system design stage (Nise et al 2004).

For power inverters, obtaining simplified models require makingassumptions and approximations. For instance, a capacitor shows parasiticresistive and inductive behaviors together with its capacitive behaviorespecially at high frequencies. These parasitic effects can be approximated by

24

an equivalent series resistance (ESR) and an equivalent series inductance(ESL). However, some inverter models use neither an ESR nor an ESL, withan ideal capacitor. Such a model will help in calculations, but will not be ableto guess any dynamics caused by the ESR or the ESL. A common assumptionmade in modeling of power inverters is ignoring the switching ripple.

Actual wave form withswitching ripple Average wave form with

switching rippleneglectedv(t)

t

Figure 2.6 Actual and average waveforms of an inverter

Converter naturalfrequency and its

harmonics Switching frequencyand its sidebands

Switchingharmonics

Spec

trum

ofv

(t)

?n ?

Figure 2.7 Harmonic spectrum of PWM inverter

25

Since power inverters are high frequency switching circuits, in awell designed inverter operating in continuous conduction mode (CCM), theswitching ripple is low and the switching frequency is much higher than thatof natural frequencies of the inverter filtering elements. Hence, a possiblesimplification of a power inverter model could be neglecting the switchingripple and averaging the circuit waveforms over the switching period (Chunget al 1998). Figure 2.6 shows both the actual and the average waveforms of atypical inverter output and Figure 2.7 shows the spectrum of this outputwaveform. For a well designed inverter, the high frequency components ofthe spectrum are small in magnitude compared to the low frequencycomponents if the switching ripple is small. Neglecting the switching ripplewill retain the low frequency component of the waveform (Jenni et al 1993,Neacsu 2006).

There are different methods used in modeling of a power converter.

Among those circuit averaging method was reported by Vorperian (1990) andstate space averaging method had been discussed by Middlebrook et al

(1977). Although the form of the end result is different for each method, theylead to the same model when ideal elements are considered by Holmes et al

(1996). In the next subsection of this chapter, a small signal model for theZ-source inverter is derived based on the state space averaging technique and

linearization of the state variables around their steady state values (Loh et al2007).

2.3.2 State Space Averaging

To derive the state model, Z-source network is assumed to be

symmetrical (split inductors and capacitors are employed), so that the currentsthrough two inductors and voltages across the capacitors are always the same

(Loh et al 2007, Liu et al 2007). With this assumption, there are three state

26

variables: the current through the inductor in Z-source network, (iL), thevoltage across the capacitor in the Z-source network, (vc), and the currentthrough the load inductor,(io). The shoot-through duty ratio is assumed as D0and ma is the modulation index. There are three switching states:

State 1

The first state occurs when S1 and S4 are simultaneously turned

on (Figure 1.1), which is the shoot-through state. During this state, theinput diode is reverse biased, the capacitors charge the inductors, and the

load current is freewheeled through the diode. The state equation for this state

is

L L

C C

oo

o

10 0Li i

1v 0 0 v

Cii R0 0

L

? ?? ?? ? ? ?? ?? ? ?? ?? ??? ? ? ?? ?? ?? ? ? ?? ?? ? ?? ?? ?? ?

???

(2.9)

where iL, vc, io and R are the inductor current, capacitor voltage, load currentand load resistance respectively. The duty ratio of this state is D0.

State 2

Second state is enabled when both the switches are turned off,which is the traditional zero state as found in traditional voltage sourceinverters. During this state, the Z-source capacitors are charged by the

Z-source inductors, and the load current is freewheeled through the diode (D).The state equation during this state could be formulated as:

27

dc

L L

C C

oo

o

1 V0 0Li i L1

v 0 0 v 0C

0ii R0 0L

? ?? ? ?? ? ? ?? ? ? ?? ? ? ?? ? ? ?? ?? ?? ?? ? ? ?? ?? ? ? ?? ? ? ?? ?? ? ? ?? ? ? ?? ?? ?

???

(2.10)

The duty ratio of this state is 1-(D0+ma)

State 3

The third state is when S1 is turned off and S4 is turned on, which isthe active state and the inverter supplies voltage to the load. The stateequation during this state is

dc

L L

C C

dcoo

o o

1 V0 0Li i L1 1

v 0 v 0C C

Vii 2 R0 LL L

? ?? ? ?? ? ? ?? ? ? ?? ? ? ?? ? ?? ?? ?? ?? ?? ? ? ?? ?? ? ? ?? ? ? ? ?? ?? ? ? ? ?? ? ? ?? ?? ?

???

(2.11)

The duty ratio of this state is ma

State model

The entire system model of the Z-source inverter could berepresented by using the state apace averaging as

0 0 a aEquation (2.9)* D Equation (2.10)*(1 D m ) Equation (2.11)* m? ? ? ?(2.12)

28

Then the Z-source inverter system model could be obtained as,

0dc

0L L

0 aC C

dcooa a

o o

2D 1 V0 0 (1 D )Li i L1 2D mv 0 v 0

C CVii 2m R m0 LL L

? ?? ? ?? ? ?? ?? ? ? ?? ? ? ?? ? ? ?? ?? ?? ?? ?? ? ? ?? ?? ? ? ?? ? ? ? ?? ?? ? ? ? ?? ? ? ?? ?? ?

???

(2.13)

In this model, iL,vc and io are the state variables; ma and D0 are

control variables and vc and vdc are the variables to be controlled.

2.3.3 Transfer Function

The transfer function of the Z-source inverter system at certain

equilibrium points can be derived by referring the state model of the system.

Small signal perturbation for a given equilibrium point ( L c 0 aI ,V ,D and m ) isperformed (Liu et al 2007); then one could have,

0dc

0L L L

aC C C

dco o Oa a

o o

2(d D ) 10 0 VL (1 (d D ))

i i I L

(m m )1 2(d D)v 0 v V 0

C C Vi i I

2(m m ) R (m m )0 LL L

? ?? ?? ? ? ?? ?? ?? ? ? ?? ? ?? ?? ? ? ?? ? ? ?? ?? ? ?? ?? ? ? ?? ? ? ?? ? ? ?? ? ??? ? ? ?? ?? ? ?? ?? ?? ?? ?? ?

???

(2.14)

L c o

i , v , i ,d,m are the perturbations of the state variables and control variables.

The above model could again be reduced by considering the steadystate relationship of the parameters and ignoring the second order effects.

Then one could have;

29

C 0 dcL c

0 a oLC L o

a c dco c o

o o o o

2V 2D 1 V

i d v dL L L

1 2D m II

v 2d i i mC C C C

2m 2V VR

i v m i mL L L L

??? ? ? ??? ?? ? ? ? ? ???? ? ? ? ??

?

?

?

(2.15)

The transfer function at any given operating point can be found by

solving the above set of equations.

Expression of the capacitor voltage perturbation from the aboveequations could be expressed as follows,

2 20 a 0 c dc L o a c dc

c

o o

(1 2D ) 2m (1 2D )(2V V ) 2SLI I m (2V V )

[S ]v d mCSL C(L S R) CSL C C(L S R)

? ?? ? ? ? ?? ?? ? ? ? ?? ?? ?? ?? ? ? ??(2.16)

Thus for a given equilibrium point,

The transfer function of capacitor voltage to shoot-through duty

ratio is

c o dc o3 2 2 2 2

o a o 0 o 0 x 0 0

v (L S R)(V 2SLI ) CLL S CLRS (2m L L 4D L 4D L )S R 4D R 4D Rd

? ?? ? ? ? ? ? ? ? ??(2.17)

The transfer function of capacitor voltage to modulation index is

c a c a c o o o

3 2 2 2 2o a o 0 o 0 x 0 0

v LS(2m V m V I R I L S)m CLL S CLRS (2m L L 4D L 4D L )S R 4D R 4D R

? ? ? ?? ? ? ? ? ? ? ? ??(2.18)

30

2.4 SUMMARY

This chapter outlined a review of the literature on Z-source inverter

fed induction motor drives. Operating modes of the Z-source inverter fedinduction motor drive systems with its equivalent circuits were briefed fromthe literature. The operation of traditional three phase Z-source inverter fed

induction motor drives was discussed and the significant drawbacks of thetraditional system were highlighted. The dynamic modeling of Z-source

inverter based system was briefed and the transfer functions; Z-sourcecapacitor voltage to shoot-through duty ratio and Z-source capacitor voltage

to modulation index were derived by state space averaging technique.