Impedance Design of Quasi-Z Source Network to Limit Double Fundamental Frequency Voltage and Current

Ripples in Single-Phase Quasi-Z Source Inverter

Dongsen Sun, Baoming Ge, Member, IEEE, Xingyu Yan School of Electrical Engineering

Beijing Jiaotong University Beijing, China

Haitham Abu-Rub, Senior Member, IEEE Dept. of Electrical & Computer Engineering

Texas A&M University at Qatar Doha, Qatar

Daqiang Bi Dept. of Electrical & Computer Engineering

Tsinghua University Beijing, China

Fang Z. Peng, Fellow, IEEE

Dept. of Electrical & Computer Engineering Michigan State University

East Lansing, USA

Abstract—This paper proposes an analytic model to calculate the double fundamental frequency (2) voltage and current ripples for single-phase quasi-Z-source inverter (qZSI). The model enables the exact calculation of the 2 components of the inductor current and the capacitor voltage, which is crucial for designing a single-phase qZSI system. A guideline for selecting the capacitance and inductance in quasi-Z-source (qZS) network to limit both the dc-link voltage and input current 2 ripples within a tolerable range is presented. Three cases are tested to verify the proposed analytic model and the design method by using simulation and experimental results. The identical analytic calculation result, the simulation result, and the experimental result validate the analytic model and the design method.

Keywords— dc-ac inverter; modeling; single phase; voltage and current ripple; Z-source inverter.

I. INTRODUCTION

The traditional voltage source inverter (VSI) is a buck converter and cannot boost the input voltage [1]. So for some renewable energy conversion such as fuel cells and solar panels, the traditional VSI has to be oversized or employs an additional DC/DC converter to cope with the variable input voltage [2], [3]. However, both the aforementioned solutions increase the system cost and degrade the conversion efficiency. The newly-proposed Z-source inverter (ZSI) and quasi-Z source inverter (qZSI) present some new attractive advantages. They can implement the voltage boosting and inversion in a single stage, and enhance the system reliability by introducing an impedance network to avoid short circuit cases [4]-[16]. Furthermore, the qZSI has more unique features, such as drawing a continuous current from the input source, lowering the capacitor voltage rating and et al. Due to the

obvious features of qZSI, it has gotten wider applications, such as motor drives, renewable energy power generation, and fuel cell vehicles.

Due to the ac power ripple in single-phase systems, the double fundamental frequency (2) voltage and current ripples in single-phase inverters have always been a troublesome problem. A huge DC link capacitor is always required to absorb the low frequency ripple in traditional H-bridge inverter. For single-phase ZSI and qZSI, both the capacitor voltage and the inductor current of the impedance network suffer the 2 ripples, which cause the oscillation of both the DC-link voltage and the input current. Some literatures focus on analyzing and eliminating the low frequency ripples [14]-[16]. In [14], the low frequency ripples of the single-phase ZSI are analyzed and the low frequency harmonic elimination PWM technique is presented. In [15], two smoothing-power circuits are employed to reduce the 2 ripple of DC-link voltage in single-phase ZSI. But the additional circuits increase the system cost and complexity. In [16], an ac equivalent model is built to analyze the 2 ripple of the cascaded qZSI PV system, but there is no exact calculation expression and no experimental verification. Up to date, there is no any detailed 2 ripple analytic calculating expression and no any exact guideline to select the capacitance and inductance for single-phase qZSI in literature. This paper will fill up this gap through building the 2 component analytic model, calculating the 2 ripples exactly, and providing the impedance design method for single-phase qZSIs.

The paper aims to present an analytic expression to calculate the 2 ripples of inductor current and the capacitor voltage for single-phase qZSIs. With the built expression, the inductance and capacitance can be selected to satisfy the

This work was supported by NPRP grant NPRP-EP No. X-033-2-007 (section II) and No. 09-233-2-096 (sections III and IV) from the QatarNational Research Fund (a member of Qatar Foundation). The statementsmade herein are solely the responsibility of the authors.

2745978-1-4799-0336-8/13/$31.00 ©2013 IEEE

preset tolerable 2 ripple range for dc-link voltage and the input current. On the basis of the proposed design method, a single-phase qZSI is prototyped to achieve experimental test. The simulation and experimental results verify the proposed analytic model and the design method.

II. MODELING AND ANALYSIS OF SINGLE-PHASE QZSI

A. Operating principle of single-phase qZSI

Fig. 1 is the structure of the single-phase qZSI, in which an LC impedance network is introduced into the traditional H-bridge inverter. There are two operating states for qZSI [10]-[12], i.e., the nonshoot-through state and the shoot-through state, and the equivalent circuits are shown in Fig. 2 (a) and (b), respectively.

During the nonshoot-through state, the power is transmitted from DC side to AC side; while, during the shoot-through state, there is no power transmission because the DC-link voltage is zero. Considering the switch frequency is greater than the AC voltage frequency, the instantaneous power balance equation is obtained as

PN PN (1 ) 0 a av i D D v i (1)

Figure 1. Single-phase qZSI.

(a)

(b)

Figure 2. Equivalent circuits of single-phase qZSI. (a) nonshoot-through state; (b) shoot-through state.

where vPN is the DC-link voltage, iPN is the equivalent current flowed into the H-bridge inverter, D is the shoot-through duty ratio, va and ia represent the output voltage and current of the H-bridge inverter, respectively. Additionally, we define va=Vasint and ia=Iasin(t-), where is the angular frequency of the output voltage, is the impedance angle, and Va and Ia refer to the amplitude of the ac output voltage and current, respectively.

Using the simple boost control method for qZS network and the unipolar double frequency SPWM technique for H-bridge inverter, the output voltage of the qZSI can be expressed as

PNav m v (2)

where m is the modulation index, i.e., m=Msint.

From (1) and (2), iPN can be obtained as

PN PN PN(1 cos 2 )2(1 )

aMIi t I i

D

(3)

As shown in (3), iPN consists of two parts: one is the DC component, i.e.,

PN cos2(1 )

aMII

D

and the other is the 2 component, i.e.,

PN cos(2 )2(1 )

aMIi t

D

.

Both IPN and PNi will have influence on the state variables of the system including iL1, iL2, vC1, and vC2, all of which also consist of two parts: the DC component and the 2 component. If x, X, and x denote the state variables of the system, the DC component, and the 2 component, respectively, the system variables can be written as

L1 L1 L1i I i , L2 L2 L2i I i ,

C1 C1 C1v V v , and C2 C2 C2v V v .

B. Equivalent model considering only the DC component

During the nonshoot-through state, the dynamic equations are

L11 DC C1

L22 C2

C11 L1 PN

C22 L2 PN

L

L

C

C

dIV V

dtdI

Vdt

dVI I

dtdV

I Idt

(4)

2746

During the shoot-through state, the capacitors discharge and the inductors are charged. There are

L11 DC C2

L22 C1

C11 L2

C22 L1

L

L

C

C

dIV V

dtdI

Vdt

dVI

dtdV

Idt

(5)

At steady state, the average value of the state variables over one switch cycle are zero. From (4) and (5), we have

L1 T1 DC C1 DC C2

L2 T2 C2 C1

C1 T1 L1 PN L2

C2 T2 L2 PN L1

L (1 ) ( ) ( ) 0

L (1 ) ( ) ( ) 0

C (1 ) ( ) ( ) 0

C (1 ) ( ) ( ) 0

d ID V V D V V

dtd I

D V D Vdt

d VD I I D I

dtd V

D I I D Idt

(6)

where <X>T represents the average value of X over one switch cycle period T.

Solving (6), the DC components are obtained as

C1 DC

C2 DC

L1

L2

1

1 2

1 2

cos2(1 2 )

cos2(1 2 )

a

a

DV V

DD

V VD

MII

D

MII

D

(7)

C. Equivalent model considering only the 2 component

The 2 component equivalent model is deduced in this section. Assuming the input DC supply power keeps constant and it does not contain 2 component, so during the nonshoot-through state, the dynamic equations are

L11 C1

L22 C2

C11 L1 PN

C22 L2 PN

L

L

C

C

div

dtdi

vdt

dvi i

dtdv

i idt

(8)

During the shoot-through state in operation, there are

L11 C2

L22 C1

C11 L2

C22 L1

L

L

C

C

div

dtdi

vdt

dvi

dtdv

idt

(9)

As x are 2 components, the average value of the variables x over one switch cycle T is not zero. To simplify the calculating equation, we assume L1=L2=L，C1=C2=C, so they are expressed as

L1 TC1 C2

L2 TC2 C1

C1 TL1 PN L2

C2 TL2 PN L1

L (1 ) ( ) ( )

L (1 ) ( ) ( )

C (1 ) ( ) ( )

C (1 ) ( ) ( )

d iD v D v

dtd i

D v D vdt

d vD i i D i

dtd v

D i i D idt

(10)

Using (8), (9), (10), and the 2 component

PN cos(2 ) 2(1 )ai MI t D

we can obtain that L1i and L2i change with a cosine function, while C1v and C2v vary in the sinusoidal way. We define

L1 L1ˆ cos(2 )i i t , L2 L2ˆ cos(2 )i i t ,

C1 C1ˆ sin(2 )v v t and C2 C2ˆ sin(2 )v v t ,

where x denote the amplitude of the 2 components x . Substituting these equations in (10), x can be obtained as

L1 L2 2 2

C1 C2 2 2

(1 2 )ˆ ˆ24LC (1 2 )

2 Lˆ ˆ

24LC (1 2 )

a

a

MIDi i

D

MIv v

D

(11)

Therefore, the 2 components x are expressed as

L1 L2 2 2

C1 C2 2 2

(1 2 )cos(2 )

24LC (1 2 )

2 Lsin(2 )

24LC (1 2 )

a

a

MIDi i t

D

MIv v t

D

(12)

Adding the DC component X and the 2 components x together, the state variables x can be written as

2747

L1 L1 L1 2 2

L2 L2 L2 2 2

C1 C1 C1 DC 2 2

C2 C2 C

1 (1 2 )cos cos(2 )

(1 2 ) 2 24LC (1 2 )

1 (1 2 )cos cos(2 )

(1 2 ) 2 24LC (1 2 )

1 2 Lsin(2 )

1 2 24LC (1 2 )

a a

a a

a

MI MIDi I i t

D D

MI MIDi I i t

D D

MIDv V v V t

D D

v V v

2 DC 2 22 L

sin(2 )1 2 24LC (1 2 )

aMIDV t

D D

(13)

As shown in (13), iL1 equals to iL2 with the same components including the DC component and the 2 component. For vC1 and vC2, the DC components are different, while, the 2 components are the same.

0.7 0.72 0.74 0.76 0.78 0.830

40

50

v C1 (

V)

0.7 0.72 0.74 0.76 0.78 0.80

50

100

v PN (

V)

0.7 0.72 0.74 0.76 0.78 0.80

5

10

i L1 (

A)

0.7 0.72 0.74 0.76 0.78 0.8-10

0

10

t (s)

i a (V

)

(a)

0.7 0.72 0.74 0.76 0.78 0.830

40

50

v C1 (

V)

0.7 0.72 0.74 0.76 0.78 0.80

50

100

v PN (

V)

0.7 0.72 0.74 0.76 0.78 0.80

5

10

i L1 (

A)

0.7 0.72 0.74 0.76 0.78 0.8-10

0

10

t (s)

i a (V

)

(b)

0.7 0.72 0.74 0.76 0.78 0.830

40

50

v C1 (

V)

0.7 0.72 0.74 0.76 0.78 0.80

50

100

v PN (

V)

0.7 0.72 0.74 0.76 0.78 0.80

5

10

i L1 (

A)

0.7 0.72 0.74 0.76 0.78 0.8-10

0

10

t (s)

i a (V

)

(c)

Figure 3. Simulation results. (a) case 1; (b) case 2; (c) case 3.

III. IMPEDANCE DESIGN OF QZS NETWORK

Theoretical analysis and experimental tests indicate that the 2 ripples are the main harmonics in single-phase qZSI system. So we mainly consider the 2 ripples and limit them within a tolerable range, when designing the LC parameters. Using the aforementioned 2 ripple analysis, the DC-link voltage of qZSI is

PN DC 2 21 4 L

sin(2 )1 2 24LC (1 2 )

aMIv V t

D D

(14)

To make sure the qZSI provides high quality output voltage, we set the ripple voltage of DC-link as

PN PNv aV (15)

By substituting (14) in (15), (15) can be rewritten as

DC2 24 L

2 1 24LC (1 2 )

aMI aV

DD

(16)

As shown in Fig. 1, the input current equals to the inductor current iL1. Likewise, to make sure the input power keep constant, we set the ripple current of qZS inductor as

L1 L1i bI (17)

By substituting (13) in (17), (17) can be rewritten as

2 2(1 2 ) cos

(1 2 )4LC (1 2 )

D b

DD

(18)

By combining (16) and (18), the capacitance and inductance can be obtained as

a DC

DC a

(1 2 )(1 cos ) (1 2 ),

2 2 cos

D b MI aV DC L

a V bMI

(19)

IV. SIMULATION AND EXPERIMENTAL VERIFICATION

A small scaled single-phase qZSI system is designed by using the proposed design method. The obtained inductance and capacitance are C1=C2=4400 F, L1=L2=1000 H to make sure that the ripples are no more than 3% (DC-link voltage ripple) and 18% (input current ripple), respectively. Other parameters of the designed system are: VDC=25 V, fs=10 kHz, D=0.25, M=0.7, f=50 Hz (Output frequency), Z=7 +1.2 mH (AC load). To verify the 2 ripple analysis and the design method, simulation and experiment are carried out. Three cases with different capacitance and inductance have been investigated: 1) C1=C2=2200 F, L1=L2=1000 H; 2) C1=C2=4400 F, L1=L2=1000 H; 3) C1=C2=4400 F, L1=L2=2000 H.

2748

TABLE I. COMPARISON OF THEORETICAL, SIMULATION AND EXPERIMENTAL RESULTS.

Parameters Theory Simulation Experiment Error

Symbol L /H C /F L1ti /A C1tv /V L1si /A C1sv /V L1ei /A C1ev /V Ls Cs Le Ce

Case 1 1000 2200 1.42 1.78 1.38 1.62 1.3 1.6 2.8% 9% 8.5% 10%

Case 2 1000 4400 0.59 0.74 0.584 0.73 0.6 0.8 1% 1.4% 1.7% 8.1%

Case 3 2000 4400 0.27 0.68 0.265 0.675 0.28 0.7 1.9% 0.7% 3.6% 2.9%

Note: Subscripts t, s, and e represent the theoretical, simulation, and experimental values, respectively. The errors are calculated by

L1t L1sLs

L1t

i i

i

, C1t C1s

CsC1t

v v

v

, L1t L1e

LeL1t

i i

i

, C1t C1e

CeC1t

v v

v

.

(a)

(b)

(c)

Figure 4. Experimental results. (a) case 1; (b) case 2; (c) case 3.

Simulation results of the three cases are shown in Fig. 3, from which we can obviously see that the 2 ripples exist in both the inductor current and the capacitor voltage. Furthermore, as shown in Fig. 3 (a) and (b), with the increasing of the capacitance, both the current and voltage 2 ripples reduce significantly. While, increasing the inductance, only the current 2 ripple reduces significantly in Fig. 3 (b) and (c).

Three cases have been tested in experiments. Fig. 4 shows the experimental waveforms including the capacitor voltage vC1, the DC-link voltage vPN, the inductor current iL1, and the load current ia, which are identical to the simulated results shown in Fig. 3. Furthermore, as shown in Fig. 4 (b) and (c), the DC-link voltage ripple and the input current ripples are limited within the preset range.

Table I presents the 2 component L1i and C1v of three cases obtained by theoretical calculation, simulation, and experiment, respectively. The comparison shows that the theoretical calculation is identical to the simulation and experiment results with an error no more than 10%.

V. CONCLUSION

This paper proposed an analytic model to analyze the 2 voltage and current ripples in single-phase qZSI. The exact equations were derived to calculate the 2 ripple components of the inductor current and the capacitor voltage. The capacitance and inductance of qZS network were designed with a preset ripple range. Simulations and experiments with different capacitances and inductances were carried out to verify the theoretical analysis and the design method. Simulation and experimental results validated the proposed model and the design method.

REFERENCES [1] B.K. Bose, Modern Power Electronics and AC Drives. Upper Saddle

River, NJ: Prentice-Hall PTR, 2002.

[2] M. Fortunato, A. Giustiniani, and G. Petrone, et al., “Maximum power point tracking in a one-cycle-controlled single-stage photovoltaic inverter,” IEEE Trans. on Industrial Electronics, vol.55, no.7, pp.2684-2693, 2008.

[3] N. Femia, G. Petrone, and G. Spagnuolo, et al., “A technique for improving P&O MPPT performances of double-stage grid-connected photovoltaic systems,” IEEE Trans. on Industrial Electronics, vol.56, no.11, pp.4473-4482, 2009.

2749

[4] Abu-Rub H., Iqbal A., Moin Ahmed Sk., Peng F. Z., Li Y., Ge B., "Quasi-Z-source inverter-based photovoltaic generation system with maximum power tracking control using ANFIS," IEEE Transactions on Sustainable Energy, vol.4, no.1, pp.11-20, Jan. 2013.

[5] D. Sun, B. Ge, F.Z. Peng, H. Abu-Rub, D. Bi, Y. Liu, " A new grid-connected PV system based on cascaded H-bridge quasi-Z source inverter," in IEEE International Symposium on Industrial Electronics, ISIE 2012, 28-31 May 2012, pp.951-956.

[6] Y. Liu, B. Ge, H. Abu-Rub, F.Z. Peng, “A modular multilevel space vector modulation for photovoltaic quasi-Z-source cascade multilevel inverters”, in 2013 Twenty-Eighth Annual IEEE Applied Power Electronics Conference and Exposition (APEC), 17-21 March 2013, pp. 714-718.

[7] Y. Liu, H. Abu-Rub, B. Ge, F. Z. Peng, "Analysis of space vector modulations for three-phase Z-Source / quasi-Z-source inverter," in IECON 2012 - 38th Annual Conference on IEEE Industrial Electronics Society, 25-28 Oct. 2012, pp.5268-5273.

[8] B. Ge, H. Abu-Rub, F. Peng, Q. Lei, de Almeida A., Ferreira F., D. Sun, Y. Liu, "An energy stored quasi-Z-source inverter for application to photovoltaic power system," IEEE Trans. Ind. Electronics, vol.60, no.10, pp.4468-4481, Oct. 2013.

[9] Y. Liu, H. Abu-Rub, B. Ge, F. Z. Peng, de Almeida A.T., Ferreira F.J.T.E., "An improved MPPT method for quasi-Z-source inverter based grid-connected photovoltaic power system," in 2012 IEEE International Symposium on Industrial Electronics (ISIE), 28-31 May 2012, pp.1754-1758.

[10] Y. Liu, B. Ge, H. Abu-Rub, Iqbal A., F. Z. Peng, "Modeling and controller design of quasi-Z-Source inverter with battery based photovoltaic power system," in 2012 IEEE Energy Conversion Congress and Exposition (ECCE), 15-20 Sept. 2012, pp.3119-3124.

[11] D. Sun, B. Ge, H. Abu-Rub, Peng F.Z., De Almeida A.T., "Power flow control for quasi-Z source inverter with battery based PV power generation system," in 2011 IEEE Energy Conversion Congress and Exposition (ECCE), 17-22 Sept. 2011, pp.1051-1056.

[12] J. Anderson and F. Z. Peng, “A class of quasi-Z-source inverters,” in Conf. Rec. IEEE IAS Annu. Meeting, Edmonton, Alta, Canada, Oct.2008, pp.1-7.

[13] B. Ge, Q. Lei, W. Qian, F. Z. Peng, "A family of Z-source matrix converters," IEEE Transactions on Industrial Electronics, vol.59, no.1, pp.35-46, Jan. 2012.

[14] Y. Yu, Q. Zhang, B. Liang, and S. Cui, “Single-phase Z-Source inverter: analysis and low-frequency harmonics elimination pulse width modulation,” In Proceeding of Energy Conversion Congress and Exposition, Phoenix, USA, 2011, pp. 2260-2267.

[15] Z. Gao, Y. Ji, Y. Sun, and J. Wang, “Suppression of voltage fluctuation on DC link voltage of Z-source,” Journal of Harbin University of Science and Technology, vol.16, no.4, pp.86-89, 2011.

[16] Y. Zhou, L. Liu, and H. Li, “A high performance photovoltaic module-integrated converter (MIC) based on cascaded quasi-Z-source inverters (qZSI) using eGaN FETs,” IEEE Trans. on Power Electronics, vol.28, no.6, pp.2727-2738, 2013.

2750