Modeling of Single Phase Four Quadrant Converter Using A Multifrequency Averaging Approach

Branimir Puskaric

KONCAR – Electrical Engineering Institute bpuskaric@koncar-institut.hr

Abstract––A frequency-selective averaging approach for modeling and analysis of the single-phase four-quadrant pulse-width modulated converter is presented. The derived model is linear and time-invariant (LTI), so that classical control design techniques can be applied. The method allows for the model to be defined by algorithm so that it can be easily expanded to arbitrary accuracy. The steady state model allows for direct calculation of steady-state values and is valuable in the design process of the converters power circuit. As an example direct calculation of disturbance progressions from the dc-link into the grid current and vice-versa is presented. Application of the linearized small-signal model for transfer function derivation is presented. The mathematical models have been verified using results obtained from simulation and system identification techniques.

I. INTRODUCTION

State-space models are effectively used to simulate and analyze the behavior of dynamic systems. Quite a few articles have been written on state-space modeling and averaging of pulse-width modulated (PWM) dc-dc converters (buck, boost and buck-boost) [1]–[4]. In these cases, generally it is justified to assume small variations of state variables around a constant steady state value, e.g. small ripple in the input current and output voltage of the converter [1]. This means that averaging the state-space equations of such models over one period of the switching function (classical averaging) results in linear-time invariant (LTI) models which are suitable for analysis of the converter dynamics up to approximately one-tenth of the switching frequency [5].

Averaging of state-space equations as it is described for dc-dc converters is not useful in case of ac-dc or resonant converters because their state variables exhibit mainly oscillatory behavior. In [6] a generalized averaging method is presented which encompasses the state-space averaging approach and is applicable even when small-ripple conditions are not satisfied by applying frequency selective averaging. In [7] the method is applied to a pulse-width modulated dc-dc converter having large ripple in the output voltage.

In this paper we apply the generalized averaging method to a model of the single-phase four-quadrant PWM converter in order to derive a linear time-invariant model applicable for small-signal analysis. The resulting transfer functions are compared against experimentally derived results, and show good agreement. The resulting LTI model is applicable for direct calculation of steady-state solution which proves to be

useful for simulations of the switched model in order to speed up extensive simulation tasks, i.e. thermal calculations, or optimization.

As an example, application of the averaging method for the analysis of higher harmonic components reflected from the dc-link into the input current and vice-versa is presented.

II. FREQUENCY-SELECTIVE AVERAGING

The considered averaging method [6] is based on the fact that a time function x(t) can be approximated on the interval (t-T, t] to arbitrary accuracy with a Fourier series representation of the form:

( ) ( )( ) njkω t T τk

kx t T τ x t e − +− + =∑ , (1)

where the sum is over all integers k, ](2 / , 0,nω π T τ T= ∈ ,

and the ( )kx t are complex Fourier coefficients. The k-th

coefficient, also to be referred to as the index-k coefficient, is determined by:

( ) ( )

0

1 ( ) n

Tjkω t T τ

kx t x t T τ e

T− − += − +∫ (2)

The derivative with respect to time of the k-th coefficient can be calculated by the following formula:

( ) ( ) ( )kn k

k

d x t dx t jkω x tdt dt

= − (3)

The idea is to apply the generalized averaging method on the equations describing the switched model of a single phase four quadrant converter which represents a periodically varying system and to derive a linear time-invariant system of equations, which is applicable for small signal analysis.

III. SWITCHED STATE SPACE MODEL

The schematic of the single phase four quadrant converter is given in Fig. 1. The voltage source (grid voltage) vs is connected in series with the input impedance Rs + jωsL which represents a simplified model of the input transformer. The load in the dc-link is modeled with resistance Rt. The IGBT semiconductor switches are modeled by means of ideal switches controlled by control function: q(t) = 1 when S1 and S4 are on (closed), q(t) = –1 when S2 and S3 are on (closed), and q(t) = 0 when either S1 and S2, or S3 and S4 are on.

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The control signal q(t) is generated by comparison of a reference sinusoidal signal and a triangular carrier signal. The capacitance in the dc-link will be denoted as C. In order to derive the nonlinear state-space model of the switched circuit we need to describe the three possible switching instances using one system of state-space equations. A natural state-space model for this converter is given with:

( ) ( ) ( ) ( )( )ss C s s

di tL v t q t v t R i t

dt= − − (4)

( ) ( ) ( ) 1 ( )Cs C

t

dv tC q t i t v t

dt R= − . (5)

IV. MULTI-FREQUENCY AVERAGED MODEL

In stationary operation the inputs and state variables of the proposed circuit are periodically varying. The voltage vC in the dc-link has a dc value and higher order terms at frequencies which are proportional to twice the fundamental grid frequency. The input current is has a fundamental component at the grid frequency, and higher order terms at uneven multiples of the grid frequency. To derive the averaged model, we first average the control function q(t) over one period of the fundamental grid frequency

2 / sT π ω= :

[ ]( ) cos( ), 0,1sq t k ω t δ k= + ∈ . (6)

Using (6) we eliminate the switching artifacts and derive a system of equations which describes the low-frequency behavior of the system:

( ) ( ) ( ) ( )

( ) ( ) ( )

1 1 ( )

1 1 ( )

s ss C s

Cs C

t

di t Rv t q t v t i t

dt L L L

dv tq t i t v t

dt C R C

= − −

= −

(7)

Each term in (7) can be expanded according to its Fourier series expansion according to (1) and each harmonic component is defined by a decoupled set of equations. Fig. 2 shows the input current waveforms of the switched and the averaged periodically varying model under voltage sag conditions.

By application of (2) and (3), we derive the general form of the index-k average system of equations:

s ks C s s s sk k k k

d iL v qv R i jkω L i

dt= − − − (8)

1C ks C s Ck k k

t

d vC qi v jkω C v

dt R= − − (9)

In this section we will expand the system up to the index-2 averaged components. For calculation purposes we need to convert (6) into its exponential form:

( ) 1 1s sjω t jω tq t q e q e−

−= + (10)

We have to apply the same procedure for grid voltage vs, input current is and dc-link voltage vC:

( ) 1 1s sjω t jω t

s s sv t v e v e−−

= + (11) ( ) 1 1

s sjω t jω ts s si t i e i e−

−= + (12)

( ) 2 20 2 2

s sj ω t j ω tC C C Cv t v v e v e−

−= + + (13)

The presented terms in (10) – (13) are generally complex valued, and for the averaged control function q(t) we have:

1 1 1

R Iq q j q− − −

= + (14)

1 1 1

R Iq q j q= + (15)

The considered real and imaginary parts are real functions, which leads to: *

1 1q q

−= . (16)

With (*) we denote the complex conjugate. The averaged products c k

qv and s kqi need to be calculated according

to the following convolution product scheme [6]:

,k k i k

ixy x y i

−= ∈∑ (17)

Fig. 1. Schematic of the analyzed single phase four quadrant converter

0.77 0.78 0.79 0.8 0.81 0.82-2000

-1500

-1000

-500

0

500

1000

1500

2000

iS

AVERAGED

vS

iS

SWITCHED

Fig. 2. Input current of the switched and averaged model under voltage sag conditions

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The products we require for our model are:

( )( )( )( )

( )

0 1 1 1 1* *

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

+

2

s s s

s s

R I R Is s

R I R Is s

R R I Is s

qi q i q i

q i q i

q j q i j i

q j q i j i

q i q i

− −= + =

= + =

= + − +

− + =

= +

(18)

1 1 1

1 1 0 1 2 1 2

1 1 0 1 2 1 2

R IC C C

R R R R I IC C C C

I I R I I RC C C C

qv qv j qv

qv q v q v q v

qv q v q v q v

= +

= + +

= + −

(19)

2 2 2

2 1 1 1 1

2 1 1 1 1

R Is s s

R R R I Is s s

I R I I Rs s s

qi qi j qi

qi q i q i

qi q i q i

= +

= −

= +

(20)

A. Calculation of Index-0 Average

The general system of equations for the index-0 average is derived from (8), and (9):

00 0 0 0

ss C s s s s

d iL v qv R i jkω L i

dt= − − − (21)

00 0 0

1Cs C s C

t

d vC qi v jkω C v

dt R= − − . (22)

After consideration of (10) – (13) and (18), we find the resulting system of equations for the index-0 average:

0 0sd iL

dt= (23)

( )01 1 1 1 0

12 R R I ICs s C

t

d vC q i q i v

dt R= + − (24)

The only dc-terms are represented in the dc-link voltage, therefore equation (23) is equal to zero, and the dynamics of the dc-link voltage are defined solely by (24).

B. Calculation of Index-1 Average

The general system of equations for the index-1 average we derive again from (8) and (9). After consideration of the approximations in (10) – (13) we derive the system of equations describing the dynamics of the components at the fundamental grid frequency:

11 1 1 1

ss C s s s s

d iL v qv R i jkω L i

dt= − − − (25)

1 0Cd vC

dt= (26)

The dc-link voltage vC has no index-1 terms, therefore equation (26) is equal to 0. Considering (25) and (19), we obtain two equations since the index-1 averages of is have both real and imaginary parts:

11 1 0

RR Rs

s C

d iL v q v

dt= − −

1 2 1 2 1 1

R R I I R IC C s s s sq v q v R i ω L i− − − + (27)

11 1 0 1 2

II I R Is

s C C

d iL v q v q v

dt= − − +

1 2 1 1

I R I RC s s s sq v R i ω L i+ − − (28)

C. Calculation of Index-2 Average

The calculation of index-2 average components is performed in the same way as for index-0 and index-1 terms. The only index-2 components are represented in the dc-link voltage (13). The equations describing the dynamics of index-2 averages are:

2

1 1 1 1

2 2

1 2

RR R I IC

s s

R IC s C

t

d vC q i q i

dt

v ω C vR

= − −

− + (29)

2

1 1 1 1

2 2

1 2

IR I I RC

s s

I RC s C

t

d vC q i q i

dt

v ω C vR

= + −

− − (30)

D. Resulting System of Differential Equations

Using (24), (27), (28), (29) and (30) we derive the complete system of equations describing the dynamics up to index-2 terms. The resulting system is given in matrix form:

ddt

= +x Ax Bu , (31)

Where the notations have meanings as follows:

0

1

1

2

2

C

Rs

Is

RC

IC

v

i

i

v

v

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

x , (32)

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1 1

1 1 1

1 1 1

1 1

1 1

2 21 0 0

10 2

10 2

R I

tR R I

ss

I I R

ss

R I

st

I R

st

q qR C C C

q q qRω

L L L Lq q qR

ωL L L L

q qω

C C R C

q qω

C C R C

⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥− − − −⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥− − − −⎢ ⎥⎢ ⎥⎢ ⎥− −⎢ ⎥⎢ ⎥⎢ ⎥

− −⎢ ⎥⎢ ⎥⎣ ⎦

A , (33)

1

1

0

00

Rs

Is

vL

vL

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Bu . (34)

V. STEADY STATE SOLUTION

The derived system (31) – (34) is linear and time invariant and therefore it is possible to apply methods used for solving systems of algebraic equations with constant coefficients for direct calculation of steady state solution. Namely, by setting the left side of (31) to zero, we have the steady state solution as: 1−= −x A Bu (35) By numerical calculation of (35) it is easy to get steady state solutions for various sets of parameters of the converter. Such calculations are particularly useful when performing circuit optimization, e.g. calculating component stresses for a broad range of component values and operating points.

The results obtained using (35) can also be used as initial conditions when simulating using the switched model to avoid transients for various operating points.

A. Analysis of Disturbance Progression from the DC-link into the Input Current

The frequency-selective averaging approach has been used to analyze the disturbance injection from the dc-side of the converter into the grid current. The schematic of the circuit being analyzed is given in Fig.3. The set of equations describing the index-k average of the disturbances is:

s ks C s s s sk k k k

d iL v qv R i jkω L i

dt= − − − (36)

1C kd s C s Ck k k k

t

d vC i qi v jkω C v

dt R= + − − (37)

Following the procedure presented for deriving (31) – (34),

we derive a linear time-invariant set of equations which exactly describes the harmonic components of the disturbances being generated. Moreover, the resulting system of equations is decoupled from (31) – (34) and allows for separate analysis of disturbance progressions.

Again, the solution can be directly calculated and allows for much faster analysis than performing FFT-s on waveforms gained from simulation. Taking in account the preparation process, this method seems useful when analyzing the system for a broad set of parameters.

VI. SMALL SIGNAL ANALYSIS

In this section we will derive the control-to-output voltage and control-to-input current transfer functions for the proposed topology (Fig.1.). The averaged system (31) describes the low frequency behavior of the considered converter, while neglecting the switching artifacts. The matrix A given in (33) can be expanded in the following way:

0 1 1

R IqR qIq q= + +A A A A (38)

For small-signal analysis the system (31) has been expanded up to index-4 averaged components. The resulting matrices A0, AqR and AqI are given in (39) – (41).

0

1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

10 0 0 2 0 0 0 0

10 0 0 2 0 0 0 0

0 0 0 0 0 3 0 0

0 0 0 0 0 3 0 0

10 0 0 0 0 0 0 4

10 0 0 0 0 0 0 4

t

ss

ss

st

st

ss

ss

st

st

R CR

ωL

Rω

L

ωR C

ωR C

Rω

LR

ωL

ωR C

ωR C

⎡ ⎤−⎢ ⎥⎢ ⎥⎢ ⎥

−⎢ ⎥⎢ ⎥⎢ ⎥

− −⎢ ⎥⎢ ⎥⎢ ⎥

−⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥− −=⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥− −⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥− −⎢ ⎥⎣ ⎦

A (39)

Fig. 3. Modeling of disturbances injected from the dc-link.

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20 0 0 0 0 0 0 0

1 10 0 0 0 0 0 0

10 0 0 0 0 0 0 0

1 10 0 0 0 0 0 0

1 10 0 0 0 0 0 0

1 10 0 0 0 0 0 0

1 10 0 0 0 0 0 0

10 0 0 0 0 0 0 0

10 0 0 0 0 0 0 0

qR

C

L L

L

C C

C C

L L

L L

C

C

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥− −⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥− −⎢ ⎥⎢ ⎥⎢ ⎥− −⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

A (40)

20 0 0 0 0 0 0 0

10 0 0 0 0 0 0 0

1 10 0 0 0 0 0 0

1 10 0 0 0 0 0 0

1 10 0 0 0 0 0 0

1 10 0 0 0 0 0 0

1 10 0 0 0 0 0 0

10 0 0 0 0 0 0 0

10 0 0 0 0 0 0 0

qI

C

L

L L

C C

C C

L L

L L

C

C

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥= −⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

A (41)

The averaged control function ( )q t is given by:

( )

( )1 1

1 1

( ) cos

=

s s

s s

s

jω t jω t

jω t jω tk k

q t k ω t δ

q e q e

k q e q e

−−

−−

= +

= +

+

(42)

In order to derive the model for small signal analysis the system (31) has to be expanded in the following way:

( )0 1 1

R IqR qIk k

d k q qdt

⎡ ⎤= + + +⎣ ⎦

x A A A x B u (43)

The steady state solution x is calculated by means of (35),

using the conditions given with the steady state values of ( )q t and the sources vector u .

For small variations around the steady state operating point we get:

( ) ( )

( ) ( )( )0

1 1 R I

qR qIk k

d ddt dt

k k q q

+ = + + +

+ + + +

x x A x x B u u

A A x x (44)

After elimination of steady state terms according to (43), and application of Laplace transform we obtain the small-signal model for control-to-input current, and control-to-output voltage transfer function calculation.

( ) ( ) ( ) ( )

( ) ( )( )0

1 1

0

+ R IqR qRk k

s s s s

q q k s k

− = + +

+ +

x x A x Bu

A A x x (45)

We obtain the resulting open-loop transfer function from (45) after division by ( )k s , and assuming ( )0 0=x [3],[4]:

( ) ( )( )

( ) ( )

1

0 1 1

1 1

R IqR qIk k

R IqR qIk k

ss

k s

s k q q

q q

−

= =

⎡ ⎤= − + + ⋅⎣ ⎦

⋅ +

xG

I A A A

A A x

(46)

Equation (46) has the form ( ) ( ) 1s s −= − +H C I A B D , which is a standard form [3, 4] to calculate transfer functions for state-space models defined as:

= +x Ax Bu (47)

= +y Cx Du (48)

The matrix G(s) contains transfer functions for the real and imaginary parts of the respective harmonic input current and output voltage components (in (46) matrix C is an identity matrix, and matrix D is zero).

In order to derive control-to-input current and control-to-output voltage transfer functions, matrix C has to be defined in such a way that it calculates the Fourier series for the desired component (input current and output voltage respectively) from its harmonic components in accordance with the following scheme:

( ) ( ) ( )01

2 cos sinR Is sξ ξ

ξx τ x x ξω τ x ξω τ

∞

=

⎡ ⎤= + +⎣ ⎦∑ (49)

( ) ( )2 2R I

ξ ξ ξx x x= + (50)

( )cosR

ξs

ξ

xξω τ

x= (51)

( )sinR

ξ ξs I

ξ

x xξω τ

x

−= (52)

The components R

ξx and I

ξx are the results of the steady-

state analysis (35), so the presented scheme is easily implemented by algorithm.

After the necessary matrices have been defined, the desired transfer functions can be easily calculated in a designated software package (e.g. MATLAB, Mathematica).

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A. Experimental Transfer Function Verification

The analytical calculations were compared against experimental transfer function identification on simulation data from a general circuit simulation software (Ansoft SIMPLORER) using a pseudo-random binary sequence (PRBS) signal added to control signal q(t) as the small-signal disturbance [8]. The PRBS signal was generated using a 16-bit shift register, and the results were sampled with 100 kHz.

As an example, a converter with the following component values has been considered:

vs(t) = ( )800 2 cos 2π50t V Ls = 4 mH C = 20 mF Rs = 50 mΩ q(t) = ( )0.8cos 2π50 - 22.07π /180t Rt = 11.25 Ω.

In Fig. 4 and Fig. 5 the resulting transfer functions are shown.

The figures show good agreement in the magnitude

response diagram. Phase response estimation using the experimental method presented in [8] was not possible.

VII. CONCLUSION

In this paper the derivation of a multi-frequency averaged state-space model of the single-phase four quadrant converter is presented. The derived model is linear and time-invariant, which makes it applicable for numerical steady-state calculation and small-signal analysis. Analysis of disturbance progression from dc-link to ac-side and vice-versa, which has been presented as one of the benefits of using this method, gives physical insight and speeds up the design process.

Calculation of control-to-output voltage and control-to-input current transfer function was presented and compared to experimental results from system identification using a PRBS signal and has shown good agreement. The methodology has been presented in a way which makes it suitable for calculations using general simulation and analysis software packages.

REFERENCES [1] G. W. Wester, R. D. Middlebrook: Low-Frequency Characterization of

Switched dc-dc Converters, IEEE Trans. On Aerospace and electronic systems vol. AES-9, May 1973, NO.3.

[2] R. D. Middlebrook, S. Cuk: A General Unified Approach to Modeling Switching Converter Power Stages, PESC’76 Record, pp. 19-34., 1976

[3] J. G. Kassakian, M. F. Schlecht, G. C. Verghese: Principles of Power Electronics, Reading, MA: Addison-Wesley, 1991.

[4] R. W. Erickson, D. Maksimović: Fundamentals of Power Electronics, 2nd ed. Dordrecht, The Netherlands.

[5] D. Maksimović, A. M. Stanković, V. J. Thottuvelil, G. C. Verghese: Modeling and simulation of power electronic converters, Proceedings of the IEEE, vol. 89, Issue 6, Pages 898 – 912, Jun 2001

[6] S. R. Sanders, J. M. Noworolski, X. Z. Liu, G. C. Verghese: Generalized averaging method for power conversion circuits, IEEE Trans. on Power Electron., vol. 6, pp. 251-259, Apr. 1991

[7] V. A. Caliskan, G. C. Verghese, A. M. Stanković: Multifrequency averaging of dc/dc converters, IEEE Trans. Power Electron., vol. 14, pp. 124-133, Jan. 1999

[8] B. Miao, R. Zane, D. Maksimović: System identification of power converters with digital control through cross-correlation methods, IEEE Trans. Power Electron, vol. 20, NO. 5, Sept. 2005.

Fig. 4 Control-to-input current transfer function

Fig. 5 Control-to-dc link voltage transfer function

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