An Alternative Approach for Modelling Power Electronics Converters

Pilar Suárez Cruz

psuarez@faraday.fie.umich.mx Jesús Rico Melgoza jerico@umich.mx

Electrical Engineering Faculty UMSNH

Morelia, Mexico

Abstract—In this paper, a systematic approach for modelling power electronics converters is shown, it is referred as the Complementarity Framework. The Complementarity Framework allows to calculate the transient and steady state responses. The single- phase diode bridge rectifier and the half-wave diode rectifier are modeled as Linear Complementarity Systems (LCS).The complementarity models can be operated in the Discontinuous Operation Mode (DCM) and the Continuous Operation Mode (CCM).

Key Words.-Power Electronics, LCS, LCP, single- phase diode bridge rectifier, the half-wave diode rectifier, DCM, CCM.

I. INTRODUCTION Most power electronics converters can be viewed as

electrical networks with linear elements(resistors, inductors, capacitors, transformers), voltage and current sources, and devices such as diodes and electronic switches(thyristors, transistors, MOSFETs, etc.). A classical approach for modeling power converters consists in assuming diodes and electronic switches to be ideal (a very detailed model can be justified when one is interested on studying any phenomena associated with the switching process), discriminating among the different modes of the converter, building for each mode a linear time-invariant dynamic model, and determining the conditions for the commutations among the different modes[1]. The resulting model is usually called switched model and it has 2n

modes with n being the number of diodes and switches. Switching models become complex when the commutation conditions depend on the state variables and the number of diodes or switches is incremented.

In order to solve many of the challenges mentioned, there have been strong efforts to provide complete models. This is, a unique model that describes all possible operating conditions in a power converter. Inside of these efforts, there are methodologies that guide the different topologies of the converters considering logical expressions [2][3]. Recently, the Complementarity Framework has been presented in

several papers providing its modeling and solution capabilities [4]. The complementarity framework is based in the theory of linear complementarity problems (LCP) of mathematical programming. It is being used in several applications [5].

To show some of the main features of the complementarity framework, the models of half-wave diode rectifier as well as the single-phase diode rectifier are developed in this paper. The resulting models can be used in transient and steady simulations regardless the conduction modes they may go through. A very interesting feature of the complementarity framework is that it provides non-iterative algorithms for steady state computation.

II. BACKGROUND FOR LINEAR COMPLEMENTARITY SYSTEMS(LCS)

Let us first introduce a continuous complementarity system and a linear complementarity problem. Definition 1: A continuous–time linear complementarity system (LCS) is the following linear system subject to complementarity constraints on z and w variables:

.

0 ( 0)

= + +

+ +

≤

=

≥⊥

C C C

C C C

x A x B z E uw C x D z F u

w z

(1 )(1 )(1 )

abc

where ∈ xNx R is the state vector, ∈ uNu R is an

exogenous input vector, ∈ zNz R and ∈ zNw R are the complementarity variables, and CA , CB , CC , CE , CF are real matrices of suitable dimensions.

The symbol, ⊥ is the orthogonality operator, it means that two real vectors z and w then 0=Tz w (the scalar product is zero). The relation (1b) implies that for each pair of scalar complementarity variables at least one of them must be zero. Now a linear complementarity problem(LCP) can be defined as follows.

Definition 2: Given a real vector q and a real matrixM , a linear complementarity problem (LCP) consists of finding a real vector z such that

( )

000

≥

+ ≥

+ =T

zq Mz

z q Mz

(2 )(2 )(2 )

abc

In the sequel conditions (2) that define the LCP ( q ,M ) will be more compactly indicated by means of the complementarity condition

00 ( ) 0= + ≥

≤ ⊥ ≥

w q Mzw z

(3 )(3 )bc

The LCP ( q ,M ) has a unique solution for any q if and only if M is a −P matrix [12]. A matrix M is called a −P matrix if all its principal minors are strictly positive.

According to the definition, every positive definite matrix is a −P matrix but the converse is not true. Therefore beingM a positive definite matrix implies uniqueness of the LCP ( q ,M ) solution.

The polarity considered to model power converters as Linear Complementarity Systems (LCS) is shown in Fig 1.

DEvDE iDE vD iD

Figure 1. Polarity for the ideal diode.

Fig 2 depicts the voltage-current characteristic of an ideal diode that can be represented as an orthogonal restriction such that the following condition

0>di and     0=dv (4) must be satisfied.

vd

id

vd

id

Figure 2. Ideal diode simbol with the corresponding current-voltage.

This implies that when the ideal diode is closed the voltage across it is zero and the current through it must be greater than zero. The following condition must be also satisfied

0>dv and   0=di . (5) Analytically, both conditions may be represented as

0 ( ) 0≤ ⊥ ≥d dv i                                                             (6)

III. OPERATION AND CIRCUIT DESCRIPTION OF HALF-WAVE DIODE RECTIFIER AS LINEAR COMPLEMENTARITY

SYSTEMS(LCS) The basic structure of a half-wave diode rectifier in Fig. 3

is shown. In this structure, the half-wave diode rectifier converts the ac supply voltage into unregulated dc voltage. An ideal diode is used to chop the positive part of the ac supply voltage.

vd

idvs

R1

L

iLd

Figure 3. The half-wave diode rectifier.

The proper tree and cotree of the half-wave diode rectifier circuit in Fig. 4 is presented. We consider a tree that contains all voltage sources, the capacitors in the network and the minimum number of inductors current sources. The switches are included either in the proper tree or in the proper cotree.

vd

idvs

R1

L

iLd

Figure 4. The half-wave diode rectifier proper tree and cotree.

With this information is possible to build the complementarity model rectifier. Where the characteristics of the diodes are modeling separately from the rectifier circuit to latter integrate them to the dynamical equations in form of complementarity constraints

Following circuit theory and the concept of proper tree [17], a loop is formed with the nodes (1,2,3,4) . The loop represents the Kirchhoff voltage law (KVL). The dynamical equation of the half-wave diode rectifier is represented by

d sLLV vdi R i

dt L L L= − + + (7.1)

For the proper cotree the Kirchhoff current law (KCL) is determined by cut set that cut the inductive element and the tree. The algebraic equation of the half-wave diode rectifier is shown:

=d Li i (7.2) and the complementarity constraints

0 ( ) ( ) 0≤ ⊥ ≥T Td di V (7.3)

Taking  equations (7.1) , (7.2) and (7.3) ,  in  reference  to  equation, (1 )a , (1 )b and (1 )c ,  yields  

( )

1 1

1 0 00 ( ) ( ) 0

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

= = =

≤ ⊥ ≥

C CC

C C C

d d

RA B EL L

i VFL

C D

The current through the inductor ai and the voltage across the diode dv , in Fig. 5 is shown. The simulation results for the half-wave rectifier, using the linear complementarity model show a good agreement with the simulation results using PSpice and Matlab.

(a)

(b)

Figure 5. a) Current in the inductor b)Negative voltage supported by the diode in the half-wave diode rectifier.

IV. OPERATION AND CIRCUIT DESCRIPTION OF SINGLE-PHASE DIODE RECTIFIER AS LINEAR COMPLEMENTARITY

SYSTEMS(LCS) The single-phase diode rectifier is a line frequency converter, which converts the ac supply voltage into unregulated dc voltage. The dc output voltage of a rectifier should be as ripple free as possible. Therefore, a large capacitor is connected as a filter on the dc side. This filter gets charged to a value close to peak of the ac input voltage. As a consequence, the current through the rectifier is very large near the peak of the input voltage, it does not flow continuously. So, these rectifiers draw highly distorted current from the utility. The basic structure of a single-phase diode bridge rectifier is presented in Fig. 6. The load is

represented by an equivalent resistance 2R . The ac source has a resistance of low value and an inductance, represented for 1R and L , respectively.

vs C R2v C

id

R1 L iL

d1d2

d4d3

Figure 6. The single-phase diode bridge rectifier.

The proper tree and cotree of the single-phase diode bridge rectifier circuit is presented in the Fig. 7. The tree is composed for a voltage source, two diodes, a capacitor, and two resistances. Four loops are obtained from the tree and five cut sets are formed from the proper cotree.

C R2vs

R1 L L

d1d2

d4d3 vd3id3vd4id4

vd2id2vd1id1

vC

id

Figure 7. The single-phase diode bridge rectifier proper tree and cotree.

LVK are applied to the loops and LCK are applied to the cut sets. Following the procedure described in the previous section it is possible to write a mathematical representation of the single-phase diode bridge rectifier,

1 4

32

1

2

= − − + +

= − + +

d d sLL

ddC C

V V vdi R idt L L L L

iidV Vdt R C C C

(8.1)

(8.2)

1 2

3 4

= − +

= +

d L d

d L d

i i i

i i i

(8.3)(8.4)

2 1

3 4

= −

= −

d C d

d C d

V V V

V V V

(8.6)(8.7)

and the complementarity constraints

1 4 2 30 ( )≤ T

d d d di i V V1 4 2 3

( ) 0⊥ ≥Td d d dV V i i (8.8) Substituting the equations (8.1) until (8.7) and (8.8) in (1 )a , (1 )b and (1 )c respectively, yields

1

2

1 10 0 0

1 1 10 0 0

1 01 1 0

0 100 1

0 0 1 00 0 0 1

01 0 0 00 1 0 0

⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟⎜ ⎟= = ⎜ ⎟⎜ ⎟ ⎜ ⎟− ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

−⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎜ ⎟= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎜ ⎟

⎝ ⎠

⎛ ⎞⎜ ⎟⎜ ⎟= =⎜ ⎟−⎜ ⎟

−⎝ ⎠

C

C

C C

C

C

RL L LA B

R C C C

E CL

D F

and the complementarity constraints

1 4 2 3

0 ( )≤ Td d d di i V V

1 4 2 3( ) 0⊥ ≥Td d d dV V i i

The steady state response in the DCM and CCM is shown

in Fig 8. The waveform of the current ( )Li t in the CCM presents a low level distortion, however the high level distortion in the current ( )Li t operating in the DCM is caused by the high capacitor value.

(a)

(b)

Figure 8. a) Waveform in the CCM b) Waveform in the DCM of the single-phase diode bridge rectifier.

When the single-phase diode bridge is operated in the DCM, there is a non-conduction period, which means that the dc side is disconnected from ac source and the current di is zero, the reason is the reverse bias in all diodes associated the large capacitor value. The results of Fig.8 show how the linear complementarity model is able to reproduce all modes operation.

V. MODEL DISCRETIZACION AND THE LINEAR COMPLEMENTARITY PROBLEM FOR TRANSIENT RESPONSE

The continuous time complementarity systems expressed by the equations (8) and equations (6) can be discretized by classical techniques for the integration of linear differential equations such as backward Euler methods or integration methods used in digital control. If the backward difference approximation method and a zero order hold with a sampling period h are applied, then it is possible to obtained the following discrete system,

1 1 1 1

1 1 1 1

( )+ + + +

+ + + +

= + + +

= + +k k k k k

k k k k

x x h Ax Bz Euw Cx Dz Fu

(9)(10)

and solving for 1+kx and substituting the variable 1+kw 1

1 1

11

11

( ( ) )

( )

( )

−+ +

−+

−+

= − +

+ −

+ + −

k k

k

k k

w C I hA hB D z

C I hA hEu

Fu C I hA x (11)

The ec.(11) has the same structure that a LCP and it is shown in the next expression

1 1

1

( ( , , , , ))( ( , , , , , , ))

+ +

+

=

+k k

k k k

w M f A B C D h zq f A C F E h u x

(12)

where

1

1

1

0 ( ( , , , , ))( ( , , , , , , ))

0

+

+

+

≤

+

⊥ ≥

k

k k k

k

M f A B C D h zq f A C F E h u xz (13)

must be satisfied for each instant of time.

The linear complementarity problem can be solved using Lemkes algorithm [13], PATH [14], and Quadratic Programing [15]. When the linear complementarity problem has large dimensions PATH is faster than the Lemkes algorithm and it would the most attractive choice.

Linear Complementarity Problems can be written as a constrained quadratic programing problem if matrix M is positive semidefinite:

( ). . 0

0

+

+ =

≥

TMinimize z q Mzs t q Mz

z

(14)

If the optimum objective value, in this quadratic program, is 0> , the linear complementarity problem has no solution.

However, if the optimum objective value in the quadratic program is zero, and z is any optimum solution for it, then z is also a unique solution for the linear complementarity problem. An advantage of postulating the linear complementarity problem as a quadratic program is that for latter, there are available algorithms in commercial packages such as Matlab and Mathematica. Due to the low order of the complementarity model discussed in this paper, a standard Matlab code has been used [15]. It implements Lemkes algorithm.

VI. PERIODIC STEADY STATE TROUGH COMPLEMENTARITY

Complementarity models of power converters can be used to reformulate the computation of the periodic steady state in a very interesting way. To show this, consider a power electronic system described by equation (7) powered by an ac periodic source { }ku of period N . Assume also that the system, exhibits a periodic response, i.e. + = ∀k N kx x k . The state evolution gives

(15) solving with respect to 0x :

01

( )−

=

= Π +∑N

N ii i

i

x A Bz Eu

(16)

where 1( )−= −Π NI A is well defined because we assume that A has nonzero eigenvalues. The matrix Π satisfies the

following properties:

Π = Π

Π = Π −N

A AA I

(17)

By using properties in (17), and (16) into (9) yields the following LCP ( Nq , NM ):

(18)

where

 

and   =N Nq P uwith  

The matrices NM and NP are block circulant matrices.

The LCP (18) cannot be decoupled into N different LCPs because for each k all components of the sequence iz for 1, ,i N= L appear into kw .

Important information may be extracted from (18), if NM is a −P matrix then the solution is unique [12]. If

matrix NM is not a −P matrix the LCP could have no solution or multiple solutions. With z computed from (18), a complete periodic of the periodic response (discretized) can be obtained from (9). As an early conclusion, the steady state computation under complementarity is very attractive, since Newton type iterations are not required, avoiding the possibility of non-convergence. Note that in the calculus of the steady-state the estimation of the periodic response is not required an starting point.

VII. SIMULATION RESULTS In order to demonstrate the validity of the proposed

method, the single- phase diode bridge rectifier shown in Fig. 8 with several sets of parameters has been simulated to show that the model reproduces the rectifier response in steady state for both conditions: CCM and DCM. In each of our experiments the solutions were compared with those obtained by using Simulink and PSpice.

A. Transient Analysis To show the transient behavior of the rectifier in different

conduction modes, two set of parameters were considered:

1 1R omhs= 1=L mH , 640 10C F−= ∗ , 2 1R ohms= and  

1 1R ohms= 1=L mH , 31.4 10C F−= ∗ ,   2 100R omhs=

in both cases the source voltage was:

120 (377 )=vs sin t Volts Fig. 9 shows the current ( )Li t obtained by the

complementarity framework, Simulink and PSpice. Only the current ( )Li t has been considered in Fig. 9 to keep the figure

simple. It can be seen that good agreement exists among the three approaches specially in the steady state.

Figure 9. Current waveforms in the CCM produced by Simulink, by PSpice and the complementarity approach for the single-phase diode bridge

rectifier.

Excellent agreement exists between simulations of the complementarity models and those obtained with Simulink and PSpice in the DCM, how is showed in Fig 10.

Figure 10. Current waveforms in the DCM produced by Simulink, by PSpice and the complementarity approach for the single-phase diode bridge

rectifier.

Similarly, Fig. 10 shows the current ( )Li t in the rectifier when it is operating in DCM. Nevertheless the reader may appreciate some differences near switching times. In this case complementarity models are more accurate.

B. Harmonic Analysis As confirmed in simulations of the previous section, the ac

side current waveforms are far from being sinusoidal which calls for a harmonic analysis of these currents. Use of the theory detailed in section yields the waveforms in the rectifier in the steady state and its harmonic analysis results from the application of the Fast Fourier Transform (FFT) over the samples of those periodic functions. The periodic steady state waveforms have been already shown in Fig. 8. The corresponding harmonic content as a percentage of the

fundamental component are shown in Fig. 11.

Figure 11. Current waveforms in the DCM produced by Simulink, by PSpice and the complementarity approach for the single-phase diode bridge rectifier.

In general, the discontinuous mode is characterized by higher values of harmonics than those in the case of the CCM, as it is appreciated in Fig 11. Therefore, in the objectives of process design, this fact must be taken into consideration.

CONCLUSIONS The   complementarity   formalism   provides   the  

possibility   of   complete   switched   models   for   power  converters.   Their   time   domain   analysis   without   prior  knowledge  of   the   sequence  of  modes  of  operation  can  be  done.   The Complementarity Framework allows a way to calculate the transient and the steady state responses in a direct manner.

The   modeling   of   single-phase diode bridge rectifier is presented under different modes of operation: DCM and CCM. This is done in a systematic way using the Complementarity Framework.

This approach is currently used in the modeling of more complex power converters including controllers [16]. The Complementarity Framework is an appealing methodology that is being used increasingly in the analysis of power converters [4] [16].

ACKNOWLEDGMENT The  authors  would  like  to  thank  CONACyT  for  granting  

scholarship   to   P.   Suarez   for   undertaking   MSc   degree  studies.

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