INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS, VOL. 24,519-528 (1996)

STEADY STATE ANALYSIS OF PERIODICALLY TIME-VARYING NETWORKS BY NEW DEVELOPMENTS IN THE SPECTRAL DOMAIN

GULAY TOHUMoCLtJ

Electrical and Electronics Engineering Department, University of Gaiiantep, 2731 0 Ga-iantep, Turkey

AND

MUHAMMET KdKSAL

Engineering Faculty, University of inonu, 44069 Malatya. Turkey

1. INTRODUCTION

The analysis of networks with time-varying elements is more complicated than the analysis of networks with constant elements. Frequency domain analysis methods are well established for the analysis of time- invariant networks. Since the complete solution can be computed for only a very narrow class of periodically time-varying linear systems, ’ general methods such as the piecewise constant approximation method (PWCAM)’ and spectral analysis334 are of interest in many cases. Although differential equations can be bypassed by the latter method, operations on matrices of infinite order are involved. K ~ r t h ’ . ~ and Gense17,* have applied it to frequency modulator circuits which are a cascade of several two-ports, namely filter and modulator, with switching time variations. Later, it was shown that the method is appropriate for the computer-aided design of linear ladder and cascade connected networks.’-” In fact, other literature on this subject can hardly be found and this paper is intended to be a complete and extended theoretical and computer-aided analysis treatment of the subject with the following contributions.

1. The use of spectral analysis is not limited to ladder or cascade networks and/or switching-type modulator circuits only. Spectral analysis of any type of periodically time-varying system is now as straightforward as the well-known classical computation of the steady state sinusoidal response of a linear time-invariant circuit. This is made possible by the improved spectral theory.

2 . The known frequency domain formulation techniques developed for time-invariant linear circuits are also suitable for the analysis of periodically time-varying linear circuits by the introduction of new spectral matrices such as didem, indem, dimodem, inmodem, demodem and others and by the validity of Kirchhoffs laws in the spectral domain.

3. Concerning the comparison with other methods, spectral analysis has advantages when the time variations of system parameters are smooth. On the other hand, the PWCAM” gives more satisfactory results in the neighbourhoods of times where discontinuities occur.

4. It is observed that spectral analysis, like the staircase approximation scheme, is a first-order process with respect to the number of harmonics considered in the computations (see Figure 3).

Briefly, the scope of this work compensates for the present theoretical and computational deficiency in the literature about spectral analysis of periodically time-varying networks; furthermore, a general and unified mathematical basis for it is set up. Some new concepts are defined and a comparison of spectral analysis with other methods is presented.

CCC 0098-9886/96/040S19-10 0 1996 by John Wiley & Sons, Ltd.

Received 31 January 1995 Revised 25 July I995

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2. SPECTRAL ANALYSIS

Spectral analysis is a frequency domain method in which the variables are represented by vectors and the relations among variables are expressed by algebraic equations containing matrices. It resembles the well- known frequency domain method where the variables are represented by complex numbers (phasors) and it is applied to linear time-invariant networks. In spectral analysis, instead of using a single frequency, the relations between different harmonic components of signals are investigated in the frequency domain; that is why this frequency domain is called the spectral domain.

In an asymptotically stable, periodically time-varying linear network excited by an exponential input u ,

u( r ) = Uep' (la)

y ( t ) = H ( p , t ) u ( t ) (lb) In this equation H(y, t ) is called a time-varying system function which is periodic with the fundamental period To of the network (wg = 2x/Tg is the network frequency). From ( la) and ( lb) the response y can be written as

where p = j w and U is a complex constant, any response y of the network can be expressed

y ( t > = U p , r)ePrr Y ( y , t ) = Y ( p , t + T o ) = H ( p , t )U (1c)

Since Y ( p , t ) is periodic, it can be represented by its Fourier series expansion

n - -_

or by the vector of its Fourier series coefficients

y = [Yo Y - , Y , , Y - , Y , , ... IT which is known as the spectral vector, where the superscript T denotes matrix transpose.

If there is a time domain relation of the form

y ( t ) = 9: x ( t ) (3a)

between the signals y ( t ) and x ( t ) , where 3 is a linear operator on x , in the spectral domain this relation can be expressed by a linear algebraic equation as"

y = F X (3b)

In this equation y (x) represents the spectral vector associated with the signal y ( t ) ( x ( t ) ) and F denotes the spectral matrix associated with the operator SF.

Here we emphasize that each vector in the spectral domain is equivalent to a time function representing some signal and a time domain equation (not necessarily algebraic) between the signals respectively; it is always possible to obtain one from the other. Therefore for the formulation and solution and even for the modelling steps of the analysis it is preferable to use the spectral domain, since all the equations are algebraic in this domain where it is also possible to use all the known frequency and time domain formulation techniques.

Before passing to a new section, it is appropriate to set up the relations between the Fourier series coefficients ( Y J and the elements ( y k ) of the spectral vector (y) of a signal which is expressed in the form (lc). These are readily derived from (2b) as

(k- 1)/2, k = l , 3 , 5 , ... k = 2,4,6, ...

2 i + l , i = O , l , 2 ,... - 2 i , i = -1, -2 , ...

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3. SPECTRAL MATRICES AND THEIR PROPERTIES

Different new spectral matrices, as well as old ones, are formally defined and their complete properties are derived. These definitions and properties are the cornerstones for the spectral analysis of general periodically time-varying networks.

3.1. Modem matrix

In a periodically time-varying network, if a signal x ( r ) is multiplied by a scalar periodic function m(t) , i.e. if the new signal y( t ) is calculated by

Y ( t ) = m(t)x(t> (5a)

Y = Mx, M n . k = mf(n)-f(k)) = M g ( f ( n ) - f ( k ) ] (5b)

then, in the spectral domain this relation is expressed as

In this equation the previously defined notation in (2b) and (4) is used for y ( t ) ; M denoted the modem matrix (with elements M n , J associated with the periodic function m(t). Equation (5b) can be obtained from (5a) by inserting the Fourier series expansion (with coefficients mi) of rn( t ) , by using the expressions in ( lc) and (2a) for y ( t ) and finally by using similar expressions for ~ ( t ) . After the coefficients of like powers of ‘e’ are equated in the resulting equation, one obtains (Sb).

In general, the modem matrix M has infinite order; however, depending on the harmonic content of the periodic signal m ( t ) and on the accuracy required in the computations, as well as on the computational cost, its upper left square submatrix with dimension nd x nd is used, where nd = nh + 1 by considering the finite number of harmonics ( nh) only.

It is important to note that the modem matrix was first defined by Gensel for the square wave;’ however, it was generalized to apply to all periodic wave shapes3 without spoiling most of its properties as stated in Reference 8

3.2. Different spectral matrices

If the operator 9 in (3a) indicates the derivative or integral with respect to time, i.e.

d

dt y ( t ) = - x ( t ) or y ( t ) = 5 x( t ) df (6a,b)

then by using ( lc) and (2a) for y ( t ) and similar expressions for x ( t ) in the above equations and equating the coefficients of like powers of ‘e’, we obtain

y, = ( p + jnwo)x,, y, = x , / ( p + jnw,), n = 0, T 1 , T 2, . . . ( 7 0 )

By using the spectral vectors for y and x, these frequency domain expressions can be written in the spectral domain as

In the above equations d,,k is the Kronecker delta, which is unity for n = k and zero otherwise. Equations (8a) and (8b) correspond to the derivative and integral operations in the time domain and we call the coefficient matrices D and S in these equations, to be coherent with the words differentiation, ktegration and m o h , the didem and indem matrices respectively. Obviously, these matrices are diagonal and inverses of each other, i.e. D = S- ’ or S = D-l.

The equation representing a time delay in the time domain, i.e. y ( r ) = x ( t - t ) , can be transformed into

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Here the spectral matrix r is called the delay matrix, which is a diagonal matrix. Another spectral domain expression is obtained from the convolution integral in the time domain or,

equivalently, from the complex frequency domain expression Y ( p ) = T ( p)X( p ) , where Y ( X ) represents the Laplace transform of the signal y (x) and T ( y ) is the transfer function between them. The transfer spectral matrix T is defined by

Y = Tx, T,,,, = S n J b + j f ( n ) o o l (10)

In the following formulae some time domain expressions and the associated spectral domain expressions are given. In the spectral domain expressions the spectral matrices DM, S, and rM are called the dirnodefn, inmodem and demodern mutrices respectively. The spectral matrices denoted by these letters with superscript ( i ) are the ith-order forms of these matrices ( v , , ~ = 1 - 6,J.

(1 la)

As a special application of cascaded networks for ladder networks, the two-port subnetworks are usually characterized by their ABCD parameter form of network equations which are functions of the complex frequency p = (J + j w in general. I t is possible to reveal the transfer properties between the sink and source ends easily under any terminating conditions. The coefficient matrix is the kernel matrix described by Gensel, who has investigated modulator circuits7 and studied square type periodic variations as well as networks containing periodically operated switches.* Kunh, on the other hand, called the same matrix simply a spectral matrix, which is not as appropriate as the name 'cascade spectral matrix' used by Bardakjian and Sablatash.9 It is important to note that the law of cascade connected networks is also valid in the spectral domain; in other words, the cascade spectral matrix of the connection of two or more two-port networks is equal to the product of the individual cascade spectral matrices. As we recall, kernel matrices do not appear only in the formulation of cascade connected networks, but also node formulation or other formulation techniques will yield such matrices as well. The terminal matrices appearing in the spectral domain terminal equation of multi-terminal components can also be called kernel matrices.

3.3. Properties of spectral matrices

The importance of the listed properties of spectral matrices appears in the spectral domain formulations of the network equations. These properties, together with the definitions of the previous sections, make it possible to extend all the complex frequency domain (p-domain) concepts into the spectral domain (D- domain). For example, the property in (15) states that the second and third postulates of system theory l 4 are

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valid in the spectral domain as well.

am([) + aM, a constant n(r) -+ M

/ / ... [m(t)(.)] dt' + S'M (140

As an example of the concepts described in the previous paragraph, consider the comparison of a time- invariant linear inductance L and a time-varying linear inductance L(t). For the first case the voltage v and current i are related in the time domain and frequency domain by

d d d

d t d t dt a=- (Li) = L - i -+ V=Lp/ y = -

respectively, where @ is the flux linkage. Similarly, for the second case the terminal equations can be written as

d d

d t d t v = - 0 = - [L(t)i(t)] -+ Y = DLi

The last equation is obtained by using the property (14e); L denotes the modem matrix associated with the periodic inductance variation L(t) and D is the didem matrix. The comparison of (16a) and (16b) shows that the scalar impedance function Lp is replaced by the spectral matrix DL; therefore DL represents the spectral impedance matrix of a periodically time-varying inductor. In a similar manner, the above properties give rise to the possibility of defining spectral admittance, resistance, conductance, reactance and susceptance matrices according to the types of dependent and independent spectral vectors in the terminal equations. An impedance matrix in the complex frequency domain, however, when it is transformed into the spectral domain, can be called a spectral impedance kernel or simply a spectral impedance matrix.

Owing to the possibility of writing the terminal equations of all linear periodically time-varying components, as well as to the validity of the second and third postulates in the spectral domain, all the time domain (state space formulation and others) and frequency domain (node, branch, chord, mixed, mesh, cutset formulations) formulations can easily be applied in the spectral domain as well. This fact can be well seen in the examples of the next section.

4. EXAMPLES

Example I

As an example of the application of the given theory about spectral analysis with the help of the prepared computer programme SPECANL, the network shown in Figure 1 is considered. The node formulation

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extended to the spectral domain is to be applied to this network; the unknown node voltages are chosen to be v,, vD and vL, which are the spectral vectors associated with the actual node voltages vB(t). vD(t) and v E ( t ) respectively. The known node voltage vA(f) is represented by the spectral vector

(17)

Although the time function v , ( f ) = 4 0 e ' ~ ~ ' = 4w cos(4ot) - j 4 0 sin(4wt) corresponding to the above spectral vector is complex, since its real part is the actual input, the actual responses will be equal to the real parts of the responses to be found; this is a fact which follows from the linearity.

v,= [ 4 0 0 0 . . . IT

Figure 1. Electrical network analysed by spectral analysis

I r , ( t ) = 4 0 cos(4wt) V, R , = 1 R, R , = R , = 2 $2

w = J ~ , i I = 1, C = l F, G , = l / R , , i = 1,2,3

i n ( f ) = [ I +0.S cos(2xt)]'", K,= l/L,, i = 1 ,2 ,3

L , = 0.5 H, L, = L, = 1 H, C, = G,G,m/(G, + G,m)

The spectral domain terminal matrix of each of the time-invariant algebraic components of the network can be written by multiplying each scalar in the time domain terminal equation by the identity matrix Z i.e. the terminal admittance matrices of the resistors R , and R , are G,I and G,Z respectively; the transfonner is represented by i, = nIiD, v, = -nIv,, where i, and i, are the spectral vectors of the currents ic(r) and iD(t) respectively. The inductors are represented by Y , = L,Di, or, equivalently, i , = K,Sv, (equations (16b) and (8)). Finally, for the time-varying resistor R , we write iR,= G3MvR,, where M is the modem matrix associated with the periodic function rn(t).

The node equations in the spectral domain can be derived as in (18a); the first and third equations are straightforward. The second one is obtained by equating the currents leaving the node D to zero, where instead of i,, i,/n = (v, - v c ) K , S = (vB + vD/n)K,S is substituted. The spectral domain node equations can also be obtained directly from the time domain node equations.

G,I + K,S + G,M K1S -G3M GJ + (K, + KJS + MD -K*S I[:]=[ ;']vA (184

- K2S (K2 + K,)S + G,M

i, = -K,Sv, + ( -Kl + K,)Sv, - (K2 + K,)SvE (18b)

The coefficient matrix of the unknown spectral vectors in the node equations (18a) is the spectral node admittance matrix or simply the node admittance kernel. The solution of these equations requires the inversion of the kernel matrix. When the spectral vectors vB, Y, and Y, are calculated, all the other unknowns of the network can be computed easily.

The above spectral domain node equations were solved using the SPECANL programme. The results are shown in Figure 2. In this figure the results corresponding to nh = 10 must be the actual results, since m ( t ) contains the 10th harmonic at most, and exhibit the minimum error. However, the error, which is

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defined by

where vDAc represents the actual values taken from Reference 2 and vDnh represents the values computed by SPECANL with n h harmonics, is not equal to zero exactly; this is a fact which is seen in Figure 3, where the error-computation time product is not zero at n h = 10. One reason for the non-zero error is the limitation of the spectral analysis to 10 harmonics only, while infinitely many harmonics are produced in the network. However, the truncation and round-off errors produced by computer become dominant as nh increases. Figure 3 reveals that with increasing Oh the error is reduced but the computation time T increases. Since the inversion of large-scale matrices takes very long computer times, after a certain harmonic number nhc the increase in nh does not become economical for the reduction of the error. This critical harmonic number can be assigned as the point where the error-time product ( E T ) curve has a minimum (nhc = 6 in this example).

Current (10 I" ).A ....

. . . . .., Figure 2. Solutions obtained by SPECANL

Figure 3. Error and computation time against number of harmonics

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Example 2

The system equations to be solved are given in the state-space representation as

.i’(t)= A ( t ) x ( t ) + B(t )u( t ) , y(t) = C ( t ) x ( r ) + D ( t ) u ( r ) where

sin r cos t + sin t 0 0 sin t cos t + sin t

-6 cos t - 6 sin t -1 1 cos t - 11 sin t -6 cos t - 5 sin 1

A ( t ) = B(t) =

C ( t ) = [cos t 0 13, D ( t ) = [ l l The exact steady state solution of this system for a cosinusoidal input function u ( t ) = cos(ot) ( w = 0-5)

is given as sin(wt) cos+ 1

y ( r ) = cos(wt) + - - esin f

w

In the spectral domain, from equations (20) and (21) we have

D,x= MAX + M ~ u , y = M ~ x + M D u (22) where the kernel matrices D,, M A , M,, M, and M, associated with the differentiation of the state vector x and with the periodically time-varying coefficient matrices A ( t ) , B ( t ) , C(t) and D ( t ) are expressed (in terms of the didem matrix D and the modem matrices S , C and E, associated with the time functions sin(o,t), cos(o,t> and e-“”(’”o’) ) respectively as

D O 0

O O D

S c + s 0

- 6 C - 6 s -11C- 11s -6C-5s , MA=[ 0 S C + S

M,=[C 0 11, M,=[I]

, M,=

Figure 4. Response function of Example 2 for various numbers of harmonics.

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The response function y ( t ) is computed from the spectral vector y at equally spaced time instants over the fundamental period and the results for various numbers of harmonics are plotted in Figure 4. It is seen that the solution for nh = 3 or 4 is the same as the exact result (up to the fourth digit) and converges to the exact one rapidly, where this fast convergence property comes from the smooth variations in the periodic time functions.

5. RESULTS AND CONCLUSIONS

A frequency domain method called spectral analysis, which is used for the analysis of periodically time- varying networks, is applied only to the steady state analysis of asymptotically stable networks excited by cissoidal inputs. However, it is quite general and offers a straightforward method.

The brute-force integration method, the Runge-Kutta method, l6 even for convergent cases is not appropriate to find the steady state solutions of networks which have a long damping period. For this kind of slowly damped system, spectral analysis is preferable, since the steady state solution is computed directly.

Another method to find the steady state solutions of periodically time-varying networks is the piecewise constant approximation technique which was developed and computerized by Koksal and Tokad.’**” With this technique both the steady state and transient responses can be computed in both the time and frequency domains. This method is similar to the method of spectral analysis in the following aspects: in the first method the accuracy increases with the number of divisions considered in one period; in the second method the same occurs with an increasing number of harmonics, while the computer time gets larger in both; the steady state solutions are obtained directly in both of these methods. Each of the methods has advantages over the other: for networks with suddenly changing or fast varying parameters (the harmonic content is high) the staircase approximation technique is best; for networks with smoothly varying parameters (the harmonic content is low) the spectral analysis technique is preferable owing to lower computer time requirements.

The spectral analysis method seems to be similar to the harmonic balance method’* (HBM) in that each state variable is represented by a Fourier series. In the HBM an optimization algorithm is then used to adjust the coefficients of the Fourier series such that the system equations are satisfied with least error. Its main disadvantage is the large number of variables that must be optimized. However, the spectral analysis method only uses the algebraic relations simply.

The scope of this paper compensates for the present lack of literature on the spectral analysis of periodically time-varying systems. One additional suggestion about the extension of spectral analysis to other areas is the investigation of jump phenomena in the network variables due to discontinuous parameter variations.

ACKNOWLEDGEMENTS

The authors would like to thank the reviewers for their useful comments and suggestions.

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