Time-domain analysis for field excited transmission lines with nonlinear loads

TIME-DOMAIN ANALYSIS OF

FIELD-EXCITED TRANSMISSION LINE SYSTEMS BY USING MODEL-BASED PARAMETER ESTIMATION M. D'Amore, IEEE Fellow M. S. Sarto, IEEE Member Department of Electrical Engineering - University of Rome "La Sapienza"

Via Eudossiana, 18 - 001 84 Rome, Italy

Abstract : The time-domain analysis of field-excited lossy transmission tine (TL) networks with nonlinear loads is based on the nodal approach in the time-domain. The computation of the nodal admittance or impedance matrix coefficients of the linear part of the network is required in the frequency-domain, in order to obtain the corresponding transient functions by the inverse Discrete Fourier Transform (DF"). The model-based parameter estimation (MBPE) technique is applied to improve the efficiency of the method in case of large-dimension configurations. The developed procedure allows to represent transcendent functions over very wide bandwidths by using only the Coefficients of their rational approximating functions.

Good accuracy and considerable reduction of the computation-time are achieved. The EMP-induced effects on three signal mesh-type networks, having increasing geometrical complexity, with nonlinear loads, are calculated in order to assess the efficiency of the proposed method.

Introduction

The analysis of multiconductor networks excited by transient electromagnetic (EM) fields, containing nonlinear protective devices, has been performed by using a combined frequency- and time- domain procedure based on the nodal approach 11-31. The developed method allows an accurate simulation of the configuration in the frequency-domain and the computation of the transient response of the system by solving nonlinear integral equation systems of Volterra second type. The transient nodal admittance matrix and source vector coefficients have been computed by means of the inverse D m . Therefore, a considerable amount of data has been handled in the frequency-domain, because of the fast transient sources exciting the system, in order to avoid the Gibb's phenomenon in the inverse DFT calculation. In case of networks having complex configurations, the described numerical procedure is quite time-consuming. The efficiency of the method should be improved by reducing the computation-time and the computer memory occupation.

The purpose of this paper is to use the MBPE approach to increase the numerical efficiency of the developed procedure for the time-domain analysis of large-dimension field-excited TL systems, containing nonlinear elements. The MBPE technique is a numerical procedure widely applied to the treatment of electromagnetic problems and based on Prony's method [4-61. This technique allows to represent with very good accuracy a generic function in a wide

frequency window by using only the coefficients of their rational approximating function. The samples of the spectral quantities for the inverse DFT computation are then obtained by evaluating rational functions, which are known analytically. As a result, the computation-time is nearly-independent of the frequency- sampling step.

The proposed procedure will be applied to the analysis of the induced effects on large-dimension field-excited TL networks. The efficiency of the method will be assessed with reference to the required computation-time.

e-Do& A&&

Let consider a field-excited network with multiconductor dissipative lines and nonlinear loads, above a lossy ground plane.

The transient response of the system is obtained by applying the nodal approach in the time-domain, having suitably represented each multiconductor line-section in the frequency-domain by an equivalent PI-type circuit containing impressed shunt current sources, induced by the extemal EM field [2].

The proposed approach leads to the following nonlinear matrix equation system of Volterra second type:

t

0 t 0

i(t) = - ^i,(t) + y(t-z) v(z) dz (la) v(t) = vo(t) + 1 ^z(t-z) i(z) dz (lb) with the nonlinear boundary conditions:

i(t) =A t v(t) 1

v(t) =f, [ i(t) 1 (2) in which all the excitations are supposed to begin after t=O.

In eqs.( 1,2) i(t) and v(t) represent the vectors of the unknown currents and voltages involving the nonlinear loads, whose characteristics are supposed to be monotone; i,(t) and vo(t) are the impressed current source and open-ended voltage vectors; y(t) and z(t) the transient nodal admittance and impedance matrices. The time-domain model expressed by eqs.( 1,2) is solved by using a numerical technique based on the trapezoidal rule, which reduces the integral matrix equation of Volterra type to a nonlinear algebraic equation system. Suitable algorithms are developed in order to achieve the solution in case of strong nonlinearities.

The coefficients of vectors i,(t), vo(t) and matrices y(t), z(t) are computed by the inverse DFT of the corresponding quantities in the frequency-domain:

C"3-1-7 $4.000 1 QB4 IEEE 258

i,(t) = F - ' [ I~(O) 1

y(t) = F-I [ Y(0) I

, vo(t) = F-' [ Vo(o) 1

z(t) = F" [ Z(0) I ⁽³⁾

.

in which Z(w) = Y -'(U).

In case of fast transient sources such as the EMP, the most time-consuming process in the numerical procedure consists in the frequency-domain evaluation of the sampled values for the calculation of the inverse DFT, because accurate results are obtained only if a wide frequency window is considered. The proposed technique, based on the MBPE method, allows a consistent reduction of the time and computer-memory in the computation of the transient quantities. In particular, the procedure is very convenient to be applied to the assessment of the v(v+1)/2 coefficients of matrix Y(o) or Z(O), in case of large-dimension networks having a great number v of nodes.

MBPE-Procedure Freauencv-Domain

Let consider the generic coefficient of matrix Y(o) or Z(W) represented by function Fij(o); in the following the subscripts i and j will be omitted for semplicity. F(w) is a transcendent function because refers to a distributed parameter system. However, since the physical structure has natural resonant frequencies, F(w) can be approximated as a rational function in the following form:

I\

N(jo) D(jw)

No + N I (io) + ... + Nn (jw)"

D, + D, (iw) + ... + Dd ( j ~ ) ~

F(o) = - = (4)

in which N(jo) and D(jo) are polynomials of jo, having order n and d, respectively. Let notice that the inverse FT of 6(0) is expected to be a real function; therefore, all its poles pi, i=l ... d, corresponding to the roots of DCjo), are supposed to be real or complex conjugated pairs and d is an even number. That means NCjw) and DOw) are polynomials with real coefficients.

As F(w) represents the transfering function of a physical system, d is assumed either equal to n or n+l. In particular, if the asymptotic value of F(o), for o approaching infinite, is equal to zero, it is set d=n+l. Otherwise, if this value differs from zero:

lim F(w)= F,,,

O+-

the following modified function is considered:

FM(o) = F(w) - F,, (6) FM(w) has a null asymptotic value and can be represented in rational form $"(w) by using d=n+l. The inverse FT of 6(w) will be then expressed as:

F-' [ P(w) ] = F [ fi'(w) J + Fhf S(t) (7) 6(t) being the Dirac's delta function. In the following, d will be always assumed equal to n+l.

The residues Ri, i=l ... d, are given by the following well-known expression:

NP,) W P l )

R, =

(8)

in which D'(pi) is the (jo)-derivative of DCjw) calculated for jo=pi;

let notice that eq.(8) is valid if p, is a single pole of @(w). Function

$(a) can be espressed in the following form:

d Ri

i = l (jw- pi)

fi(w)=X

(9)

The MBPE allows the complete representation of a generic function by estimating the coefficients of the rational approximation, obtained by using only few samples of the function itself. The technique is applied following the algorithm described in Appendix, which involves a point-matching procedure and leads to a 2d-order linear equation system, whose solutions are the terms No ... ^{Nn, Do} ...

Dd-, in eq.(4), having assumed D,=l and d samples of F(o) as forcing quantities.

Difficulties can arise in the treatment of the 2d-order equation system because the 2d x 2d matrix of the coefficients involves terms like wk with k=l ... d. Therefore, computer overflow problems occur in case of high values of w. Moreover, if F(w) contains a great number of resonance peaks, it should be approximated by a rational function with high value of d, over a wide frequency-window; as a consequence, the full 2d x 2d matrix, including elements with very different values, is ill-conditioned and the computation of the unknown coefficients is not accurate at all.

In order to avoid computer overflow problems, the calculation is carried out by using the normalized frequency a* = w/wo, coo being minimum or maximum value of the sampled points. In this case, the poles and residues are:

pi=p:w, , R i = R ; w , , i=l ... d (10) where p; and R; are the normalized ones.

The ill-conditioning problem is overcome by breaking up the whole frequency window into several sub-domains, containing only few resonance frequencies, in which F(o) can be approximated by a low-order rational function. The frequency-range of the generic kIh sub-domain is Amk = ok - wk-', wk and o ~ . ~ being the maximum and minimum value, respectively. The obtained approximation of F(w) in the kth sub-domain is the rational function Gk(w) given in eq.(9).

Therefore, @a) is expressed in the following form:

where q is the number of the frequency-windows and U is the step- function.

The choice of d=10 to represent function $,(U) having only two peaks over the window allows to overcome both the problems of overflow and ill-conditioning.

Critical Points of the Procedure

The main problem in the use of the described procedure consists in the choice of the sub-domains for the rational approximation of F(o) and of the d sampling points inside each frequency-window.

Concerning the first point, let begin with the case of a single-conductor lossless line, which can be considered as a very simple network constituted by only one 1 long line-section and two ports.

Let enforce a current pulse 6,(t) at one end of the line. The

observed voltage at the other side will consist of a pulse SY(t-7)

having a time-delay T = l/c = 2 d 0 , with c = 3.108 m s-'. Therefore,

the fundamental resonance frequency of the line is or and is related

to the path the current and voltage waves have to cover between the two sides. The Fourier transform of the 2-delayed pulse GV(t-.r), is the complex function exp(-jurs) = cos(o2) - j sin(w) with period equal to r=2d.r=or, and is just the transfering function of the line between the two ports.

As a result, the frequency-spectra of the real and imaginary parts of the transfering function F(w), representing the generic mutual coefficient of matrix Z(o), will b e even and odd functions, respectively, showing two peaks in each period r. Therefore, the rational approximation of F(w) can be computed by using ten-order expressions over each r-wide interval.

Let now observe the voltage waveform at the same side where the current pulse is enforced. In case of mismatched line, the transient response will be constituted of a peak at time t d , followed by an other one after a 2.r-delay. In fact, the current and voltage waves leaving from one side of the Line, have to cover a 21-long path to reach again the starting point.

Consequently, if F ( o ) represents the generic proper term of matrix Z(w), the frequency-spectra of the real and imaginary parts will be even and odd functions, respectively, having two peaks in each period r / 2 . In this case, the width of the windows for the rational approximation of F(w) have to be chosen equal to r/2, if it is set d=10.

Finally, let consider a more complex network constituted by L line-sections with different lenghts I,, 1, ... I,. A current pulse enforced at node j will get the node k by following different paths.

The observed voltage at node k will then consists of several pulses having a different delay .r,=l,lc. The corresponding spectra of the real and imaginary parts of F(o) are obtained by the superposition of different functions exp(-jmi). The period r of the resulting function, which is the same for all the coefficients of matrix Z(w), will be equal to the minimum common multiple among the periods r,, T2 ...

rL, ^with ri=2x/.ri=ori. The fundamental resonance frequency or=2xc/A of the network corresponds to the maximum factor A common to I,, I, ... IL, and is equal to r. In this case, function F(o) is characterized by several peaks inside each q-wide interval. Eq.(lI) is then used by considering sub-intervals of r with the same width Am, equal to the breadth of the smallest windows containing only two peaks of F(w). In case of tree-type networks, it correspond to the longest path connecting pairs of nodes. If the system has a loop-structure this path, including the same node only once, can be calculated by considering the maximum number of line sections which connect the farest ports.

The last remark concerns the computation of the poles and residues of the rational approximating function $(a). It is important to point out that the computed roots of the denominator of the rational function are just curve-fitting poles and do not represent the actual poles of the system. In fact, in each Aw-wide window a two-peak complex function is approximated as a ten-order rational function; the corresponding ten poles are situated whatever along the frequency-axis and their contribution in any point external to the considered frequency-window is not null. In other words, each ten-pole set referring to a Am-wide window produces a good approximation of the transcendent function only inside that interval, whereas outside other ten-pole sets are used,

The previous considerations can also be extended to case of lossy networks, still representing valid criteria for the choice of Am.

In conclusion, the developed procedure consists in the following determination of the period r of the real and imaginary part of the coefficients of Z(O) or Y(o);

determination of the window width Ao;

computation of the coefficients of the rational approximating function fik(o) over each sub-domain by using only ten equidistant samples;

reconstruction of the frequency-spectrum of $(a) from the computed parameters.

steps:

The proposed procedure is applied t o the analysis of three field-excited TL networks having different configurations, in order to verify the efficiency of the method, assessed in terms of the required time for the computation of the sampled values of the nodal impedance matrix coefficients in the frequency-domain.

At first, the four-port network sketched in Fig.1 is considered.

Each line-section is constituted by three parallel 2 m long wires with 1 mm radius, 1 cm apart, 70 c m distant from the aluminium reference plane. T h e network i s stressed b y an impinging electromagnetic plane wave having incident angle 6=30° and azimuth angle w=lOo, represented in the time-domain as a double exponential function with time-constants 2, = 250 ns, T~ = 2 ns, and peak value E,, = 50 kV/m. The ports of the network are closed on 400 R resistances parallel connected t o varistors, whose static characteristic is described by the following function:

v = V, [ i / % 1 l'a (12) with V, = 5 kV, I, = 0.1 A, a = 30. The analysis of the configuration is performed by using the voltage formulation (lb). The MBPE method is applied to the computation of the elements of the modified twelve-order impedance matrix ZM(o), which is obtained by subtracting the high-frequency nodal impedance matrix Z,, from Z(o) = Y(o)-'. As all the line-sections have the same length, the maximum common factor A is equal to 2m and the period is r/2x=clA=150 MHz. The width A d 2 n of the generic sub-interval is assumed to be of 30 MHz, corresponding to the maximum extension of the network. Over each 30 MHz-wide interval, tenth-order rational functions are computed by using ten equidistant samples.

.Y

3

* &

Fig.1- Sketch of the four-port field-excited multiconductor network.

100 -

150

frequency [MHz] 300

40

I

-40 '

^I

0 150 300

frequency [MHz]

(b)

Fig.2 - Frequency-spectra of the real (-) and imaginary (---) parts of

ky,l(l.l) (a) and ky,ql,3) (b) for the network in Fig.1.

Figs.2a and 2b show the frequency spectra of the real and imaginary parts of the generic proper and mutual coefficients zy,l(l,l)

and ky,4(l,3K respectively, computed by using the proposed procedure. Z y , l ( l , l ) j s the coefficient ( 1 , l ) of the input matrix impedance at port 1; Zy,4(13) is the coefficient (1,3) of the transfering matrix impedance between port 2 and 4. Let notice that all the curves in Figs.2a,b have the period of 150 MHz; moreover, it is interesting to point out that the real parts (continuous line) are even functions, whereas the imaginary parts (hidden line) odd functions.

The waveforms of the corresponding frequency-spectra of the exact impedance matrix ZM are nearly coincident. The maximum error is not greater than few units per cent as shown in Fig.3, with reference to the real and imaginary parts of ?y.4(l,3).

1

0 150

frequency [MHz] 300

Fig.3 - Percentage error in the computation of the real (-) and imaginary (---) parts of ?:,%,>) for the network in Fig.1.

~ I""

0 150 300

frequency [MHz]

Fig.4 - Frequency-spectra of the real (-) and imaginary (---) parts of

ky,l(l,l) for the network in Fig.1 with 3 m long line-sections.

1 17

(a) (b)

Fig.5 - Geometrical configuration of the multiconductor networks with seven- (a) and nineteen- (b) ports.

0 150

frequency [MHz] 300

Fig.6 - Frequency-spectra of the real parts ky,,,,,3, ^(-) and 2y.7(1.3) (---) for the seven-port network in FigSa.

The 150 MHz periodicity of the impedance matrix coefficients disappears in Fig.4, where the real and imaginary frequency-spectra of k ~ , l ( l + l ) are represented when all the line-sections in the network are 3 m long; the waveforms have period r/2n = c/3 = 1 0 0 MHz.

Successively. the seven-port network in FigSa is analyzed. The

structure is obtained by adding other four elementary triangle-

networks to the previous configuration; therefore, it keeps the same

geometrical characteristics of the network in Fig. 1 but an increased

complexity. The periodicity of the frequency-spectra of the

impedance matrix coefficients is always l-/2~=150 MHz, but the

width of each sub-interval is Awl211 = 25 MHz, because of the

increased dimension of the network. F i g . 6 shows the

frequency-spectra of the real part of both the approximated and exact

coefficients ky,7(l,3) and 2y.,(,.3); t h e two curves cannot be

distinguished and are very similar to the waveform in Fig.2b

referring to the real part of ?y,4(l,3), ^{except for} two additional peaks

261

around 75 MHz, in each period. These peaks give evidence of the increased dimension of the network.

The computation-times required by the MBPE method and by the exact procedure are reported in the graph of Fig.7 as functions of the assessed samples of a generic coefficient of the impedance matrix, with reference to both the first and second configuration. The shown data are obtained by running the programs on a RISC-based processor with 33 MHz-clock. The hidden lines, corresponding to the obtained curves for the exact procedure, increase linearly with the number of samples. At the contrary, the continuous lines, referring to the MBPE method, are nearly horizontal; infact, whatever is the wished number of samples, the algorithm always requires the same computation-time to evaluate the coefficients of the rational approximation. The sampled values are then obtained by using the achieved analytical representation. Let notice that good accuracy in the inverse DFT computation is reached by using at least 2000 samples; that means 910 seconds with the exact method and only 52 seconds via MBPE, for the network in FigSa.

At last, the large dimension network in FigSb, with 19 ports and 42 three-conductor line-sections is considered. The structure is obtained by joining 24 elementary triangle-networks. The period r / 2 n is always 150 MHz, but Aw/2n is set equal to 10 MHz. The frequency spectra of the real part of coefficients ky,19(l,3) ^and

Zy,19(1,3) are represented in Fig.8. The maximum error is of only few unit per cent.

I

I

I 1 I I 1 1 1 1 1 I 1 I I 1 1 l 1 1

100 lo00 loo00

number of samples

Fig.7 - Computation-time using the MBPE-method (-) and the exact procedure (---), for the networks in Figs.1 (a) and 5a (b).

-10 ₀ ' ₅₀

₁₀₀ ^I

time Ins]

(b)

Fig9 - Induced voltages on conductor 1 at ports 1 , 4 , 6 , 15 of the network in Fig.Sb, terminated on 400 Q resistances parallel connected (b) or not (a) to 5 kV varistors.

It it very interesting to note that in this case the computation-time of 3000 samples required by the exact procedure is equal to 4346 seconds, that is more than one hour, whereas it is only 689 seconds by using the MBPE method. These times are obtained by running both the programs on a RISC-based processor with 66 MHz-clock.

Finally the transient voltages at each port of the three described networks were computed. Figs.9a,b show the induced voltages on conductor 1 at ports 1,4, 6, 15 of the largest network of FigSb, with or without varistors.

Conclusions

-" I

I

0 150 300

frequency [MHz]

Fig.8 - Frequency-spectra of the real parts of 2:,19(1,3) (-) and Zy,,9(l,3) (---) for the nineteen-port network.

...

._.- . ..__ ..

10 I

0

E ^-10

s m

> -20 -30 4 0

0 50

time [ns]

(a)

100

l o

The developed procedure based on the MBPE method allows a consistent reduction of the computation-time in the transient analysis of transmission line networks, containing nonlinear loads, excited by fast transient electromagnetic sources. The method allows to compute the rational approximation of the coefficients of the nodal admittance or impedance matrix by using only few samples in the frequency-domain. In fact, especially in case of large-dimension configurations the evaluation of the sampled values of these terms represents the most time-consuming process in the whole procedure, because a considerable number of samples is necessary in order to achieve good accuracy in the inverse DFT calculation.

Practical criteria, based on the geometrical configuration of the network, are given for the choice of the sampling-frequencies.

262

Accurate results and very little error are obtained by breaking up the whole frequency bandwidth in several sub-domains. T h e transcendent functions are then approximated over each sub-domain by using low-order rational expressions. The efficiency of the procedure has been evaluated with reference to the required computation-time, expressed as function of the number of the calculated samples. In case of a nineteen-port network with 4 2 three-conductor line-sections, the time saving for the computation of 3000 samples by using the MBPE technique was about the 84%.

Further investigations will be devoted to the possibility of estimating the actual poles and residues of the system. In this case, it will possible to express analitycally the inverse DFT of the transcendent functions as a sum of exponential terms.

A = Acknowledmnent

F, F, s1 ... F, sp' -1 -sI ,.. -sq - F, F,s, ... F, si-' -1 -sa ... -si

... ... ... (17a)

... ... ...

- F, Fm sm ... Fm s:' -1 -sm ... - s i -

This work was performed by the economic support of the Commission of the European Communities to the Science Project

"Electromagnetic Compatibility: Fast Transients i n Telecommunication and Power Apparatus and Systems".

References

M. D' Amore, M. S . Sarto, "EMP-coupling to multiconductor dissipative lines with nonlinear loads above a lossy ground",

---

Proc. 10* Int. Zurich on EMC, Zurich, March 9-1 1, 1993.

M. D'Amore, M. S . Sarto, "Time response of a network containing field-excited multiconductor lossy lines with nonlinear loads", Proc. 1993 IEEE EMC Int. Symp,, Dallas, August 9-13, 1993.

M. D' Amore, M. S . Sarto, "Transient analysis of field-excited multiconductor lossy networks with nonlinear dynamic loads: an integral equation approach", Proc. ICEEA, Turin, September

1417,1993.

S . Chakrabarti, K. R. Demarest, E. K. Miller, "An extended frequency-domain Prony's method for transfer function parameter estimation", Int. Journal of Numerical Modelling, Vol.

6, pp. 269-281, Nov., 1993.

E. K. Miller, G. J. Burke, "Using model-based parameter estimation to increase the physical interpretability and numerical efficiency of computational electromagnetics", Computer Phvsics Communication, Vo1.68, pp.43-75, 1991.

J. N. Brittingham, E. K. Miller, J. L. Willows, "Pole extraction from real-frequency information", ELQE. m, Vo1.68, No.2, Feb., 1980.

ApDendix Rational Aporoximation of a Function by .MBPE

A

Let F(s) the pole-residue approximation of the generic function F(s) defined in Laplace domain:

N(s) D(s)

No + ^{N I} s + ... + N,, s"

F(s) = - = (13)

Do + D, s + ... + D, sd

in which No ... Nn, Do ... D are unknown complex coefficients. For physical causal systems, F(s) is R a proper rational function, that it

means the degree n of the numerator is not greater than the degree d of the denominator.

In (13) the unknown coefficients of N(s) and D(s) can be calculated by using a point- matching algorithm, that is forcing the approximated function @(s) to be equal to the exact function F(s) only for few values of the complex frequency

+ jw. By setting D, = 1, m=n+d+l sampled values of F(s) are necessary to compute the n+d+l unkown coefficients in eq.(13). Therefore, the following linear equation system is obtained:

N(s.) D(sJ

No + ^{N, si} + ... + ^{Nn sin}

F(si) = F(sJ = = , i=l ... m (14) Do + D, si + ^... + ^S:

Eq.( 14) can be rewritten in the following form:

F(s,) D(si) = N(sJ , i=l ... m (15) or by using a matrix expression:

A X = B where:

is an mxm matrix and

X = [ D o D, ... D,, No N, ... Nn It (17b) B = [ -F, S; -F, si ... -Fd s: ... ... ..

are the m-dimensional vectors of the unknown coefficients and the sampled values, respectively. In eqs.(l7a-c) Fi is the ith sampled value of F(s), calculated for s=si. The algebraic equation system (16) is solved by using well known numerical techniques, such as the low-up (LU) decomposition.

Let consider the case in which F(w) is a function defined in the frequency-domain. Let suppose that its poles and residues are real or complex pairs and the degree of D u o ) is an even number. The equation system (15) can be rewritten by using a real formulation. In fact, by setting si=jwi:

F(oi) = F, = FR(oi) + j F'(wi) = F : + j F/ (18) with F: and F: real numbers and

(-1)h o?h , k=2h

, k=2h+ 1 (19)

{ ^{j (-11,}

( i o , ) k =

the real and the imaginary part in eq.( 15) can be separated:

d/2

h=I C (-Uh (QZh ^[ F r D,,, - N,,, - wi F: Dd+l-2h I = - (14~; ^(2Oa)

d/2

h=l (-l)h [ - Fit D,,, + ^W, Nd+,-,, - ai F r D,+,-,, ] = (14 F; (20b) with i=l ... d. Eqs.(20a,b) are obtained by supposing that n=d-1.

However, similar expressions can be achieved for whatever n. Let notice that in this case only d sampled values of F ( o ) are necessary to calculate the real unknown coefficients Do, D, ... D,.,, No, N , ...

N,.,, having assumed D,=l. In fact, by using expression (15), d sampled values lead to 2d forcing quantities.

263