SIMULATION OF MULTICONDUCTOR TRANSMISSION LINE CIRCUITS COMBINING 1D AND 2D LAPLACE TRANSFORMATIONS

Lubomir BranCik

Bmo University of Technology, Purkyaova 118,612 00 Bmo, Czech Republic, E-mail: brancik@feec.vutbr.cz

Abstract

The paper presents an innovative way of a simulation of multiconductor transmission line (MTL) circuits which combines the Laplace transformations in one and in two variables. This approach enables to compute both nodal voltages (branch currents) in lumped-element parts of a circuit and voltage (current) waves distributions along MTLs’ wires effectively. The first one is performed by the numerical inversion of ID Laplace transforms when the solution in frequency d o m a i n is formulated using a modified nodal admittance equation method (MNA). The second one is done by the numerical inversion of 2D Laplace transforms when the solution appertaining to distributed parts of the circuit is formulated in the (q,s)-domain. Both NILT methods utilize the FFT in conjuction with quotient-difference algorithm to provide the high speed of calculation and precission of results.

1. INTRODUCTION

In previously published works, see at least [I-81, several methods that enable to simulate the transient phenomena in circuits containing multiconductor transmission lines (MTL) have been proposed. A possible block diagram of such a circuit is shown in Fig.1.

section with lumped-parameter components

I Figure 1. Linear circuit containing MTL sections

If linear circuits are only considered the solution can be formulated in the frequency s-domain when Laplace transformation with respect to time f is performed. In this way possible frequency dependences of per-unit- length matrices of MTLs can also be taken into account if necessary. In [&6] the solution in the s-domain is formulated by the modified nodal admittance equation

method (MNA) while a method for the numerical inversion of Laplace transforms in one variable (ID- NILT) is applied to obtain the solution in the time domain. This approach can be regarded as conventional one because only different ways of frequency-domain circuits descriptions and finding time-domain solutions are used. In [7], however, an unconventional approach for solving simple MTL circuits using two-dimensional Laplace transformation was proposed. There is namely often such a case that uniform MTLs form distributed parts of the circuit. Consequently second partial Laplace transformation, with respect to a geometric coordinate x, can be performed and by this hvwdimensional Laplace transform of the solution in the (q,sAomain is found. In so doing the boundaty conditions for the individual MTLs must be incorporated into the solution. In [7] this is made by using generalized Norton or ThCvenin equivalents to model MTL terminating networks. That is way only quite simple MTL systems could be taken into account. Herein, however, solutions obtained just as the result of the modified nodal admittance equation method in the s-domain are utilized to find necessary boundary conditions. By reason that the MNA method is very general one the arbitrarily complex MTL circuits can be considered at simulations. The final step lies in using a method for a numerical inversion of Laplace transforms in two variables (ZD-NILT) to find the solution in the (x,f)Aomain. Both ID-NILT and 2D-NILT methods are recently developed techniques utilizing the FFT in conjuction with the quotient-difference algorithm of Rutishauser. These techniques provide a relatively high speed of calculation and precission of results [8-101. If only the nodal voltage and/or branch current time behaviours in lumped-element parts of the circuit are wanted the solution finishes after ID-NILT method is applied. If moreover (only) voltage and/or current distributions along MTLs’ wires are needed to visualize the 2D-NILT method is applied as another (the only) and final step of the solution.

2. CONVENTIONAL METHOD SUMMARY

AS is shown e. g. in [&6] a modified nodal admittance matrix equation in the time domain has the form as

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where C, and G , are N x N constant matrices with entries determined by the lumped memory and memoryless components, respectively, vM (f) is the N x 1 vector of nodal voltages appended by currents of independent voltage sources and inductors, i, (t) is the N x l vector of source waveforms, it(t) is the nk x l vector of currents entering the k-th MTL, and D, is the N x nk selector matrix with entries d,,j E {0,1} mapping the vector i,(t) into the node space of the network. After Laplace transformation with respect to time t is applied the frequency-domain representation has the form as

P

[GM + c~lv~(s) + cDklk(s)=I,(s)+c,v,(o)0) .(2) 1-1

The MTLs consist of N, =n, /2 active conductors, i.e. they can be regarded as 2Nk -ports. Then the I, (s) in (2) is formed to contain vectors of currents entering the input and output ports as I,@) = [ I f ’ ( ~ ) , l ~ ’ ( s ) ] ~ and they are determined t?om a basic MTL matrix equation as follows.

Suppose a uniform MTL of a length 1 , with per-unit- length matrices R, L, G and C. In (x,twomain the basic MTL matrix equation is ofthe form [l]

a v(.%f) 0 -R V(X,O 0 L a v ( 4 r[i(x,tJ=[-G 0 ].[i(x,tJ-[c o]-a[i(x,t,]. (3)

After partial Laplace transformation with respect to time f is done the equation (3) results in

$:;;]=[-;(s) 0 ]-[I(x,s)j+[c o].[i(.qqj’ (4)

where V(x,s) = L[v(x,t)] and I(x,s) = L[i(x,t)] are column vectors of Laplace transforms of voltages and currents at distance x from MTL left end, respectively, v(x,O) and i(x,O) are column vectors of initial voltage and current distributions, respectively, and 0 is a zero matrix. Further Z(s) = R + sL and Y(s) = G + sC are series impedance and shunting admittance matrices, respectively. Clearly, the equation (4) can formally he rewritten as

-as) V(W) 0 L <&0)

( 5 ) d

-W(x,s)=M(s)W(~,s)+Nw(x,O). m

Then taking W(O, s) as the solution at x = 0 (MTL’s input) the solution at x coordinate can he written [l 11

W(x,s) =O(x,s)W(O,s)+ l@(x-t,s)NW(t,O)dt, (6)

where O(x,s) is an integral matrix defined in the special case of a uniform MTL (i. e. M(s) is constant), by

0

O(x,s) = 6.M‘S).I . (7)

In terms of a multiport theory the integral matrix O(/,s) acts as the MTL chain matrix O(s) when a coordinate x = I (MTL‘s output) is considered. Thus after denoting

W(0,s) = W“’(s) =[V“’(s),1(~)(s)]~, (8)

W(1,s) = W‘2’(s) =[,,(2’(s),--1(2’(s)]T. (9) I

p(l-&s)Nw(&O)dc = W’”(s) +V(o’(s),I‘o’(s)]T, (10)

the MTL can be described by the equation as

0

From the (11) the admittance equations taking nonzero initial conditions into account have a form as

and submatrices Y,,(s)=-O;~(s)@,,(s), Y,,(s)=O;i(s),

Yzz(s)=42z(s)@;~(s), and YzI(s)=Yi(s) due to the reciprocity of the MTL. A s the MTLs are uniform the equality Y,l(s)=Y,,(s) is also valid. Further 0 and E mean the zero and identiry matrix, respectively. Now considering the k-th MTL of the circuit the (12) can he rewritten in a compact matrix form

I~(s)=Y~(s)V~(~)-X~(~)W:O’(S) , (13)

and substituted into the MNA equation (2). This leads to the resultant MNA matrix equation of the form

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[ k d

I ’ (14)

I’ V,(s) = G , +sc, +CI),Yk(s)D:

’ I,W +c,v,(o)+ C o , x , ( ~ ) w : O ~ ( s ) [ k=l

In this point the s-domain solution is prepared to be used in the 1 P N I L T resulting in nodal voltage and/or branch current time behaviours. However, it can also serve as the basis for MTLs‘ boundary conditions assesment enabling to apply the Z&NLT method when the MTLs‘ voltage and/or current wave distributions shall he the results.

3. USAGE OF 2D LAPLACE TRANSFORMATION

In the case the voltage and/or current distributions along MTLs’ wires are wanted to visualize them the 2D Laplace transformation is used as follows. Performing second partial Laplace transformation with respect to a coordinate x on the equation ( 5 ) the result is

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qW(q,s)-W(O,s) =M(s)W(q,s)+NW(q,O) (15)

and finally

W(q,s) = [qE - M(s)]-’[W(O,s) + NW(q,O)l

where E means 2N-order identity matrix.

Now taking the (8) into account, denoting

W(q,O) = Wdq) =[Vll(q),~,(q)l‘ (17)

as a vector of the q-domain Laplace transforms of initial voltage and current distributions, and marking

D(q,s) = [qE - M(s)]-’ , (18)

the equation (16) can he rewritten into the form

W(q,s) = D(q,s)W‘’’(s) + W,’o’(q,s) . (19)

It is evident that the term

W,’o’(q,s) = D(q,s)NWo(q) (20)

will affect the result only if the nonzero initial conditions exist, i. e. the (17) takes a nonzero value.

To determine W“’(s) its items have to he assessed under

the (8). Namely, using the result (14) the voltage V“’(s) appertaining to the k-th MTL can he extracted from the equation

V,(s) = D:V,(s) , (21)

and the current I‘”(s) then from the equation (13)

When the partial inverse Laplace transformation of the (19) w. r. to x is done and x =I substituted we have

L;[w(q,s)], = L;[D(q,s)], w”(s)+LI;[w’(q,s)L, (22)

and finally

W(2’(s)=@(s)W(’’(s)+W~o’(s). (23)

This equation is, of course, again the equation (1 I), hut in its compact form. The proof can also he performed using the matrix analogies of well-known relations for Laplace transforms of the exponential function and the convolution integral.

From the last two equations follows that if there are nonzero initial conditions then not only the convolution integral but also partial inverse Laplace transformation L:[.] can he used to obtain the W“’(s) needed in the equation (14). This can be advantageous e. g. in the case when such a procedure figures as a part of the 2ITNILT technique.

Note: The above used partial Laplace transformations are defined as [12]

01 I

L,[f(x,t)]= jf(x,t)e-‘% , L,[f(x,t)]= jf(x,l)e-‘dr, 0 0

because the MTL has a finite length / and for x > I the function f(x,f) is supposed as zero.

4. EXAMPLES

Consider a linear circuit with two uniform lossy coupled multiconductor transmission lines, two-conductor MTLl and four-conductor MTLz, in Fig.2 [13].

A A L

U

10onQ

Figure 2. Linear circuit with two MTLs

~ 0 t h MTLS are of the same lengths / = o . l m , per-unit- length matrices are stated in [13]. A 1V pulse with 0.4 ns rise/fall time and 5 ns duration is applied at the input.

The time-domain solution vy(f) =L;’oI,(s)} is found by the ID-NET method [8], see an example in Fig.3.

I 0.5 1 1.5 2

.lo“ T i m [SI

Figure 3. Waveforms of input and output voltages

The voltage and current wave distributions on MTLs’ wires w(x,f) =Li{W(q,s)}are found by the 2DNILT method [9,10], see examples inFig.4.

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5. CONCLUSION

In the paper the nontraditional approach to a Simulation of transient phenomena in MTL circuits combining 1D and 2D Laplace transformations was elaborated. If only time behaviours in circuit nodes and/or branches are needed a 1D-NILT method is applied to get the f-domain solution. If voltage and/or current wave distributions on MTLs’ wires are moreover to be found then a 2D-NILT method is used to get the (x,twomain solution. Newly boundary conditions into the (qswornain solution are incorporated utilizing just the MNA equation method which enables to simulate arbitrarily complic,ded circuits. It has also been shown how the method can very effectively be realized in terms of universal scientific--technical language Matlab.

6. REFERENCES

[I] C. R. Paul, Analysis of Mui’ticonductor Trammission Lines, J. Wiley & Sons, New York, 1994. 121 A. K. Goel, High-speed VLSI Interconnections: Modeling, Anolysis, andSimulotion, John Wiley & Sons, New York, 1994 [3] J.A.B. Faria, Multiconducfor Transmission-Line Shuchrres; Modal Analysis Techniques. I. Wiley & Sons, New York, 1993. [4] S. Lum, M.S. Nakhla, and (2. J. Zhang, “Sensitivity Analysis of Lossy Coupled Transmission Lines,” IEEE Tramactiom on Microwave Themy and Techniques, vol. 39, no. 12, pp. 2089- 2099, December 1991. [5] E.Chiprout, M.S. Nakhla, Asymptotic Wavefom Evaluation and Moment Matching for Interconnect Analysis. Carleton University, Kluwer Academic Publisher, Boston, 1994. 161 L. BranEik, “Time-Domaiii Simulation of Multiconductor Transmission Line Systems under Nonzero Initial Conditions,” in Proc. The IS” ECCTD’OI, Espoo, Finland, August 2001, vol. l,pp.l3-76. [7] L. BranEik, “Simulation of Multiconductor Transmission Lines Using Two-Dimensional Laplace Transformation,” in Proc. The IS“ ECCTD’OI, Espoo, Finland, August 2001, “01.2, pp. 133 - 136. 181 L. BranEik, “Improved Numerical Inversion of Laplace Transfoms Applied to Simulation of Distributed Circuits,” in Proc. XI ISTETOI, Linz, Auslria, August 2001, pp. SI - 54. [9] L. BranEik, ”Numerical Inversion of Two-Dimensional Laplace Transforms Based on FFT and Quotient-Difference Algorithm,” in hoc. ICFS‘2ilO2, Tokyo, Japan, March 2002, pp. 13-15- 13-20. [IO] L. BranEik, “Improved Method of Numerical Inversion of Two-Dimensional Laplace Tnmsfonns for Dynamical Systems Simulation,” in Proc. The Y“ IEEE ICECS’2002, Lhbrovnik, Croatia, September 2002, pp. 385 - 388. [ I l l F. R. Ganbnacher, The Theory ofhfalrices. Chelsea, New York, 1917. 1121 V.A. Ditkin, A.P. Prudnik~ov, Operotiona! Calmlvs in Two Variables and its Applications, New York, Pergamon 1962. [I31 R. Sanaie, E. Chiprout, 1vI.S. Nakhla, “A Fast Method for Frequency and Time Domain Simulation of High-speed VLSI Interconnects,” IEEE Tronsactionr on Microwave Theory and Techniques, vol. 42, no. 12, 1994, pp. 2562.2571.

ACKNOWL,EDGEMENTS

The work was supported by grant GACR No.102/03/0241 and realized within the scope of Research programmes MSM 26220001 1 and MSM 262200022.

Voltage on the 1”wire of the MTL, , . . . . . . : .... ... .. 0.6, . .

Voltage on the Znd wire of the MTLz . . .,.. . ... . . . . .

... . 0.04 ,,,, ... .... :

Voltage on the 3‘d wire of the,,MTL, . . . .. .

I .. ... .. . .. 0.6, .... .

, ‘1.5 2

x[ml 0 0.5 ’ x t 14

Voltage on the @wire of the MTL, . . , . . . . :.

. . . . 0.02 , ,, . . ..:..’ ’ ’

Figure 4. Voltage wave distributions on MTL, Wires

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