Transient Analysis of Interconnects by Means of Time-Domain Scattering Parameters

Transcription

1 Transient Analysis of Interconnects by Means of Time-Domain Scattering Parameters Wojciech Bandurski, Poznań Univ. of Technology Poznań, Piotrowo 3a, Poland, INTRODUCTION Time-domain transmission line analysis has been a subject of interest in recent microwave studies and in interconnect analysis and simulation. For the analysis of linear microwave circuits, we use frequency-domain scattering parameters to evaluate circuit performance. The time-domain responses of such a circuit are obtained by taking the inverse Laplace transforms of corresponding frequency domain functions. However, when the transmission lines are terminated with nonlinear loads, the above technique is no longer valid and the only possible solution is the time-domain analysis. A similar situation takes place in the field of interconnect analysis and simulation, where the line is characterized by a set of two-port parameters in timedomain. Among the two-port parameters the most suitable are scattering parameters. The advantages of scattering parameters reside in the short time of duration and an exponential like form of curves [2,5]. These two features allow for applying of very fast and low memory consuming the recursive convolution algorithm to calculate transients in the presence of nonlinear loads. Recently similar approach has been proposed in [7]. The authors approximated characteristic impedance and propagation function of the line in frequency-domain and next by inverse FFT found the time-domain functions. In the last step these functions were fitted with exponential functions and cubic polynomials. Finally they applied efficient recursive convolution [6,7]. In this paper, basing on integral eqs. of the transmission line [3], an approach for calculation of scattering parameters of the nonuniform lossy line in the time-domain is presented. It allows for obtaining an analytical form of these parameters in some cases ( for instance lossless exponential line ) and numerically in other cases. In the uniform lossy line case the approximation by means of elementary functions is possible and the order of approximation is estimated by admissible error and line parameters. THEORY In [3] basing on the method of characteristics a new formulation of eqs. of the nonuniform transmission line has been introduced. Instead of voltage and current as the dependent variable the forward and backward current waves are used in these eqs.. The current waves have the form (1) where: u, i - voltage and current along the line, y - space coordinate normalized with respect to the line length l, τ - temporal coordinate normalized with respect to the line delay, Y c (y) - characteristic admittance of the line. Derivation of scattering parameters requires of introducing normalized current waves in the following form:

2 (2) in different abbreviations we have (3) Finally integral eqs. for normalized current waves are (4a) (4b) where l - length of the line, R(y), G(y), C(y), L(y) - line parameters. Eqs. (4) are solved by the method of successive approximation. The solution has the following operator form (5) where Supplying the transmission line as it is shown in Fig. 1, and taking into account eqs. (3) we obtain Fig. 1 Quasi matched line excited by the current source j(τ) = 2 Y c ( 0 ) δ(τ) a 1 (τ) =δ(τ) and a 2 (τ) = 0. The following operator expressions for scattering parameters result

3 from eqs. (5) (6) The expressions (6) represent the integral operator series (7) The nonuniform lossy transmission line is reciprocal what implies s 21 =s 12. Parameter s 22 can be obtained from eqs. (5) when supplying the line from right termination and poses the form (8) In general case n-fold integration in formulas (7) and (8) are not possible to perform analytically, however in the case of nonuniform lossless transmission lines with constant velocity of propagation mentioned formulas can be applied in some cases. The first term in series (7) and (8) for scattering parameters of lossless nonuniform transmission line with constant velocity of propagation are (9) where It can be proved [3] that series (7) and (8) are fast convergent when (10) Final form of series(7) and (8) is following: (11) Pulse functions p 2n+1 (τ), p 2n (τ) and q 2n+1 (τ) follow from n-fold integration of successive terms in series (7) and (8). For lossy uniform line it can be proved that the number N of terms, needed to approximate series (11) with error, fulfill the following condition: (12) where l - length of the line, a, b - previously define parameters.

4 RESULTS 1) Exponential line In the papers [1,4] authors consider an exponential line. In [1] in the first step parameters in s domain are derived. Next applying inverse Laplace transform three first terms of parameters s 11 (t) and s 21 (t) have been obtained. Every term in these expressions is rather complicated and contains Bessel functions or their second derivatives. In [4] authors apply Allen's method to obtain step response of exponential line. Here we consider exponential line with characteristic impedance The first three calculated terms of series s 11 (τ) and s 21 (τ) are: Fig. 2 The successive terms p k (τ) k = 1,2,...,8 of series (7), s 11 (τ) p 1 (τ) +p 3 (τ) +p 5 (τ) +p 7 (τ), and s 21 (τ) p 2 (τ) +p 4 (τ) +p 6 (τ) +p 8 (τ) Fig. 3 Scattering parameters s 11 (τ), s 21 (τ) ofthe exponential line

5 All terms p k (τ) k=1,2,...8 are plotted in Fig. 2. Calculation are very simple and can be done by hand or any program for symbolic calculations as Mathematica or Derive. One can observe that terms p k (τ) have decreasing amplitude and are delayed to the right. In Fig. 3 we have scattering parameters s 11 (τ) and s 21 (τ), as a superposition of pulse functions p k (τ), which are in good agreement with results obtained in [4]. 2) Uniform lossy line Example 1: uniform line with parameters R=2.5Ω/cm, G=0.5mS/cm, C=4pF/cm, L=10nH/cm, l=10 cm basing on formula (12) we find N=1 for error =0.02. In this case we have the following scattering parameters: The line was excited by a pulse with amplitude E=1V, rise and fall time T r =T f =0.5ns pulse width PW=2ns. The line was terminated by resistor R L =50 Ω (Z c =50 Ω). Input and output currents are shown in Fig. 4. Fig. 5 Output voltages in example 2, subscript d denotes exact results Fig. 4 Input and output currents of the line, subscript d denotes exact results. Example 2: uniform line with parameters R=10.53Ω/cm, G=0. ms/cm, C=1.17pF/cm, L=2.9nH/cm, l=1cm. The line was excited by a ramp with amplitude E=5V, rise time T r =20ps, and was terminated by capacitor C L =0.1pF. Output voltages are shown in Fig. 5. The formulas for scattering parameters were as in first example. The results in both examples are good, as one can see in Figs. 4 and 5. In the both examples recursive convolution was applied.

6 CONCLUSIONS The presented approach permits for calculation of scattering parameters of lossy nonuniform transmission line. Instead of approximation by a chain of segments of transmission line with constant parameters as in [2], calculation of successive terms of series (7) and (8) is proposed. The terms p 2n+1 (0,τ), p 2n (1,τ), q 2n+1 (1,τ) of that series represents n-fold integral. In some cases e.g. exponential line, uniform lossy line these n-fold integral expressions are calculated analytically. In other cases integration have to be perform numerically. When the series (11) are fast convergent only a few first terms of that series are required. More over successive terms "begin later" and in finite period of time the truncated series (11) my be "near" the exact solution. It is important to notice that as result of the matching at the input and output ports of the line s 11 (τ) and s 21 (τ) have very short duration-period. Figs. 2 and 3 confirm this observation. For the uniform lossy line case, very simple scattering parameters formulas give results, which present sufficient level of exactness (see Figs. 4 and 5). On the other side simplicity of these formulas guarantees low memory and short time of simulation. REFERENCES [1] C.W.Hsue,"Time-domain Scattering parameters of an exponential transmission line," IEEE Trans. Microwave Theory Tech., vol. MTT-39, pp , Nov [2] J.E.Schutt-Aine,"Transient analysis of nonuniform transmission lines," IEEE Trans. Circuit Theory, vol. CAS-I-39, pp , May [3] W.Bandurski,"Transient analysis and simulation of nonuniform lossy transmission lines,"int. J. of Microwave and Millimeter-Wave Computer-Aided Engineering, vol. 2, pp , [4] P.Bouchard et all,"transients on lossless exponential transmission lines using Allen's method,"ieee Trans. Microwave Theory Tech., vol. MTT-41, pp , Nov [5] I.Maio et all,"influence of characterization on transient analysis of nonlinearly loaded lossy transmission lines,"ieee Trans. Circuit Theory, vol. CAS-I-41, pp , March [6] A.Semlyen et all,"fast an accurate switching transient calculation on transmission lines with ground return using recursive convolution," IEEE Trans. on Power Apparatus Systems, vol. PAS-94, pp , [7] Q.Yu, S.Kuh, " An accurate time domain interconnect model of transmission line networks" IEEE Trans. on Circuits Sys. I, vol. 43, pp , March 1996.