[IEEE 1991., IEEE International Sympoisum on Circuits and Systems - Singapore (11-14 June 1991)] 1991., IEEE International Sympoisum on Circuits and Systems - An efficient SPICE model for lossy and dispersive coupled transmission lines

AN EFFICIENT SPICE MODEL FOR LOSSY AND DISPERSIVE COUPLED TRANSnISSIoN LINES Josd I . Alonso, Jose Borja and Ftlix P&ez Departamento de Sefiales, Sistemas y Radiocomunicaciones ETSI. Telecomunicacibn. Ciudad Universitaria. 28040 Madrid. SPAIN. ABSTRACT A simple circuit model for the time domain study of multiconductor tranmission lines with CAD standard software such as SPICE is presented. The model consists of a group of decoupling networks. implemented with dependent voltage and current sources, which are obtained from a nodal analysis in the frequency domain and decoupled dispersive lines. These lines are modeled with ordinary lossless TEM lines and analogical filters, which are designed to have the required propagation characteristics. INTRODUCTION Evaluation of the time domain response of multiconductor transmission lines is of great importance in the analysis of the crosstalk in fast digital circuit interconnections frequently encountered in the design of digital computers and communication systems. . Several techniques for the computation of the line response have been proposed. The conventional technique for evaluation of the time domain response starting from data in the frequency domain using Fast Fourier Transform (FFT). is not valid when the line has nonlinear terminations 111. Another alternative approach i s obtaining an equivalent circuit model for this kind of structures. In this approach, the coupled transmission line system is decoupled into a set of independent transmission lines by linear transformation (normal-mode theory). In this way, coupled lossy dispersive lines are modeled as an interconnection of uncoupled lossy dispersive lines and mode coupling and decoupling networks. These networks have been realized either in terms of congruent transformer banks [21 or linear dependent sources [31 making them compatible with CAD programs such as SPICE. This technique has been used in [SI to obtain a circuit model, but its utility is limited for complexity of the model. In this paper a simple and efficient circuit model for coupled dispersive lines compatible with SPICE is presented. The technique is based on the previous approach. but single dispersive transmissibn lines are modeled by lossless TEM lines and analog filters, which are designed to have the required propagation characteristics. BASIC PRINCIPLES In a system of N coupled dispersive and lossy lines the relationship between voltages and currents can be written in the frequency domain as: where: v(x.w) = [Vi (x,o) . . . VN(X,0ll' i(x,w) = [il(X,W) . . . iN(X,W)lT t vector of voltages in the lines. L vector of currents in the lines. [Z(o)] = matrix of series impedances. [Y(w)l s matrix of shunt admitances. x s distance to the end of the lines. The elements of [Z(o)l and [Y(w)l outside the principal diagonal take into account the coup1 ing. Under certain conditions matrixes [Svl and [SI] can be found such that the auxiliary voltages and currents e(x,o) and j(x,o) defined as: are related by the following: ( 4 ) ( 5 ) where [rl and [Yo] are diagonal matrixes. Equations (4 ) and ( 5 ) represent a system of N decoupled transmission lines where the nonzero elements of [Yo]-' and [rl, Zoi and 73 represent a j (x,w) = -[rI[Yole(x,w) the characteristic impedances and propagation constants of the lines. In fact this will be possible, as long as [Z(o)l[Y(o)l can be diagonalized and has the same eigenvalues as [Y(o)l [Z(w)l. being 7i the N corresponding eigenvalues. In that case, [Sv]=[aijI must contain the N independent eigenvectors of [Z(o)l[Y(o)l and [SI]=[bi~l-~ the N independent eigenvectors of [Y(o)l[Z(o)l. associated to the eigenvalues 71 and [YOI=[SII-'[Z(~)I [SVI [ri Since the eigenvalue problem depends upon frequency. so will happen with the 71. 201, aij and bij although this dependence is not important in many useful cases. I ~ A T I O N OF THE CIRCUIT Once the matrixes [SvI,[S~l,[rl and [Yo] are found, the relationships (2) and (3) can be sinthetised in circuital terms by a network of coupling generators as shown in fig.1. I' i +I ,ih I 1 The transmission lines can be implemented as described in [41, where filters that take into account the dispersive effects are obtained: and N 2 3 s +gijs+hij i js+ki j zap = c 2 (7) j=i s +e The coefficients of the previous expressions are initially found by specific methods and afterwards a least-squares matching procedure from available data in the frequency domain, for the effective dielectric constants and the characteristic impedances. allows for a more exact solution. Stability considerations must be taken into account on the minimisation procedure. The lumped elements of the model are obtained synthesizing the functions (6) and (7) by conventional techniques. In order to validate the accuracy of the modeling technique and demonstrate the usefulness of the circuits models. the transient response of two microstrip coupled lines is computed. A Gaussian pulse is chosen with a 3dE3 pulsewidth of r-50psg. For this example. the length of the line is 40mm, the microstrips are on a GaAs substrate (cr =12.2) of H=O.635mm thick, with W=O.Smm and S=l.Omm. Both the even and the odd mode line length in the model are divided into 5 sections and each section was modeled in the same manner. Fig.2 shows the effect of dispersion on the effective even and odd mode dielectric constants from [SI and for the model. Fig.3 shows the SPICE circuit and the values of the elements for each section. The time domain response of the signal line (the line with the impressed signal) and the sense line (the line adjacent) using SPICE and DFT techniques are shown in figs. 4 and 5. REFERENCES Fig.]. Circuit model for N coupled microstrip lines. The N equivalent decoupled lines [11 A.R. Djordjevic. T.K. Sarkar and R.F. of represent the propagation of the N modes and Harrington* "Time-domain response they are coupled at the terminations by the multiconductor transmission lines". Proc. dependent sources ZEEE. Vo1.75, No.6, pp.743-764. June 1987. 2714 [21 F. Y. Chang. "Transient analysis of lossless coupled transmission lines in a nonhomogeneous dielectric medium". IEEE Trans. Microwave Theory Tech. Vol. MTT-18, pp. 616-626, Sept. 1970. [31 V.K. Tripathi and A. Hill. "Equivalent circuit modeling of losses and dispersion in single and coupled lines for microwave and millimeter-wave integrated circuits". IEEE Trans. Microwave Theory Tech., Vol. MTT-36, pp. 256-232, February 1988. B nl- .SUBCKT SIMUL 1 2 3 4 V1 1 51 0 V2 2 52 0 V 3 3 5 3 0 v 4 4 54 0 Frequency -GHz- +A F1 0 5 V1 .5 F2 0 6 V1 .5 F3 0 5 V2 .5 F4 6 8 V2 .5 F5 70V3 .5 F6 0 8 V3 .5 F7 0 7 V4 .5 F8 0 8 V4 .5 0 E l 5 1 9 5 0 1 E!!!; + E2 52 10 5 O 1 . "-$I- E39 0 6 0 1 E4 10 0 0 6 1 E5 1 1 0 0 7 1 E6 12 0 7 0 1 E753 1 1 8 0 1 E8 54 12 8 0 1 X I M P 6 7 I M P 0 XPAR 5 a PAR .ENDS SIMUL , C i T = 66,0259 psg T = 61,9122 psg Zoe = 54.2776 R Ce = 98,9084 fF Le = .3912 Cle = .lo30 pF Lle = .5709 nH C2e = . 1978 pF Zoo = 35,9624 R CO = .1419 pF Lo = .1995 nH Clo = 27,0387 fF Llo = 1.1472 nH C20 = .2838 pF Fig.3. Values of the elements of one section. [41 J . I . Alonso. "AnBlisis en el dominio del tiempo de circuitos de microondas mediante la transformada Z. Aplicacidn a1 estudio de efectos dispersivos en lineas de transmisidn" . M. S. Thesis. UPM. September 1989. [SI R. Garg and I .J . Bahl. "Characteristics of coupled microstriplines". IEEE Trans. Microwave Theory Tech. , Vol. MTT-27, July 1979, pp. 700-705. Also see correction in IEEE Trans. MTT-28. 1980, p.272. . 0 0 0 - 0 ***** COUPLING NETWORK ***** 0 * d' a nl- R ni- Effect of dispersion on the effective even an odd mode. - From 151. ----- This model. 2715 :-I - SPICE a d‘ a d‘ E! 10 #I # ‘ Tima -nog- Fig.4 . Pulse dispersion on coupled lines, signal line response. 8 d‘ a d‘ E! # - SPICE - T i n e -nog- 10 9 - 1 I 1 I I a4o 0.4 aso ass 0.00 Fig.5 . Pulse dispersion on coupled lines, sense line response. COtKLUSIowS AMNoULEDG- A simple circuit model for the time domain study This work was supported by the project of multiconductor transmission lines has been TIC-0023/14 of the National Programme on presented. The developed model is compatible Information and Communication Technologies with time domain comercial.programs, such as (l”TIC). SPICE. The case test presented shows good agreement between the results obtained with our model and those obtained with DFT technique. 2716