october1999 p68

Direct s-matrix Analysis of Nonuniform Transmission Lines Here is an analysis that allows arbitrarily varied characteris...

1 downloads 48 Views 215KB Size
Direct s-matrix Analysis of Nonuniform Transmission Lines Here is an analysis that allows arbitrarily varied characteristic impedances and propagation constants

By B. Yu. Kapilevich Siberia State University of Telecommunications and Informatics his paper discusses the direct analysis of nonuniform transmission lines (NTL) with arbitrarily varied characteristic impedances and propagation constants. The s-matrix approach is applied to determine a reflection coefficient. The Wronskian determinant is used to control the accuracy of calculations. This technique is free of any limitations concerning longitudinal profile of the char- ▲ Figure 1. A section of nonuniform transmission line, ζ= x/L. acteristic impedance and effective dielectric constant. Some numerical examples are presented to illustrate the method An alternative solution is reached by using for lossless and lossy NTL. scattering parameters directly. This approach The wide applications of NTL are encourage- has been successfully applied to the determinament to develop efficient analysis methods. tion of reflection and transmission coefficients Using nonlinear Riccati equations, an analytical in inhomogeneous dielectric filled waveguides solution for a reflection coefficient can be [5-7]. The idea can be applied to NTL with arbiobtained for limited types of NTL, primarily trary laws describing longitudinal impedance exponential [1] and low power [2, 3]. Different and propagation coefficients. As shown in an numerical techniques have been suggested for article by T. W. Kao [5], the Wronskian determian arbitrary impedance profile. They are based nant from Figure 1 can aid in checking the accuon the representation of NTL by many small racy of the solutions. In this paper, the numericascaded uniform transmission lines. cal technique demonstrates facilities of a direct To improve the efficiency of this technique, s-matrix analysis of NTL, including a line with an ideal linearly varied NTL has been recently arbitrary loss. applied using exact an ABCD matrix description [4]. As a result, the stepped-type of impedance Basic problem formulation profile can be replaced with fewer ideal linear Consider an arbitrary NTL characterized by varied NTL sections. However, the exact ABCD position dependent characteristic impedance matrix can be written in closed analytical form, Zc(ζ) and a propagation coefficient β(ζ) where assuming that a propagation coefficient is not ζ=x/L is a normalized longitudinal coordinate, positioned dependently. To avoid unfavorable and L is a length of a NTL section, as shown in error accumulation, the number of linearly var- Figure 1. The voltage in NTL within the interval ied NTL sections should be controlled in a process of numerical calculations using addi- 0 Zend: = 100; ‘impedance at the end of NTL in ohms’ > Z:=Zst – (Zst-Zend) × (14 × (0.5-zeta)^2); ‘parabolic NTL impedance in ohms’ > plot(Z,zeta = 0..1); ‘impedance plot’ Calculating eeff and w/h of microstrip line: > epsilon: = 10; ‘substrate dielectric constant’ > R: = Z × sqrt(epsilon)/377; > A: = ln((–1.386+Pi/R)) + 1; > P: = 0.637 × A+1/R; ‘w/h ratio as a function of z’ > Eps:=(epsilon +1)/2+((epsilon-1)/2) × (1+10/P)^ (-0.5); ‘eeff as a function of z’ > plot({Eps,P},zeta=0..1); ‘plotting eeff and w/h’ Solving differential equation: > DZ:=diff(Z,zeta); ‘differentiating impedance function’ > E0:=7.793; ‘initial eeff ‘ > f0:=0.1; ‘starting frequency point in GHz’ > L:=35; ‘length of NTL in mm’ > M:=100; ‘number of frequency points to be calculated a reflection coefficient’ > Ec:=11/2+9/2 × 1/((1+10 × 1/(–.637 × ln (–1.386+377/10 × Pi × sqrt(10)/(100-320) × (.5-zeta)^2)) -.637+377/10 × sqrt(10)/(100-320 × (.5-zeta)^2)))^.5): ‘eeff as a function of z for the above specified NTL configuration’ > for i from 1 to M do i: = i; f: = 0.1 × i; > deqn: = diff(y (zeta),zeta$2) + (DZ( zeta)/Z ( zeta)) × diff(y (zeta), zeta)+((.2 × Pi × (f) × L/30)^2) × (Ec – 0.67 × I) × y (zeta): ‘differential equation for lossy NTL’ > init: = y(0) = 1,D(y)(0) = 0: ‘the first initial condition’ > F1: = dsolve({deqn,init},y( zeta),numeric): > F1(1): f1: = eval(y(zeta),%): f1d: = eval(diff(y(zeta),zeta),%%): ‘solutions for the first initial condition’ > init: = y(0) = 0,D(y)(0) = 1: ‘the second initial condition’ > F2:=dsolve ({deqn,init},y ( zeta),numeric): F2 (1): > f2: = eval (y(zeta),%): f2d: = eval (diff(y(zeta),zeta),%%): ),%%): ‘solutions for the second initial condition’ > b1: = 0.2 × Pi × (f) × L × sqrt (E0)/30: ‘propagation constant at the NTL input’ > b2: = b1: ‘propagation constant at the NTL output’ > S11: = (f1d-I × f2d × b1+I × b2 × f1+b1 × b2 × f2)/(–f1d-I × f2d × b1-I × b2 × f1+b1 × b2 × f2): ‘the reflection coefficient’ > w[i]: = 1.–abs((f1 × 2d-f2 × f1d)); ‘Wronskian evaluating’ > evalf(S11): SS[i]: = evalf(abs(%)); ‘the module of the reflection coefficient’ > od: with(plots): listplot([seq([i,SS[i]],i = 1..M)]); ‘plotting S11’ > listplot([seq([i,w[i]],i = 1..M)]); ‘plottingWronskian’

76 · APPLIED MICROWAVE & WIRELESS