Using the Transfer Function LTI objects

(Uses Matlab): tf, step

(Sample):
Lumped Distributed Delay Line - large delay-line circuit normally require hundreds of differential equations to sufficiently simulate them. It is often regarded as a "lumped" approximation to a continuously distributed delay line described by a partial differential equation. Here we are considering a sample line with 13 inductors and 12 capacitors and the circuit is doubly-terminated by resistive values. The simulation results clearly depict the wave on the line corresponding to (a) matched and (b) unmatched line where the reflecting wave occurs.

Rather than using State-Space Representation that require system of first order differential equations in matrix form, we will use tf to create the Transfer Function LTI objects instead, along with the 2-Port Transmission Matrix known as the ABCD. Please refer to ABCD for 2-Port Network Transmission concepts and derivation of the ABCD for SERIES and SHUNT element.

The voltage transfer function, vL / vS, m-Script of the unmatched delay line,

rS=50;
rL=10;
L=4;
C=0.14E-3;
SERIES=[1,tf([L/2,0],1); 0,1];
SHUNT=[1,0; tf([C,0],1),1];
LC=SERIES*SHUNT;
TF=[1,rS;0,1]*LC*LC*LC*LC*LC*LC*LC*LC*LC*LC*LC*LC*SERIES*[1,0;1/rL,1];
% display for 0 to 2 sec
T=0: .01: 2;
step(1/TF(1,1),T)

If you want to see a rational expression in s of any line on the script shown above, simply remove the semi-colon character at the end of the line, save the m-file and select "Run Script" in Matlab.

Given next the Step Response, observe the reflection on the wave created due to unmatched terminations.

For the matched terminations, rS=rL=50, we have,

Expression of the Rational Transfer Function of a Matched Lumped-Delay-Line is available in Maple V, under the title "Matrices and ABCD-Network Paramter".