Figure 1: Equivalent Representation of the Transmission Line
Figure 2: N-port S-parameters and the Port Voltage Transfer Function
Time-domain Impedance, S-parameters and the Volt-
age Transfer Function
Let us consider a simple circuit where the voltage source E(t) is connected through the reference impedance
Z
0
to the transmission line, see Figure 1. The line here is represented by its equivalent circuit enclosed into
the pink box. Essentially, the model consists of two parts affecting each other through the propagation
operator W(s), that contains the line’s delay and possibly dispersion. The component Z denotes the line’s
characteristic impedance, possibly frequency dependent if the line is lossy. With zero initial conditions, a
step voltage E(t) produces an immediate change in the input current i
1
(t) and voltage v
1
(t).The backward
current i
b1
will appear at the front end only after the wave travels twice through the propagation operator,
i.e. not before two times the line’s delay. During all this period, the ratio of the input voltage to input current
remains equal the line’s characteristic impedance - if it is constant - or slightly varies if the line has skin or
dielectric losses [1]. When the reflected current comes back this ratio may change if the load impedance
differs from the line’s characteristic impedance. If the passive load matches characteristic impedance, the
backward current i
b1
is zero because i
f2
= i
z2
= −i
2
and the sum (i
f2
+ i
2
) remains zero.
Since the reference resistance and source voltage are known, the line’s impedance can be determined
from the input voltage and the current defined as i
1
(t) = [E (t) − v
1
(t)] /Z
0
. The ratio becomes:
2
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