Input Impedance of a Terminated Lossless Transmission Line
Consider Figure 3.15.1, which shows a lossless transmission line being driven from the left and which is terminated by an impedance
on the right. If
is equal to the characteristic impedance
of the transmission line, then the input impedance
will be equal to
depends on both
and the characteristics of the transmission line. In this section, we determine a general expression for
in terms of
, the phase propagation constant
, and the length
of the line.
Using the coordinate system indicated in Figure 3.15.1, the interface between source and transmission line is located at
. Impedance is defined at the ratio of potential to current, so:
Now employing expressions for
from Section 3.13 with
, we find:
Multiplying both numerator and denominator by
in the above expression is:
Equation 3.15.3 is the input impedance of a lossless transmission line having characteristic impedance
and which is terminated into a load
. The result also depends on the length and phase propagation constant of the line.
is periodic in
. Since the argument of the complex exponential factors is
, the frequency at which
; and since
, the associated period is
. This is very useful to keep in mind because it means that all possible values of
are achieved by varying
. In other words, changing
by more than
results in an impedance which could have been obtained by a smaller change in
. Summarizing to underscore this important idea:
The input impedance of a terminated lossless transmission line is periodic in the length of the transmission line, with period
is also the period of the standing wave (Section 3.13). This is because – once again – the variation with length is due to the interference of incident and reflected waves.
Also worth noting is that Equation 3.15.3 can be written entirely in terms of
depends only on these two parameters. Here’s that version of the expression:
Finally, note that the argument
appearing Equations 3.15.3 and 3.15.5 has units of radians and is referred to as electrical length. Electrical length can be interpreted as physical length expressed with respect to wavelength and has the advantage that analysis can be made independent of frequency.
Get the latest tools and tutorials, fresh from the toaster.