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Calculate the reflectance of the terminated waveguide.
That is, find the Laplace transform of the return wave divided by the
Laplace transform of the input wave going into the waveguide. In general,
the reflectance of an impedance step for force waves (voltage waves in
the electrical case) is
|
(N.1) |
This is easily derived from continuity constraints across the
junction. Specifically, referring to Fig. N.1b, let
denote the physical force and its traveling-wave
components within the ``pseudo-infinitesimal-generalized-waveguide''
defined by the element impedance , with the `' superscript
denoting a right-going wave.N.1 Similarly, let
denote the velocity and its component wave variables on
the side of the junction at impedance , and let
denote the corresponding quantities on the
element-side of the junction at impedance . Again, the `'
superscript denotes travel to the right. Then the physical continuity
constraints imply
By the definition of wave impedance in a waveguide, we have
Thus,
Defining
and
, we have
|
(N.2) |
Now that we've solved for the junction force , the outgoing
waves are simply obtained from the force continuity constraint,
:
Finally, the force-wave reflectance of an impedance step from to
can be found by solving Eq. (N.3) and (N.2) for
with
set to zero:
as claimed.
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