In this section, scattering relations will be derived for the general case of N waveguides meeting at a load. When a load is present, the scattering is no longer lossless, unless the load itself is lossless. (i.e., its impedance has a zero real part). For , will denote a velocity wave traveling into the junction, and will be called an ``incoming'' velocity wave as opposed to ``right-going.''C.9
Consider first the series junction of
waveguides
containing transverse force and velocity waves. At a series junction,
there is a common velocity while the forces sum. For definiteness, we
may think of
ideal strings intersecting at a single point, and the
intersection point can be attached to a lumped load impedance
, as depicted in Fig.C.29 for
. The presence of
the lumped load means we need to look at the wave variables in the
frequency domain, i.e.,
for velocity waves and
for force waves, where
denotes
the Laplace transform. In the discrete-time case, we use the
transform instead, but otherwise the story is identical. The physical
constraints at the junction are
(C.90) | |||
(C.91) |
The parallel junction is characterized by
(C.92) | |||
(C.93) |
The scattering relations for the series junction are derived as
follows, dropping the common argument `
' for simplicity:
(C.94) | |||
(C.95) | |||
(C.96) |
Similarly, the scattering relations for the loaded parallel junction
are given by
It is interesting to note that the junction load is equivalent to an st waveguide having a (generalized) wave impedance given by the load impedance. This makes sense when one recalls that a transmission line can be ``perfectly terminated'' (i.e., suppressing all reflections from the termination) using a lumped resistor equal in value to the wave impedance of the transmission line. Thus, as far as a traveling wave is concerned, there is no difference between a wave impedance and a lumped impedance of the same value.