Loaded Waveguide Junctions
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
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
where the reference direction for the load force is taken to be opposite that for the . (It can be considered the ``equal and opposite reaction'' force at the junction.) For a wave traveling into the junction, force is positive pulling up, acting toward the junction. When the load impedance is zero, giving a free intersection point, the junction reduces to the unloaded case, and signal scattering will be energy preserving. In general, the loaded junction is lossless if and only if re, and it is memoryless if and only if im.
The parallel junction is characterized by
For example, could be pressure in an acoustic tube and the corresponding volume velocity. In the parallel case, the junction reduces to the unloaded case when the load impedance goes to infinity.
The scattering relations for the series junction are derived as
follows, dropping the common argument `' for simplicity:
where is the wave impedance in the th waveguide, a real, positive constant. Bringing all terms containing to the left-hand side, and solving for the junction velocity gives
(written to be valid also in the multivariable case involving square impedance matrices ), where
Finally, from the basic relation , the outgoing velocity waves can be computed from the junction velocity and incoming velocity waves as
Similarly, the scattering relations for the loaded parallel junction
are given by
where is the Laplace transform of the force across all elements at the junction, is the load admittance, and are the branch admittances.
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.
In the unloaded case, , and we can return to the time domain and define (for the series junction)
These we call the alpha parameters, and they are analogous to those used to characterize ``adaptors'' in wave digital filters (§F.2.2). For unloaded junctions, the alpha parameters obey
In the unloaded case, the series junction scattering relations are given (in
the time domain) by
The alpha parameters provide an interesting and useful parametrization of waveguide junctions. They are explicitly the coefficients of the incoming traveling waves needed to compute junction velocity for a series junction (or junction force or pressure at a parallel junction), and losslessness is assured provided only that the alpha parameters be nonnegative and sum to . Having them sum to something less than simulates a ``resistive load'' at the junction.
Note that in the lossless, equal-impedance case, in which all waveguide impedances have the same value , (C.102) reduces to
When, furthermore, is a power of two, we have that there are no multiplies in the scattering relations (C.105). This fact has been used to build multiply-free reverberators and other structures using digital waveguide meshes [430,518,396,520].
Two Coupled Strings
Properties of Passive Impedances