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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 $ N>2$, $ v^{+}_i$ will denote a velocity wave traveling into the junction, and will be called an ``incoming'' velocity wave as opposed to ``right-going.''C.9

Figure C.29: Four ideal strings intersecting at a point to which a lumped impedance is attached. This is a series junction for transverse waves.
\includegraphics[width=\twidth]{eps/fNstrings}

Consider first the series junction of $ N$ 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 $ N$ ideal strings intersecting at a single point, and the intersection point can be attached to a lumped load impedance $ R_J(s)$, as depicted in Fig.C.29 for $ N=4$. The presence of the lumped load means we need to look at the wave variables in the frequency domain, i.e., $ V(s) = {\cal L}\{v\}$ for velocity waves and $ F(s) = {\cal L}\{f\}$ for force waves, where $ {\cal L}\{\cdot\}$ denotes the Laplace transform. In the discrete-time case, we use the $ z$ transform instead, but otherwise the story is identical. The physical constraints at the junction are

$\displaystyle V_1(s) = V_2(s) = \cdots = V_N(s)$ $\displaystyle \isdef$ $\displaystyle V_J(s)$ (C.90)
$\displaystyle F_1(s) + F_2(s) + \cdots + F_N(s)$ $\displaystyle =$ $\displaystyle V_J(s) R_J(s) \isdef F_J(s)$ (C.91)

where the reference direction for the load force $ F_J$ is taken to be opposite that for the $ F_i$. (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 $ R_J(s)$ 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$ \left\{R_J(j\omega)\right\}\equiv0$, and it is memoryless if and only if im$ \left\{R_J(j\omega)\right\}\equiv0$.

The parallel junction is characterized by

$\displaystyle F_1(s) = F_2(s) = \cdots = F_N(s)$ $\displaystyle \isdef$ $\displaystyle F_J(s)$ (C.92)
$\displaystyle V_1(s) + V_2(s) + \cdots + V_N(s)$ $\displaystyle =$ $\displaystyle F_J(s)/R_J(s) \isdef V_J(s)$ (C.93)

For example, $ F_i(s)$ could be pressure in an acoustic tube and $ V_i(s)$ the corresponding volume velocity. In the parallel case, the junction reduces to the unloaded case when the load impedance $ R_J(s)$ goes to infinity.

The scattering relations for the series junction are derived as follows, dropping the common argument `$ (s)$' for simplicity:

$\displaystyle R_J V_J$ $\displaystyle =$ $\displaystyle \sum_{i=1}^N F_i = \sum_{i=1}^N (F^+_i + F^-_i)$ (C.94)
  $\displaystyle =$ $\displaystyle \sum_{i=1}^N (R_i V^+_i - R_i \underbrace{V^-_i}_{V_J-V^+_i})$ (C.95)
  $\displaystyle =$ $\displaystyle \sum_{i=1}^N (2 R_i V^+_i - R_i V_J)$ (C.96)

where $ R_i$ is the wave impedance in the $ i$th waveguide, a real, positive constant. Bringing all terms containing $ V_J$ to the left-hand side, and solving for the junction velocity gives
$\displaystyle V_J$ $\displaystyle =$ $\displaystyle 2\left(R_J + \sum_{i=1}^N R_i\right)^{-1} \sum_{i=1}^N R_i V^+_i$ (C.97)
  $\displaystyle \isdef$ $\displaystyle \sum_{i=1}^N{\cal A}_i(s) V^+_i(s)$ (C.98)

(written to be valid also in the multivariable case involving square impedance matrices $ R_i$ [433]), where

$\displaystyle {\cal A}_i(s) \isdef \frac{2R_i}{R_J(s) + R_1 + \cdots + R_N}$ (C.99)

Finally, from the basic relation $ V_J = V_i = V^+_i + V^-_i$, the outgoing velocity waves can be computed from the junction velocity and incoming velocity waves as

$\displaystyle V^-_i(s) = V_J(s) - V^+_i(s)$ (C.100)

Similarly, the scattering relations for the loaded parallel junction are given by

$\displaystyle F_J(s)$ $\displaystyle =$ $\displaystyle \sum_{i=1}^N{\cal A}_i(s) F^+_i(s), \quad {\cal A}_i(s) \isdef \frac{2\Gamma _i}{\Gamma _J(s) + \Gamma _1 + \cdots + \Gamma _N}$  
$\displaystyle F^-_i(s)$ $\displaystyle =$ $\displaystyle F_J(s) - F^+_i(s)$ (C.101)

where $ F_J(s)$ is the Laplace transform of the force across all elements at the junction, $ \Gamma _J(s)$ is the load admittance, and $ \Gamma _i=1/R_i$ are the branch admittances.

It is interesting to note that the junction load is equivalent to an $ N+1$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, $ R_J(s)=0$, and we can return to the time domain and define (for the series junction)

$\displaystyle \alpha_i = \frac{2R_i}{R_1 + \cdots + R_N}$ (C.102)

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

$\displaystyle 0\leq\alpha_i \leq 2$ (C.103)

and

$\displaystyle \sum_{i=1}^N\alpha_i = 2$ (C.104)

In the unloaded case, the series junction scattering relations are given (in the time domain) by

$\displaystyle v_J(t)$ $\displaystyle =$ $\displaystyle \sum_{i=1}^N \alpha_i v^+_i(t) \protect$ (C.105)
$\displaystyle v^-_i(t)$ $\displaystyle =$ $\displaystyle v_J(t) - v^+_i(t)$ (C.106)

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 $ 2$. Having them sum to something less than $ 2$ simulates a ``resistive load'' at the junction.

Note that in the lossless, equal-impedance case, in which all waveguide impedances have the same value $ R_i=R$, (C.102) reduces to

$\displaystyle \alpha_i = \frac{2}{N}$ (C.107)

When, furthermore, $ N$ 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].

An elaborated discussion of $ N=2$ strings intersection at a load is given in in §9.2.1. Further discussion of the digital waveguide mesh appears in §C.14.

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Two Coupled Strings
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Properties of Passive Impedances