The Lossless Junction

A scattering junction of waveguide sections is characterized by its scattering matrix ${\mbox{\boldmath$A$}}$. The relationship between the $N$ incoming and outgoing traveling waves is given by:

$${\mbox{\boldmath$p$}}^-={\mbox{\boldmath$A$}}{\mbox{\boldmath$p$}}^+$$ (23)

where ${\mbox{\boldmath$p$}}^+$ is the vector of incoming waves (assumed scalar here) and ${\mbox{\boldmath$p$}}^-$ is the vector of outgoing waves relative to the junction (see Fig. 2). We say that the junction is $N$-way (or it has $N$ branches) if $N$ is the dimension of the incoming and outgoing wave vectors.

Figure: A schematic depiction of the 3-way waveguide junction.

We now consider the case of a constant scattering matrix ${\mbox{\boldmath$A$}}$. The more general case of scattering matrices as functions of $z$ will be considered in Section 10.

The net complex power entering the junctions is

$${\cal P}= {\mbox{\boldmath$u$}}^* {\mbox{\boldmath$p$}} = {\mbox{\boldmath$p$}}^{+*}{\mbox{\boldmath$\Gamma$}}^*(1/z^*) {\m... ...+- {\mbox{\boldmath$p$}}^{-*}{\mbox{\boldmath$\Gamma$}}{\mbox{\boldmath$p$}}^-+$$

$${\mbox{\boldmath$p$}}^{+*}{\mbox{\boldmath$\Gamma$}}^*(1/z^*) {\m... ...^-- {\mbox{\boldmath$p$}}^{-*}{\mbox{\boldmath$\Gamma$}}{\mbox{\boldmath$p$}}^+$$

$$= ({\cal P}^+- {\cal P}^-) + ({\cal P}^\times - {\cal P}^{\times*})$$ (24)

where ${\mbox{\boldmath$\Gamma$}}$ is the diagonal matrix containing the $N$ wave admittances of all the branches meeting at the junction. Assuming the branch admittances are Hermitian and nonzero, we have that ${\mbox{\boldmath$\Gamma$}}$ has positive real elements along its diagonal and zeros elsewhere. The quantity ${\cal P}^+= {\mbox{\boldmath$p$}}^{+*}{\mbox{\boldmath$\Gamma$}}{\mbox{\boldmath$p$}}^+\geq 0$ is incoming active power, and ${\cal P}^-= {\mbox{\boldmath$p$}}^{-*}{\mbox{\boldmath$\Gamma$}}{\mbox{\boldmath$p$}}^-\geq 0$ is then the outgoing active power relative to the junction. The term ${\cal P}^+- {\cal P}^-$ is the absorbed active power, while the term ${\cal P}^\times - {\cal P}^{\times*}$, containing the mixed incoming and outgoing waves, is called the reactive power.

A scattering junction is said to be passive when the absorbed active power is nonnegative, i.e., when

$${\cal P}^+\geq {\cal P}^-$$ (25)

for $\vert z\vert \geq 1$. In other terms, the outgoing active power does not exceed the incoming active power.