Medium Passivity

Following the classical definition of passivity [4,133], a medium is said to be passive if

$${\cal P}^++ {\cal P}^-\geq 0$$ (17)

for $\vert z\vert \geq 1$. Thus, a sufficient condition for ensuring passivity in a medium is that each traveling active-power component be real and non-negative.

To derive a definition of passivity in terms of the wave impedance, consider a perfectly reflecting interruption in the transmission line, such that ${\mbox{\boldmath$u$}}^-= {\mbox{\boldmath$u$}}^+$. For a passive medium, using (16), (17) becomes

$${\mbox{\boldmath$R$}}(z) + {\mbox{\boldmath$R$}}^*(1/z^*) \geq 0$$ (18)

for $\vert z\vert \geq 1$. The wave impedance ${\mbox{\boldmath$R$}}(z)$ is an $m$-by-$m$ function of the complex variable $z$. Condition (18) is essentially the same thing as saying ${\mbox{\boldmath$R$}}(z)$ is positive real [128], except that it is allowed to be complex (but Hermitian), even for real $z$.⁵The matrix ${\mbox{\boldmath$R$}}^*(1/z^*)$ is the paraconjugate of ${\mbox{\boldmath$R$}}$, i.e., the unique analytic continuation (when it exists) of the Hermitian transpose of ${\mbox{\boldmath$R$}}$ from the unit circle to the complex $z$ plane [110]. Since the inverse of a positive-real function is positive real, the corresponding generalized wave admittance ${\mbox{\boldmath$\Gamma$}}(z) = {\mbox{\boldmath$R$}}^{-1}(z)$ is positive real (and hence analytic) in $\vert z\vert \geq 1$. In other terms, the sum of the wave impedance and its paraconjugate is positive semidefinite.

We say that wave propagation in the medium is lossless if the wave impedance matrix satisfies

$${\mbox{\boldmath$R$}}(z)={\mbox{\boldmath$R$}}^*(1/z^*)$$ (19)

i.e., if ${\mbox{\boldmath$R$}}(z)$ is para-Hermitian (which implies its inverse ${\mbox{\boldmath$\Gamma$}}(z)$ is also).

Most applications in waveguide modeling are concerned with nearly lossless propagation in passive media. In this paper, we will state results for ${\mbox{\boldmath$R$}}(z)$ in the more general case when applicable, while considering applications only for constant and diagonal impedance matrices ${\mbox{\boldmath$R$}}$. As shown in Section 2.3, coupling in the wave equation (5) implies a non-diagonal impedance matrix, since there is usually a proportionality between the speed of propagation ${\mbox{\boldmath$C$}}$ and the impedance ${\mbox{\boldmath$R$}}$ through the non-diagonal matrix ${\mbox{\boldmath$M$}}$.

The wave components of equations (11) travel undisturbed along each axis. This propagation is implemented digitally using $m$ bidirectional delay lines, as depicted in Fig. 1. We call such a collection of delay lines an $m$-variable waveguide section. Waveguide sections are then joined at their endpoints via scattering junctions (discussed further in following sections) to form a DWN.

Figure: An $m$-variable waveguide section.