Medium Passivity

Following the classical definition of passivity [40,41], a medium is said to be passive if

$$\hbox{Re}\{P^++ P^-\} \geq 0$$ (22)

for $\vert z\vert \geq 1$. Thus, a sufficient condition for ensuring passivity in a medium is that each traveling active-power component is 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 (22), the inequality (23) becomes

$${\mbox{\boldmath$R$}}(z) + {\mbox{\boldmath$R$}}^*(1/z^*) \geq 0$$ (23)

for $\vert z\vert \geq 1$. I.e., the sum of the wave impedance and its paraconjugate is positive semidefinite.

The wave impedance ${\mbox{\boldmath$R$}}(z)$ is an $m$-by-$m$ function of the complex variable $z$. Condition (24) is essentially the same thing as saying ${\mbox{\boldmath$R$}}(z)$ is positive real [42], except that it is allowed to be complex, even for real $z$.

The matrix ${\mbox{\boldmath$R$}}^*(1/z^*)$ is the paraconjugate of ${\mbox{\boldmath$R$}}$. Since ${\mbox{\boldmath$R$}}^*(1/z^*)$ generalizes $\overline{{\mbox{\boldmath$R$}}(e^{j\omega})}^T$, to the entire complex plane, we may interpret $[{\mbox{\boldmath$R$}}(z) + {\mbox{\boldmath$R$}}^*(1/z^*)]/2$ as generalizing the Hermitian part of ${\mbox{\boldmath$R$}}(z)$ to the $z$-plane, viz., the para-Hermitian part.

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 $\vert z\vert \geq 1$.

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

$${\mbox{\boldmath$R$}}(z)={\mbox{\boldmath$R$}}^*(1/z^*)$$ (24)

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 II-C, coupling in the wave equation (7) implies a non-diagonal impedance matrix, since there is usually a proportionality between the speed of propagation ${\mbox{\boldmath$C_p$}}$ and the impedance ${\mbox{\boldmath$R$}}$ through the non-diagonal matrix ${\mbox{\boldmath$M$}}$ (see eq. 19).