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### Medium Passivity

Following the classical definition of passivity [40,41], a medium is said to be passive if (22)

for . 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 . For a passive medium, using (22), the inequality (23) becomes (23)

for . I.e., the sum of the wave impedance and its paraconjugate is positive semidefinite.

The wave impedance is an -by- function of the complex variable . Condition (24) is essentially the same thing as saying is positive real , except that it is allowed to be complex, even for real .

The matrix is the paraconjugate of . Since generalizes , to the entire complex plane, we may interpret as generalizing the Hermitian part of to the -plane, viz., the para-Hermitian part.

Since the inverse of a positive-real function is positive real, the corresponding generalized wave admittance is positive real (and hence analytic) in .

We say that wave propagation in the medium is lossless if the impedance matrix is such that (24)

i.e., if is para-Hermitian (which implies its inverse is also).

Most applications in waveguide modeling are concerned with nearly lossless propagation in passive media. In this paper, we will state results for in the more general case when applicable, while considering applications only for constant and diagonal impedance matrices . 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 and the impedance through the non-diagonal matrix (see eq. 19).

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