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### Medium Passivity

Following the classical definition of passivity [4,133], a medium is said to be passive if (17)

for . 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 . For a passive medium, using (16), (17) becomes (18)

for . The wave impedance is an -by- function of the complex variable . Condition (18) is essentially the same thing as saying is positive real , except that it is allowed to be complex (but Hermitian), even for real .5The matrix is the paraconjugate of , i.e., the unique analytic continuation (when it exists) of the Hermitian transpose of from the unit circle to the complex plane . Since the inverse of a positive-real function is positive real, the corresponding generalized wave admittance is positive real (and hence analytic) in . 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 (19)

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 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 and the impedance through the non-diagonal matrix .

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