On the Passivity and Stability of Propagating Electromagnetic Waves

We consider the uniform transmission line network that represents electromagnetic waves, with propagation constants
k(p)
and characteristic impedances
Z(p)
(
p
is the complex frequency variable), that are propagating in uniform homogeneous waveguides. A wave is said to be passive (active) if the net flow of electromagnetic energy associated with it is into (out of) the material through which the wave is propagating. On the other hand, a wave is stable if it does not grow, either spatially or temporally, as it propagates. It is shown that in order for a wave to be passive it is necessary that
k(p)
be analytic,
Z(p)
be analytic and nonzero,
[Re k(p)] \{Re [1/Z(p)]\} \geq O, Z(p^{\ast})=Z^{\ast}(p)
, and
k(p^{\ast})=k^{\ast}(p)
, all in
Re p>O
. In the special case where
Z(p)=p/k(p)
, we prove that the passivity conditions guarantee that
Re k(p)\neq 0
and
Re [k(p)/p] \neq O
in
Re p>O
, which enables us to demonstrate that a passive wave is in fact stable, even though a stable wave may be active. We also develop an algebraic test to determine if any roots of a polynomial in
k
(the socalled dispersion equation) are propagation constants of active waves.