It is well known that a real impedance
(in Ohms, for example) is
passive so long as
. A passive impedance cannot
create energy. On the other hand, if
, the impedance is
active and has some energy source. The concept of
passivity can be extended to complex frequency-dependent impedances
as well: A complex impedance
is
passive if
is positive real, where
is the
Laplace-transform variable. The positive-real property is discussed in
§C.11.2 below.
This section explores some implications of the positive real condition for passive impedances. Specifically, §C.11.1 considers the nature of waves reflecting from a passive impedance in general, looking at the reflection transfer function, or reflectance, of a passive impedance. To provide further details, Section C.11.2 derives some mathematical properties of positive real functions, particularly for the discrete-time case. Application examples appear in §9.2.1 and §9.2.1.