Properties of Passive Impedances

It is well known that a real impedance $R$ (in Ohms, for example) is passive so long as $R\geq 0$ . A passive impedance cannot create energy. On the other hand, if $R<0$ , the impedance is active and has some energy source. The concept of passivity can be extended to complex frequency-dependent impedances $R(j\omega)$ as well: A complex impedance $R(j\omega)$ is passive if $R(s)$ is positive real, where $s$ 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.