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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.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2017-05-16 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University