Properties of Passive Impedances

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.

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Subsections

• Passive Reflectances
  • Reflectance and Transmittance of a Yielding String Termination
  • Power-Complementary Reflection and Transmission
  
  
• Positive Real Functions
  • Relation to Stochastic Processes
  • Relation to Schur Functions
  • Relation to -plane Positive-Real Functions
  • Special cases and examples
  • Minimum Phase (MP) polynomials in
  • Miscellaneous Properties

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.