Passive Impedances

It is well known that a real impedance (in Ohms, e.g.) is passive so long as . A passive impedance cannot create energy. On the other hand, if , the impedance is said to be active, and it must be connected to some energy source. The concept of passivity can be extended to complex impedances as well: We say that a complex impedance is passive if is positive real, where is the Laplace-transform variable. In the discrete-time case, must be positive real using an analogous definition (given in §M.4 below).

This appendix explores some implications of the positive real condition for passive impedance. Section M.1 considers the nature of waves reflecting from a passive impedance, while §M.2 considers the particular passive impedance created by a moving termination. Next, §M.3 looks at the acoustic guitar bridge, including a look at some laboratory measurements. Finally, §M.4 provides a review of some mathematical properties of positive real functions in the

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Subsections

• Passive Reflectances
• Passive String Terminations
  • Reflection Transfer Function
  
  
• The Guitar Bridge
  • Building a Synthetic Guitar Bridge Admittance
    • Method 1
    • Method 2
    • Matlab for Method 1
    • Matlab for Method 2
  
  
• 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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written by 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.