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Physical Audio Signal Processing
    Lumped Models
       One-Port Network Theory
          Passive One-Ports

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Passive One-Ports

It is well known that the impedance of every passive one-port is positive real (see §C.11.2). The reciprocal of a positive real function is positive real, so every passive impedance corresponds also to a passive admittance.

A complex-valued function of a complex variable $ \Gamma (s)$ is said to be positive real (PR) if

1)
$ \Gamma (s)$ is real whenever $ s$ is real
2)
$ \Re\{\Gamma(s)\} \geq 0$ whenever $ \Re\{s\} \geq 0$.

A particularly important property of positive real functions is that the phase is bounded between plus and minus $ 90$ degrees, i.e.,

$\displaystyle -\frac{\pi}{2} \leq \angle{\Gamma(j\omega)} \leq \frac{\pi}{2} $

This is a significant constraint on the rational function $ \Gamma (s)$. One implication is that in the lossless case (no dashpots, only masses and springs--a reactance) all poles and zeros interlace along the $ j\omega $ axis, as depicted in Fig.7.14.

Figure 7.14: Poles and zeros of a lossless immittance (reactance or suseptance) must interlace along the $ j\omega $ Axis. Left: Pole-zero plot. Right: Phase response. The ``spring/mass'' labels along the frequency axis correspond to the case of a lossless admittance (susceptance) in which a spring admittance ( $ \Gamma _k(j\omega )=j\omega /k$) gives a $ +\pi /2$ phase shift, while that of a mass ( $ \Gamma _m(j\omega )=-j/(m\omega )$) gives a $ -\pi /2$ phase shift between the input driving-force and output velocity.
\includegraphics[width=\twidth]{eps/interlace}

Referring to Fig.7.14, consider the graphical method for computing phase response of a reactance from the pole zero diagram [449].8.4Each zero on the positive $ j\omega $ axis contributes a net 90 degrees of phase at frequencies above the zero. As frequency crosses the zero going up, there is a switch from $ -90$ to $ +90$ degrees. For each pole, the phase contribution switches from $ +90$ to $ -90$ degrees as it is passed going up in frequency. In order to keep phase in $ [-\pi/2,\pi/2]$, it is clear that the poles and zeros must strictly alternate. Moreover, all poles and zeros must be simple, since a multiple poles or zero would swing the phase by more than $ 180$ degrees, and the reactance could not be positive real.

The positive real property is fundamental to passive immittances and comes up often in the study of measured resonant systems. A practical modeling example (passive digital modeling of a guitar bridge) is discussed in §9.2.1.

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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.


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