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
is said to be
*positive real* (PR) if

- 1)
- is real whenever is real
- 2)
- whenever .

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

This is a significant constraint on the rational function . One implication is that in the lossless case (no dashpots, only masses and springs--a

Referring to Fig.7.14, consider the graphical method for
computing phase response of a reactance from the pole zero diagram
[453].^{8.4}Each zero on the positive
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
to
degrees. For each
pole, the phase contribution switches from
to
degrees as
it is passed going up in frequency. In order to keep phase in
, it is clear that the poles and zeros must strictly
alternate. Moreover, all poles and zeros must be simple, since a
repeated pole or zero would swing the phase by more than
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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University