Passive One-Ports

It is well known that the impedance of every passive one-port is positive real [528]. 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 $\Re\{s\} \geq 0$.

Properties of positive-real functions are discussed further in §L.4. A particularly important property of positive real functions is that the phase is bounded between plus and minus $90$ degrees, i.e.,

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

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.K.14.

Figure K.14: Poles and zeros of a reactance (or suseptance) must interlace along the $j\omega$ Axis. Left: Pole-zero plot. Right: Phase response.

Referring to Fig.K.14, consider the graphical method for computing phase response of a reactance from the pole zero diagram [455].^K.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. We will return to properties of positive real immittances later in the context of a practical modeling example (§L.3).