Positive-real function

Positive-real functions, often abbreviated to PR function, are a kind of mathematical function that first arose in electrical network analysis. They are complex functions, Z(s), of a complex variable, s. A function is defined to have the PR property if it has a positive real part and is analytic in the right halfplane of the complex plane and takes on real values on the real axis.

In symbols the definition is,

${\begin{aligned}&\Re [Z(s)]>0\quad {\text{if}}\quad \Re (s)>0\\&\Im [Z(s)]=0\quad {\text{if}}\quad \Im (s)=0\end{aligned}}$

In electrical network analysis, Z(s) represents an impedance expression and s is the complex frequency variable, often expressed as its real and imaginary parts;

$s=\sigma +i\omega \,\!$

in which terms the PR condition can be stated;

${\begin{aligned}&\Re [Z(s)]>0\quad {\text{if}}\quad \sigma >0\\&\Im [Z(s)]=0\quad {\text{if}}\quad \omega =0\end{aligned}}$

The importance to network analysis of the PR condition lies in the realisability condition. Z(s) is realisable as a one-port rational impedance if and only if it meets the PR condition. Realisable in this sense means that the impedance can be constructed from a finite (hence rational) number of discrete ideal passive linear elements (resistors, inductors and capacitors in electrical terminology).

History

The condition was first proposed by Wilhelm Cauer (1926) who determined that it was a necessary condition. Otto Brune (1931) coined the term positive-real for the condition and proved that it was both necessary and sufficient for realisability.

Properties

• The sum of two PR functions is PR.
• The composition of two PR functions is PR. In particular, if Z(s) is PR, then so are 1/Z(s) and Z(1/s).
• All the poles and zeros of a PR function are in the left half plane or on its boundary the imaginary axis.
• Any poles and zeroes on the imaginary axis are simple (have a multiplicity of one).
• Any poles on the imaginary axis have real strictly positive residues, and similarly at any zeroes on the imaginary axis, the function has a real strictly positive derivative.
• Over the right half plane, the minimum value of the real part of a PR function occurs on the imaginary axis (because the real part of an analytic function constitutes a harmonic function over the plane, and therefore satisfies the maximum principle).
• For a rational PR function, the number of poles and number of zeroes differ by at most one.