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.