Isserlis' theorem

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 rational 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,

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

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;

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

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).

Definition

The term positive-real function was originally defined by Otto Brune to describe any function Z(s) which

• is rational (the quotient of two polynomials),
• is real when s is real
• has positive real part when s has a positive real part

Many authors strictly adhere to this definition by explicitly requiring rationality, or by restricting attention to rational functions, at least in the first instance. However, a similar more general condition, not restricted to rational functions had earlier been considered by Cauer, and some authors ascribe the term positive-real to this type of condition, while other consider it to be a generalization of the basic definition.

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.

Generalizations

A couple of generalizations are sometimes made, with intention of characterizing the immittance functions of a wider class of passive linear electrical networks.

Irrational functions

The impedance Z(s) of a network consisting of an infinite number of components (such as a semi-infinite ladder), need not be a rational function of s, and in particular may have branch points on the negative real s-axis. To accommodate such functions in the definition of PR, it is therefore necessary to relax the condition that the function be real for all real s, and only require this when s is positive. Thus, a possibly irrational function Z(s) is PR if and only if

• Z(s) is analytic in the open right half s-plane (Re[s] > 0)
• Z(s) is real when s is positive and real
• Re[Z(s)] ≥ 0 when Re[s] ≥ 0

Some authors start from this more general definition, and then particularize it to the rational case.

Matrix-valued functions

Linear electrical networks with more than one port may be described by impedance or admittance matrices. So by extending the definition of PR to matrix-valued functions, linear multi-port networks which are passive may be distinguished from those that are not. A possibly irrational matrix-valued function Z(s) is PR if and only if

• Each element of Z(s) is analytic in the open right half s-plane (Re[s] > 0)
• Each element of Z(s) is real when s is positive and real
• The Hermitian part (Z(s) + Z^†(s))/2 of Z(s) is positive semi-definite when Re[s] ≥ 0

References

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