Positive-real functions, often abbreviated toPR function orPRF, are a kind of mathematical function that first arose in electricalnetwork synthesis. They arecomplex functions,Z(s), of a complex variable,s. Arational function is defined to have the PR property if it has a positive real part and is analytic in the right half of the complex plane and takes on real values on the real axis.

In symbols the definition is,

In electrical network analysis,Z(s) represents animpedance expression ands is thecomplex frequency variable, often expressed as its real and imaginary parts;

${\displaystyle s=\sigma +i\omega }$

in which terms the PR condition can be stated;

The importance to network analysis of the PR condition lies in the realisability condition.Z(s) is realisable as aone-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 idealpassive linear elements (resistors,inductors andcapacitors in electrical terminology).^[1]

The termpositive-real function was originally defined by^[1]Otto Brune to describe any functionZ(s) which^[2]

• isrational (the quotient of twopolynomials),
• is real whens is real
• has positive real part whens has a positive real part

Many authors strictly adhere to this definition by explicitly requiring rationality,^[3] or by restricting attention to rational functions, at least in the first instance.^[4] However, a similar more general condition, not restricted to rational functions had earlier been considered by Cauer,^[1] and some authors ascribe the termpositive-real to this type of condition, while others consider it to be a generalization of the basic definition.^[4]

The condition was first proposed byWilhelm Cauer (1926)^[5] who determined that it was a necessary condition.Otto Brune (1931)^[2][6] coined the term positive-real for the condition and proved that it was both necessary and sufficient for realisability.

• The sum of two PR functions is PR.
• Thecomposition of two PR functions is PR. In particular, ifZ(s) is PR, then so are 1/Z(s) andZ(1/s).
• All thezeros and poles of a PR function are in the left half plane or on its boundary of the imaginary axis.
• Any poles and zeroes on the imaginary axis aresimple (have amultiplicity of one).
• Any poles on the imaginary axis have real strictly positiveresidues, 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 aharmonic function over the plane, and therefore satisfies themaximum principle).
• For arational PR function, the number of poles and number of zeroes differ by at most one.

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

The impedanceZ(s) of a network consisting of an infinite number of components (such as a semi-infiniteladder), need not be a rational function ofs, and in particular may havebranch points in the left halfs-plane. To accommodate such functions in the definition of PR, it is therefore necessary to relax the condition that the function be real for all reals, and only require this whens is positive. Thus, a possibly irrational functionZ(s) is PR if and only if

• Z(s) is analytic in the open right halfs-plane (Re[s] > 0)
• Z(s) is real whens 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.

Linear electrical networks with more than oneport may be described byimpedance oradmittance 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 functionZ(s) is PR if and only if

• Each element ofZ(s) is analytic in the open right halfs-plane (Re[s] > 0)
• Each element ofZ(s) is real whens is positive and real
• TheHermitian part (Z(s) +Z^†(s))/2 ofZ(s) ispositive semi-definite when Re[s] ≥ 0

• Wilhelm Cauer (1932)The Poisson Integral for Functions with Positive Real Part,Bulletin of the American Mathematical Society 38:713–7, link fromProject Euclid.
• W. Cauer (1932)"Über Funktionen mit positivem Realteil",Mathematische Annalen 106: 369–94.

• Complex analysis
• Electronic engineering
• Types of functions

• Use dmy dates from July 2015