Positive-real function

Positive-real functions, often abbreviated to PR function or PRF, are a kind of mathematical function that first arose in electrical network synthesis. 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 half of the complex plane and takes on real values on the real axis.

In symbols the definition is,

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

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;

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

in which terms the PR condition can be stated;

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

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).^[1]

Definition

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

• 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,^[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 term positive-real to this type of condition, while others consider it to be a generalization of the basic definition.^[4]

History

The condition was first proposed by Wilhelm 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.

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 zeros 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 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 in the left half s-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 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

1. E. Cauer, W. Mathis, and R. Pauli, "Life and Work of Wilhelm Cauer (1900 – 1945)", Proceedings of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems (MTNS2000), Perpignan, June, 2000. Retrieved online 19 September 2008.
2. Brune, O, "Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency", Doctoral thesis, MIT, 1931. Retrieved online 3 June 2010.
3. Bakshi, Uday; Bakshi, Ajay (2008). Network Theory. Pune: Technical Publications. ISBN 978-81-8431-402-1.
4. Wing, Omar (2008). Classical Circuit Theory. Springer. ISBN 978-0-387-09739-8.
5. Cauer, W, "Die Verwirklichung der Wechselstromwiderst ände vorgeschriebener Frequenzabh ängigkeit", Archiv für Elektrotechnik, vol 17, pp355–388, 1926.
6. Brune, O, "Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency", J. Math. and Phys., vol 10, pp191–236, 1931.

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