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

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

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]

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 other 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

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

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

References

  1. 1 2 3 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. 1 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. 1 2 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.
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