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| '''Positive-real functions''', often abbreviated to '''PR function''', are a kind of mathematical function that first arose in [[Network analysis (electrical circuits)|electrical network analysis]]. They are [[Complex analysis#Complex functions|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.
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| In symbols the definition is,
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| :<math> \begin{align}
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| & \Re[Z(s)]>0 \quad\text{if}\quad \Re(s) > 0 \\
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| & \Im[Z(s)]=0 \quad\text{if}\quad \Im(s)=0
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| \end{align} </math>
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| In electrical network analysis, ''Z''(''s'') represents an [[electrical impedance|impedance]] expression and ''s'' is the [[complex frequency]] variable, often expressed as its real and imaginary parts;
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| :<math>s=\sigma+i\omega \,\!</math>
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| in which terms the PR condition can be stated;
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| :<math> \begin{align}
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| & \Re[Z(s)]>0 \quad\text{if}\quad \sigma > 0 \\
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| & \Im[Z(s)]=0 \quad\text{if}\quad \omega=0
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| \end{align} </math>
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| 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 [[Passivity (engineering)|passive]] linear elements ([[resistor]]s, [[inductor]]s and [[capacitor]]s in electrical terminology).<ref name=CauerMathisPauli>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. [http://www.cs.princeton.edu/courses/archive/fall03/cs323/links/cauer.pdf Retrieved online] 19 September 2008.</ref>
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| ==Definition==
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| The term ''positive-real function'' was originally defined by<ref name=CauerMathisPauli /> [[Otto Brune]] to describe any function ''Z''(''s'') which<ref name=Brune1931a>Brune, O, "Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency", Doctoral thesis, MIT, 1931. [http://dspace.mit.edu/bitstream/handle/1721.1/10661/36311006.pdf?sequence=1 Retrieved online] 3rd June 2010.</ref>
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| *is [[rational function|rational]] (the quotient of two [[polynomials]]),
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| *is real when ''s'' is real
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| *has positive real part when ''s'' has a positive real part
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| Many authors strictly adhere to this definition by explicitly requiring rationality,<ref>{{cite book |title=Network Theory |last=Bakshi |first=Uday |last2=Bakshi |first2=Ajay |year=2008 |publisher=Technical Publications |location=Pune |isbn=978-81-8431-402-1}}</ref> <!--U.A.Bakshi, J.S.Chitode --> or by restricting attention to rational functions, at least in the first instance<!-- Kendall Ling-chiao Su + -->.<ref name=Wing>{{cite book |title=Classical Circuit Theory |last=Wing |first=Omar |year=2008 |publisher=Springer |isbn=978-0-387-09739-8}}</ref> However, a similar more general condition, not restricted to rational functions had earlier been considered by Cauer,<ref name=CauerMathisPauli /> and some authors ascribe the term ''positive-real'' to this type of condition<!-- Wai-Kai Chen -->, while other consider it to be a generalization of the basic definition.<ref name=Wing />
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| ==History==
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| The condition was first proposed by [[Wilhelm Cauer]] (1926)<ref>Cauer, W, "Die Verwirklichung der Wechselstromwiderst ände vorgeschriebener Frequenzabh ängigkeit", ''Archiv für Elektrotechnik'', '''vol 17''', pp355–388, 1926.</ref> who determined that it was a necessary condition. [[Otto Brune]] (1931)<ref name=Brune1931a /><ref name=Brune1931b>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.</ref> coined the term positive-real for the condition and proved that it was both necessary and sufficient for realisability.
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| ==Properties==
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| *The sum of two PR functions is PR.<!-- too obvious to mention? -->
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| *The [[function composition|composition]] of two PR functions is PR. In particular, if ''Z''(''s'') is PR, then so are 1/''Z''(''s'') and ''Z''(1/''s'').
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| *All the [[pole (complex analysis)|poles]] and [[zero (complex analysis)|zeros]] of a PR function are in the left half plane or on its boundary the imaginary axis.
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| *Any poles and zeroes on the imaginary axis are [[zero (complex analysis)#Multiplicity_of_a_zero|simple]] (have a [[multiplicity (mathematics)|multiplicity]] of one).
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| *Any poles on the imaginary axis have real strictly positive [[residue (complex analysis)|residues]], and similarly at any zeroes on the imaginary axis, the function has a real strictly positive derivative.
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| *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]]).
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| *For a [[rational function|rational]] PR function, the number of poles and number of zeroes differ by at most one.
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| ==Generalizations==
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| A couple of generalizations are sometimes made, with intention of characterizing the [[immittance]] functions of a wider class of passive linear electrical networks.
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| ===Irrational functions=== | |
| The impedance ''Z''(''s'') of a network consisting of an infinite number of components (such as a semi-infinite [[Ladder_network#Ladder_topologies|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
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| *''Z''(''s'') is analytic in the open right half ''s''-plane (Re[''s''] > 0)
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| *''Z''(''s'') is real when ''s'' is positive and real
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| *Re[''Z''(''s'')] ≥ 0 when Re[''s''] ≥ 0
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| Some authors start from this more general definition, and then particularize it to the rational case.
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| ===Matrix-valued functions===
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| Linear electrical networks with more than one port may be described by [[impedance parameters|impedance or]] [[admittance parameters|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
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| *Each element of ''Z''(''s'') is analytic in the open right half ''s''-plane (Re[''s''] > 0)
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| *Each element of ''Z''(''s'') is real when ''s'' is positive and real
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| *The [[Hermitian matrix|Hermitian]] part (''Z''(''s'') + ''Z''<sup>†</sup>(''s''))/2 of ''Z''(''s'') is [[Positive-definite matrix|positive semi-definite]] when Re[''s''] ≥ 0
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| ==References==
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| {{reflist}}
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| [[Category:Complex analysis]]
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| [[Category:Electronic engineering]]
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| [[Category:Types of functions]]
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