Positive-real function

Positive-real functions, often abbreviated to PR function or PRF, are a kind of mathematical function that first arose in electrical <a href="/facts/Network_synthesis/Ulst8XaJ">network synthesis</a>. They are <a href="/facts/Complex_analysis/hgCoQlH0">complex functions</a>, Z(s), of a complex variable, s. A <a href="/facts/Rational_function/2nJGu0Ko">rational function</a> 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,

ℜ
                [
                Z
                (
                s
                )
                ]
                >
                0
                
                
                  if
                
                
                ℜ
                (
                s
                )
                >
                0
              
            
            
              
              
                ℑ
                [
                Z
                (
                s
                )
                ]
                =
                0
                
                
                  if
                
                
                ℑ
                (
                s
                )
                =
                0
              
            
          
        
      
    
    {\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 <a href="/facts/Electrical_impedance/I06NSXtN">impedance</a> expression and s is the <a href="/facts/Complex_frequency/GUPlfVuC">complex frequency</a> variable, often expressed as its real and imaginary parts;

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

in which terms the PR condition can be stated;

ℜ
                [
                Z
                (
                s
                )
                ]
                >
                0
                
                
                  if
                
                
                σ
                >
                0
              
            
            
              
              
                ℑ
                [
                Z
                (
                s
                )
                ]
                =
                0
                
                
                  if
                
                
                ω
                =
                0
              
            
          
        
      
    
    {\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 <a href="/facts/One-port/g2mokqv0">one-port</a> 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 <a href="/facts/Passivity_(engineering)/7oDeoqd3">passive</a> linear elements (<a href="/facts/Resistor/JrsGReM6">resistors</a>, <a href="/facts/Inductor/YnfAXB3x">inductors</a> and <a href="/facts/Capacitor/hyowqv7n">capacitors</a> in electrical terminology).

Positive-real function open-in-new

Positive-real function