Richards' theorem

Richards' theorem is a mathematical result due to <a href="/facts/Paul_I._Richards/K3C5FsCA">Paul I. Richards</a> in 1947. The theorem states that for,

R
        (
        s
        )
        =
        
          
            
              k
              Z
              (
              s
              )
              −
              s
              Z
              (
              k
              )
            
            
              k
              Z
              (
              k
              )
              −
              s
              Z
              (
              s
              )
            
          
        
      
    
    {\displaystyle R(s)={\frac {kZ(s)-sZ(k)}{kZ(k)-sZ(s)}}}

if 
 
 
 
 Z
 (
 s
 )
 
 
 {\displaystyle Z(s)}
 
 is a <a href="/facts/Positive-real_function/1r3CxwU7">positive-real function</a> (PRF) then 
 
 
 
 R
 (
 s
 )
 
 
 {\displaystyle R(s)}
 
 is a PRF for all real, positive values of 
 
 
 
 k
 
 
 {\displaystyle k}
 
.
The theorem has applications in electrical <a href="/facts/Network_synthesis/Ulst8XaJ">network synthesis</a>. The PRF property of an impedance function determines whether or not a passive network can be realised having that impedance. Richards' theorem led to a new method of realising such networks in the 1940s.

Richards' theorem open-in-new

Richards' theorem