Richards' theorem

<h2 id="proof">Proof</h2>

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)}}}

where 
 
 
 
 Z
 (
 s
 )
 
 
 {\displaystyle Z(s)}
 
 is a PRF, 
 
 
 
 k
 
 
 {\displaystyle k}
 
 is a positive real constant, and 
 
 
 
 s
 =
 σ
 +
 i
 ω
 
 
 {\displaystyle s=\sigma +i\omega }
 
 is the <a href="/facts/Complex_frequency/GUPlfVuC">complex frequency</a> variable, can be written as,

R
        (
        s
        )
        =
        
          
            
              
                1
                −
                W
                (
                s
                )
              
              
                1
                +
                W
                (
                s
                )
              
            
          
        
      
    
    {\displaystyle R(s)={\dfrac {1-W(s)}{1+W(s)}}}

where,

W
        (
        s
        )
        =
        
          
            
              1
              −
              
                
                  
                    
                      Z
                      (
                      s
                      )
                    
                    
                      Z
                      (
                      k
                      )
                    
                  
                
              
            
            
              1
              +
              
                
                  
                    
                      Z
                      (
                      s
                      )
                    
                    
                      Z
                      (
                      k
                      )
                    
                  
                
              
            
          
        
        
          (
          
            
              
                k
                +
                s
              
              
                k
                −
                s
              
            
          
          )
        
      
    
    {\displaystyle W(s)={1-{\dfrac {Z(s)}{Z(k)}} \over 1+{\dfrac {Z(s)}{Z(k)}}}\left({\frac {k+s}{k-s}}\right)}

Since 
 
 
 
 Z
 (
 s
 )
 
 
 {\displaystyle Z(s)}
 
 is PRF then

1
        +
        
          
            
              
                Z
                (
                s
                )
              
              
                Z
                (
                k
                )
              
            
          
        
      
    
    {\displaystyle 1+{\dfrac {Z(s)}{Z(k)}}}

is also PRF. The <a href="/facts/Zero_(complex_analysis)/gnA3U8VP">zeroes</a> of this function are the <a href="/facts/Pole_(complex_analysis)/gnA3U8VP">poles</a> of 
 
 
 
 W
 (
 s
 )
 
 
 {\displaystyle W(s)}
 
. Since a PRF can have no zeroes in the right-half <a href="/facts/S-plane/GUPlfVuC">s-plane</a>, then 
 
 
 
 W
 (
 s
 )
 
 
 {\displaystyle W(s)}
 
 can have no poles in the right-half s-plane and hence is analytic in the right-half s-plane.
Let

Z
        (
        i
        ω
        )
        =
        r
        (
        ω
        )
        +
        i
        x
        (
        ω
        )
      
    
    {\displaystyle Z(i\omega )=r(\omega )+ix(\omega )}

Then the magnitude of 
 
 
 
 W
 (
 i
 ω
 )
 
 
 {\displaystyle W(i\omega )}
 
 is given by,

|
          
            W
            (
            i
            ω
            )
          
          |
        
        =
        
          
            
              
                
                  (
                  Z
                  (
                  k
                  )
                  −
                  r
                  (
                  ω
                  )
                  
                    )
                    
                      2
                    
                  
                  +
                  x
                  (
                  ω
                  
                    )
                    
                      2
                    
                  
                
                
                  (
                  Z
                  (
                  k
                  )
                  +
                  r
                  (
                  ω
                  )
                  
                    )
                    
                      2
                    
                  
                  +
                  x
                  (
                  ω
                  
                    )
                    
                      2
                    
                  
                
              
            
          
        
      
    
    {\displaystyle \left|W(i\omega )\right|={\sqrt {\dfrac {(Z(k)-r(\omega ))^{2}+x(\omega )^{2}}{(Z(k)+r(\omega ))^{2}+x(\omega )^{2}}}}}

Since the PRF condition requires that 
 
 
 
 r
 (
 ω
 )
 ≥
 0
 
 
 {\displaystyle r(\omega )\geq 0}
 
 for all 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
 then 
 
 
 
 
 |
 
 W
 (
 i
 ω
 )
 
 |
 
 ≤
 1
 
 
 {\displaystyle \left|W(i\omega )\right|\leq 1}
 
 for all 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
. The maximum magnitude of 
 
 
 
 W
 (
 s
 )
 
 
 {\displaystyle W(s)}
 
 occurs on the 
 
 
 
 i
 ω
 
 
 {\displaystyle i\omega }
 
 axis because 
 
 
 
 W
 (
 s
 )
 
 
 {\displaystyle W(s)}
 
 is analytic in the right-half s-plane. Thus 
 
 
 
 
 |
 
 W
 (
 s
 )
 
 |
 
 ≤
 1
 
 
 {\displaystyle |W(s)|\leq 1}
 
 for 
 
 
 
 σ
 ≥
 0
 
 
 {\displaystyle \sigma \geq 0}
 
.
Let 
 
 
 
 W
 (
 s
 )
 =
 u
 (
 σ
 ,
 ω
 )
 +
 i
 v
 (
 σ
 ,
 ω
 )
 
 
 {\displaystyle W(s)=u(\sigma ,\omega )+iv(\sigma ,\omega )}
 
, then the real part of 
 
 
 
 R
 (
 s
 )
 
 
 {\displaystyle R(s)}
 
 is given by,

ℜ
        (
        R
        (
        s
        )
        )
        =
        
          
            
              
                1
                −
                
                  |
                
                W
                (
                s
                )
                
                  
                    |
                  
                  
                    2
                  
                
              
              
                (
                1
                +
                u
                (
                σ
                ,
                ω
                )
                
                  )
                  
                    2
                  
                
                +
                
                  v
                  
                    2
                  
                
                (
                σ
                ,
                ω
                )
              
            
          
        
      
    
    {\displaystyle \Re (R(s))={\dfrac {1-|W(s)|^{2}}{(1+u(\sigma ,\omega ))^{2}+v^{2}(\sigma ,\omega )}}}

Because 
 
 
 
 W
 (
 s
 )
 ≤
 1
 
 
 {\displaystyle W(s)\leq 1}
 
 for 
 
 
 
 σ
 ≥
 0
 
 
 {\displaystyle \sigma \geq 0}
 
 then 
 
 
 
 ℜ
 (
 R
 (
 s
 )
 )
 ≥
 0
 
 
 {\displaystyle \Re (R(s))\geq 0}
 
 for 
 
 
 
 σ
 ≥
 0
 
 
 {\displaystyle \sigma \geq 0}
 
 and consequently 
 
 
 
 R
 (
 s
 )
 
 
 {\displaystyle R(s)}
 
 must be a PRF.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
Richards' theorem can also be derived from <a href="/facts/Schwarz%27s_lemma/70DeQvPJ">Schwarz's lemma</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<h2 id="uses">Uses</h2>
The theorem was introduced by <a href="/facts/Paul_I._Richards/K3C5FsCA">Paul I. Richards</a> as part of his investigation into the properties of PRFs. The term PRF was coined by <a href="/facts/Otto_Brune/1tkYVryc">Otto Brune</a> who proved that the PRF property was a <a href="/facts/Necessary_and_sufficient/G45aDCY9">necessary and sufficient</a> condition for a function to be realisable as a passive electrical network, an important result in <a href="/facts/Network_synthesis/Ulst8XaJ">network synthesis</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> Richards gave the theorem in his 1947 paper in the reduced form,<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

R
        (
        s
        )
        =
        
          
            
              Z
              (
              s
              )
              −
              s
              Z
              (
              1
              )
            
            
              Z
              (
              1
              )
              −
              s
              Z
              (
              s
              )
            
          
        
      
    
    {\displaystyle R(s)={\frac {Z(s)-sZ(1)}{Z(1)-sZ(s)}}}

that is, the special case where 
 
 
 
 k
 =
 1
 
 
 {\displaystyle k=1}

The theorem (with the more general casse of 
 
 
 
 k
 
 
 {\displaystyle k}
 
 being able to take on any value) formed the basis of the <a href="/facts/Network_synthesis/Ulst8XaJ">network synthesis</a> technique presented by <a href="/facts/Raoul_Bott/5JHkhPx8">Raoul Bott</a> and <a href="/facts/Richard_Duffin/E67hY3dn">Richard Duffin</a> in 1949.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> In the Bott-Duffin synthesis, 
 
 
 
 Z
 (
 s
 )
 
 
 {\displaystyle Z(s)}
 
 represents the electrical network to be synthesised and 
 
 
 
 R
 (
 s
 )
 
 
 {\displaystyle R(s)}
 
 is another (unknown) network incorporated within it (
 
 
 
 R
 (
 s
 )
 
 
 {\displaystyle R(s)}
 
 is unitless, but 
 
 
 
 R
 (
 s
 )
 Z
 (
 k
 )
 
 
 {\displaystyle R(s)Z(k)}
 
 has units of impedance and 
 
 
 
 R
 (
 s
 )
 
 /
 
 Z
 (
 k
 )
 
 
 {\displaystyle R(s)/Z(k)}
 
 has units of admittance). Making 
 
 
 
 Z
 (
 s
 )
 
 
 {\displaystyle Z(s)}
 
 the subject gives

Z
        (
        s
        )
        =
        
          
            (
            
              
                
                  
                    R
                    (
                    s
                    )
                  
                  
                    Z
                    (
                    k
                    )
                  
                
              
              +
              
                
                  k
                  
                    s
                    Z
                    (
                    k
                    )
                  
                
              
            
            )
          
          
            −
            1
          
        
        +
        
          
            (
            
              
                
                  1
                  
                    Z
                    (
                    k
                    )
                    R
                    (
                    s
                    )
                  
                
              
              +
              
                
                  s
                  
                    k
                    Z
                    (
                    k
                    )
                  
                
              
            
            )
          
          
            −
            1
          
        
      
    
    {\displaystyle Z(s)=\left({\frac {R(s)}{Z(k)}}+{\frac {k}{sZ(k)}}\right)^{-1}+\left({\frac {1}{Z(k)R(s)}}+{\frac {s}{kZ(k)}}\right)^{-1}}

Since 
 
 
 
 Z
 (
 k
 )
 
 
 {\displaystyle Z(k)}
 
 is merely a positive real number, 
 
 
 
 Z
 (
 s
 )
 
 
 {\displaystyle Z(s)}
 
 can be synthesised as a new network proportional to 
 
 
 
 R
 (
 s
 )
 
 
 {\displaystyle R(s)}
 
 in parallel with a capacitor all in series with a network proportional to the inverse of 
 
 
 
 R
 (
 s
 )
 
 
 {\displaystyle R(s)}
 
 in parallel with an inductor. By a suitable choice for the value of 
 
 
 
 k
 
 
 {\displaystyle k}
 
, a resonant circuit can be extracted from 
 
 
 
 R
 (
 s
 )
 
 
 {\displaystyle R(s)}
 
 leaving a function 
 
 
 
 
 Z
 ′
 
 (
 s
 )
 
 
 {\displaystyle Z'(s)}
 
 two degrees lower than 
 
 
 
 Z
 (
 s
 )
 
 
 {\displaystyle Z(s)}
 
. The whole process can then be applied iteratively to 
 
 
 
 
 Z
 ′
 
 (
 s
 )
 
 
 {\displaystyle Z'(s)}
 
 until the degree of the function is reduced to something that can be realised directly.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
The advantage of the Bott-Duffin synthesis is that, unlike other methods, it is able to synthesise any PRF. Other methods have limitations such as only being able to deal with two kinds of <a href="/facts/Electrical_element/CenJ4IkU">element</a> in any single network. Its major disadvantage is that it does not result in the minimal number of elements in a network. The number of elements grows exponentially with each iteration. After the first iteration there are two 
 
 
 
 
 Z
 ′
 
 
 
 {\displaystyle Z'}
 
 and associated elements, after the second, there are four 
 
 
 
 
 Z
 ″
 
 
 
 {\displaystyle Z''}
 
 and so on.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
Hubbard notes that Bott and Duffin appeared not to know the relationship of Richards' theorem to Schwarz's lemma and offers it as his own discovery,<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> but it was certainly known to Richards who used it in his own proof of the theorem.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>

<h2 id="bibliography">Bibliography</h2>
<ul><li><a href="/facts/Raoul_Bott/5JHkhPx8">Bott, Raoul</a>; <a href="/facts/Richard_Duffin/E67hY3dn">Duffin, Richard</a>, <a href="https://aip.scitation.org/doi/10.1063/1.1698532">"Impedance synthesis without use of transformers"</a>, Journal of Applied Physics, vol. 20, iss. 8, p. 816, August 1949.</li>
<li>Cauer, Emil; Mathis, Wolfgang; Pauli, Rainer, <a href="http://www.cs.princeton.edu/courses/archive/fall03/cs323/links/cauer.pdf">"Life and Work of Wilhelm Cauer (1900 – 1945)"</a>, Proceedings of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems (MTNS2000), Perpignan, June, 2000.</li>
<li><a href="/facts/John_H._Hubbard/FTMDUtRC">Hubbard, John H.</a>, "The Bott-Duffin synthesis of electrical circuits", pp. 33–40 in, Kotiuga, P. Robert (ed), A Celebration of the Mathematical Legacy of Raoul Bott, American Mathematical Society, 2010 <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780821883815.</li>
<li>Hughes, Timothy H.; Morelli, Alessandro; <a href="/facts/Malcolm_C._Smith/z86dQJhY">Smith, Malcolm C.</a>, <a href="https://link.springer.com/chapter/10.1007/978-3-319-67068-3_21">"Electrical network synthesis: A survey of recent work"</a>, pp. 281–293 in, Tempo, R.; Yurkovich, S.; Misra, P. (eds), Emerging Applications of Control and Systems Theory, Springer, 2018 <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9783319670676.</li>
<li><a href="/facts/Paul_I._Richards/K3C5FsCA">Richards, Paul I.</a>, <a href="http://projecteuclid.org/euclid.dmj/1077474311">"A special class of functions with positive real part in a half-plane"</a>, Duke Mathematical Journal, vol. 14, no. 3, 777–786, 1947.</li>
<li>Wing, Omar, Classical Circuit Theory, Springer, 2008 <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0387097406.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Wing, p. 122 <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Wing, pp. 122–123 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Hubbard, p. 33 <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Cauer et al., pp. 6–7 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Richards, p. 779 <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Wing, p. 122 <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Wing, pp. 123–125Hughes et al., pp. 284–285 <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Wing, p. 115 <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Hubbard, p. 33 <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Richards, p. 779 <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
</ol>

Richards' theorem open-in-new

Richards' theorem