Duality (electrical circuits)

<h2 id="history">History</h2>
<p>The use of duality in <a href="/facts/Circuit_theory/IjmVqbG8">circuit theory</a> is due to Alexander Russell who published his ideas in 1904.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<h2 id="examples">Examples</h2>
<h3>Constitutive relations</h3>
<ul><li>Resistor and conductor (Ohm's law) 
  
    
      
        v
        =
        i
        R
        
        ⟺
        
        i
        =
        v
        G
        
      
    
    {\displaystyle v=iR\iff i=vG\,}
  
</li>
<li>Capacitor and inductor – differential form 
  
    
      
        
          i
          
            C
          
        
        =
        C
        
          
            d
            
              d
              t
            
          
        
        
          v
          
            C
          
        
        
        ⟺
        
        
          v
          
            L
          
        
        =
        L
        
          
            d
            
              d
              t
            
          
        
        
          i
          
            L
          
        
      
    
    {\displaystyle i_{C}=C{\frac {d}{dt}}v_{C}\iff v_{L}=L{\frac {d}{dt}}i_{L}}
  
</li>
<li>Capacitor and inductor – integral form 
  
    
      
        
          v
          
            C
          
        
        (
        t
        )
        =
        
          V
          
            0
          
        
        +
        
          
            1
            C
          
        
        
          ∫
          
            0
          
          
            t
          
        
        
          i
          
            C
          
        
        (
        τ
        )
        
        d
        τ
        
        ⟺
        
        
          i
          
            L
          
        
        (
        t
        )
        =
        
          I
          
            0
          
        
        +
        
          
            1
            L
          
        
        
          ∫
          
            0
          
          
            t
          
        
        
          v
          
            L
          
        
        (
        τ
        )
        
        d
        τ
      
    
    {\displaystyle v_{C}(t)=V_{0}+{1 \over C}\int _{0}^{t}i_{C}(\tau )\,d\tau \iff i_{L}(t)=I_{0}+{1 \over L}\int _{0}^{t}v_{L}(\tau )\,d\tau }
  
</li></ul>
<h3>Voltage division — current division</h3>
<p>
  
    
      
        
          v
          
            
              R
              
                1
              
            
          
        
        =
        v
        
          
            
              R
              
                1
              
            
            
              
                R
                
                  1
                
              
              +
              
                R
                
                  2
                
              
            
          
        
        
        ⟺
        
        
          i
          
            
              G
              
                1
              
            
          
        
        =
        i
        
          
            
              G
              
                1
              
            
            
              
                G
                
                  1
                
              
              +
              
                G
                
                  2
                
              
            
          
        
      
    
    {\displaystyle v_{R_{1}}=v{\frac {R_{1}}{R_{1}+R_{2}}}\iff i_{G_{1}}=i{\frac {G_{1}}{G_{1}+G_{2}}}}

</p>
<h3>Impedance and admittance</h3>
<ul><li>Resistor and conductor 
  
    
      
        
          Z
          
            R
          
        
        =
        R
        
        ⟺
        
        
          Y
          
            G
          
        
        =
        G
      
    
    {\displaystyle Z_{R}=R\iff Y_{G}=G}
  
 
  
    
      
        
          Z
          
            G
          
        
        =
        
          
            1
            G
          
        
        
        ⟺
        
        
          Y
          
            R
          
        
        =
        
          
            1
            R
          
        
      
    
    {\displaystyle Z_{G}={1 \over G}\iff Y_{R}={1 \over R}}
  
</li>
<li>Capacitor and inductor 
  
    
      
        
          Z
          
            C
          
        
        =
        
          
            1
            
              C
              s
            
          
        
        
        ⟺
        
        
          Y
          
            L
          
        
        =
        
          
            1
            
              L
              s
            
          
        
      
    
    {\displaystyle Z_{C}={1 \over Cs}\iff Y_{L}={1 \over Ls}}
  
 
  
    
      
        
          Z
          
            L
          
        
        =
        L
        s
        
        ⟺
        
        
          Y
          
            c
          
        
        =
        C
        s
      
    
    {\displaystyle Z_{L}=Ls\iff Y_{c}=Cs}
  
</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Duality_(electricity_and_magnetism)/wnUElqzv">Duality (electricity and magnetism)</a></li>
<li><a href="/facts/Duality_(mechanical_engineering)/N7vNQvAY">Duality (mechanical engineering)</a></li>
<li><a href="/facts/Dual_impedance/uhE1WgdN">Dual impedance</a></li>
<li><a href="/facts/Dual_graph/1t14hhEF">Dual graph</a></li>
<li><a href="/facts/Mechanical%25E2%2580%2593electrical_analogies/VGGhVJec">Mechanical–electrical analogies</a></li>
<li><a href="/facts/List_of_dualities/TN2A0YUV">List of dualities</a></li></ul>

<ul><li>Turner, Rufus P, <i>Transistors Theory and Practice</i>, Gernsback Library, Inc, New York, 1954, Chapter 6.</li></ul>

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

<ol>
<li id="fn:1"><p>Belevitch, V, "Summary of the history of circuit theory", Proceedings of the IRE, vol 50, Iss 5, pp. 848–855, May 1962 doi:10.1109/JRPROC.1962.288301. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Alexander Russell, A Treatise on the Theory of Alternating Currents, volume 1, chapter XVII, Cambridge: University Press 1904 OCLC 264936988. <a href="/wiki/OCLC_(identifier)" target="_blank">/wiki/OCLC_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Duality (electrical circuits) open-in-new

Duality (electrical circuits)