Star-mesh transform

The star-mesh transform, or star-polygon transform, is a mathematical <a href="/facts/Circuit_analysis/IjmVqbG8">circuit analysis</a> technique to transform a <a href="/facts/Network_analysis_(electrical_circuits)/IjmVqbG8">resistive network</a> into an equivalent network with one less node. The equivalence follows from the <a href="/facts/Schur_complement/n5Frtl3p">Schur complement</a> identity applied to the <a href="/facts/Kirchhoff_matrix/Iwea2mm9">Kirchhoff matrix</a> of the network.

The equivalent impedance betweens nodes A and B is given by:

z
          
            AB
          
        
        =
        
          z
          
            A
          
        
        
          z
          
            B
          
        
        ∑
        
          
            1
            z
          
        
        ,
      
    
    {\displaystyle z_{\text{AB}}=z_{\text{A}}z_{\text{B}}\sum {\frac {1}{z}},}

where 
 
 
 
 
 z
 
 A
 
 
 
 
 {\displaystyle z_{\text{A}}}
 
 is the impedance between node A and the central node being removed.
The transform replaces N resistors with 
 
 
 
 
 
 1
 2
 
 
 N
 (
 N
 −
 1
 )
 
 
 {\textstyle {\frac {1}{2}}N(N-1)}
 
 resistors. For 
 
 
 
 N
 >
 3
 
 
 {\textstyle N>3}
 
, the result is an increase in the number of resistors, so the transform has no general inverse without additional constraints. 
It is possible, though not necessarily efficient, to transform an arbitrarily complex two-terminal resistive network into a single equivalent resistor by repeatedly applying the star-mesh transform to eliminate each non-terminal node.

Star-mesh transform open-in-new

Star-mesh transform