Editing Y-Δ transform (section)

==Simplification of networks==
Resistive networks between two terminals can theoretically be [[Equivalent impedance transforms|simplified]] to a single equivalent resistor (more generally, the same is true of impedance). Series and parallel transforms are basic tools for doing so, but for complex networks such as the bridge illustrated here, they do not suffice. 

The Y-Δ transform can be used to eliminate one node at a time and produce a network that can be further simplified, as shown. 

[[Image:wye-delta bridge simplification.svg|center|thumb|480px|Transformation of a bridge resistor network, using the Y-Δ transform to eliminate node ''D'', yields an equivalent network that may readily be simplified further.]]

The reverse transformation, Δ-Y, which adds a node, is often handy to pave the way for further simplification as well.

[[Image:delta-wye bridge simplification.svg|center|thumb|336px|Transformation of a bridge resistor network, using the Δ-Y transform, also yields an equivalent network that may readily be simplified further.]]

Every two-terminal network represented by a [[planar graph]] can be reduced to a single equivalent resistor by a sequence of series, parallel, Y-Δ, and Δ-Y transformations.<ref>{{Cite journal|doi=10.1002/jgt.3190130202|title=On the delta-wye reduction for planar graphs|year=1989|last1=Truemper|first1=K.|journal=[[Journal of Graph Theory]]|volume=13|issue=2|pages=141–148}}</ref>  However, there are non-planar networks that cannot be simplified using these transformations, such as a regular square grid wrapped around a [[torus]], or any member of the [[Petersen family]].