Last modified on 20 November 2014, at 14:10

On 2D Inverse Problems/Y-Δ and star-mesh transforms

The effective conductivities b/w boundary nodes and the Dirichlet-to-Neumann operator of a network are invariant under the following star-mesh and Y-Δ transforms, illustrated by the following drawings from Wikipedia:

Star-mesh transform
Exercise (**). Let d be a diagonal entry of the Kirchhoff matrix K of a network G, corresponding to an interior node. Use the Schur complement formula \Lambda(G) = K/C = (K/d)/(C/d) for the Dirichlet-to-Neumann operator to prove the invariance.

The rules for replacing conductors in series or parallel connection by a single electrically equivalent conductor follow from the invariance property of the Y-Δ transform and can be viewed as its special cases, as also erasing an edge w/an end point of degree 1 and erasing a loop.

The Y-Δ transform is a special case of the star-mesh transform in which the center node is not a boundary node and has the degree 3.