The YΔ transform, also written Ydelta, Wyedelta, Kennelly’s deltastar transformation, starmesh transformation, TΠ or Tpi transform, is a mathematical technique to simplify the analysis of an electrical network. The name derives from the shapes of the circuit diagrams, which look respectively like the letter Y and the Greek capital letter Δ. In the United Kingdom, the wye diagram is sometimes known as a star. This circuit transformation theory was published by Arthur Edwin Kennelly in 1899.^{[1]}
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The transformation is used to establish equivalence for networks with 3 terminals. Where three elements terminate at a common node and none are sources, the node is eliminated by transforming the impedances. For equivalence, the impedance between any pair of terminals must be the same for both networks. The equations given here are valid for real as well as complex impedances.
The general idea is to compute the impedance R_{y} at a terminal node of the Y circuit with impedances R', R'' to adjacent nodes in the Δ circuit by
where R_{Δ} are all impedances in the Δ circuit. This yields the specific formulae
The general idea is to compute an impedance R_{Δ} in the Δ circuit by
where R_{P} = R_{1}R_{2} + R_{2}R_{3} + R_{3}R_{1} is the sum of the products of all pairs of impedances in the Y circuit and R_{opposite} is the impedance of the node in the Y circuit which is opposite the edge with R_{Δ}. The formula for the individual edges are thus
In graph theory, the YΔ transform means replacing a Y subgraph of a graph with the equivalent Δ subgraph. The transform preserves the number of edges in a graph, but not the number of vertices or the number of cycles. Two graphs are said to be YΔ equivalent if one can be obtained from the other by a series of YΔ transforms in either direction. For example, the Petersen graph family is a YΔ equivalence class.
To relate {R_{a},R_{b}, R_{c}} from Δ to {R_{1},R_{2},R_{ 3}} from Y, the impedance between two corresponding nodes is compared. The impedance in either configuration is determined as if one of the nodes is disconnected from the circuit.
The impedance between N_{1} and N_{2} with N_{3} disconnected in Δ:
To simplify, let's call R_{T} the sum of {R_{a},R_{b}, R_{c}}.
Thus,
The corresponding impedance between N_{1} and N_{2} in Y is simple:
hence:
Repeating for R(N_{2},N_{3}):
and for R(N_{1},N_{3}):
From here, the values of {R_{1},R_{2},R_{ 3}} can be determined by linear combination (addition and/or subtraction).
For example, adding (1) and (3), then subtracting (2) yields
thus,
where R_{T} = R_{a} + R_{b} + R_{c}
For completeness:
Let
We can write the Δ to Y equations as
Multiplying the pairs of equations yields
and the sum of these equations is
Factor R_{a}R_{b} R_{c} from the right side, leaving R_{T} in the numerator, canceling with an R_{T} in the denominator.
Note the similarity between (8) and {(1),(2),(3)}
Divide (8) by (1)
which is the equation for R_{c}. Dividing (8) by R_{2} or R_{3} gives the other equations.
