Y-Δ transform

 Basic Y-Δ transformation

Y and Δ circuits with the labels which are used in this article.

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.

Equations for the transformation from Δ-load to Y-load 3-phase circuit

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

,

,

.

Equations for the transformation from Y-load to Δ-load 3-phase circuit

The general idea is to compute an impedance R_Δ in the Δ circuit by

where R_P = R₁R₂ + R₂R₃ + R₃R₁ 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 formulae for the individual edges are thus

,

,

.

 Graph theory

In graph theory, the Y-Δ 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 Y-Δ equivalent if one can be obtained from the other by a series of Y-Δ transforms in either direction. For example, the Petersen graphs are a Y-Δ equivalence class.

 Demonstration

 Δ-load to Y-load transformation equations

Let us know the values of R_b, R_c and R_a from the Δ configuration. We want to obtain the values of R₁, R₂ and R₃ of the equivalent Y configuration. In order to do that, we will calculate the equivalent impedances of both configurations in N₁N₂, N₁N₃ and N₂N₃, supposing in each case that the omitted node is unconnected, and we will equal both expressions, since the resistance must be the same.

The resistance between N₁ and N₂ when N₃ is not connected in the Δ configuration is

.

In the Y configuration, we have

R(N₁,N₂) = R₁ + R₂;

hence we have

  (1)

By similar calculations we obtain

  (2)

and

.   (3)

The impedances for the Y configuration can be derived from these equations by adding two equations and subtracting the third. For example, adding (1) and (3), then subtracting (2) yields

and hence

and

.

 Y-load to Δ-load transformation equations

Let R_T = R_a + R_b + R_c. We can write the Δ to Y equations as

  (1)

  (2)

.   (3)

Multiplying the pairs of equations yields

  (4)

  (5)

  (6)

and the sum of these equations is

  (7)

Now we divide each side of (7) by R₁, leaving

.   (8)

Using (1) in (8), we have

and by definition of R_T

which is the equation for R_c. Dividing (7) by R₂ and R₃ gives the other equations.

References

• William Stevenson, “Elements of Power System Analysis 3rd ed.”, McGraw Hill, New York, 1975, 

 External links

• Star-Triangle Conversion: Knowledge on resistive networks and resistors

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