# Y-Δ transform: Wikis

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# Encyclopedia

The Y-Δ transform, also written Y-delta, Wye-delta, Kennelly’s delta-star transformation, star-mesh transformation, T-Π or T-pi 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]

## Basic Y-Δ transformation

Δ and Y 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.

The general idea is to compute the impedance Ry at a terminal node of the Y circuit with impedances R', R'' to adjacent nodes in the Δ circuit by

$R_y = \frac{R'R''}{\sum R_\Delta}$

where RΔ are all impedances in the Δ circuit. This yields the specific formulae

$R_1 = \frac{R_aR_b}{R_a + R_b + R_c},$
$R_2 = \frac{R_bR_c}{R_a + R_b + R_c},$
$R_3 = \frac{R_aR_c}{R_a + R_b + R_c}.$

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

$R_\Delta = \frac{R_P}{R_\mathrm{opposite}}$

where RP = R1R2 + R2R3 + R3R1 is the sum of the products of all pairs of impedances in the Y circuit and Ropposite is the impedance of the node in the Y circuit which is opposite the edge with RΔ. The formula for the individual edges are thus

$R_a = \frac{R_1R_2 + R_2R_3 + R_3R_1}{R_2}$
$R_b = \frac{R_1R_2 + R_2R_3 + R_3R_1}{R_3}$
$R_c = \frac{R_1R_2 + R_2R_3 + R_3R_1}{R_1}$

## 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 graph family is a Y-Δ equivalence class.

## Demonstration

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

To relate {Ra,Rb, Rc} from Δ to {R1,R2,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 N1 and N2 with N3 disconnected in Δ:

\begin{align} R_\Delta(N_1, N_2) &= R_b \parallel (R_a+R_c) \ &= \frac{1}{\frac{1}{R_b}+\frac{1}{R_a+R_c}} \ &= \frac{R_b(R_a+R_c)}{R_a+R_b+R_c}. \end{align}

To simplify, let's call RT the sum of {Ra,Rb, Rc}.

RT = Ra + Rb + Rc

Thus,

$R_\Delta(N_1, N_2) = \frac{R_b(R_a+R_c)}{R_T}$

The corresponding impedance between N1 and N2 in Y is simple:

RY(N1, N2) = R1 + R2

hence:

$R_1+R_2 = \frac{R_b(R_a+R_c)}{R_T}$   (1)

Repeating for R(N2,N3):

$R_2+R_3 = \frac{R_c(R_a+R_b)}{R_T}$   (2)

and for R(N1,N3):

$R_1+R_3 = \frac{R_a(R_b+R_c)}{R_T}.$   (3)

From here, the values of {R1,R2,R 3} can be determined by linear combination (addition and/or subtraction).

For example, adding (1) and (3), then subtracting (2) yields

$R_1+R_2+R_1+R_3-R_2-R_3 = \frac{R_b(R_a+R_c)}{R_T} + \frac{R_a(R_b+R_c)}{R_T} - \frac{R_c(R_a+R_b)}{R_T}$
$2R_1 = \frac{2R_bR_a}{R_T}$

thus,

$R_1 = \frac{R_bR_a}{R_T}.$

where RT = Ra + Rb + Rc

For completeness:

$R_1 = \frac{R_bR_a}{R_T}$ (4)
$R_2 = \frac{R_bR_c}{R_T}$ (5)
$R_3 = \frac{R_aR_c}{R_T}$ (6)

Let

RT = Ra + Rb + Rc.

We can write the Δ to Y equations as

$R_1 = \frac{R_aR_b}{R_T}$   (1)
$R_2 = \frac{R_bR_c}{R_T}$   (2)
$R_3 = \frac{R_aR_c}{R_T}.$   (3)

Multiplying the pairs of equations yields

$R_1R_2 = \frac{R_aR_b^2R_c}{R_T^2}$   (4)
$R_1R_3 = \frac{R_a^2R_bR_c}{R_T^2}$   (5)
$R_2R_3 = \frac{R_aR_bR_c^2}{R_T^2}$   (6)

and the sum of these equations is

$R_1R_2 + R_1R_3 + R_2R_3 = \frac{R_aR_b^2R_c + R_a^2R_bR_c + R_aR_bR_c^2}{R_T^2}$   (7)

Factor RaRb Rc from the right side, leaving RT in the numerator, canceling with an RT in the denominator.

$R_1R_2 + R_1R_3 + R_2R_3 = \frac{(R_aR_bR_c)(R_a+R_b+R_c)}{R_T^2}$
$R_1R_2 + R_1R_3 + R_2R_3 = \frac{R_aR_bR_c}{R_T}$ (8)

-Note the similarity between (8) and {(1),(2),(3)}

Divide (8) by (1)

$\frac{R_1R_2 + R_1R_3 + R_2R_3}{R_1} = \frac{R_aR_bR_c}{R_T}\frac{R_T}{R_aR_b},$
$\frac{R_1R_2 + R_1R_3 + R_2R_3}{R_1} = R_c,$

which is the equation for Rc. Dividing (8) by R2 or R3 gives the other equations.

## Notes

1. ^ A.E. Kennelly, Equivalence of triangles and stars in conducting networks, Electrical World and Engineer, vol. 34, pp. 413-414, 1899.

## References

• William Stevenson, “Elements of Power System Analysis 3rd ed.”, McGraw Hill, New York, 1975, ISBN 0070612854