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 Home Electronics lessons General electronics Y-DELTA transform , star to delta transformation

 Y-DELTA transform , star to delta transformation
Written by Bogdan

Y-Δ transform

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 known as a star.

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.

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

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

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 formulae 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 graphs are a Y-Δ equivalence class.

Demonstration

Let us know the values of Rb, Rc and Ra 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

$R(N_1, N_2) = R_b \parallel (R_a+R_c) = \frac{R_b(R_a+R_c)}{R_b+R_c+R_a} = \frac{R_bR_a+R_bR_c}{R_b+R_c+R_a}$.

In the Y configuration, we have

R(N1,N2) = R1 + R2;

hence we have

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

By similar calculations we obtain

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

and

$R_1+R_3 = \frac{R_aR_b+R_aR_c}{R_b+R_c+R_a}$.   (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

$R_1+R_2+R_1+R_3-R_2-R_3 = \frac{R_bR_a+R_bR_c}{R_b+R_c+R_a} + \frac{R_aR_b+R_aR_c}{R_b+R_c+R_a} - \frac{R_cR_a+R_cR_b}{R_b+R_c+R_a}$

and hence

$2R_1 = \frac{2R_bR_a}{R_b+R_c+R_a}$

and

$R_1 = \frac{R_bR_a}{R_b+R_c+R_a}$.

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)

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

$\frac{R_1R_2 + R_1R_3 + R_2R_3}{R_1} = \frac{1}{R_1}\frac{R_aR_b^2R_c + R_a^2R_bR_c + R_aR_bR_c^2}{R_T^2}$.   (8)

Using (1) in (8), we have

$\frac{R_1R_2 + R_1R_3 + R_2R_3}{R_1} = \frac{R_c(R_b + R_a + R_c)}{R_T}$

and by definition of RT

$\frac{R_1R_2 + R_1R_3 + R_2R_3}{R_1} = R_c$

which is the equation for Rc. 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,