Y-Δ transform

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For the device which transforms three-phase electric power without a neutral wire into three-phase power with a neutral wire, see delta-wye transformer.

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. This circuit transformation theory was published by Arthur Edwin Kennelly in 1899. [1]

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[edit] Basic Y-Δ transformation

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

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

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}.

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

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 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}.

[edit] 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.

[edit] Demonstration

[edit] Δ-load to Y-load transformation equations

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

Given the values of Rb, Rc and Ra from the Δ configuration, we want to obtain the values of R1, R2 and R3 in 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}.

[edit] Y-load to Δ-load transformation equations

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.

[edit] References

  • William Stevenson, “Elements of Power System Analysis 3rd ed.”, McGraw Hill, New York, 1975, ISBN 0070612854
  1. ^ A.E. Kennelly, Equivalence of triangles and stars in conducting networks, Electrical World and Engineer, vol. 34, pp. 413-414, 1899.

[edit] See also

[edit] External links