Y-Δ transform

For application in statistical mechanics see Yang-Baxter equation.
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 wye-delta and known by many other names, 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. This circuit transformation theory was published by Arthur Edwin Kennelly in 1899.[1]

Contents

Names

[[Image:Theoreme de kennelly.png|right|thumb|400px|Illustration of the transform in its T-Π representation, ]

The Y-Δ transform is known by a variety of other names, mostly based upon the two shapes involved, listed in either order. The Y, spelt out as wye, can also be called T or star; the Δ, spelt out as delta, can also be called triangle, Π (spelt out as pi), or mesh. Thus, common names for the transformation include wye-delta or delta-wye, star-delta, star-mesh, or T-Π.

Basic Y-Δ transformation

The transformation is used to establish equivalence for networks with three 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 complex as well as real 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

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

where R_\Delta are all impedances in the Δ circuit. This yields the specific formulae

R_1 = \frac{R_bR_c}{R_a %2B R_b %2B R_c},
R_2 = \frac{R_aR_c}{R_a %2B R_b %2B R_c},
R_3 = \frac{R_aR_b}{R_a %2B R_b %2B R_c}.

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

The general idea is to compute an impedance R_\Delta in the Δ circuit by

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

where R_P = R_1R_2%2BR_2R_3%2BR_3R_1 is the sum of the products of all pairs of impedances in the Y circuit and R_\mathrm{opposite} is the impedance of the node in the Y circuit which is opposite the edge with R_\Delta. The formula for the individual edges are thus

R_a = \frac{R_1R_2 %2B R_2R_3 %2B R_3R_1}{R_1},
R_b = \frac{R_1R_2 %2B R_2R_3 %2B R_3R_1}{R_2},
R_c = \frac{R_1R_2 %2B R_2R_3 %2B R_3R_1}{R_3}.

Usefulness

Resistive networks between two terminals can theoretically be simplified to a single equivalent resistor (more generally, the same is true of impedance). Series and parallel transforms are basic tools for doing so, but for complex networks such as the bridge illustrated here, they do not suffice.

The Y-Δ transform can be used to eliminate one node at a time and produce a network that can be further simplified, as shown.

The reverse transformation, Δ-Y, which adds a node, is often handy to pave the way for further simplification as well.

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-load to Y-load transformation equations

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 N1 and N2 with N3 disconnected in Δ:

\begin{align} R_\Delta(N_1, N_2) &= R_c \parallel (R_a%2BR_b) \\[8pt] &= \frac{1}{\frac{1}{R_c}%2B\frac{1}{R_a%2BR_b}} \\[8pt] &= \frac{R_c(R_a%2BR_b)}{R_a%2BR_b%2BR_c}. \end{align}

To simplify, let R_T be the sum of \{R_a, R_b, R_c\}.

R_T = R_a %2B R_b %2B R_c

Thus,

R_\Delta(N_1, N_2) = \frac{R_c(R_a%2BR_b)}{R_T}

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

R_Y(N_1, N_2) = R_1 %2B R_2

hence:

R_1%2BR_2 = \frac{R_c(R_a%2BR_b)}{R_T}   (1)

Repeating for R(N_2,N_3):

R_2%2BR_3 = \frac{R_a(R_b%2BR_c)}{R_T}   (2)

and for R(N_1,N_3):

R_1%2BR_3 = \frac{R_b(R_a%2BR_c)}{R_T}.   (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

R_1%2BR_2%2BR_1%2BR_3-R_2-R_3 = \frac{R_c(R_a%2BR_b)}{R_T} %2B \frac{R_b(R_a%2BR_c)}{R_T} - \frac{R_a(R_b%2BR_c)}{R_T}
2R_1 = \frac{2R_bR_c}{R_T}

thus,

R_1 = \frac{R_bR_c}{R_T}.

where R_T = R_a %2B R_b %2B R_c

For completeness:

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

Y-load to Δ-load transformation equations

Let

R_T = R_a%2BR_b%2BR_c.

We can write the Δ to Y equations as

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

Multiplying the pairs of equations yields

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

and the sum of these equations is

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

Factor R_aR_bR_c from the right side, leaving R_T in the numerator, canceling with an R_T in the denominator.

R_1R_2 %2B R_1R_3 %2B R_2R_3 = \frac{(R_aR_bR_c)(R_a%2BR_b%2BR_c)}{R_T^2}
R_1R_2 %2B R_1R_3 %2B 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 %2B R_1R_3 %2B R_2R_3}{R_1} = \frac{R_aR_bR_c}{R_T}\frac{R_T}{R_bR_c},
\frac{R_1R_2 %2B R_1R_3 %2B R_2R_3}{R_1} = R_a,

which is the equation for R_a. Dividing (8) by (2) or (3) (expressions for R_2 or R_3) gives the remaining equations.

See also

Notes

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

References

External links