Y-Δ transform

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The Y-Δ transform (also written Y-delta or Wye-delta), Kennelly's delta-star transformation, star-mesh transformation or T-Π (or T-pi) transform is a mathematical technique to simplify 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 UK the wye diagram is known as a star.

(A Δ-Y transformer, on the other hand, is a transformer that converts three-phase electric power without a neutral wire into 3-phase power with a neutral wire. )

1 Basic Y-Δ transformation
- 1.1 Delta-to-Star transformation equations
- 1.2 Star-to-Delta transformation equations
2 In graph theory
3 Demonstration
- 3.1 Delta-to-Star transformation equations
- 3.2 Star-to-delta transformation equations
4 See also
5 References

[edit] Basic Y-Δ transformation

The transformation is used to establish equivalence for networks with 3 terminals. Where three elements terminate at one point (node) and none is a source, the node is eliminated by transforming the impedances.

For equivalence, the impedance between any pair of terminals must be the same for both networks.

[edit] Delta-to-Star transformation equations

General Idea: $R_y = {{R_{\Delta adjacent 1} \times R_{\Delta adjacent 2}} \over {\Sigma R_{\Delta}} }$

$R_1 = \left( \frac{R_aR_b}{R_a + R_b + R_c} \right)$

$R_2 = \left( \frac{R_bR_c}{R_a + R_b + R_c} \right)$

$R_3 = \left( \frac{R_aR_c}{R_a + R_b + R_c} \right)$

Balanced System: $R_{\Delta} = 3 \times R_y$

[edit] Star-to-Delta transformation equations

General Idea: $R_{\Delta} = { \Sigma (R_{y i} R_{y j})_{all pairs} \over R_{y opposite} }$

$R_a = \left( \frac{R_1R_2 + R_2R_3 + R_3R_1}{R_2} \right)$

$R_b = \left( \frac{R_1R_2 + R_2R_3 + R_3R_1}{R_3} \right)$

$R_c = \left( \frac{R_1R_2 + R_2R_3 + R_3R_1}{R_1} \right)$

Note: that the equations are equally valid for impedances expressed in complex form.

[edit] In graph theory

In graph theory, the Y-Δ transform is used in contexts where there are no resistances labeling the edges, so it simply means replacing a wye subgraph of a graph with the delta subgraph. A Y-Δ 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 and their inverses, Δ-Y transforms.

The Petersen graph family is an example of a Y-Δ equivalence class.

[edit] Demonstration

[edit] Delta-to-Star transformation equations

Let us know the values of R_b, R_c and R_a from the delta configuration in the figure above; we want to obtain the values of R₁, R₂ and R₃ of the equivalent wye configuration. In order to do that, we will calculate the equivalent impedance of both configurations between the points N₁-N₂, N₁-N₃ and N₂-N₃ leaving the other node as if it was 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 delta configuration is

$R( N_1 , N_2 )= {R}_{b} || ( {R}_{a}+ {R}_{c} ) = \frac{{R}_{b}( {R}_{a}+{R}_{c})}{{R}_{b}+{R}_{c}+{R}_{a}} = \frac{{R}_{b}{R}_{a}+{R}_{b}{R}_{c}}{{R}_{b}+{R}_{c}+{R}_{a}}$

On the other hand, the resistance between N₁ and N₂ in the wye configuration is R(N₁,N₂)= R₁+R₂; hence,

$(1) R( N_1 , N_2 )= {R}_{1} + {R}_{2} = \frac{{R}_{b}{R}_{a}+{R}_{b}{R}_{c}}{{R}_{b}+{R}_{c}+{R}_{a}}$

Calculating the equivalent resistance between the two other nodes in the same way we obtain the expressions

$(2) R ( N_2 , N_3 )= {R}_{2} + {R}_{3} = \frac{{R}_{c}{R}_{a}+{R}_{c}{R}_{b}}{{R}_{b}+{R}_{c}+{R}_{a}}$

$(3) R ( N_1 , N_3 )= {R}_{1} + {R}_{3} = \frac{{R}_{a}{R}_{b}+{R}_{a}{R}_{c}}{{R}_{b}+{R}_{c}+{R}_{a}}$

The equations for the delta-to-star transform can be derived from equations (1), (2) and (3) in the following way:

R₁: Adding equations (1) and (3) and subtracting equation (2)
R₂: Adding equations (1) and (2) and subtracting equation (3)
R₃: Adding equations (2) and (3) and subtracting equation (1)

We will develop the first case as an example:

$(1) + (3) - (2) = ({R}_{1} + {R}_{2}) + ({R}_{1} + {R}_{3}) - ({R}_{2} + {R}_{3}) = \left( \frac{{R}_{b}{R}_{a}+{R}_{b}{R}_{c}}{{R}_{b}+{R}_{c}+{R}_{a}} \right) + \left( \frac{{R}_{a}{R}_{b}+{R}_{a}{R}_{c}}{{R}_{b}+{R}_{c}+{R}_{a}} \right) - \left( \frac{{R}_{c}{R}_{a}+{R}_{c}{R}_{b}}{{R}_{b}+{R}_{c}+{R}_{a}} \right)$

${R}_{1} + {R}_{2} + {R}_{1} + {R}_{3} - {R}_{2} - {R}_{3} = \frac{{R}_{b}{R}_{a}+{R}_{b}{R}_{c} + {R}_{a}{R}_{b}+{R}_{a}{R}_{c} - {R}_{c}{R}_{a}-{R}_{c}{R}_{b}}{{R}_{b}+{R}_{c}+{R}_{a}}$

$2 {R}_{1} = \frac{2 {R}_{b}{R}_{a}}{{R}_{b}+{R}_{c}+{R}_{a}}$

${R}_{1} = \frac{ {R}_{b}{R}_{a}}{{R}_{b}+{R}_{c}+{R}_{a}}$

Which is the expression for R₁ for the delta-to-star transform.

[edit] Star-to-delta transformation equations

Let

R T = R a + R b + R c

We can rewrite the delta-to-star transfomation equations as

$(1) R_1 = \frac{R_aR_b}{R_T}$

$(2) R_2 = \frac{R_bR_c}{R_T}$

$(3) R_3 = \frac{R_aR_c}{R_T}$

Multiplying the expressions in pairs:

$(1).(2)= R_1 R_2 = \frac{R_a R_b^2 R_c}{R_T^2} (4)$

$(1).(3)= R_1 R_3 = \frac{R_a^2 R_b R_c}{R_T^2} (5)$

$(2).(3)= R_2 R_3 = \frac{R_a R_b R_c^2}{R_T^2} (6)$

Adding the expressions (4), (5) and (6) we have

$(1).(2) + (1).(3) + (2).(3) = R_1 R_2 + R_1 R_3 + R_2 R_3 = \frac{R_a R_b^2 R_c + R_a^2 R_b R_c + R_a R_b R_c^2}{R_T^2} (7)$

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

$\frac{R_1 R_2 + R_1 R_3 + R_2 R_3}{R_1} = \frac{1}{R_1} \frac{R_a R_b^2 R_c + R_a^2 R_b R_c + R_a R_b R_c^2}{R_T^2} (8)$

From expression (1) we now that

$(1) R_1 = \frac{R_aR_b}{R_T}$

If we use this in the right side of expression (8) we have

$\frac{R_1 R_2 + R_1 R_3 + R_2 R_3}{R_1} = \frac{R_T}{R_aR_b} \frac{R_a R_b^2 R_c + R_a^2 R_b R_c + R_a R_b R_c^2}{R_T^2} = \frac{R_b R_c + R_a R_c + R_c^2}{R_T} = \frac{R_c (R_b + R_a + R_c)}{R_T} (9)$

But we defined

R T = R a + R b + R c

Which will simplify the right side of expression (9), leading

$\frac{R_1 R_2 + R_1 R_3 + R_2 R_3}{R_1} = R_c$

which is the expression of R_c for the star-to-delta transform. The expressions for R_a and R_b can be obtained in a similar way dividing expression (7) by R₂ and R₃, respectively.