Y-Δ transform

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. )

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.

File:Delta wye circ.PNG

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

Delta-to-Star transformation equations

General Idea: $R_{y}={{R_{\Delta adjacent1}\times R_{\Delta adjacent2}} \over {\Sigma R_{\Delta }}}$

$R_{1}=\left({\frac {R_{a}R_{b}}{R_{a}+R_{b}+R_{c}}}\right)$

$R_{2}=\left({\frac {R_{b}R_{c}}{R_{a}+R_{b}+R_{c}}}\right)$

$R_{3}=\left({\frac {R_{a}R_{c}}{R_{a}+R_{b}+R_{c}}}\right)$

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

Star-to-Delta transformation equations

General Idea: $R_{\Delta }={\Sigma (R_{yi}R_{yj})_{allpairs} \over R_{yopposite}}$

$R_{a}=\left({\frac {R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{2}}}\right)$

$R_{b}=\left({\frac {R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{3}}}\right)$

$R_{c}=\left({\frac {R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{1}}}\right)$

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

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.

Demonstration

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:

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

Star-to-delta transformation equations

Let

${R}_{T}={R}_{a}+{R}_{b}+{R}_{c}$

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

$(1)R_{1}={\frac {R_{a}R_{b}}{R_{T}}}$

$(2)R_{2}={\frac {R_{b}R_{c}}{R_{T}}}$

$(3)R_{3}={\frac {R_{a}R_{c}}{R_{T}}}$

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

$(1)R_{1}={\frac {R_{a}R_{b}}{R_{T}}}$

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

${\frac {R_{1}R_{2}+R_{1}R_{3}+R_{2}R_{3}}{R_{1}}}={\frac {R_{T}}{R_{a}R_{b}}}{\frac {R_{a}R_{b}^{2}R_{c}+R_{a}^{2}R_{b}R_{c}+R_{a}R_{b}R_{c}^{2}}{R_{T}^{2}}}={\frac {R_{b}R_{c}+R_{a}R_{c}+R_{c}^{2}}{R_{T}}}={\frac {R_{c}(R_{b}+R_{a}+R_{c})}{R_{T}}}(9)$

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.

See also

• Star-Triangle Conversion: Knowledge on resistive networks and resistors.
• Analysis of resistive circuits
• Electrical network: single phase electric power, alternating-current electric power, Three-phase power, polyphase systems for examples of wye and delta connections
• Electric motors for a discussion of wye-delta starting technique

References

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