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 an electrical device that converts three-phase electric power without a neutral wire into 3-phase power with a neutral wire. It is generally built from 3 independent transformers.)

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.

Image:Delta wye circ.PNG

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

Transformation equations

General Idea: <math> R_y = {{R_{\Delta adjacent 1} \times R_{\Delta adjacent 2}} \over {\Sigma R_{\Delta}} }</math>

<math>R_1 = \left( \frac{R_aR_b}{R_a + R_b + R_c} \right)</math>

<math>R_2 = \left( \frac{R_bR_c}{R_a + R_b + R_c} \right)</math>

<math>R_3 = \left( \frac{R_aR_c}{R_a + R_b + R_c} \right)</math>

Balanced System: <math> R_{\Delta} = 3 \times R_y </math>

Wye-to-Delta transformation equations

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

<math>R_a = \left( \frac{R_1R_2 + R_2R_3 + R_3R_1}{R_2} \right)</math>

<math>R_b = \left( \frac{R_1R_2 + R_2R_3 + R_3R_1}{R_3} \right)</math>

<math>R_c = \left( \frac{R_1R_2 + R_2R_3 + R_3R_1}{R_1} \right)</math>

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.

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 0070612854ca:Teorema de Kennelly

cs:Přepočet hvězda trojúhelník de:Stern-Dreieck-Umwandlung es:Teorema de Kennelly fr:Théorème de Kennelly nl:Ster-driehoektransformatie