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

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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.

Delta-to-Wye transformation equations

General Idea: $R\_y\; =\; \backslash over\; \{\backslash Sigma\; R\_\{\backslash Delta\}\}\; \}$

$R\_1\; =\; \backslash left(\; \backslash frac\{R\_aR\_b\}\{R\_a\; +\; R\_b\; +\; R\_c\}\; \backslash right)$

$R\_2\; =\; \backslash left(\; \backslash frac\{R\_bR\_c\}\{R\_a\; +\; R\_b\; +\; R\_c\}\; \backslash right)$

$R\_3\; =\; \backslash left(\; \backslash frac\{R\_aR\_c\}\{R\_a\; +\; R\_b\; +\; R\_c\}\; \backslash right)$

Balanced System: $R\_\{\backslash Delta\}\; =\; 3\; \backslash times\; R\_y$

Wye-to-Delta transformation equations

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

$R\_a\; =\; \backslash left(\; \backslash frac\{R\_1R\_2\; +\; R\_2R\_3\; +\; R\_3R\_1\}\{R\_2\}\; \backslash right)$

$R\_b\; =\; \backslash left(\; \backslash frac\{R\_1R\_2\; +\; R\_2R\_3\; +\; R\_3R\_1\}\{R\_3\}\; \backslash right)$

$R\_c\; =\; \backslash left(\; \backslash frac\{R\_1R\_2\; +\; R\_2R\_3\; +\; R\_3R\_1\}\{R\_1\}\; \backslash right)$

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

In graph theory

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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

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• 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

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

Electrical circuits | Electric power | Graph theory

Teorema de Kennelly | Přepočet hvězda trojúhelník | Stern-Dreieck-Transformation | Teorema de Kennelly | Théorème de Kennelly | Trasformazioni stella-triangolo | Ster-driehoektransformatie