YΔ transformThe YΔ transform, also written Ydelta, Wyedelta, Kennelly’s deltastar transformation, starmesh transformation, TΠ or Tpi transform, 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. In the United Kingdom, the wye diagram is known as a star. Basic YΔ transformation Y and Δ circuits with the labels which are used in this article. The transformation is used to establish equivalence for networks with 3 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 real as well as complex impedances. Equations for the transformation from Δload to Yload 3phase 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 where R_{Δ} are all impedances in the Δ circuit. This yields the specific formulae  ,
 ,
 .
Equations for the transformation from Yload to Δload 3phase circuit The general idea is to compute an impedance R_{Δ} in the Δ circuit by where R_{P} = R_{1}R_{2} + R_{2}R_{3} + R_{3}R_{1} is the sum of the products of all pairs of impedances in the Y circuit and R_{opposite} is the impedance of the node in the Y circuit which is opposite the edge with R_{Δ}. The formulae for the individual edges are thus  ,
 ,
 .
Graph theory In graph theory, the YΔ 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 YΔ equivalent if one can be obtained from the other by a series of YΔ transforms in either direction. For example, the Petersen graphs are a YΔ equivalence class. Demonstration Δload to Yload transformation equations Let us know the values of R_{b}, R_{c} and R_{a} from the Δ configuration. We want to obtain the values of R_{1}, R_{2} and R_{3} of the equivalent Y configuration. In order to do that, we will calculate the equivalent impedances of both configurations in N_{1}N_{2}, N_{1}N_{3} and N_{2}N_{3}, supposing in each case that the omitted node is unconnected, and we will equal both expressions, since the resistance must be the same. The resistance between N_{1} and N_{2} when N_{3} is not connected in the Δ configuration is  .
In the Y configuration, we have  R(N_{1},N_{2}) = R_{1} + R_{2};
hence we have  (1)
By similar calculations we obtain  (2)
and  . (3)
The impedances for the Y configuration can be derived from these equations by adding two equations and subtracting the third. For example, adding (1) and (3), then subtracting (2) yields and hence and  .
Yload to Δload transformation equations Let R_{T} = R_{a} + R_{b} + R_{c}. We can write the Δ to Y equations as  (1)
 (2)
 . (3)
Multiplying the pairs of equations yields  (4)
 (5)
 (6)
and the sum of these equations is  (7)
Now we divide each side of (7) by R_{1}, leaving  . (8)
Using (1) in (8), we have and by definition of R_{T} which is the equation for R_{c}. Dividing (7) by R_{2} and R_{3} gives the other equations. References  William Stevenson, “Elements of Power System Analysis 3rd ed.”, McGraw Hill, New York, 1975,
External links
From Wikipedia, the free encyclopedia
