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Y-? transform
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The Y-? transform, also written Y-delta, Wye-delta, Kennelly’s delta-star transformation, star-mesh transformation, T-? or T-pi 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 ?. In the United Kingdom, the wye diagram is sometimes known as a star.
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Encyclopedia
The Y-? transform, also written Y-delta, Wye-delta, Kennelly’s delta-star transformation, star-mesh transformation, T-? or T-pi 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 ?. In the United Kingdom, the wye diagram is sometimes known as a star. This circuit transformation theory was published by Arthur Edwin Kennelly in 1899.
Basic Y-? transformation
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 ?-load to Y-load 3-phase circuit
The general idea is to compute the impedance at a terminal node of the Y circuit with impedances , to adjacent nodes in the ? circuit by
where are all impedances in the ? circuit. This yields the specific formulae
Equations for the transformation from Y-load to ?-load 3-phase circuit
The general idea is to compute an impedance in the ? circuit by
where is the sum of the products of all pairs of impedances in the Y circuit and is the impedance of the node in the Y circuit which is opposite the edge with . The formula for the individual edges are thus
Graph theory
In graph theory, the Y-? transform means replacing a Y subgraph of a graph with the equivalent ? 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 Y-? equivalent if one can be obtained from the other by a series of Y-? transforms in either direction. For example, the Petersen graphs are a Y-? equivalence class.
Demonstration
?-load to Y-load transformation equations
Given the values of , and from the ? configuration, we want to obtain the values of , and in the equivalent Y configuration. In order to do that, we will calculate the equivalent impedances of both configurations in N1N2, N1N3 and N2N3, 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 N1 and N2 when N3 is not connected in the ? configuration is
In the Y configuration, we have
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
Y-load to ?-load transformation equations
Let . We can write the ? 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 , leaving
(8)
Using (1) in (8), we have
and by definition of
which is the equation for . Dividing (7) by and gives the other equations.
See also
External links
- : Knowledge on resistive networks and resistors
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