Y- transform
1
Y- transform
The Y- transform, also written Y-delta, Wye-delta, Kennellys delta-star transformation, star-delta, 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.
[1]
Basic Y- transformation
and Y circuits with the labels which are used in this article.
The transformation is used to establish
equivalence for networks with three
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 complex as well as
real 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
Y- transform
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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 graph family is a Y- equivalence class.
Demonstration
-load to Y-load transformation equations
and Y circuits with the labels that are used in
this article.
To relate from to from Y, the impedance between two corresponding nodes is
compared. The impedance in either configuration is determined as if one of the nodes is disconnected from the
circuit.
The impedance between N
1
and N
2
with N
3
disconnected in :
To simplify, let be the sum of .
Thus,
The corresponding impedance between N
1
and N
2
in Y is simple:
hence:
(1)
Repeating for :
Y- transform
3
(2)
and for :
(3)
From here, the values of can be determined by linear combination (addition and/or subtraction).
For example, adding (1) and (3), then subtracting (2) yields
thus,
where
For completeness:
(4)
(5)
(6)
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)
Y- transform
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Factor from the right side, leaving in the numerator, canceling with an in the denominator.
(8)
-Note the similarity between (8) and {(1),(2),(3)}
Divide (8) by (1)
which is the equation for . Dividing (8) by (2) or (3) (expressions for or ) gives the remaining equations.
Notes
[1] A.E. Kennelly, Equivalence of triangles and stars in conducting networks, Electrical World and Engineer, vol. 34, pp. 413414, 1899.
References
William Stevenson, Elements of Power System Analysis 3rd ed., McGraw Hill, New York, 1975, ISBN
0070612854
External links
Star-Triangle Conversion (http:/ / www. designcabana. com/ knowledge/ electrical/ basics/ resistors): Knowledge
on resistive networks and resistors
Calculator of Star-Triangle transform (http:/ / www. stud. feec. vutbr. cz/ ~xvapen02/ vypocty/ transfigurace.
php?language=english)
Article Sources and Contributors
5
Article Sources and Contributors
Y- transform Source: http://en.wikipedia.org/w/index.php?oldid=429198300 Contributors: A. Carty, Abdull, Alejo2083, Ap, Apparition11, Bkell, Blotwell, Bodanger, Cbdorsett, Charles
Matthews, DMahalko, Damian Yerrick, DavidCary, Davr, Dicklyon, Draksis314, Fresheneesz, Giftlite, Greudin, Heron, Ideal gas equation, JHunterJ, Jamelan, Karada, L'Aquatique, Linas,
Lmdemasi, Michael Hardy, MonoAV, MonteChristof, Mwarren us, Oleg Alexandrov, PV=nRT, Pebkac, Phil Boswell, R.e.b., Reddi, Saippuakauppias, Shyam, Slandete, Tbhotch, The Anome,
Twri, Unraveled, Wdl1961n, Wtshymanski, Xyzzy n, Zzyzx11, 98 anonymous edits
Image Sources, Licenses and Contributors
Image:Wye-delta.svg Source: http://en.wikipedia.org/w/index.php?title=File:Wye-delta.svg License: GNU Free Documentation License Contributors: Xyzzy n
License
Creative Commons Attribution-Share Alike 3.0 Unported
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