In electrical engineering, the Y-Δ transform, also written wye-delta and also known by many other names, 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. This circuit transformation theory was published by Arthur Edwin Kennelly in 1899.[1] It is widely used in analysis of three-phase electric power circuits.
The Y-Δ transform can be considered a special case of the star-mesh transform for three resistors. In mathematics, the Y-Δ transform plays an important role in theory of circular planar graphs.[2]
The Y-Δ transform is known by a variety of other names, mostly based upon the two shapes involved, listed in either order. The Y, spelled out as wye, can also be called T or star; the Δ, spelled out as delta, can also be called triangle, Π (spelled out as pi), or mesh. Thus, common names for the transformation include wye-delta or delta-wye, star-delta, star-mesh, or T-Π.
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. Complex impedance is a quantity measured in ohms which represents resistance as positive real numbers in the usual manner, and also represents reactance as positive and negative imaginary values.
The general idea is to compute the impedance
RY
R'
R''
RY=
R'R'' | |
\sumR\Delta |
where
R\Delta
\begin{align} R1&=
RbRc | |
Ra+Rb+Rc |
\\[3pt] R2&=
RaRc | |
Ra+Rb+Rc |
\\[3pt] R3&=
RaRb | |
Ra+Rb+Rc |
\end{align}
The general idea is to compute an impedance
R\Delta
R\Delta=
RP | |
Ropposite |
where
RP=R1R2+R2R3+R3R1
Ropposite
R\Delta
\begin{align} Ra&=
R1R2+R2R3+R3R1 | |
R1 |
\\[3pt] Rb&=
R1R2+R2R3+R3R1 | |
R2 |
\\[3pt] Rc&=
R1R2+R2R3+R3R1 | |
R3 |
\end{align}
Or, if using admittance instead of resistance:
\begin{align} Ya&=
Y3Y2 | |
\sumYY |
\\[3pt] Yb&=
Y3Y1 | |
\sumYY |
\\[3pt] Yc&=
Y1Y2 | |
\sumYY |
\end{align}
Note that the general formula in Y to Δ using admittance is similar to Δ to Y using resistance.
The feasibility of the transformation can be shown as a consequence of the superposition theorem for electric circuits. A short proof, rather than one derived as a corollary of the more general star-mesh transform, can be given as follows. The equivalence lies in the statement that for any external voltages (
V1,V2
V3
N1,N2
N3
I1,I2
I3
1 | |
3 |
\left(I1-I2\right),-
1 | |
3 |
\left(I1-I2\right),0
0, | 1 |
3 |
\left(I2-I3\right),-
1 | |
3 |
\left(I2-I3\right)
- | 1 |
3 |
\left(I3-I1\right),0,
1 | |
3 |
\left(I3-I1\right)
The equivalence can be readily shown by using Kirchhoff's circuit laws that
I1+I2+I3=0
R3+R1=
\left(Rc+Ra\right)Rb | |
Ra+Rb+Rc |
,
R3 | |
R1 |
=
Ra | |
Rc |
.
Though usually six equations are more than enough to express three variables (
R1,R2,R3
Ra,Rb,Rc
In fact, the superposition theorem establishes the relation between the values of the resistances, the uniqueness theorem guarantees the uniqueness of such solution.
Resistive networks between two terminals can theoretically be simplified to a single equivalent resistor (more generally, the same is true of impedance). Series and parallel transforms are basic tools for doing so, but for complex networks such as the bridge illustrated here, they do not suffice.
The Y-Δ transform can be used to eliminate one node at a time and produce a network that can be further simplified, as shown.
The reverse transformation, Δ-Y, which adds a node, is often handy to pave the way for further simplification as well.
Every two-terminal network represented by a planar graph can be reduced to a single equivalent resistor by a sequence of series, parallel, Y-Δ, and Δ-Y transformations.[3] However, there are non-planar networks that cannot be simplified using these transformations, such as a regular square grid wrapped around a torus, or any member of the Petersen family.
In graph theory, the Y-Δ 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 Y-Δ equivalent if one can be obtained from the other by a series of Y-Δ transforms in either direction. For example, the Petersen family is a Y-Δ equivalence class.
To relate
\left\{Ra,Rb,Rc\right\}
\left\{R1,R2,R3\right\}
\begin{align}R\Delta\left(N1,N2\right) &=Rc\parallel(Ra+Rb)\\[3pt] &=
1 | |||||||
|
\\[3pt] &=
Rc\left(Ra+Rb\right) | |
Ra+Rb+Rc |
\end{align}
To simplify, let
RT
\left\{Ra,Rb,Rc\right\}
RT=Ra+Rb+Rc
Thus,
R\Delta\left(N1,N2\right)=
Rc(Ra+Rb) | |
RT |
The corresponding impedance between N1 and N2 in Y is simple:
RY\left(N1,N2\right)=R1+R2
hence:
R1+R2=
Rc(Ra+Rb) | |
RT |
Repeating for
R(N2,N3)
R2+R3=
Ra(Rb+Rc) | |
RT |
and for
R\left(N1,N3\right)
R1+R3=
Rb\left(Ra+Rc\right) | |
RT |
.
From here, the values of
\left\{R1,R2,R3\right\}
For example, adding (1) and (3), then subtracting (2) yields
\begin{align} R1+R2+R1+R3-R2-R3&=
Rc(Ra+Rb) | |
RT |
+
Rb(Ra+Rc) | |
RT |
-
Ra(Rb+Rc) | |
RT |
\\[3pt] {} ⇒ 2R1&=
2RbRc | |
RT |
\\[3pt] {} ⇒ R1&=
RbRc | |
RT |
. \end{align}
For completeness:
R1=
RbRc | |
RT |
R2=
RaRc | |
RT |
R3=
RaRb | |
RT |
Let
RT=Ra+Rb+Rc
We can write the Δ to Y equations as
R1=
RbRc | |
RT |
R2=
RaRc | |
RT |
R3=
RaRb | |
RT |
.
Multiplying the pairs of equations yields
R1R2=
| |||||||||||||
|
R1R3=
| |||||||||||||
|
R2R3=
| ||||||||||
|
and the sum of these equations is
R1R2+R1R3+R2R3=
| |||||||||||||||||||||
|
Factor
RaRbRc
RT
RT
\begin{align} R1R2+R1R3+R2R3 &={}
\left(RaRbRc\right) \left(Ra+Rb+Rc\right) | ||||||
|
\\ &={}
RaRbRc | |
RT |
\end{align}
Note the similarity between (8) and
Divide (8) by (1)
\begin{align}
R1R2+R1R3+R2R3 | |
R1 |
&={}
RaRbRc | |
RT |
RT | |
RbRc |
\\ &={}Ra, \end{align}
which is the equation for
Ra
R2
R3
During the analysis of balanced three-phase power systems, usually an equivalent per-phase (or single-phase) circuit is analyzed instead due to its simplicity. For that, equivalent wye connections are used for generators, transformers, loads and motors. The stator windings of a practical delta-connected three-phase generator, shown in the following figure, can be converted to an equivalent wye-connected generator, using the six following formulas:
\begin{align} &Zs1Y=\dfrac{Zs1Zs3
The resulting network is the following. The neutral node of the equivalent network is fictitious, and so are the line-to-neutral phasor voltages. During the transformation, the line phasor currents and the line (or line-to-line or phase-to-phase) phasor voltages are not altered.
If the actual delta generator is balanced, meaning that the internal phasor voltages have the same magnitude and are phase-shifted by 120° between each other and the three complex impedances are the same, then the previous formulas reduce to the four following:
\begin{align} &ZsY=\dfrac{Zs
where for the last three equations, the first sign (+) is used if the phase sequence is positive/abc or the second sign (−) is used if the phase sequence is negative/acb.