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Incircuit design, theY-Δ transform, also writtenwye-delta and also known by many other names, is a mathematical technique to simplifythe analysis of anelectrical network. The name derives from the shapes of thecircuit diagrams, which look respectively like the letter Y and the Greek capital letterΔ. This circuit transformation theory was published byArthur Edwin Kennelly in 1899.^[1] It is widely used in analysis ofthree-phase electric power circuits.

The Y-Δ transform can be considered a special case of thestar-mesh transform for threeresistors. In mathematics, the Y-Δ transform plays an important role in theory of circularplanar graphs.^[2]

TheY-Δ transform is known by a variety of other names, mostly based upon the two shapes involved, listed in either order. TheY, spelled out aswye, can also be calledT orstar; theΔ, spelled out asdelta, can also be calledtriangle,Π (spelled out aspi), ormesh. Thus, common names for the transformation includewye-delta ordelta-wye,star-delta,star-mesh, orT-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 inohms which represents resistance aspositive real numbers in the usual manner, and also representsreactance as positive and negativeimaginary values.

The general idea is to compute the impedance${\displaystyle R_{\text{Y}}}$ at a terminal node of the Y circuit with impedances${\displaystyle R^{\prime }}$,${\displaystyle R^{″}}$ to adjacent nodes in the Δ circuit by

${\displaystyle R_{\text{Y}}=\frac{R^{\prime }{R}^{″}}{\sum R_{\mathrm{\Delta }}}}$

where${\displaystyle R_{\mathrm{\Delta }}}$ are all impedances in the Δ circuit. This yields the specific formula

The general idea is to compute an impedance${\displaystyle R_{\mathrm{\Delta }}}$ in the Δ circuit by

${\displaystyle R_{\mathrm{\Delta }}=\frac{R_P}{R_{\text{opposite}}}}$

where${\displaystyle R_P=R_1{R}_2+R_2{R}_3+R_3{R}_1}$ is the sum of the products of all pairs of impedances in the Y circuit and${\displaystyle R_{\text{opposite}}}$ is the impedance of the node in the Y circuit which is opposite the edge with${\displaystyle R_{\mathrm{\Delta }}}$. The formulae for the individual edges are thus

Or, if usingadmittance instead of resistance:

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 thesuperposition theorem for electric circuits. A short proof, rather than one derived as a corollary of the more generalstar-mesh transform, can be given as follows. The equivalence lies in the statement that for any external voltages (${\displaystyle V_1,V_2}$ and${\displaystyle V_3}$) applying at the three nodes (${\displaystyle N_1,N_2}$ and${\displaystyle N_3}$), the corresponding currents (${\displaystyle I_1,I_2}$ and${\displaystyle I_3}$) are exactly the same for both the Y and Δ circuit, and vice versa. In this proof, we start with given external currents at the nodes. According to the superposition theorem, the voltages can be obtained by studying the superposition of the resulting voltages at the nodes of the following three problems applied at the three nodes with current:

1. ${\displaystyle \frac{1}{3}\left(I_1-I_2\right),-\frac{1}{3}\left(I_1-I_2\right),0}$
2. ${\displaystyle 0,\frac{1}{3}\left(I_2-I_3\right),-\frac{1}{3}\left(I_2-I_3\right)}$ and
3. ${\displaystyle -\frac{1}{3}\left(I_3-I_1\right),0,\frac{1}{3}\left(I_3-I_1\right)}$

The equivalence can be readily shown by usingKirchhoff's circuit laws that${\displaystyle I_1+I_2+I_3=0}$. Now each problem is relatively simple, since it involves only one single idealcurrent source. To obtain exactly the same outcome voltages at the nodes for each problem, the equivalent resistances in the two circuits must be the same, this can be easily found by using the basic rules ofseries and parallel circuits:

${\displaystyle R_3+R_1=\frac{\left(R_{\text{c}}+R_{\text{a}}\right)R_{\text{b}}}{R_{\text{a}}+R_{\text{b}}+R_{\text{c}}},\frac{R_3}{R_1}=\frac{R_{\text{a}}}{R_{\text{c}}}.}$

Though usually six equations are more than enough to express three variables (${\displaystyle R_1,R_2,R_3}$) in term of the other three variables(${\displaystyle R_{\text{a}},R_{\text{b}},R_{\text{c}}}$), here it is straightforward to show that these equations indeed lead to the above designed expressions.

In fact, the superposition theorem establishes the relation between the values of the resistances, theuniqueness theorem guarantees the uniqueness of such solution.

Resistive networks between two terminals can theoretically besimplified 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 aplanar 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 atorus, or any member of thePetersen family.

Ingraph theory, the Y-Δ transform means replacing a Ysubgraph 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 ofcycles. Two graphs are said to beY-Δ equivalent if one can be obtained from the other by a series of Y-Δ transforms in either direction. For example, thePetersen family is a Y-Δequivalence class.

To relate${\displaystyle \left\{R_{\text{a}},R_{\text{b}},R_{\text{c}}\right\}}$ from Δ to${\displaystyle \left\{R_1,R_2,R_3\right\}}$ 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 betweenN₁ andN₂ withN₃ disconnected in Δ:

To simplify, let${\displaystyle R_{\text{T}}}$ be the sum of${\displaystyle \left\{R_{\text{a}},R_{\text{b}},R_{\text{c}}\right\}}$.

${\displaystyle R_{\text{T}}=R_{\text{a}}+R_{\text{b}}+R_{\text{c}}}$

Thus,

${\displaystyle R_{\mathrm{\Delta }}\left(N_1,N_2\right)=\frac{R_{\text{c}}(R_{\text{a}}+R_{\text{b}})}{R_{\text{T}}}}$

The corresponding impedance between N₁ and N₂ in Y is simple:

${\displaystyle R_{\text{Y}}\left(N_1,N_2\right)=R_1+R_2}$

hence:

${\displaystyle R_1+R_2=\frac{R_{\text{c}}(R_{\text{a}}+R_{\text{b}})}{R_{\text{T}}}}$   (1)

Repeating for${\displaystyle R(N_2,N_3)}$:

${\displaystyle R_2+R_3=\frac{R_{\text{a}}(R_{\text{b}}+R_{\text{c}})}{R_{\text{T}}}}$   (2)

and for${\displaystyle R\left(N_1,N_3\right)}$:

${\displaystyle R_1+R_3=\frac{R_{\text{b}}\left(R_{\text{a}}+R_{\text{c}}\right)}{R_{\text{T}}}.}$   (3)

From here, the values of${\displaystyle \left\{R_1,R_2,R_3\right\}}$ can be determined by linear combination (addition and/orsubtraction).

For example, adding (1) and (3), then subtracting (2) yields

For completeness:

${\displaystyle R_1=\frac{R_{\text{b}}{R}_{\text{c}}}{R_{\text{T}}}}$ (4)

${\displaystyle R_2=\frac{R_{\text{a}}{R}_{\text{c}}}{R_{\text{T}}}}$ (5)

${\displaystyle R_3=\frac{R_{\text{a}}{R}_{\text{b}}}{R_{\text{T}}}}$ (6)

Let

${\displaystyle R_{\text{T}}=R_{\text{a}}+R_{\text{b}}+R_{\text{c}}}$.

We can write the Δ to Y equations as

${\displaystyle R_1=\frac{R_{\text{b}}{R}_{\text{c}}}{R_{\text{T}}}}$   (1)

${\displaystyle R_2=\frac{R_{\text{a}}{R}_{\text{c}}}{R_{\text{T}}}}$   (2)

${\displaystyle R_3=\frac{R_{\text{a}}{R}_{\text{b}}}{R_{\text{T}}}.}$   (3)

Multiplying the pairs of equations yields

${\displaystyle R_1{R}_2=\frac{R_{\text{a}}{R}_{\text{b}}{R}_{\text{c}}^2}{R_{\text{T}}^2}}$   (4)

${\displaystyle R_1{R}_3=\frac{R_{\text{a}}{R}_{\text{b}}^2{R}_{\text{c}}}{R_{\text{T}}^2}}$   (5)

${\displaystyle R_2{R}_3=\frac{R_{\text{a}}^2{R}_{\text{b}}{R}_{\text{c}}}{R_{\text{T}}^2}}$   (6)

and the sum of these equations is

${\displaystyle R_1{R}_2+R_1{R}_3+R_2{R}_3=\frac{R_{\text{a}}{R}_{\text{b}}{R}_{\text{c}}^2+R_{\text{a}}{R}_{\text{b}}^2{R}_{\text{c}}+R_{\text{a}}^2{R}_{\text{b}}{R}_{\text{c}}}{R_{\text{T}}^2}}$   (7)

Factor${\displaystyle R_{\text{a}}{R}_{\text{b}}{R}_{\text{c}}}$ from the right side, leaving${\displaystyle R_{\text{T}}}$ in the numerator, canceling with an${\displaystyle R_{\text{T}}}$ in the denominator.

(8)

Note the similarity between (8) and {(1), (2), (3)}

Divide (8) by (1)

which is the equation for${\displaystyle R_{\text{a}}}$. Dividing (8) by (2) or (3) (expressions for${\displaystyle R_2}$ or${\displaystyle R_3}$) gives the remaining equations.

During the analysis of balancedthree-phasepower systems, usually an equivalent per-phase (or single-phase) circuit is analyzed instead due to its simplicity. For that, equivalent wye connections are used forgenerators,transformers, loads andmotors. 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^[a]:

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:

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.

• Star-mesh transform
• Network analysis (electrical circuits)
• Electrical network,three-phase power,polyphase systems for examples of Y and Δ connections
• AC motor for a discussion of the Y-Δ starting technique

• William Stevenson,Elements of Power System Analysis 3rd ed., McGraw Hill, New York, 1975,ISBN 0-07-061285-4

• Star-Triangle ConversionArchived 2009-08-30 at theWayback Machine: Knowledge on resistive networks and resistors
• Calculator of Star-Triangle transform

• Electrical circuits
• Electric power
• Graph operations
• Circuit theorems
• Three-phase AC power

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