Y-Δ transform

Last updated • 6 min readFrom Wikipedia, The Free Encyclopedia

In circuit design, 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.

Contents

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]

Names

Illustration of the transform in its T-P representation. Theoreme de kennelly2.svg
Illustration of the transform in its T-Π representation.

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.

Basic Y-Δ transformation

D and Y circuits with the labels which are used in this article. Wye-delta-2.svg
Δ 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. 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.

Equations for the transformation from Δ to Y

The general idea is to compute the impedance at a terminal node of the Y circuit with impedances , to adjacent nodes in the Δ circuit by

where are all impedances in the Δ circuit. This yields the specific formula

Equations for the transformation from Y to Δ

The general idea is to compute an impedance in the Δ 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 formulae for the individual edges are thus

Or, if using admittance instead of resistance:

Note that the general formula in Y to Δ using admittance is similar to Δ to Y using resistance.

A proof of the existence and uniqueness of the transformation

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 ( and ) applying at the three nodes ( and ), the corresponding currents ( and ) 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. and

The equivalence can be readily shown by using Kirchhoff's circuit laws that . Now each problem is relatively simple, since it involves only one single ideal current 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 of series and parallel circuits:

Though usually six equations are more than enough to express three variables () in term of the other three variables(), 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, the uniqueness theorem guarantees the uniqueness of such solution.

Simplification of networks

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.

Transformation of a bridge resistor network, using the Y-D transform to eliminate node D, yields an equivalent network that may readily be simplified further. Wye-delta bridge simplification.svg
Transformation of a bridge resistor network, using the Y-Δ transform to eliminate node D, yields an equivalent network that may readily be simplified further.

The reverse transformation, Δ-Y, which adds a node, is often handy to pave the way for further simplification as well.

Transformation of a bridge resistor network, using the D-Y transform, also yields an equivalent network that may readily be simplified further. Delta-wye bridge simplification.svg
Transformation of a bridge resistor network, using the Δ-Y transform, also yields an equivalent network that may readily be simplified further.

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.

Graph theory

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.

Demonstration

Δ-load to Y-load transformation equations

D and Y circuits with the labels that are used in this article. Wye-delta-2.svg
Δ and Y circuits with the labels that are used in this article.

To relate from Δ 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 N1 and N2 with N3 disconnected in Δ:

To simplify, let be the sum of .

Thus,

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

hence:

  (1)

Repeating for :

  (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

For completeness:

(4)
(5)
(6)

Y-load to Δ-load transformation equations

Let

.

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)

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.

Δ to Y transformation of a practical generator

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 [a] :

Practical generator connected in delta/triangle/pi. The quantities shown are phasor voltages and complex impedances. Click on image to expand it. Practical generator connected in delta-triangle (version 2).png
Practical generator connected in delta/triangle/pi. The quantities shown are phasor voltages and complex impedances. Click on image to expand it.

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.

Equivalent practical generator connected in wye/star/tee. Click on image to expand it. Equivalent practical generator connected in wye-star (version 2).png
Equivalent practical generator connected in wye/star/tee. Click on image to expand it.

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.

See also

Related Research Articles

<span class="mw-page-title-main">Cauchy–Riemann equations</span> Chacteristic property of holomorphic functions

In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which form a necessary and sufficient condition for a complex function of a complex variable to be complex differentiable.

<span class="mw-page-title-main">Electrical impedance</span> Opposition of a circuit to a current when a voltage is applied

In electrical engineering, impedance is the opposition to alternating current presented by the combined effect of resistance and reactance in a circuit.

<span class="mw-page-title-main">Moment of inertia</span> Scalar measure of the rotational inertia with respect to a fixed axis of rotation

The moment of inertia, otherwise known as the mass moment of inertia, angular/rotational mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is defined relative to a rotational axis. It is the ratio between the torque applied and the resulting angular acceleration about that axis. It plays the same role in rotational motion as mass does in linear motion. A body's moment of inertia about a particular axis depends both on the mass and its distribution relative to the axis, increasing with mass & distance from the axis.

In geodesy, conversion among different geographic coordinate systems is made necessary by the different geographic coordinate systems in use across the world and over time. Coordinate conversion is composed of a number of different types of conversion: format change of geographic coordinates, conversion of coordinate systems, or transformation to different geodetic datums. Geographic coordinate conversion has applications in cartography, surveying, navigation and geographic information systems.

In electrical engineering and electronics, a network is a collection of interconnected components. Network analysis is the process of finding the voltages across, and the currents through, all network components. There are many techniques for calculating these values; however, for the most part, the techniques assume linear components. Except where stated, the methods described in this article are applicable only to linear network analysis.

<span class="mw-page-title-main">Two-port network</span> Electric circuit with two pairs of terminals

In electronics, a two-port network is an electrical network or device with two pairs of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them satisfy the essential requirement known as the port condition: the current entering one terminal must equal the current emerging from the other terminal on the same port. The ports constitute interfaces where the network connects to other networks, the points where signals are applied or outputs are taken. In a two-port network, often port 1 is considered the input port and port 2 is considered the output port.

<span class="mw-page-title-main">Microstrip</span> Conductor–ground plane electrical transmission line

Microstrip is a type of electrical transmission line which can be fabricated with any technology where a conductor is separated from a ground plane by a dielectric layer known as "substrate". Microstrip lines are used to convey microwave-frequency signals.

In mathematics, the Fubini–Study metric is a Kähler metric on a complex projective space CPn endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study.

<span class="mw-page-title-main">Attenuator (electronics)</span> Type of electronic component

An attenuator is a passive broadband electronic device that reduces the power of a signal without appreciably distorting its waveform.

A Wilson current mirror is a three-terminal circuit that accepts an input current at the input terminal and provides a "mirrored" current source or sink output at the output terminal. The mirrored current is a precise copy of the input current.

Impedance parameters or Z-parameters are properties used in electrical engineering, electronic engineering, and communication systems engineering to describe the electrical behavior of linear electrical networks. They are also used to describe the small-signal (linearized) response of non-linear networks. They are members of a family of similar parameters used in electronic engineering, other examples being: S-parameters, Y-parameters, H-parameters, T-parameters or ABCD-parameters.

<span class="mw-page-title-main">Prototype filter</span> Template for electronic filter design

Prototype filters are electronic filter designs that are used as a template to produce a modified filter design for a particular application. They are an example of a nondimensionalised design from which the desired filter can be scaled or transformed. They are most often seen in regard to electronic filters and especially linear analogue passive filters. However, in principle, the method can be applied to any kind of linear filter or signal processing, including mechanical, acoustic and optical filters.

<span class="mw-page-title-main">Mild-slope equation</span> Physics phenomenon and formula

In fluid dynamics, the mild-slope equation describes the combined effects of diffraction and refraction for water waves propagating over bathymetry and due to lateral boundaries—like breakwaters and coastlines. It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is often used in coastal engineering to compute the wave-field changes near harbours and coasts.

<span class="mw-page-title-main">Velocity</span> Speed and direction of a motion

Velocity is the speed in combination with the direction of motion of an object. Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies.

The transmission-line matrix (TLM) method is a space and time discretising method for computation of electromagnetic fields. It is based on the analogy between the electromagnetic field and a mesh of transmission lines. The TLM method allows the computation of complex three-dimensional electromagnetic structures and has proven to be one of the most powerful time-domain methods along with the finite difference time domain (FDTD) method. The TLM was first explored by Raymond Beurle while working at English Electric Valve Company in Chelmsford. After he had been appointed professor of electrical engineering at the University of Nottingham in 1963 he jointly authored an article, "Numerical solution of 2-dimensional scattering problems using a transmission-line matrix", with Peter B. Johns in 1971.

<span class="mw-page-title-main">Primary line constants</span> Parameters of transmission lines

The primary line constants are parameters that describe the characteristics of conductive transmission lines, such as pairs of copper wires, in terms of the physical electrical properties of the line. The primary line constants are only relevant to transmission lines and are to be contrasted with the secondary line constants, which can be derived from them, and are more generally applicable. The secondary line constants can be used, for instance, to compare the characteristics of a waveguide to a copper line, whereas the primary constants have no meaning for a waveguide.

<span class="mw-page-title-main">RLC circuit</span> Resistor Inductor Capacitor Circuit

An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.

<span class="mw-page-title-main">Star-mesh transform</span> Mathematical circuit analysis technique

The star-mesh transform, or star-polygon transform, is a mathematical circuit analysis technique to transform a resistive network into an equivalent network with one less node. The equivalence follows from the Schur complement identity applied to the Kirchhoff matrix of the network.

In mathematics, a Janet basis is a normal form for systems of linear homogeneous partial differential equations (PDEs) that removes the inherent arbitrariness of any such system. It was introduced in 1920 by Maurice Janet. It was first called the Janet basis by Fritz Schwarz in 1998.

Reciprocity in electrical networks is a property of a circuit that relates voltages and currents at two points. The reciprocity theorem states that the current at one point in a circuit due to a voltage at a second point is the same as the current at the second point due to the same voltage at the first. The reciprocity theorem is valid for almost all passive networks. The reciprocity theorem is a feature of a more general principle of reciprocity in electromagnetism.

References

  1. Kennelly, A. E. (1899). "Equivalence of triangles and three-pointed stars in conducting networks". Electrical World and Engineer. 34: 413–414.
  2. Curtis, E.B.; Ingerman, D.; Morrow, J.A. (1998). "Circular planar graphs and resistor networks". Linear Algebra and Its Applications. 283 (1–3): 115–150. doi: 10.1016/S0024-3795(98)10087-3 .
  3. Truemper, K. (1989). "On the delta-wye reduction for planar graphs". Journal of Graph Theory . 13 (2): 141–148. doi:10.1002/jgt.3190130202.

Notes

  1. For a demonstration, read the Talk page.

Bibliography