Star To Delta Conversion Formula Pdf
Cori Riska
In the previous chapter, we discussed about the conversion of delta network into an equivalent star network. Now, let us discuss about the conversion of star network into an equivalent delta network. This conversion is called as Star to Delta Conversion.
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 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.
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.
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.
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]:
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:
The relation between star to delta equivalent impedance is clear from the given equation. The sum of the two-product of all star-impedances divide by the star impedance of the corresponding terminal is equal to the delta impedance connected with the opposite terminal.
Resistors are the most fundamental components in building of any electrical circuit, because of this most circuits constitutes of multiple resistors and they have to be simplified to obtain the net resistance for the circuit analysis. The resistances are grouped in either star/wye or delta topology and for the complete network resolution they have to be inter-converted into one another as there is no other transformation.
In this book, the three-phase ac systems are considered as a balanced circuit, made up of a balanced three-phase source, a balanced line, and a balanced three-phase load. Therefore, a balanced system can be studied using only one-third of the system, which can be analyzed on a line to neutral basis.
The star-delta (Y-Δ) or delta-star (Δ-Y) conversion (Fig. 3-15) is required in three-phase ac systems to simplify the circuits and ease their analysis. If a three-phase supply or a three-phase load is connected in delta, it can be transformed into an equivalent star-connected supply or load. After the analysis, the results are converted back into their original delta equivalent.
Since the load is balanced, the impedance per phase of the star-connected load will be one-third of the impedance per phase of the delta-connected load. Hence the equivalent impedances can be given by
One of the common uses of these transformations is in power system transmission line modeling and in three-phase transformer analysis. Circuit analysis involving three-phase transformers under balanced conditions can be performed on a per-phase basis. When Δ-Y or Y-Δ connections are present, the parameters refer to the Y side. In Δ-Δ connections, the Δ-connected impedances are converted to equivalent Y-connected impedances.
The objective of the following VI is to study these transformation concepts and provide an easy calculation tool using the complex impedances. The front panel of Star Delta Transformations.vi is given in Fig. 3-16 and is capable of transforming balanced or unbalanced three-phase impedance loads.
The circuit shown in Fig. 3-17 is called an unbalanced Wheatstone Bridge. Find the equivalent resistance between terminals A and D, which then can be used to calculate the source current for a given supply voltage.
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