# Star Delta Transformation

Star Delta Conversion and Delta Star Conversion allow us to convert interconnected impedances-resistors from one type of connection to another in a 3-phase configuration.

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We can now solve simple series, parallel or bridge-type resistant networks using Kirchhoff's Circuit Laws, mesh current analysis or node voltage analysis techniques, but we can use different mathematical techniques to simplify the analysis of the circuit in a balanced 3-phase circuit.

Standard 3-phase circuits or networks take two main forms with names that represent the way resistors are connected , the **Star-linked**network with the symbol of the letter Υ (y), and the **Delta**(Triangle) connected network with the symbol Δ (delta).

If a 3-phase, 3-wire feed or even a 3-phase load is connected in a configuration type, it can be easily converted using star **delta conversion** or delta **star conversion,** or replaced into an equivalent configuration of the other type.

A resilient network of three impedances can be connected to create a T or "Tee" configuration, but the network can also be redrawn to create a **Star** or Υ type network, as shown below.

### T-Linked and Equivalent Star Network

As we have already seen, we can redraw the above T resistance network to produce an electrically equivalent **Star** or Υ type network .However, we can also convert a Pi or π type resistance network to an electrically equivalent **Delta** or Δ type network, as shown below .

### Pi Connected and Equivalent Delta Network

Now it is possible to make **Star** and **Delta** transformations after defining two different circuits.

This process allows us **to** produce a mathematical relationship between various resistors that give us a Delta Star Transformation as well as a **Star Delta Transformation.**

These circuit transformations allow us to replace the three connected resistances (or impedance) for a star or triangular connected circuit with their equivalents measured between terminals 1-2, 1-3 or 2-3.However, the resulting networks are equivalent only to voltages and currents outside the star or triangular networks, since internally the voltages and currents are different, but each network will consume the same amount of power and have the same power factor to each other.

## Delta Star Conversion

To convert a delta network to an equivalent star network, we need to derive a conversion formula to synchronize various resistors between various terminals.Consider the circuit below.

### Delta to Star Network

Resistances between Terminals 1 and 2:

Resistances between terminals 2 and 3:

Resistances between terminals 1 and 3:

If we simplify these three equations:

Next, rewriting Equation 1 will give us:

Combining equation 1 with the above result of equation 3 minus equation 2 returns:

This gives us the last equation for resistance P:

Similarly, finding Q resistance in a star network is the result of equation 2 plus equation 1 minus equation 3 or Eq2 + (Eq1 – Eq3), which gives us the transformation of Q as follows:

and again, in order to find resistance R in a Star network, equation 3 plus equation 2 minus equation 1 or Eq3 + (Eq2 – Eq1) is the result, which gives us the transformation of R as follows:

When converting a delta network to a star network, all conversion formulas have the same denominators: A + B + C and ALL delta resistors are the sum.Then we can summarize the above transformation equations to convert any delta-connected network to an equivalent star network as follows:

### Delta-to-Star Conversion Equations

If all three resistances in the delta network are equal in value, the resulting resistances in the equivalent star network will be equal to a third of the value of delta resistors.This gives each resistant slae in the star network the following value: R _{STAR} = 1/3*R _{DELTA} , which is the same as the following expression: (R _{DELTA} )/3

### Delta Star Conversion Question Example 1

Turn the following Delta Resistant Network into an equivalent Star Network.

## Star Delta Transformation

The Star Delta transformation is the opposite of the one above.When converting from a delta network to an equivalent star network, we found that the resistance attached to one terminal is the product of two delta resistances connected to the same terminal.

By rewriting previous formulas, we can also find conversion formulas that give us a way to produce a star delta transformation, as shown below, to transform a resilient star network into an equivalent delta network.

### Star-to-Delta Transformation

The value of resistance on any side of the delta is the sum of all two-product resistance combinations in the Δ network, star resistance found "directly opposite" the found delta resistance.For example, resistance A is given as follows:

According to terminal 3 and resistance B, it is given as follows:

According to terminal 2 with C resistance:

Regarding Terminal 1.

By dividing each equation by denominator value, we get three separate conversion formulas that can be used to transform any Delta-resistant network into an equivalent star network given below.

### Star Delta Conversion Equations

One last point about turning a star-resistant network into an equivalent delta network.If the value of all resistances in the star network is equal, the resulting resistances in the equivalent delta network will be three times and equal to the value of star resistors, which gives: R _{DELTA} = 3*R _{STAR}

### Star Delta Transformation Question Example 2

Convert the following Star-Resistant Network to an equivalent Delta Network.

Both **Star Delta Conversion** and Delta Star **Conversion**allow us to convert one type of circuit connection to another type so that we can easily analyze the circuit.These conversion techniques can be used to simplify star or triangular circuits containing resistors or impedances.