# Star Delta Transformation

We can now solve simple series, parallel or bridge type resistive networks using Kirchhoff´s Circuit Laws, mesh current analysis or nodal voltage analysis techniques but in a balanced 3-phase circuit we can use different mathematical techniques to simplify the analysis of the circuit and thereby reduce the amount of math’s involved which in itself is a good thing.

Standard 3-phase circuits or networks take on two major forms with names that represent the way in which the resistances are connected, a

**Star**connected network which has the symbol of the letter, Υ (wye) and a

**Delta**connected network which has the symbol of a triangle, Δ (delta).

If a 3-phase, 3-wire supply or even a 3-phase load is connected in one type of configuration, it can be easily transformed or changed it into an equivalent configuration of the other type by using either the

**Star Delta Transformation**or

**Delta Star Transformation**process.

A resistive network consisting of three impedances can be connected together to form a T or “Tee” configuration but the network can also be redrawn to form a

**Star**or Υ type network as shown below.

### T-connected and Equivalent Star Network

### As we have already seen, we can redraw the T resistor network above to produce an electrically equivalent

**Star**or Υ type network. But we can also convert a Pi or π type resistor network into an electrically equivalent

**Delta**or Δ type network as shown below.

### Pi-connected and Equivalent Delta Network.

### Having now defined exactly what is a Star and Delta connected network it is possible to transform the Υ into an equivalent Δ circuit and also to convert a Δ into an equivalent Υ circuit using a the transformation process. This process allows us to produce a mathematical relationship between the various resistors giving us a Star Delta Transformation as well as a Delta Star Transformation.

## Delta Star Transformation

To convert a delta network to an equivalent star network we need to derive a transformation formula for equating the various resistors to each other between the various terminals. Consider the circuit below.### Delta to Star Network.

### Compare the resistances between terminals 1 and 2.

### Resistance between the terminals 2 and 3.

### Resistance between the terminals 1 and 3.

### This now gives us three equations and taking equation 3 from equation 2 gives:

### Then, re-writing Equation 1 will give us:

Adding together equation 1 and the result above of equation 3 minus equation 2 gives:

From which gives us the final equation for resistor P as:

Similarly, resistor Q and R can be found :-

When converting a delta network into a star network the denominators of all of the transformation formulas are the same: A + B + C, and which is the sum of ALL the delta resistances.

## Star Delta Transformation

Star Delta transformation is simply the reverse of above. We have seen that when converting from a delta network to an equivalent star network that the resistor connected to one terminal is the product of the two delta resistances connected to the same terminal, for example resistor P is the product of resistors A and B connected to terminal 1.The value of the resistor on any one side of the delta, Δ network is the sum of all the two-product combinations of resistors in the star network divide by the star resistor located “directly opposite” the delta resistor being found. For example, resistor A is given as:

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