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# Star Delta Transformation and Delta Star Transformation

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Sometimes, in an electric circuit, resistors are connected either in Wye (or tee) form or Delta (or pie) form.

Above wye and Tee networks are electrically identical (same), means that we can replace wye circuit with tee circuit and vice versa.

Similarly, Delta and Pi networks are identical electrically and can be replaced one with another.

In order to convert wye circuit into Delta one or vice versa, a set of equations is used.

## Delta to Star (Wye) Transformation

Wye circuit is commonly known as star circuit. In order to convert delta connected resistors to wye network, we use the following set of equations:

Delta Star Transformation

${{R}_{1}}=\frac{{{R}_{b}}{{R}_{c}}}{{{R}_{a}}+{{R}_{b}}+{{R}_{c}}}$

${{R}_{2}}=\frac{{{R}_{a}}{{R}_{c}}}{{{R}_{a}}+{{R}_{b}}+{{R}_{c}}}$

${{R}_{3}}=\frac{{{R}_{a}}{{R}_{b}}}{{{R}_{a}}+{{R}_{b}}+{{R}_{c}}}$

The subscripts used in above equations can be seen in above network figures.

## Star (Wye) to Delta Transformation

In this case, to convert wye connected resistors to Delta connection, the following set of equations can be used:

Star Delta Transformation

${{R}_{a}}=\frac{{{R}_{1}}{{R}_{2}}+{{R}_{2}}{{R}_{3}}+{{R}_{3}}{{R}_{1}}}{{{R}_{1}}}$

${{R}_{b}}=\frac{{{R}_{1}}{{R}_{2}}+{{R}_{2}}{{R}_{3}}+{{R}_{3}}{{R}_{1}}}{{{R}_{2}}}$

${{R}_{c}}=\frac{{{R}_{1}}{{R}_{2}}+{{R}_{2}}{{R}_{3}}+{{R}_{3}}{{R}_{1}}}{{{R}_{3}}}$

Special Case

Sometimes, we encounter a situation where all wye connected resistors are equal and similarly Delta connected resistors are equal such as:

${{R}_{1}}={{R}_{2}}={{R}_{3}}={{R}_{Y}}$

And

${{R}_{a}}={{R}_{b}}={{R}_{c}}={{R}_{\Delta }}$

In this particular case, the transformation is quite easier and can be done using the simple relation between wye connected and delta connected resistors.

${{R}_{\Delta }}=3{{R}_{Y}}$

Using above equation, wye to delta and delta to wye transformation can be done with signification case.

## Example of Delta to Wye (Star) transformation

For example, we have following delta connected resistors network:

We can convert above circuit to wye connection using delta to wye transformation equation using delta to wye transformation equations:

${{R}_{1}}=\frac{{{R}_{b}}{{R}_{c}}}{{{R}_{a}}+{{R}_{b}}+{{R}_{c}}}$

${{R}_{2}}=\frac{{{R}_{a}}{{R}_{c}}}{{{R}_{a}}+{{R}_{b}}+{{R}_{c}}}$

${{R}_{3}}=\frac{{{R}_{a}}{{R}_{b}}}{{{R}_{a}}+{{R}_{b}}+{{R}_{c}}}$

So, the values of the resistors from above equations will be:

${{R}_{1}}=5~\Omega$

${{R}_{2}}=7.5~\Omega$

${{R}_{3}}=3~\Omega$

After connecting R1, R2 and R3 in Wye manner, we came up with following circuit (network).

Fig.1: Delta- Star (Wye) Transformation Example

## Delta to Wye (Star) transformation using Matlab

Here, we will solve the same above example using Matlab code.

% This code is written according to Figure 1 in the text. You can change
% it in case having different Delta-Wye Configurations
%%
% Here, Insert Delta-Connection Resistance Values
R_a=input('R_a=');
R_b=input('R_b=');
R_c=input('R_c=');
% Output is &amp;quot;Star Connection Resistances&amp;quot; as given in the text
[R_1,R_2,R_3]=delta2star(R_a,R_b,R_c);


Function delta2star.m

function[R_1,R_2,R_3]=delta2star(R_a,R_b,R_c)
R_1=(R_b*R_c)/(R_a+R_b+R_c);
R_2=(R_a*R_c)/(R_a+R_b+R_c);
R_3=(R_a*R_b)/(R_a+R_b+R_c);
fprintf('Delta Resistances \t Star Resistances\n ')
fprintf('R_a=%.2f \t R_1=%.2f \n',R_a,R_1);
fprintf('R_b=%.2f \t R_2=%.2f \n',R_b,R_2);
fprintf('R_c=%.2f \t R_3=%.2f \n',R_c,R_3);
end


Results:

Delta Resistances              Star Resistances
R_a=15.00                               R_1=5.00
R_b=10.00                              R_2=7.50
R_c=25.00                               R_3=3.00

Star delta transformation is mainly used in bridge networks where resistors are neither in series manner nor in parallel, such as given below:

Circuit Analysis techniques such as superposition, Nodal analysis, Mesh analysis and Millions theorem are not of any help in solving such circuits. Therefore we use such transformation to convert bridge networks into a simplified form.

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