# Star To Delta Or Delta To Star Transformation

Star to delta or Delta to star Transformation is used to simplify the circuit when we cant not able to solve the circuit by using series or parallel combination. Star and delta can be transformed from one form to another form easily with the help of driven parameters.

A star connection consists of three branches of a network often connected to a mutual point in Y -model.

A delta connection consists of the three branches of a network is connected in a closed loop in the form of a delta model.

## Delta(Δ) To Wye (Y) Or Star Conversion

It consist of three three- terminal network of three resistance R_{AB},R_{BC},R_{CA} are connected in delta. We need to convert this Delta to star of resistance R_{A},R_{B},R_{C} in wye or star network.

Lets solve the circuit and calculate the equivalent resistance between the two terminals

Equivalent resistance between A and C

R_A+R_C=\frac { R_{CA}(R_{BC}+R_{AB}) }{ R_{AB}+R_{BC}+R_{CA} }

Equivalent resistance between B and A

R_A+R_B=\frac { R_{AB}(R_{BC}+R_{CA}) }{ R_{AB}+R_{BC}+R_{CA} }

Equivalent resistance between C and B

R_B+R_C=\frac { R_{BA}(R_{CA}+R_{AB}) }{ R_{AB}+R_{BC}+R_{CA} }

Combining all the equations then

R_A+R_B+R_C=\frac { R_{AB}R_{BC}+R_{BC}R_{CA}+R_{CA}R_{AB} }{ R_{AB}+R_{BC}+R_{CA} }

By solving above equations we can get the the unknown resistance of why or star circuit

R_A=\frac { R_{CA}R_{AB} }{ R_{AB}+R_{BC}+R_{CA} }

R_B=\frac { R_{AB}R_{BC} }{ R_{AB}+R_{BC}+R_{CA} }

R_C=\frac {R_{BC}R_{CA} }{ R_{AB}+R_{BC}+R_{CA} }

## Wye (Y) Or Star To Delta(Δ) Conversion

Lets us consider a three- terminal network of three resistance R_{AB},R_{BC},R_{CA} are connected in star or wye . We need to convert this star to delta of resistance R_{A},R_{B},R_{C} in network.

Current equation through the R_A can be written as

I_A=\frac { (V_A-V_N) }{ R_A }

By apply KCL at node N for star connection

\frac { (V_A-V_N) }{ R_A } +\frac { (V_B-V_N) }{ R_B } +\frac { (V_C-V_N) }{ R_C } =0

By solving this equation we get V_N=\frac { \left( \frac { V_A }{ R_A } +\frac { V_B }{ R_B } +\frac { V_C }{ R_C } \right) }{ \left( \frac { 1 }{ R_A } +\frac { 1 }{ R_B } +\frac { 1 }{ R_C } \right) }

As we know that current entering from one node= current leaving to that node

I_A=\frac { V_{AB} }{ R_{AB} }+\frac { V_{AC} }{ R_{AC} } (for delta network)

From above two current equation we can compare both then

\frac { V_{AB} }{ R_{AB} }+\frac { V_{AC} }{ R_{AC} }=\frac { (V_A-V_N) }{ R_A }

Put the value of V_N in the above equation

\frac { V_{AB} }{ R_{AB} }+\frac { V_{AC} }{ R_{AC} }=\frac { V_A-\left( \frac { \frac { V_A }{ R_A } +\frac { V_B }{ R_B } +\frac { V_C }{ R_C } }{ \frac { 1 }{ R_A } +\frac { 1 }{ R_B } +\frac { 1 }{ R_C } } \right) }{ R_A }

By solving this equation and equating coefficient of V_{AB} and V_{AC} both side we get

\frac { 1 }{ R_{AB} } =\frac { 1 }{ R_AR_B\left( \frac { 1 }{ R_A } +\frac { 1 }{ R_B } +\frac { 1 }{ R_B } \right) }

R_{AB}=R_A+R_B+\frac { R_AR_B }{ R_C }

\frac { 1 }{ R_{AC} } =\frac { 1 }{ R_AR_C\left( \frac { 1 }{ R_A } +\frac { 1 }{ R_B } +\frac { 1 }{ R_C } \right) }

R_{AC}=R_A+R_C+\frac { R_AR_C }{ R_B }

similary we can drive the resistance for R_{BC}

steps for calculating R_{BC}

1. Calculate the current I_B for star circuit as we find above
2. Calculate the current I_B for delta circuit as we find above
3. Equating the above two equations and using the value of V_N

Then we get

\frac { 1 }{ R_{BC} } =\frac { 1 }{ R_BR_C\left( \frac { 1 }{ R_A } +\frac { 1 }{ R_B } +\frac { 1 }{ R_C } \right) }

R_{BC}=R_B+R_C+\frac { R_BR_C }{ R_A }

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