Star to Delta and Delta to Star Transformations:

Star to Delta and Delta to Star Transformations:

 In a pattern series and parallel connections, electrical components may be connected in Star or Delta configurations as shown in the figure below (with some Resistances). Many a times circuits have to be transformed from Star to equivalent Delta and Delta to equivalent Star configurations such that the network terminal Resistances (  or Impedance's ) across the terminals are the same. We will show this transformation methodologies and the resulting configurations for both Delta to Star and Star to Delta one by one.

 

 

 

Delta to Star Transformation: 

The circuit configurations are identically provided then the network net resistances across the terminal pairs XY , YZ and ZX in both connections are the same. 

In Star Connection they are:

 RX-Y = RX+RY ----------------------(a) 

 RY-Z = RY+RZ -----------------------(b) 

 RZ-X = RZ+RX -----------------------(c)

 Similar way in Delta connection they are:

 RX-Y = R1//(R2+R3) = R1(R2+R3) / [R1+R2+R3] ------------------(d)  

 RY-Z= R2//(R1+R3) = R2(R1+R3) / [R1+R2+R3]  -------------------(e)

 RZ-X= R3//(R1+R2) = R3(R1+R2) / [R1+R2+R3] --------------------(f) 

 By equating these respective equation's, we get

RX+ RY = R1(R2+R3) / [R1+R2+R3] ------------------(g) 

RY+ RZ = R2(R1+R3) / [R1+R2+R3]  -------------------(h)

 RZ+ Rx = R3(R1+R2) / [R1+R2+R3] --------------------(i)

 By subtracting equation h from equation g given above,we get

 

𝑅𝑋−𝑅𝑍= [[𝑅1𝑅2+𝑅1𝑅3]/[𝑅1+𝑅2+𝑅3]]− [[𝑅2𝑅1+𝑅2𝑅3]/[𝑅1+𝑅2+𝑅3]]---------(j)

 Then adding this equation to equation i above i.e. (RZ+RX) we get

 2𝑅𝑋= [𝑅1𝑅2+𝑅1𝑅3−𝑅2𝑅1−𝑅2𝑅3+𝑅3𝑅1+𝑅3𝑅2] / [𝑅1+𝑅2+𝑅3] 

2𝑅𝑋=2𝑅1𝑅3 / [𝑅1+𝑅2+𝑅3]

𝑅𝑋=𝑅1𝑅3 / [𝑅1+𝑅2+𝑅3]

And in a similar way we can get:

 𝑅𝑌=𝑅1𝑅2 /[𝑅1+𝑅2+𝑅3]

𝑅𝑍=𝑅2𝑅3 /[𝑅1+𝑅2+𝑅3]

Where RX , RY and RZ are the equivalent resistances in the Star Circuit connection corresponding to the Delta Circuit connection with resistances R1, R2 and R3.

 Star to Delta Transformation: 

Now we have to get the equivalent values of R1, R2 and R3 in Delta connection in terms of the three resistances RX, RY and RZ in Star connection. Let us, use the equations we got earlier i.e. RX, RY and RZ in terms of R1, R2 and R3 and get the sum of the three product pairs i.e. RXRY+ RYRZ+ RZRX as

 𝑅𝑋𝑅𝑌+𝑅𝑌𝑅𝑍+𝑅𝑍𝑅𝑋 = 𝑅1²𝑅2𝑅3+ 𝑅2²𝑅1𝑅3+ 𝑅3²𝑅1𝑅2 / (𝑅1+𝑅2+𝑅3)²

 Here let us divide this equation by RX to get

 𝑅𝑌+𝑅𝑍+ [𝑅𝑌𝑅𝑍] / 𝑅𝑋 = 𝑅1𝑅2𝑅3(𝑅1+𝑅2+𝑅3) / 𝑅𝑋(𝑅1+𝑅2+𝑅3)²

                            =𝑅1𝑅2𝑅3 / 𝑅𝑋(𝑅1+𝑅2+𝑅3)

Now substituting the value of RX= (R1+R2+R3) / R1.R3 from the earlier equations into the above equation we will get

 𝑅𝑌+𝑅𝑍+𝑅𝑌𝑅𝑍𝑅𝑋=[𝑅1𝑅2𝑅3 / (𝑅1+𝑅2+𝑅3)] × [(𝑅1+𝑅2+𝑅3) / 𝑅1𝑅3] =𝑅2

 Then after similarly dividing the same equation by RY and RZ we get the other two relations as:

 𝑅𝑋+𝑅𝑍+𝑅𝑋𝑅𝑍 / 𝑅𝑌 = R3

 𝑅𝑌+𝑅𝑋+𝑅𝑋𝑅𝑌 / 𝑅𝑍 = 𝑅1

Thus we get the three equivalent resistances R1, R2 and R3 in Delta connection in terms of the three resistances RX, RY and RZ in Star connection as 

 𝑅𝑌+𝑅𝑋+𝑅𝑋𝑅𝑌 / 𝑅𝑍 = 𝑅1 

𝑅𝑌+𝑅𝑍+𝑅𝑌𝑅𝑍 / 𝑅𝑋 = 𝑅2 

𝑅𝑋+𝑅𝑍+𝑅𝑋𝑅𝑍 / 𝑅𝑌 = R3