## Star Delta Transformation – how it works

The star-delta conversion is a tool for simplifying complex resistor networks. Here, the three resistors are reconnected from delta to star - and vice versa, with correspondingly altered resistance values, so that the ratios between the pins remain the same in both circuit variants.

__Delta to Star Transformation__

__Delta to Star Transformation__

We proof this:

Condition for the calculation of the resistors RS1, RS2, RS3: Current and voltage values must be the same between the terminals in both circuit types!

__We consider the pins 1 and 2:__

__U _{D12}__ =

__U__=> resulting in R

_{S12}_{D12}= R

_{S12}

I

_{D12 }I

_{S12}

__We consider the pins ____2 und 3:__

__U _{D23}__ =

__U__=> resulting in R

_{S23}_{D23}= R

_{S23}

I

_{D23 }I

_{S23}

__We consider the pins ____1 und 3:__

__U _{D13}__ =

__U__=> resulting in R

_{S13}_{D13}= R

_{S13}

I

_{D13 }I

_{S13}

Calculation of the resistances between the pins:

**R _{D12}** = R

_{D2}ΙΙ (R

_{D1}+ R

_{D3}) =

__R__

_{D2}

__(R__

_{D1}

__+ R__

_{D3}

__)__must be equal to

**R**= R

_{S12}_{S1}+ R

_{S2}

. R

_{D1}+ R

_{D2}+ R

_{D3}

Resulting in:

**R _{D12}** =

__R__

_{D1}

__R__

_{D2}

__+ R__

_{D2}

__R__

_{D3}=

**R**= R

_{S12 }_{S1}+ R

_{S2 }…Eq.(1)

. R

_{D1}+ R

_{D2}+ R

_{D3}

also:

**R _{D31}** =

__R__

_{D1}

__R__

_{D2}

__+ R__

_{D1}

__R__

_{D3}=

**R**= R

_{S31}_{S3}+ R

_{S1}…Eq.(2)

. R

_{D1}+ R

_{D2}+ R

_{D3}

**R _{D23}** =

__R__

_{D3}

__R__

_{D1}

__+ R__

_{D3}

__R__

_{D2}=

**R**= R

_{S23}_{S2}+ R

_{S3}…Eq.(3)

. R

_{D1}+ R

_{D2}+ R

_{D3}

We add Eq.(1) + Eq.(2) - Eq.(3):

__R___{D1}__R___{D2}__ + R___{D2}__R___{D3}__ + R___{D1}__R___{D}_{2}__ + R___{D1}__R___{D3}__ - (R___{D3}__R___{D1}__ + R___{D3}__R___{D2}__)__ = R_{S1} + R_{S2 }+ R_{S1 }+ R_{S3} - R_{S2} - R_{S3} = 2R_{S1} . R_{D1} + R_{D2} + R_{D3}

We get as a result:

R_{S1} = __ R___{D1}__ R___{D2} _{.}

. R_{D1} + R_{D2} + R_{D3}

The same way we get:

R_{S2} = __ R___{D2}__ R___{D3} _{.}

. R_{D1} + R_{D2} + R_{D3}

R_{S3} = __ R___{D1}__ R___{D3} _{.}

. R_{D1} + R_{D2} + R_{D3}

If the delta connected system has the same resistance R_{D} at its three sides then the equivalent star resistance R_{S} will be:

**R _{S} = R_{D} R_{D} . = R_{D} .**

**. R**

_{D}+ R_{D}+ R_{D}3__Star to Delta Transformation__

__Star to Delta Transformation__

We proceed in the same way for the transformation from star to triangle. The resistors RS1, RS2 and RS3 of the star circuit are converted into the resistors RD1, RD2 and RD3 of the delta circuit with the corresponding resistance values in such a way that the current and voltage values between terminals 1 to 3 are identical.

We proof this:

In the star connection, the equivalent resistance between the connection points 1 and 2 (resp.3) is obtained by short-circuiting the connections 2 and 3. Thus, an equivalent conductance is formed from R_{S1} and the parallel connection of R_{S2} with R_{S3}.

If the same points are short-circuited in the equivalent delta circuit, the total conductance results from the parallel connection of the resistors R_{D1} and R_{D2}.

Left side results in a common denominator:

The same applied to terminal 3 and short-circuited terminals 1 and 2:

The same applied to terminal 2 and short-circuited terminals 1 and 3:

We add: Eq. (1) + Eq. (2) - Eq. (3):

Solve this equation for R_{D1}:

The same procedure applies to the remaining branches or resistors.