Transformação Estrela-Triângulo

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

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_S12    =>   resulting in R_D12 = R_S12
I_D12        I_S12

We consider the pins 2 und 3:

U_D23  =  U_S23    =>   resulting in R_D23 = R_S23
I_D23        I_S23

We consider the pins 1 und 3:

U_D13  =  U_S13    =>   resulting in R_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_S12 = R_S1 + R_S2
.                                               R_D1 + R_D2 + R_D3

Resulting in:

R_D12 =  R_D1R_D2 + R_D2R_D3  = R_S12 = R_S1 + R_S2                   …Eq.(1)
.           R_D1 + R_D2 + R_D3

also:

R_D31 = R_D1R_D2 + R_D1R_D3    =  R_S31 = R_S3 + R_S1           …Eq.(2)
.         R_D1 + R_D2 + R_D3

R_D23 = R_D3R_D1 + R_D3R_D2   =  R_S23 = R_S2 + R_S3            …Eq.(3)
.         R_D1 + R_D2 + R_D3

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

R_D1R_D2 + R_D2R_D3 + R_D1R_D2 + R_D1R_D3 - (R_D3R_D1 + R_D3R_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

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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.

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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.