Star-Delta and Delta-Star Transformations BEF 12503 Star-Delta and Delta-Star Transformations

Learning Outcomes: After completing this study unit you will be able to understand: How to convert a delta circuit into a star circuit, How to convert a star circuit into a delta circuit, How to solve a complex circuit using the star-delta transformation together with the voltage divider and current divider rules.

Star-Delta Transformation In some circuits, we cannot apply the voltage divider rule or the current divider rule directly because of the perculiar geometry of the circuit. In some cases, we may be able to transform the circuit in such a way so that after the transformation we can finally apply the voltage or the current divider rule to solve the circuit. The following worked examples show how we can use the delta-star transformation technique together with the voltage divider and current divider rules to solve a complex circuit.

Worked Example Find I, I1. I2, I3, I4, and V0 in the circuit given. I VS = 300°V Z1 Z2 Z1 = j4 Ω _ + V0 VS 3 2 Z2 = -j3 Ω Z5 Z3 Z3 = 5 – j2 Ω Z4 I3 I4 Z4 = 10 Ω 4 Z5 = 8 + j5 Ω

Solution Observation We cannot apply the voltage divider rule directly here because of the presence of the delta connection in the upper part (or lower part, depending on how you look at it) of the circuit. We note that by transforming the upper delta connection (say) into a star, we can transform the bridge circuit into a series-parallel circuit. This will allow us to calculate I3 and I4 using voltage and current divider rules. Likewise, by transforming the lower delta into a star we can again transform the bridge circuit into a series-parallel circuit. This will then allow us to calculate I1 and I2 using voltage and current divider rules. 1

Consider transforming the upper delta connection into a star Consider transforming the upper delta connection into a star. The resulting circuit is shown below. I 1 1 2 3 4 V0 I1 I2 I4 I3 Z1 Z2 Z3 Z4 + _ VS I Z5 ZA ZB ZC VS 3 + V0 2 Z3 Z4 I3 I4 4

The formulae for transforming a delta-connected circuit into a star-connected circuit are as follows: Z1 Z2 Z5 1 2 3 1 ZA ZB ZC 3 2

Therefore, with Z1 = j4 Ω, Z2 = -j3 Ω and Z5 = 8 + j 5 Ω,

The transformed circuit is shown below. ZA ZC ZB Z4 Z3 1 2 3 4 I4 I3 I The transformed circuit is shown below.

ZA ZC ZB Z4 Z3 1 2 3 4 I4 I3 I Let

Then, ZX ZA ZY 1 4 I I4 I3

Consider transforming the lower delta connection into a star Consider transforming the lower delta connection into a star. The resulting circuit is shown below. I 1 I1 I2 Z1 Z2 1 2 3 4 V0 I1 I2 I4 I3 Z1 Z2 Z3 Z4 + _ VS I Z5 3 V0 + 2 ZA ZB VS ZC

The formulae for transforming the delta-connected circuit into a star-connected circuit are as follows: Z5 Z3 Z4 4 2 3 ZA ZB ZC

Therefore, with Z3 = 5 – j2 Ω, Z4 = 10 Ω, and Z5 = 8 + j 5 Ω

The transformed circuit is shown below. Z2 ZB Z1 ZC ZA 1 2 3 4 I2 I1 I The transformed circuit is shown below.

Z2 ZB Z1 ZC ZA 1 2 3 4 I2 I1 I Let

Z2 ZB Z1 ZC ZA 1 2 3 4 I2 I1 I

Z2 ZB Z1 ZC ZA 1 2 3 4 I2 I1 I

and 1 2 3 4 V0 I1 I2 I4 I3 Z1 Z2 Z3 Z4 + _ VS I Z5

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