Y-Δ transform

The star-delta transformation, or delta-star transformation, known in English as Delta -Star transformation and as Kennelly theorem after Arthur Edwin Kennelly, is in electrical engineering a circuitry forming of three electrical resistors, the circuit analysis of resistive networks serves. The star-delta transformation is a special case of the star - polygon transformation.

General

To illustrate is to serve adjacent figure: In the star -delta transformation, the star-shaped is ( star ) right arrangement of the resistors in a triangular (delta) resistor arrangement, shown on the left reshaped. The delta-star transformation, the counterpart and allows the reverse transformation. The electrical connection is given on the marked terminals a, b and c remain exactly the same. There are only three resistance values ​​exchanged in this transformation by appropriate substitute values ​​for the new circuit.

By appropriate application of these two transformations and the rules for parallel and series connection of resistors simplified equivalent resistances complicated resistor networks can be formed as part of the circuit analysis.

Transformation rules

For the delta-star transformation following calculations are necessary to determine the equivalent resistances:

For the reverse star-delta transformation following calculations are necessary to determine the equivalent resistances:

Derivation of the transformation rules

To understand why the star-delta transformation works, it is advisable to consider the derivation of the transformation rules.

For our purposes it is important that the terminal behavior between the respective terminals ( ab, bc, ac) does not change after transformation.

USAB, the voltage at the terminals from the star and in a triangle Udab. Similarly, of course, also apply to the other terminals bc and ac

Referring now to the sketch of the triangle or star connection, can be used to determine the rules of the series connection and parallel connection of the resistances between the terminals.

Brings you the double fracture to the same denominator, we arrive at the following equation:

The same is done with the Star connection:

And set equal to the delta connection.

If you repeat these steps for the terminals bc and ac, we obtain the following two formulas:

Solving this system of equations by Ra, Rb and Rc, we obtain the above mentioned transformation rules.

Mnemonic and return transformation

There is a slight Wish - rule for the forward or reverse transformation:

Application in the AC circuit analysis

The star-delta transformation can also be applied in the complex AC circuit analysis, as long as the components used, such as capacitors, inductors and resistors exhibit linear behavior. The complex impedances Z are used in the equations instead of the purely ohmic resistors R. The transformation is the same.