Star-delta transformation

The star-triangle transformation or triangle-star transformation , referred to in English as delta-star transformation and Kennelly theorem according to Arthur Edwin Kennelly , is a circuit-related transformation of three electrical resistances in electrical engineering , which is used for the circuit analysis of resistor networks serves. The star-triangle transformation is a special case of the star-polygon transformation .

General

Star-delta transformation of resistors

The illustration on the right should serve to clarify: With the star-triangle transformation, the star-shaped ( star ) right-hand arrangement of the resistors is transformed into a triangular ( delta ) resistor arrangement , shown on the left. The triangle-star transformation is the counterpart to this and enables the reverse transformation. The electrical connection values at the terminals a , b and c shown remain exactly the same. During this transformation, only the three resistance values ​​are exchanged for suitable substitute values ​​for the new circuit arrangement.

By appropriately applying these two transformations and the rules for parallel connection and series connection of resistors, simplified equivalent resistances of complicated resistor networks can be formed within the scope of the circuit analysis.

The star-delta transformation is identical to the pi-t transformation between the π-circuit and the T-circuit , which graphically arranges the resistors differently and is used in the field of telecommunications for filter circuits .

Transformation rules

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

${\displaystyle R_{a}={\frac {R_{ac}R_{ab}}{R_{ac}+R_{ab}+R_{bc}}}}$

${\displaystyle R_{b}={\frac {R_{ab}R_{bc}}{R_{ac}+R_{ab}+R_{bc}}}}$

${\displaystyle R_{c}={\frac {R_{ac}R_{bc}}{R_{ac}+R_{ab}+R_{bc}}}}$

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

${\displaystyle R_{ac}={\frac {R_{a}R_{b}+R_{b}R_{c}+R_{c}R_{a}}{R_{b}}}}$

${\displaystyle R_{ab}={\frac {R_{a}R_{b}+R_{b}R_{c}+R_{c}R_{a}}{R_{c}}}}$

${\displaystyle R_{bc}={\frac {R_{a}R_{b}+R_{b}R_{c}+R_{c}R_{a}}{R_{a}}}}$

Derivation of the transformation rules

In order to understand why the star-triangle transformation works, it is advisable to look at 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 the transformation.

${\displaystyle {\frac {U_{dab}}{I_{dab}}}={\frac {U_{sab}}{I_{sab}}}}$

U _sab is the voltage at terminals ab in the star and U _dab in the triangle. The other terminals bc and ac also apply analogously.

If you now look at the sketch of the delta or star connection, you can determine the resistances between the terminals using the rules for series connection and parallel connection .

${\displaystyle {\tilde {R}}_{ab}={\frac {1}{{\frac {1}{R_{ab}}}+{\frac {1}{R_{ac}+R_{bc}}}}}}$

If you bring the double fraction to the same denominator, you get the following equation:

${\displaystyle {\tilde {R}}_{ab}={\frac {R_{ab}R_{ac}+R_{ab}R_{bc}}{R_{ab}+R_{ac}+R_{bc}}}}$

The same is done with the star connection:

${\displaystyle {\tilde {R}}_{ab}=R_{a}+R_{b}}$

and equated with the delta connection.

${\displaystyle {\frac {R_{ab}R_{ac}+R_{ab}R_{bc}}{R_{ab}+R_{ac}+R_{bc}}}=R_{a}+R_{b}}$

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

${\displaystyle {\frac {R_{ac}R_{ab}+R_{ac}R_{bc}}{R_{ab}+R_{ac}+R_{bc}}}=R_{a}+R_{c}}$

${\displaystyle {\frac {R_{bc}R_{ab}+R_{bc}R_{ac}}{R_{ab}+R_{ac}+R_{bc}}}=R_{b}+R_{c}}$

Solving this system of equations for R _a , R _b and R _c , one obtains the transformation rules mentioned above.

An alternative derivation that is also valid for the special star-triangle case dealt with here is given under star-polygon transformation . There the transformation equations follow from a simple comparison of coefficients.

Memory aid forward and backward transformation

There is a simple rule of thumb for the forward and backward transformation:

${\displaystyle {\text{Dreieck-Stern-Transformation wird für }}\Delta \to {\text{Y}}}$

${\displaystyle R_{c}={\frac {R_{ac}\cdot R_{bc}}{R_{ab}+R_{bc}+R_{ac}}}={\frac {\text{Produkt der Anliegerwiderstände }}{\text{Maschenumlaufwiderstand}}}}$

${\displaystyle {\text{Stern-Dreieck-Transformation wird für Y}}\to \Delta }$

${\displaystyle R_{ac}={\frac {R_{a}\cdot R_{c}}{R_{b}}}+R_{a}+R_{c}={\frac {\text{Produkt der Anliegerwiderstände}}{\text{verbleibender Widerstand}}}+{\text{Beide Anliegerwiderstände}}}$

Application in the AC bill

The star-delta transformation is also used in the complex AC calculation . However, with the restriction that it only applies to one frequency for any linear impedances in the branches. The star-triangle transformation is valid for all frequencies if all branches contain only capacitors , only inductors or only resistances . The star and delta connections are therefore not equivalent circuits in AC technology, but can be used for calculating networks with only one frequency (e.g. 50 Hz). Instead of the purely ohmic resistances, the complex impedances are used in the equations. The transformation takes place analogously. ${\displaystyle R}$ ${\displaystyle Z}$

literature

• Dieter Nührmann: The large work book electronics. Volume 1: Tables, mathematics, formulas, AC technology, mechanics, SMD technology, passive components, batteries, solar cells, EMC technology. 6th, revised and expanded edition. Franzis, Poing 1994, ISBN 3-7723-6546-9 , p. 389.