Star-polygon transformation

Star connection Each connection is connected to the star point via a resistor.

Polygon circuit Each connection is connected to every other connection via a resistor.

The star-polygon transformation is a generalization of the star-delta transformation and is used in electrical engineering to convert a star connection of electrical resistances into a polygon connection ${\displaystyle n\geq 3;n\in \mathbb {N} }$

${\displaystyle {\frac {n\,\left(n-1\right)}{2}}}$to convert electrical resistances, which behave the same with regard to the connections . However, the reverse conversion is only possible in the case (i.e. with the star-delta connection ). ${\displaystyle X_{1},\dots ,X_{n}}$ ${\displaystyle n=3}$

The conversion takes place from the relationship between the conductance values

${\displaystyle G_{i,j}={\frac {G_{i,0}\,G_{j,0}}{G_{s}}}}$

with the total conductance

${\displaystyle G_{s}=\sum _{i=1}^{n}{G_{i,0}}}$. Here is the conductance of the resistor from the connection to the connection in the polygon circuit or are the conductance of the resistance from the connection or to the star point in the star connection. ${\displaystyle G_{i,j}}$ ${\displaystyle X_{i}}$ ${\displaystyle X_{j}}$ ${\displaystyle G_{i,0}}$ ${\displaystyle G_{j,0}}$ ${\displaystyle X_{i}}$ ${\displaystyle X_{j}}$

It does not apply to frequency-dependent complex impedances.

Derivation

The transformation equations can be derived from the condition that the polygon network should absorb the same currents at its connection points up to (correspondingly up to the sketches) as the star network, if the connection points of both networks are impressed with the same arbitrarily predeterminable potentials . This could practically be achieved with the help of voltage sources connected to form a star. The sum of the currents flowing to the star point is equal to zero according to Kirchhoff's node theorem. The neutral point potential follows from this . This denotes the sum of all star conductance values ​​as above. ${\displaystyle j=1}$ ${\displaystyle n}$ ${\displaystyle {\text{X}}_{1}}$ ${\displaystyle {\text{X}}_{n}}$ ${\displaystyle \varphi _{j}}$ ${\displaystyle n}$ ${\displaystyle \sum _{j=1}^{n}G_{j,0}(\varphi _{j}-\varphi _{0})}$ ${\displaystyle \varphi _{0}={\frac {1}{G_{s}}}\sum _{j=1}^{n}G_{j,0}\varphi _{j}}$ ${\displaystyle G_{s}}$ ${\displaystyle n}$

The current flowing to the star point through a selected conductance has the value . The external conductor current entering the corresponding connection point of the polygon network is equal to the sum of all currents flowing from the connection point through the polygon conductance values . ${\displaystyle G_{i,0}}$ ${\displaystyle I_{i}=G_{i,0}(\varphi _{i}-\varphi _{0})}$ ${\displaystyle I'_{i}=\sum _{j=1,~j\neq i}^{n}G_{i,j}(\varphi _{i}-\varphi _{j})}$ ${\displaystyle G_{i,j}}$

With the equality of the currents required as a transformation condition (see above) and follows ${\displaystyle I_{i}}$ ${\displaystyle I'_{i}}$

${\displaystyle G_{i,0}(\varphi _{i}-{\frac {1}{G_{S}}}\sum _{j=1}^{n}G_{j,0}\varphi _{j})=\sum _{j=1,~j\neq i}^{n}G_{i,j}(\varphi _{i}-\varphi _{j})}$.

On the left and right side of the equation there is a linear combination of all potentials that can be freely used according to the approach. The equation is fulfilled for all possible potential values if each coefficient on the left side coincides with the corresponding one on the right side. Equating the coefficients of directly yields the transformation equation given above ${\displaystyle \varphi }$ ${\displaystyle \varphi _{j}}$

${\displaystyle {\frac {G_{i,0}G_{j,0}}{G_{s}}}=G_{i,j}}$.

literature

• H. Haase, H. Garbe, H. Gerth: Fundamentals of electrical engineering . 4th edition. Schöneworth-Verlag Dehre, 2018, ISBN 978-3-9808805-5-8 .