Chaining factor

Star connection in a three-phase system

Delta connection in a three-phase system

The chaining factor in a multi-phase systems, the ratio of the electric voltage between two adjacent outer conductors to the value of the rating voltage between any phase conductor and the neutral point. With symmetrical loading , this also applies to the currents .

Terms

Different voltages occur in a multi-phase system:

• The voltage between two consecutive outer conductors is referred to as a linked voltage or (outer) conductor voltage , in three-phase systems also known as triangular voltage . Example in a three-phase system: U _L3L1 is the voltage between the outer conductors L3 and L1.
• The voltage between any outer conductor and the neutral point of the network is referred to as star voltage or phase voltage . Example: U _L1 is the voltage between the outer conductor L1 and the (mostly earthed ) star point. It is lower than the linked voltage by the linkage factor.

Three phase system

The concatenation factor is based on the time offset (= phase shift ) of the alternating voltages and is in three-phase systems

${\displaystyle {\sqrt {3}}\approx 1{,}732}$.

From this, for example, the line voltage of 230 V, which is common in low-voltage networks in Europe,  results in the linked voltage

${\displaystyle U=230\,\mathrm {V} \cdot {\sqrt {3}}\approx 400\,\mathrm {V} }$.

In high-voltage networks , the line-to-line voltage is usually given as the nominal value. Example: With a 110 kV line, the triangular voltage between two outer conductors is 110 kV and the star voltage between an outer conductor and the earth potential, which is usually given by the potential of the star point . ${\displaystyle {\tfrac {110\,\mathrm {kV} }{\sqrt {3}}}\approx 63{,}5\,\mathrm {kV} }$

The concatenation factor is also already included in the reference values for the per-unit system (pu) used in electrical power engineering .

Derivation

Voltage vector diagram in the three-phase system in the complex number plane . The three star voltages are shown in red, one of the three linked voltages is shown in black.

The chaining factor for a three-phase system is calculated using the phasor diagram shown on the right : ${\displaystyle {\sqrt {3}}}$

${\displaystyle {\frac {1}{2}}\cdot U_{\text{L3L1}}=U_{\text{L1}}\cdot \cos 30^{\circ }={\frac {\sqrt {3}}{2}}\cdot U_{\text{L1}}}$

it follows:

${\displaystyle {U_{\text{L3L1}}}={\sqrt {3}}\cdot {U_{\text{L1}}}}$

m -phase systems

The concatenation factor results in symmetrical multiphase system, depending on the number of phases , generally to: ${\displaystyle m}$ ${\displaystyle m\geq 2}$

${\displaystyle {U_{\text{L1L2}}}=2\cdot \sin \left({\frac {\pi }{m}}\right)\cdot U_{\text{L1}}}$   (Calculation in radians )

This results in the chaining factor in different phase systems as follows:

Number of phases ${\displaystyle m}$	Chaining factor
2	${\displaystyle 2}$
3	${\displaystyle {\sqrt {3}}\approx 1{,}732}$
4th	${\displaystyle {\sqrt {2}}\approx 1{,}414}$
5	${\displaystyle {\frac {\sqrt {10-2{\sqrt {5}}}}{2}}\approx 1{,}176}$
6th	${\displaystyle 1}$
≥ 7	${\displaystyle <1}$

In phase systems with more than six phases, the linked voltage between two adjacent outer conductors is therefore always lower than the star voltage.

literature

• René Flosdorff , Günther Hilgarth: Electrical energy distribution . 9th edition. Vieweg-Teubner, Stuttgart 2005, ISBN 978-3-519-36424-5 .