Symmetrical components

In electrical engineering , the method of symmetrical components is used in order to be able to carry out a simplified analysis using symmetrical subsystems in asymmetrical multi-phase systems, usually three-phase systems. An asymmetrically loaded system of phasors is divided into several superimposed subsystems. The usual three-phase systems are divided into a symmetrical positive sequence system whose pointers move with the rotating field , a counter system with an opposing rotating field and a zero system .

The method of symmetrical components represents a special case of general modal transformation and the method of modal components that is important in practice and is used, among other things, in the analysis of asymmetrical errors in three-phase systems and in the investigation of electrical machines , especially multi-phase machines .

Historical development

Charles Legeyt Fortescue showed in a work presented in 1918 under the title English Method of Symmetrical Co-Ordinates Applied to the Solution of Polyphase Networks that every asymmetrically loaded three-phase system can be represented as the sum of three symmetrical phasor sets. This analysis was subsequently taken up and improved by engineers at General Electric and Westinghouse . After the Second World War , the symmetric component method was expanded into a general method for analyzing asymmetric errors.

method

Every asymmetrical set of phasors that does not add up to zero can be clearly separated into an asymmetrical set that adds up to zero and a system of identical phasors. Furthermore, each asymmetrical set of phasors, which however add to zero, can be divided into two symmetrical sets with opposite directions of rotation of the rotating fields. This means that any unsymmetrical phasor set can always be clearly divided. The method enables, for example, a symmetrically constructed asynchronous motor which is fed asymmetrically to be broken down into a superposition of two asynchronous motors rotating in opposite directions but symmetrically fed.

Example two-phase system

Asymmetrical two-phase system

In the simplest case, there is a two-phase system , represented by two phasors A and B , as shown in the adjacent sketch. This can be in two subsystems disassemble: the positive sequence ( English positive sequence component ) in red, it is of the two phasors A _m and B _m are formed to be rotary field has the same direction of rotation as the original system. The counter system ( English negative sequence component ) is in green, with the two phasors A _g and B _g shown to be rotating field has an opposite direction as the original system. The phasors in each subsystem have the same amount and are normal to each other in the two-phase system:

${\displaystyle A_{m}={\frac {A-\mathrm {j} B}{2}}}$

${\displaystyle A_{g}={\frac {A+\mathrm {j} B}{2}}}$

and

${\displaystyle B_{m}=\mathrm {j} A_{m}}$

${\displaystyle B_{g}=-\mathrm {j} A_{g}}$

with j as the imaginary unit .

Calculation in the three-phase system

Unbalanced three-phase system (A, B and C) and its symmetrical components

geometric representation of the complex pointer ${\displaystyle {\underline {a}}}$

With the aid of the coefficient matrix which can phasors in a three phase system of the symmetrical components (positive sequence) (NPS), (zero System) from the phase currents , , of the three-phase system can be calculated. In the zero System ( English zero sequence component ) phasors the same direction and same length. The zero system occurs in the asymmetrical three-phase system and offsets the "non-addition" of the original system to zero. ${\displaystyle T}$ ${\displaystyle {\underline {I}}_{m}}$ ${\displaystyle {\underline {I}}_{g}}$ ${\displaystyle {\underline {I}}_{0}}$ ${\displaystyle {\underline {I}}_{L1}}$ ${\displaystyle {\underline {I}}_{L2}}$ ${\displaystyle {\underline {I}}_{L3}}$

The electrical currents as a physical quantity are selected as an example in the following equations; the method of symmetrical components can be applied analogously to all quantities such as electrical voltages or magnetic fluxes.

The complex pointer is a rotation operator for linking the external conductor currents. Multiplication by means a rotation by 120 ^o counterclockwise: ${\displaystyle {\underline {a}}}$ ${\displaystyle {\underline {a}}}$

${\displaystyle {\underline {a}}=e^{j120^{o}}=e^{j{\frac {2\pi }{3}}}=-{\frac {1}{2}}+j{\frac {\sqrt {3}}{2}}}$

${\displaystyle {\underline {a}}^{2}=e^{j240^{o}}=e^{j{\frac {4\pi }{3}}}=-{\frac {1}{2}}-j{\frac {\sqrt {3}}{2}}}$

${\displaystyle {\underline {a}}^{3}={\underline {a}}^{0}=e^{j0^{o}}={1}}$

There are also the following theorems:

${\displaystyle {\underline {a}}^{4}={\underline {a}},\quad {1}+{\underline {a}}+{\underline {a}}^{2}={0},\quad {\underline {a}}^{2}={\underline {a}}^{-1}}$

The coefficient matrix is ​​obtained:

${\displaystyle T={\begin{pmatrix}1&1&1\\{\underline {a}}^{2}&{\underline {a}}&1\\{\underline {a}}&{\underline {a}}^{2}&1\end{pmatrix}}={\begin{pmatrix}1&1&1\\\mathrm {e} ^{\mathrm {j} {\frac {-2\pi }{3}}}&\mathrm {e} ^{\mathrm {j} {\frac {2\pi }{3}}}&1\\\mathrm {e} ^{\mathrm {j} {\frac {2\pi }{3}}}&\mathrm {e} ^{\mathrm {j} {\frac {-2\pi }{3}}}&1\end{pmatrix}}}$

${\displaystyle {\begin{pmatrix}{\underline {I}}_{1m}\\{\underline {I}}_{1g}\\{\underline {I}}_{10}\end{pmatrix}}=T^{-1}\cdot {\begin{pmatrix}{\underline {I}}_{L1}\\{\underline {I}}_{L2}\\{\underline {I}}_{L3}\end{pmatrix}}={\frac {1}{3}}\cdot {\begin{pmatrix}1&{\underline {a}}&{\underline {a}}^{2}\\1&{\underline {a}}^{2}&{\underline {a}}\\1&1&1\end{pmatrix}}\cdot {\begin{pmatrix}{\underline {I}}_{L1}\\{\underline {I}}_{L2}\\{\underline {I}}_{L3}\end{pmatrix}}}$

This results in the positive sequence:

${\displaystyle {\underline {I}}_{1m}={\frac {1}{3}}\cdot ({\underline {I}}_{L1}+{\underline {I}}_{L2}\cdot {\underline {a}}+{\underline {I}}_{L3}\cdot {\underline {a}}^{2})}$

${\displaystyle {\underline {I}}_{2m}={\underline {I}}_{1m}\cdot {\underline {a}}^{2}}$

${\displaystyle {\underline {I}}_{3m}={\underline {I}}_{1m}\cdot {\underline {a}}}$

The following applies to the negative system:

${\displaystyle {\underline {I}}_{1g}={\frac {1}{3}}\cdot ({\underline {I}}_{L1}+{\underline {I}}_{L2}\cdot {\underline {a}}^{2}+{\underline {I}}_{L3}\cdot {\underline {a}})}$

${\displaystyle {\underline {I}}_{2g}={\underline {I}}_{1g}\cdot {\underline {a}}}$

${\displaystyle {\underline {I}}_{3g}={\underline {I}}_{1g}\cdot {\underline {a}}^{2}}$

And for the zero system:

${\displaystyle {\underline {I}}_{10}={\frac {1}{3}}\cdot ({\underline {I}}_{L1}+{\underline {I}}_{L2}+{\underline {I}}_{L3})}$

${\displaystyle {\underline {I}}_{20}={\underline {I}}_{10}}$

${\displaystyle {\underline {I}}_{30}={\underline {I}}_{10}}$

With the expansion of a single-pole display to show the positive, negative and negative systems of generators, three-phase transformers and other electrical components, the analysis of unbalanced circumstances such as earth faults is greatly simplified. The division into symmetrical components can also be extended to higher phase orders.

literature

• Bernd R. Oswald: Calculation of three-phase networks - calculation of stationary and non-stationary processes with symmetrical components and space vectors . Vieweg + Teubner, 2009, ISBN 978-3-8348-0617-8 .

Norms

• DIN EN 60909-0 (VDE 0102): 2016-12 Short-circuit currents in three-phase networks - Part 0: Calculation of the currents

Individual evidence

1. ↑ Stephen E. Marx: Symmetrical components 1 & 2. 2012, accessed on August 30, 2016 . 
2. ^ Charles LeGeyt Fortescue: Method of Symmetrical Co-Ordinates Applied to the Solution of Polyphase Networks . AIEE Transactions 37 (II), 1918, pp. 1027 to 1140 .