1

Symmetrical Components

Prof. Dr.-Ing. Ralph Kennel

(ralph.kennel@tum.de)

Technische Universität München

Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik

München, 06. März 2018

Symmetrical Components

2

symmetrical 3phase system

U1

U2

U2 = U1 e-j2/3π

U3

U3 = U1 e-j4/3π

versors

U2 = U1 a²

U3 = U1 a

the basic idea is

that an asymmetrical set of N phasors

can be expressed

as a linear combination

of N symmetrical sets of phasors

by means of a complex linear transformation

3Dr. rer. nat. Erika Mustermann (TUM) | kann beliebig erweitert werden | Infos mit Strich trennen

Symmetrical Components

usual strategies

in electrical engineering

Real World

(time domain)initial situation

Dream World

(xxx-domain)initial situation

transformation

Dream World

(xxx-domain)resultback

transformation

calculation(s)

Real World

(time domain)result

calculation(s)

might be

possible

as well

usually

it is easier

this way!

but more complex

Symmetrical Components

5all components are rotating in the same direction (within the complex plane) !!!

6Prof. Dr.-Ing. Ralph Kennel (TUM)

any 3phase system

can be split into the

symmetrical components

7Prof. Dr.-Ing. Ralph Kennel (TUM)

what is the purpose ?

8Prof. Dr.-Ing. Ralph Kennel (TUM)

what is the purpose ?

1st step :

split the system into the

symmetrical components

2nd step :

calculate the response for

each of the

symmetrical components

3rd step :

recombine the

symmetrical components

into real system

9Prof. Dr.-Ing. Ralph Kennel (TUM)

what is the purpose ?

1st step :

split the system into the

symmetrical components

2nd step :

calculate the response for

each of the

symmetrical components

3rd step :

recombine the

symmetrical components

into real system

10Prof. Dr.-Ing. Ralph Kennel (TUM)

what is the purpose ?

1st step :

split the system into the

symmetrical components

2nd step :

calculate the response for

each of the

symmetrical components

3rd step :

recombine the

symmetrical components

into real system

11Prof. Dr.-Ing. Ralph Kennel (TUM)

what is the purpose ?

1st step :

split the system into the

symmetrical components

2nd step :

calculate the response for

each of the

symmetrical components

3rd step :

recombine the

symmetrical components

into real system

12Prof. Dr.-Ing. Ralph Kennel (TUM)

what is the purpose ?

1st step :

split the system into the

symmetrical components

2nd step :

calculate the response for

each of the

symmetrical components

3rd step :

recombine the

symmetrical components

into real system

13

Symmetrical Components

14Prof. Dr.-Ing. Ralph Kennel (TUM)

what is the purpose ? electrical AC machines

show different impedances

in symmetrical components

Graphic Decomposition in Symmetrical Components15

16Graphic Decomposition in Symmetrical Components

17Graphic Decomposition in Symmetrical Components

18Graphic Decomposition in Symmetrical Components

all in one

19Prof. Dr.-Ing. Ralph Kennel (TUM)

Multiphase Systems in General

… the number of phases is not really fixed to 3 …

20Prof. Dr.-Ing. Ralph Kennel (TUM)

Symmetrical Components

… for 3 phase systems

21Prof. Dr.-Ing. Ralph Kennel (TUM)

Symmetrical Components

… for 4 phase systems

22Prof. Dr.-Ing. Ralph Kennel (TUM)

Symmetrical Components

… for 5 phase systems

23Prof. Dr.-Ing. Ralph Kennel (TUM)

Symmetrical Components

… for 6 phase systems

24Prof. Dr.-Ing. Ralph Kennel (TUM)

What about 2phase Systems ???

the so-called 2phase system

is not really a 2phase system

because the phase shift is not 180°

based on the phase shift of 90° between the phases

it is a 4phase system with only 2 phases used

symmetrical components for 4phase systems

25Prof. Dr.-Ing. Ralph Kennel (TUM)

Symmetrical Components

Example: Transformer

U1 = Z1 I1

U2 = Z2 I2

U0 = Z0 I0

in rotating electrical machines Z1, Z2 and Z0 are different

in stationary electrical machines at least Z1 and Z2 are equal

in a transformer Z1 = Z2 = ZK2

equivalent circuit

26Prof. Dr.-Ing. Ralph Kennel (TUM)

Symmetrical Components

Example: Transformer

in a transformer the zero impedance Z0 depends on the design of the transformer

U0 = Z0 I0

in case of a star connection on primary and secondary side

there cannot be any zero sequence current on the primary side (due to Krichhoff‘s law)

the equivalent circuit contains the secondary side only

the quantity of the mutual inductance depends on the design

3-leg-core mutual inductance is low

(magnetic flux has to go through the air)

27Prof. Dr.-Ing. Ralph Kennel (TUM)

Symmetrical Components

Example: Transformer

in a transformer the zero impedance Z0 depends on the design of the transformer

U0 = Z0 I0

in case of a star connection on primary and secondary side

there cannot be any zero sequence current on the primary side (due to Krichhoff‘s law)

the equivalent circuit contains the secondary side only

the quantity of the mutual inductance depends on the design

5-leg-core mutual inductance is higher

(magnetic flux goes through the outer legs)

28Prof. Dr.-Ing. Ralph Kennel (TUM)

Symmetrical Components

Example: Transformer

in a transformer the zero impedance Z0 depends on the design of the transformer

U0 = Z0 I0

in case of a delta connection on primary and a star connection secondary side

a zero sequence current on the primary side is possble

the equivalent circuit contains primary and secondary side

The quantity of the mutual inductance is negligible

in comparison to the leakage inductances

29Prof. Dr.-Ing. Ralph Kennel (TUM)

Symmetrical Components

Example: single phase load of a transformer

Uu = Uz = Iz Z

Iu = - Iz

Iu = I1 + I2 + I0 = - Iz

Iv = Iw = 0

Uu = U1 + U2 + U0 = Uz

U1 = U20u + Z1 I1

U2 = Z2 I2

U1 = Z0 I0

… we assume, that the

grid source voltage contains

a U20u component only

30Prof. Dr.-Ing. Ralph Kennel (TUM)

Symmetrical Components

Example: single phase load of a transformer

Uu = Uz = Iz Z

Iu = - Iz

Iu = I1 + I2 + I0 = - Iz

Iv = Iw = 0

Uu = U1 + U2 + U0 = Uz

U1 = U20u + Z1 I1

U2 = Z2 I2

U1 = Z0 I0

… we assume, that the

grid source voltage contains

a U20u component only

I1 = 1/3 (Iu + Iv a + Iv a²)

I1 = I2 = I0 = 1/3 Iu = - 1/3 Iz

U1 = Z1 I1

U2 = Z2 I2

U0 = Z0 I0

U1 = Uz = U20u + Z1 I1 + Z2 I2 + Z0 I0

U1 = Uz = U20u + 1/3 (Z1 + Z2 + Z0) Iu

Iu = - IzU1 = Uz = U20u - 1/3 (Z1 + Z2 + Z0) Iz

31Prof. Dr.-Ing. Ralph Kennel (TUM)

Symmetrical Components

Example: single phase load of a transformer

U1 = Uz = U20u + 1/3 (Z1 + Z2 + Z0) Iu

U1 = Uz = U20u - 1/3 (Z1 + Z2 + Z0) Iz

in case of a delta connection : Z1 = Z2 Z0

in case of a double star connection :

Z1 = Z2 Z0

… and consequently :

Z1 = Z2 1/3 (Z1 + Z2 + Z0)

with respect to high 1/3 (Z1 + Z2 + Z0)

U1 = Uz = U20u - 1/3 (Z1 + Z2 + Z0) Izwill be very low

32Prof. Dr.-Ing. Ralph Kennel (TUM)

Symmetrical Components

Conclusion

symmetrical components

simplify the investigation

in unsymmetrical loads

Thank you !