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Page 1: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

1

Symmetrical Components

Prof. Dr.-Ing. Ralph Kennel

([email protected])

Technische Universität München

Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik

München, 06. März 2018

Page 2: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

Page 3: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

Page 4: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

Page 5: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

Symmetrical Components

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

Page 6: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

any 3phase system

can be split into the

symmetrical components

Page 7: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

what is the purpose ?

Page 8: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

Page 9: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

Page 10: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

Page 11: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

Page 12: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

Page 13: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

13

Symmetrical Components

Page 14: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

what is the purpose ? electrical AC machines

show different impedances

in symmetrical components

Page 15: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

Graphic Decomposition in Symmetrical Components15

Page 16: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

16Graphic Decomposition in Symmetrical Components

Page 17: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

17Graphic Decomposition in Symmetrical Components

Page 18: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

18Graphic Decomposition in Symmetrical Components

all in one

Page 19: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

Multiphase Systems in General

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

Page 20: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

Symmetrical Components

… for 3 phase systems

Page 21: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

Symmetrical Components

… for 4 phase systems

Page 22: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

Symmetrical Components

… for 5 phase systems

Page 23: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

Symmetrical Components

… for 6 phase systems

Page 24: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

Page 25: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

Page 26: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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)

Page 27: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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)

Page 28: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

Page 29: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

Page 30: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

Page 31: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

Page 32: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

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

Symmetrical Components

Conclusion

symmetrical components

simplify the investigation

in unsymmetrical loads

Page 33: Symmetrical Components - eal.ei.tum.de · 1 Symmetrical Components Prof. Dr.-Ing. Ralph Kennel (ralph.kennel@tum.de) Technische Universität München Lehrstuhl für Elektrische Antriebssysteme

Thank you !