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Symmetrical Components
Prof. Dr.-Ing. Ralph Kennel
Technische Universität München
Lehrstuhl für Elektrische Antriebssysteme und Leistungselektronik
München, 06. März 2018
Symmetrical Components
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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
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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 !
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