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2 / 48

Symmetrical system

Voltages

Currents

a U

a I

a e

- p

=- +

=- -

Phase notation: A, B, C lub R, S,

Vector diagram of voltages in three-

phase system

complex rotational

phasors:

3 / 48

Linear component

’



ELEMENT

TRÓJFAZOWY

OPISANY

MACIERZĄ Z



Equipment in power systems is represented by
equivalent circuits, which are designed for the
individual tasks of power system analysis. For the
calculation of short-circuits, the equivqalent
circuits iclude resistance and inductive reactance.
Shunt elements, i.e. capacitive reactance are
omitted.



The equivalent circuit has couplings in the three-
phase system which are of an inductive type.

4 / 48

The equations of linear component

RR R

RS S

RT T

Z I

= -

SR R

SS S

ST T

Z I

D = -

TR R

TS S

TT T

Z I

D =

U ZI

�

� �

�

� �

�

D = D

�

� �

�

� �

�

� �

�

� �

=��

� �

I
I

�

=�

�

The ralationship between voltages and currents of a
three-phase system can be described by the
following equations:

In a matrix notation

where

Matrix Z is the matrix of self-impedances and mutual
(coupling)  impedances  of  the  element.  Self-
impedances Zii (i = R, S, T) are located on the main
diagonal,  and  mutual  impedances  outside  it.  For
symmetrical  elements,  the  self-impedances  of
individual phases are equal.

5 / 48

The equations of linear component

Mutual impedances between individual phases
depend on the type of element.



Static element (line, transformer, reactor) –
impedances are reciprocal:



Rotating element (electric motor, generator) –
impedances are circular:

Mutual impedances of a given phase in relation
to the other phases are not equal because
windings of these phases are differently
positioned in relation to the rotor.

Z Z

�

6 / 48

Impedance matrix

�

=�

�

= -

�

Thus the matrices of impedances can be written

if the form:

for a static component

for a

rotating component

Sign – (minus) indicates that currents in the other
phases induce in the given phase EMFs having the
opposite sense in relation to self-induction EMFs,
which results from the mutual location of stator
phase windings.

In the equations of 3-phase circuit the voltage loss in
each phase of the component is dependent on all
phase currents, and on the self-impedance of this
phase and the mutual impedances between this
phase and the other phases.

7 / 48

The equations of linear component

(

)

D =

s R

m R

Z I

Z a I

Z aI

Z Z I

(

)

D =

s S

m S

s S

m S

Z I

Z aI

Z I

Z a I

Z Z I

(

)

D =

s T

m T

s T

m T

Z I

Z a I

Z aI

Z I

Z Z I

For balanced load the set of equations
receives the form:

Voltage drop in each
phase depends only on
the current of this
phase – the equations
are
mutually independent
and Z impedance
matrix is diagonal.

�

Z Z

9 / 48

The symmetrical component

transformation

The transformation diagonalising the matrix Z is introduced

D =

S U SZS SI

�

1 1 1

1 a a ;

1 a

1 a a

The matrix S is a third-order nonsingular
(determinant S ≠ 0) matrix, whose elements shall be
so selected that as a result of transformation S Z S

-1

the impedance matrix is diagonalised. Matrix S is
called

the

operator

matrix

symmetrical

components, and its element a – the complex
operator

- p

=- +

=- -

a e

10 / 48

Symmetrical components method

where I

– vector of sequence

currents,
ΔU

– vector of sequence

voltages:

D =

p p

U Z I

(0)

(1)

(2)

1 1 1

I I

1 a a

a I

1 a

a I

� �

+ +

�

��

�

� �

�

��

�

= =

� =

=� �

�

��

�

� �

�

��

�

��

�

� � �

I SI

(0)

(1)

(2)

1 1 1

1 a a

a U a U

1 a

a U a U

�

D +D +D

�

��

�

��

�

D = D =

�D

D + D + D

= D

�

��

�

��

�

D + D + D

�

��

�

� �

�

U S U

As a result of transformation the following
equation is obtained:

11 / 48

Transformation of impedances

(0)

(1)

(2)

1 1 1

1 a a

1 a

1 a a

Z 2Z

Z Z

0 Z

�

��

�

��

�

��

�

��

�

��

�

=�

�

� �

�

Z SZS



For a static component:

12 / 48

Transformation of impedances

(0)

(1)

(2)

1 1 1

1 a a

1 a

1 a a

Z Z

Z a Z

a Z

Z a Z

a Z

0 Z

�

�-

�

=�

�

Z SZS



For a rotating component

13 / 48

Symmetrical component method

Finally, the equations of 3-phase circuit may be
written in the form:

(0)

(1)

(2)

0 Z

�

� �

��

�

� �

��

�

� �

��

�

� �

��

�

��

�

(0)

(1)

(2)

(0)

(1)

(2)

0 Z

�

��

�

� �

��

�

��

�

� �

��

�

� �

��

�

The impedance matrix of symmetrical components is
diagonal.

Thus the coupled set of equations which described
the 3-phase circuit in phase variables becomes an
uncoupled set of equations in the symmetrical
components.

14 / 48

Interpretation of the symmetrical

components system

The reverse transformation of the 012-system to
the RST-system is achieved by the matrix S

-1

accordance with:

I S I U S U

(2)

(2)R

(2)

(2)S

(2)

(0)

(0)R

(0)

(0)S

(0)

(

(1)

(1)R

(1)

(1)S

(1)

(

(2)T

)

I
I

a I

= + + =

+ =

(2)

(2)R

(2)

(2)S

(2)

(0)

(0)R

(0)

(0)S

(0)

(

(1)

(1)R

(1)

(1)S

(1)

(

(2)T

)

a U

Reference

phase

The vectors of phase currents and voltages are

the linear combination of sequence

components.

15 / 48

The invariance of power in 012-system

S U I

U I

U I

* *

T T

U I

U S S I

U I

( )

T T

U S

( )

* *

S I

( ) ( )

(

)

S 3

3 U I

U I

U I

The transformation from RST-system to 012-

system is not invariant with respect to power.

In 012-system:

In RST-system:

because

Thus,

16 / 48

Equivalent circuits for symmetrical

components

’



ELEM

TRÓJFAZOW

OPISAN

ACIERZĄ



= +D +D

E U

Earthing impedance

The voltage equation of the

circuit:

17 / 48

Equivalent circuits for symmetrical

components

� �

=� �

� �

I
I

(

)

D =

+ +

u R

Z I

� �

=� �

� �

E
E

� �

=� �

� �

U
U

� � �

��

� � �

��

D = D

� � �

��

� � �

��

� �

�

� �

= + +

E U ZI Z I

SE SU SZS S I SZ S S I

= +

P P

uP P

Z I

012-transformation yields:

The voltage matrix equation
receive the form:

18 / 48

Equivalent circuits for symmetrical

components

The source EMF symmetrical components:

( )

� �

( )( )

�

0 0

1 1

2 2

SZS

�

=�

�

0 0

SZ S

The transformation of impedances:

for the

element

for the

earthing

19 / 48

Equivalent circuits for symmetrical

components

We assume that the source is

symmetrical:

a E

+ + =

(0)

a E

aE )

E (1 a

a) 0

(1)

aa E

a aE )

+ +

2 2

(2)

a a E

aaE )

E (1 a a ) 0

In such a case the symmetrical component of EMF are the following:

( )

� �

=� �

� �

Thus:

The EMF of sources form the

positive sequence system.

20 / 48

Equivalent circuits for symmetrical

components

( )

( )( ) ( )

(

)

( )

0 0 0

m 0

Z 2Z I

( )

( )( ) ( )

(

)

( )

1 1 1

m 1

Z Z I

( )

( )( ) ( )

(

)

( )

2 2 2

m 2

Z Z I

( )

u 0

3Z I

( )

u 1

( )

u 2

The equations of symmetrical components circuits:

( )

( )( ) ( )

( )

0 0 0

u 0

0 U

3Z I

( )

( )( ) ( )

1 1 1

( )

( )( ) ( )

2 2 2

0 U

( )

( )( )

(

)

( )

0 0

3Z I

( )

( )( ) ( )

1 1 1

( )

( )( ) ( )

2 2 2

Taking into account the Z

and Z

matrices yield the following equations for

voltage drop ΔU and ΔU

in symmetrical components:

21 / 48

Single-phase sequence networks

 

(1)

 

(2)

 

(0)

( )

(

)

( )

( )( )

1 1

( )

( )( )

2 2

( )

( ) ( )

0 0

Z I

( )

( ) ( )

1 1

Z I

( )

( ) ( )

2 2

Z I

(0)

, Z

(1)

, Z

(2)

– equivalent impedances of network for zero, positive and

negative sequence components, respectively

Using the notation:

We can describe sub-

networks for each

symmetrical

component

22 / 48

Single line-to-earth short-circuit



Assumptions:

•

The network is unloaded before a fault

•

Phase R is short-circuited



Boundary conditions are:

( )

)

= =

( )

)

Using the inverse of

symmetrical

component

transformation

Using the

transformation

itself

These two expressions describe the sequence

network connection.

23 / 48

Single line-to-earth short-circuit

( )

( ) ( ) ( )

( )

= +

( )

= =

Using the sequence

network equations:

( )

( ) ( )

0 0

Z I

( )

( ) ( )

1 1

Z I

( )

( ) ( )

2 2

Z I

The condition 1) gives:

( )

( ) ( )

1 1

2 2

0 0

Z I

from

here:

Phase current can be

expressed by

Initial current

24 / 48



 



 

(1)

(2)

 

(0)

Single line-to-earth short-circuit

Sequence network
connection for single
phase short-circiut has
all three sequence
networks connected in
series.

25 / 48

Single line-to-earth short-circuit

( )

( ) ( )

( )

0 0

Z E

Z I

( )

( ) ( )

( )

1 1

Z I

( )

( ) ( )

( )

2 2

Z E

Z I

Voltage symmetrical

components:

Phase voltages:

( )

(

)

( )

a U

a a Z

a 1 Z

�

( )

(

)

( )

(

)

( )

a U

a a Z

a 1 Z

�

+ -

�

26 / 48

Single line-to-earth short-circuit

 

E 

 



 

a 

 

a

 

a

 

a 

 

Vector diagram of currents
and voltages for single phase
short-circuit (network
resistance is omitted)

27 / 48

Single line-to-earth short-circuit

 

E 

 



 

a 

 

a

 

a

 

a 

 

Vector diagram of currents
and voltages for single phase
short-circuit (network
resistance is taken into
account).

28 / 48

Single line-to-earth short-circuit

through

an impedance

Z R

Z I

= =

( )

Z 1

3Z I

( )

( ) ( )

( )

1 1

2 2

0 0

Z 1

Z I

3Z I

( )

( ) ( ) ( )

( )

= +

Boundary conditions:

)

The condition 1) gives:

From

here:

29 / 48

Single line-to-earth short-circuit

through

an impedance

( )

( ) ( )

( )

0 0

Z E

Z I

( )

( ) ( )

( )

1 1

Z I

( )

( ) ( )

( )

2 2

Z E

Z I

Voltage symmetrical components:

30 / 48

Single line-to-earth short-circuit

through

an impedance

( )

a U

( )

(

)

( )

a a Z

a 1 Z

3a Z

�

( )

a U

( )

(

)

( )

(

)

( )

a a Z

a 1 Z

3aZ

�

+ -

�

( )

u 0

3Z I

D =

Phase voltages:

31 / 48

Single line-to-earth short-circuit

through

an impedance

 

E 

 



 

a 

 

a

 

a

 

a 

 

Vector diagram of currents
and voltages for single phase
short-circuit through an
impedance (network
resistance is omitted)

32 / 48

Line-to-line short-circuit

( )

(

)

a a I

( )

(

)

a a I

( )

(

)

1 1 I

( )



Assumptions:

•

Load currents are omitted.

•

The fault involves the phases S and T



Boundary conditions:

)

Expressing these relationships in terms of the

symmetrical components:

33 / 48

Line-to-line short-circuit

( )

(

)

(

)

+ +

( )

(

)

(

)

a U

( )

(

)

(

)

a U

( )

Voltage sequence components:

34 / 48

Line-to-line short-circuit

 

(1)

(2)

 

(0)

( )

( ) ( )

1 1

2 1

Z I

( )

j 3

=- =-

( )

= = =

Sequence network connection for
line-to-line short-circiut has two
sequence networks connected in
parallel.

35 / 48

Line-to-line short-circuit

( )

( ) ( )

( )

1 1

2 2

Z I

( )

( ) ( )

( )

Z E

a U

Phase voltages:

36 / 48

Line-to-line short-circuit

 

E 

 

a 

 

a

 

a

 

a 

 

U 

 

a 

 

a

 

a

Vector diagram of
currents and voltages
for line-to-line short-
circuit (network
resistance is omitted)

37 / 48

Line-to-line short-circuit

through an

impedance

Z S

Z I

( )

(

) (

)

( )

(

)

( )

a U

a a U

( )

(

)

( )

Z S

Z 1

j 3 U

Z I

j 3Z I

( )

Z 1

Z I

( )

( ) ( )

( )

1 1

2 1

Z 1

Z I

( )

j 3

=- =-

Boundary conditions

)

38 / 48

Double line-to-ground short-circuit

( ) ( ) ( )

( )

= =

( )

( ) ( )

1 1

2 2

Z I

( )

( ) ( )

1 1

0 0

Z I

( )

( ) ( )

( )

Z Z

( )

The short-circuit involves phases S, T and ground.
Initial relationship at the point of the fault:

From solving the equations it follows that:

Using the sequence

transformation

39 / 48

Double line-to-ground short-circuit

 

(1)

(2)

 

(0)

Sequence network connection
for double line-to-ground short-
circuit has three sequence
networks connected in parallel.

40 / 48

Double line-to-ground short-circuit

( )

a I

a a

�

( )

(

)

( )

(

)

j 3 X

2 X

�

( )

(

)

( )

(

)

j 3 X

2 X

�

Phase currents:

41 / 48

Double line-to-ground short-circuit

( )

(

)

( )

(

)

( )

( ) ( )

( )

(

)

X X

3 X

3 1

2 X

= = =

Phase current magnitudes:

Voltages:

( )

( ) ( )

( )

( ) ( )

( )

Z Z

( )

42 / 48

Double line-to-ground short-circuit

 



 



 

Vector diagram of
currents and voltages
for double line-to-
ground short-circuit
(network resistance is
omitted)

43 / 48

Double line-to-ground short-circuit

through an impedance

(

)

Z Z

Z S

Z I

( )

Z Z

Z 0

Z I

3Z I

( ) ( ) ( )

Boundary

conditions:

44 / 48

Three-phase, three-phase with ground

short-circuit

+ + =

( )

= =

( )

= =

Boundary conditions:

45 / 48

Initial current

Positive sequence initial current for different

types of short-circuits

 

 
 

I 

 





 





 

   

 





Three-phase short-circuit

Line-to-line

short-circuit

Single phase line-to-earth short-circuit

Double line-

to-earth short-circuit

46 / 48

Initial current

Z 



 

Z 



 







   

 







General

formula:

 







where

Three-phase short-

circuit

Line-to-line short-

circuit

Single phase line-to-earth

short-circuit

Double line-to-earth short-

circuit

47 / 48

Initial current

 

I 

 

I 

 

I 

 





 







 

I 

Three-phase short-

circuit

Line-to-line short-

circuit

Single phase line-to-earth short-

circuit

Double line-to-earth

short-circuit

General

expression:

48 / 48

Currents and voltages in relation with

( )

2fz

3 1

1 2

+a +a

+ a

( )

a =

Assumptio

( )

3 1

+a +a

( )

2fz

1 2

+ a

If X

(0)

(1)

the 3-phase short-circuit current is the biggest one.

If X

(0)

(1)

the single-phase line –to-earth current is the biggest

one.

If X

(0)

(1)

voltages of unfaulted (healthy) phases for single-phase

and line-to-line short-circuits are > than EMF.

If X

(0)

(1)

voltages of unfaulted (healthy) phases for single-phase

and line-to-line short-circuits are < than EMF.

Document Outline