1

Symmetrical components

Symmetrical components

transformation

transformation

2 / 59

Positive sequence transformation

Positive sequence transformation

jN

jN

g

g

6

6

d

d

U

U

e

e

U

U

p

p

J =

=

=J

( )

( )
( )

1 g

1

1d

U
U

J

=

=J

( )

( )

( )

jN

6

1 g

1d

1d

U

U

U

e

p

=

J =

J

( )

( )

( )

jN

6

1d

1 g

1 g

1

1

U

U

U

e

p

-

=

=

J

J

Complex transformation ratio:

N – a numer labelling the

phase shift between

voltage vectors on the

high and low sides of a

transformer
Nπ/6 – shift angle

measured in clockwise

rotation

voltage

current

Positive sequence

transformation of:

( )
( )

1d

1 g

I
I

*

J =

( )

( )

( )

jN

6

1 g

1d

1d

1

1

I

I

I

e

p

*

=

=

J

J

( )

( )

( )

jN

6

1d

1 g

1 g

I

I

I

e

p

-

*

=

J =

J

3 / 59

Negative sequence transformation

Negative sequence transformation

voltage

current

Negative sequence

transformation of:

( )
( )

2 g

2 d

U
U

*

J =

( )

( )

( )

jN

6

2 g

2 d

2 d

U

U

U

e

P

-

*

=

J =

J

( )

( )

( )

jN

6

2 d

2 g

2 g

1

1

U

U

U

e

P

*

=

=

J

J

( )
( )

2 d

2 g

I
I

J =

( )

( )

( )

jN

6

2 g

2 d

2 d

1

1

I

I

I

e

p

-

=

=

J

J

( )

( )

( )

jN

6

2 d

2 g

2 g

I

I

I

e

p

=

J =

J

The transformer shift angle for negative sequence

is a completion to twelve of shift angle for positive

sequence. It means the replacement of the

complex transformer ratio with its conjugate value.

4 / 59

Sequence components transformation

Sequence components transformation

r

I

s

I

t

I

r

s

t

R

u

I

R

I

S

I

T

I

R

S

T

S

u

I

T

u

I

Zero sequence component is transformed

through YNyn transformers only.

Example for YNd11 transformer

Wiring diagram for a three-phase transformer connected Y- Δ

5 / 59

Sequence currents transformation

Sequence currents transformation

Positive sequence

Negative

sequence

R

I

S

I

T

I

R

u

I

R

u

I



S

u

I

S

u

I



T

u

I

T

u

I



r

I

s

I

t

I

R

I

S

I

T

I

R

u

I

R

u

I



S

u

I

S

u

I



T

u

I

T

u

I



r

I

s

I

t

I

6 / 59

Transformation of power and

Transformation of power and

impedances

impedances

( )

( )

( )

( )

( )

( )

( )

( )

1 g

1d

1d

1 g

1 g

1d

1d

1d

1

S

S

*

* *

*

*

=

=

J

=

=

J

U

I

U

I

U

I

( )

( )

( )

( )

( )

( )

( )

( )

( )

2 g

2 d

2 d

2 g

2 g

2 d

2 d

2 d

1

S

S

*

*

*

*

*

=

=

J

=

=

J

U

I

U

I

U

I

( )
( )

( )

( )

( )

( )

( )

( )

( )

( )

N

j

6

1 g

1 g 1d

1 g

1 g

2

N

j

1d

1 g

1 g

1d

6

1 g

U

U

Z

I

I

e

Z

I

U

I

1

U

e

p

-

p

-

J

=

=

=J

J

( )
( )

( )

( )

( )

( )

( )

( )

( )

( )

N

j

6

2 g

2 g 2 d

2 g

2 g

2

N

j

2 d

2 g

2 g

2 d

6

2 g

U

U

Z

I

I

e

Z

I

U

I

1

U

e

p

p

J

=

=

=J

J

7 / 59

Short-circuit current transformation

Short-circuit current transformation



Line-to-line short-circuit, YNyn transformator

I

I



I



I

R

S

T

r

s

r

t

In computing unbalanced currents, the Δ-Δ or Y-Y transformers
present no special problems since the currents are transformed
as mirror images. Thus, a line-to-line short-circuit between
phases s and t on one side of the transformer appears as a
similar pair of currents on the other side and will also flow in
lines S and T.

8 / 59

Short-circuits current transformation

Short-circuits current transformation



Line-to-line short circuit, YNd11 transformer

I

I

I

3

1

I

3

1

I

3

2



3

3

I



3

3

I



3

3

2I

s

t

r

R

S

T

Wiring diagram for a three-phase transformer
connected Y-Δ

In Y-Δ transformation a line-to-line short-circuit between
phases s and t on the Δ side appears as a current flow in three
lines on the Y side.

9 / 59

Short-circuit currents transformation

Short-circuit currents transformation



Line-to-line short-circuit, Dyn5 transformer

I

I



3

I



3

2I

r

s

t

R

S

T



3

I



3

I



3

I

Wiring diagram for a three-phase transformer
connected Δ-Y.
The opposite location of coil terminals is assumed.

In Δ-Y transformation a line-to-line short-circuit between
phases s and t on the Y side appears as a current flow in three
lines on the Δ side.

10 / 59

Short-circuit currents transformation

Short-circuit currents transformation



Single phase with ground short-circuit, Dyn5 transformer

I

I

r

s

t

R

S

T



3

I



3

I



3

I

Wiring diagram for a three-phase transformer
connected Δ-Y.
The opposite location of coil terminals is assumed.

Here, a single-phase-to-ground short-circuit on phase r on the Y
side appears as a current flow in both lines R and S on the Δ
side. Thus, the generator views this short-circuit as a fault in
two phases.