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Behaviour of Current Transformers (CT's) under severe saturation conditions Behaviour of Current Transformers (CT's) under severe saturation conditions

H�ctor. O. Pascual
Electrical Engineering Department
National University of La Plata
1900, La Plata, Argentina
hpascual@volta.ing.unlp.edu.ar
Jorge L. Damp�
Electrical Engineering Department
National University of La Plata
1900, La Plata, Argentina
dampe@volta.ing.unlp.edu.ar
Jos� A. Rapallini
Electrical Engineering Department
National University of La Plata
1900, La Plata, Argentina
josrap@ing.unlp.edu.ar


Abstract - Modern protective systems require a faithful
reproduction of primary short circuit current. Often,
specially in high power installations, an important part
of the current, during a few cycles at least, is the d.c.
component, which causes severe saturation conditions,
if the current transformer is not correctly selected and
employed.
Prediction of the behaviour of these devices during
the first 20-40 ms, when d.c. component is higher,
becomes a must.
Many models have been presented to simulate
current transformers, but only some of them are well
suited for transient conditions. This paper presents a
comparison between predicted results, from accepted
models, and real conditions ones, from high power
laboratory tests. Significative differences, that tend to
disappear with time, have been found in certain cases.

Keywords:
Current transformer, saturation.


1. INTRODUCTION

It is important to be able to determine the behaviour of
a CT within a certain range of accuracy when it is applied
a primary current which contains a d.c. component that
may cause its saturation, since this will allow to predict
the behaviour of related equipment, such as that aimed at
protecting power electric systems, which due to this
situation might make an incorrect operation within the
period involved.
This paper shows the theoretical and experimental
results obtained from typical CT's, emphasising on the
first cycles of the transient event, and states some
considerations on their applicability.


2. EQUIVALENT CIRCUIT OF A CURRENT
TRANSFORMER.

Fig. 1 shows the typical equivalent circuit of a
transformer.

1
R

1
L

2
R

2
L



p
i

fe
i

mag
i

s
i

bur
R


fe
R

mag
L


bur
L


Fig. 1. Equivalent circuit of a transformer.

1
R
,
1
L
: primary leakage impedance.
2
R
,
2
L
: secondary leakage impedance.
bur
R
,
bur
L
: burden impedance.
fe
R
,
mag
L
: derivation branch.
p
i
: primary current.
s
i
: secondary current.
fe
i
: current derived by the branch representing the core
losses.
mag
i
: current derived by the magnetising branch.

As usual, all magnitudes must be referred to one side
of the transformer. In this work they are considered as
referred to the secondary side.


2-1. Considerations on the equivalent circuit.

In order to incorporate the hysteresis loop into the
suggested model, two alternative ways can be taken. On
one hand, considering the iron core losses (by means of a
variable resistance
fe
R
) separately from the magnetising
current (by means of a variable inductance
mag
L
). On the
other hand, introducing the iron core losses into the
magnetising branch and considering the hysteresis loop
dynamics in this branch. The latter has been chosen to
perform the analysis hereby presented, since in this way
the possibility to introduce the models which predict the
hysteresis loop dynamics in the CT core during the
transient situation is more straightforward
([1],[2],[3],[4],[5] or [6]).
According to usual considerations for these cases, it is
assumed that
constant
R
=
2
, and since the primary
leakage impedance does not affect the behaviour of the
CT, the following simplified equivalent circuit is obtained:

v
2
R

2
L



p
i

m
i

s
i

bur
R


m
L


bur
L


Fig. 2. Simplified equivalent circuit of the a current
transformer.

2-2. Mathematical model representing the equivalent
circuit of the CT.

Fig. 2, shows that the primary current is the sum of
two components:

s
m
p
i
i
i
+
=
(1)

Based on [7] and considering that the path taken by
flux as a function of current
m
i
is available (by means
of a test performed on the secondary side of the CT), and
stating a linear trajectory between the points used as data,
it is possible to posit (2).

)
(
1
data
new
data
m
new
m
P
i
i = (2)

In equation (2) the subscript data is assigned to
m
i

and values, which correspond to the start of the linear
segment of the current-flux curve in which the simulation
calculus is situated and P is the slope of such segment,
which changes when flux value exceeds either limit of
such segment. Subscript new refers the values obtained at
present simulation time (t).

m
new
new
m
K
P
i
+
= 1
(3)

For which

data
m
data
m
i
P
K
+ = 1


Combining the secondary impedance of the CT with
the burden we get:

)
(
)
(
2
2
bur
bur
s
s
L
L
j
R
R
L
j
R +
+
+
=
+
(4)

Taking into account the aforementioned, voltage v
shown in Fig. 2, will be (5)

t
i
L
i
R
v
s
s
s
s
+
=
(5)

furthermore:

t
v
=
(6)

where
N
=
, is the total linked flux, N being the
number of turns and the equivalent flux per turn,
hence:

t
i
L
i
R
t
s
s
s
s
+
=
(7)

by approximating the derivatives by means of a difference
quotient,

where subscripts new and old refer to the values
at the present time step (t) and the preceding time step
)
(
t
t
, (8) is obtained .

t
i
i
L
i
i
R
t
old
s
new
s
s
old
s
new
s
s
old
new
+
+
=
)
(
2
)
(
)
(
(8)

from which we get (9) in which
s
J
is a constant value
throughout simulation, and
old
s
h
is a history variable,
(where history variable is the one whose value for
simulation corresponds to time step
t
t
).

old
s
new
s
new
s
h
J
i
+
=
(9)



+ =
s
s
s
L
t
R
J
2
1


old
s
s
old
s
old
s
i
d
J
h
= (10)

where
s
d
is another constant:



=
s
s
s
s
L
t
R
J
d
2


Considering the equations developed so far and
relating (1), (3) and (9):

new
s
new
m
new
p
i
i
i
+
=


by substituting terms it is possible to relate the flux value
to the primary current, which is data:

s
old
s
m
new
p
new
J
P
h
K
i
+
=
1 (11)

With the result from (11) and substituting in (9),
secondary current
new
s
i
is obtained. Once these values are
obtained, variable
s
h
is updated, so that these variables
are considered as old values for the next simulation time
step. Thus, with calculated values
new and
new
s
i
the
value of
new
s
h
is obtained, which will be the
old
s
h
for
the next simulation step.

new
s
s
new
s
new
s
i
d
J
h
=

Considering the aforementioned, variable
m
K
will be
modified when slope P changes its value.

This algorithm permits a fast solution for the
representation of a CT; it can therefore be used in real
time applications to obtain the current signals to feed
protection relays in order to analyse their behaviour.


3. MEASUREMENTS ON THE CURRENT
TRANSFORMER.

The necessary data to be introduced in the
mathematical model have been obtained from two CT's
and they are:

Transformer 1:

Ratio: 400/5
Accuracy class: 10P
Rated burden: 10 VA
2
R
(secondary winding resistance): 0.187 .
bur
R
(burden resistance connected to secondary
winding): 0.34
bur
L
(burden inductance connected to secondary
winding): 0.

Transformer 2:

Ratio: 1000/5
Accuracy class: 0.5
Rated burden: 30 VA
2
R
(secondary winding resistance): 0.183 .
bur
R
(burden resistance connected to secondary
winding): 1.6
bur
L
(burden inductance connected to secondary
winding): 0.


3-1. Obtaining the hysteresis loop feeding the CT through
the secondary winding.

The CT's have been fed from the secondary side, with
the primary side open circuited. By integrating the voltage
across secondary terminals, and considering the value of
2
R
, we get:
=
t
m
dt
R
i
v
0
2
int
or:
=
t
m
dt
R
i
v
0
2
sec
)
(

in both equations current
m
i
is the measured current,
while
int
v
is the integral of the voltage in connection
terminals
sec
v
.
The curves obtained are those in Figs. 3 and 4, which
show that for these transformers the hysteresis loop area is
negligible for the current amounts involved.


Fig. 3: Flux-current curve of transformer 1.


Fig. 4: Flux-current curve of transformer 2.

Fig. 5 shows the hysteresis loop obtained from another
test performed on transformer 1, with current values about
100 times lower than tho