G. Ziegler. Digital Differential Protection
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G. Ziegler, 10/2005 page55
Differential Protection Symposium
Belo HorizonteNovember 2005
t
t
t
I
P
φ
I
S
saturation flux
CT saturation
Currents and magnetising
G. Ziegler, 10/2005 page56
Differential Protection Symposium
Belo HorizonteNovember 2005
Φ
Φ
S
I
SC
I
m
DC flux non-saturated
AC flux
Magnetising current
Short circuit current
Transient CT saturation due to DC component
G. Ziegler, 10/2005 page57
Differential Protection Symposium
Belo HorizonteNovember 2005
Course of CT-flux during off-set short-circuit current
Ι
P
primarycurrent
T
p
Φ
a.c.flux
transient
d.c.flux
t
t
d.c.component
Ι
P
primarycurrent
T
p
Φ
Total flux
a.c.flux
transient
d.c.flux
t
t
d.c.component
−⋅⋅= ωt)cos(
Tp
t
-
e
p
I 2(t)
p
i
ωtsin -)e-(e
s
T-
p
T
s
T
p
ωT
.c.a
ˆ
Ts
t
Tp
t
-
-
=
Φ
Φ
1)e-(e
s
T-
p
T
s
T
p
ωT
.c.a
ˆ
max
(t)
td
K
Ts
t
Tp
t
-
-
+=
Φ
Φ
=
p
p
pmaxtd
R
X
1T1K +=ω+=
−
K
td
(t)
.c.a
ˆ
max
Φ
Φ
.c.a
ˆ
Φ
G. Ziegler, 10/2005 page58
Differential Protection Symposium
Belo HorizonteNovember 2005
Theoretical CT over-dimensioning factor K
TF
K
TF
50100150200
10
20
30
40
50
60
100
250
500
1000
5000
∞ (K
TF
≈1+ωT
N
)
T
N
[ms]
NS
S
TT
T
S
N
STF ω1
−
+=
T
T
TK
T
S
[ms]
Closed iron core
Linear core
T
N
=network time constant
(short-circuit time constant)
T
S
= CT secondary time constant
G. Ziegler, 10/2005 page59
Differential Protection Symposium
Belo HorizonteNovember 2005
CT with closed iron core,
Over-dimensioning factor K
TF
‘for specified time to saturation (t
M
)
−⋅+=
−
p
M
T
t
ptd
eTK 11
'
ω
0 20 40 60 80 100
5
10
15
0.5 cycles
1.0 cycles
1.5 cycles
2.0 cycles
2.5 cycles
t
M
K‘
td
∞→
M
t
20
8
T
p
(ms)
G. Ziegler, 10/2005 page60
Differential Protection Symposium
Belo HorizonteNovember 2005
15
0
Ktd0tM,()
Ktd80tM,()
Ktd90tM,()
Ktd_EnveloptM()
100
0
tM
0 20 40 60 80 100
0
5
10
15
10
0
tM
0 1 2 3 4 5 6 7 8 9 10
0
0.5
1
1.5
2
2.5
3
−⋅ω+=
−
−
p
T
M
t
pEnveloptd
e1T1K
() ()
Θ+ω−Θ+
−⋅θ⋅⋅ω=Θ
−
M
p
T
M
t
pM
''
td
tsinsine1cosTt,K
Transient dimensioning factor K’’
td
for short time to saturation t
M
G. Ziegler, 10/2005 page61
Differential Protection Symposium
Belo HorizonteNovember 2005
020406080100
0
20
40
60
80
100
120
140
160
180
020406080100
0
0.5
1
1.5
2
2.5
3
3.5
4
K
TF
T
N
in ms
10 ms
8 ms
6 ms
4 ms
2 ms
2 ms
6 ms
4 ms
8 ms
10 ms
t
M
t
M
Θ (t
M
,TN)
(el. degree)
CT over-dimensioning factor K
TF
(t
M
,T
N
)
in the case of short time to saturation (t
M
)
T
N
in ms
G. Ziegler, 10/2005 page62
Differential Protection Symposium
Belo HorizonteNovember 2005
Ι
P
B
B
Max
Ι
m
B
R
t
Ι
m
B
Max
B
R
B
t
Current transformer
Magnetising and de-magnetising
G. Ziegler, 10/2005 page63
Differential Protection Symposium
Belo HorizonteNovember 2005
−
−
−
−
⋅⋅
++
+
−
⋅
−
−
−
−
⋅⋅
+= ))
ˆ
S
F2
N
F2
SN
SN
S
F2DT
S
F1
N
F1
SN
SN
T
t
T
t
(
ω
1
T
tt
T
t
T
t
(
ω
1
~
max
ee
TT
TT
eee
TT
TT
B
B
Current transformer
Course of flux in the case of non-successful auto-reclosure
t
F1
= duration of 1st fault
t
DT
dead time
t
F2
=duration of 2nd fault
t
F1
t
DT
t
F2
B
t
B
max
G. Ziegler, 10/2005 page64
Differential Protection Symposium
Belo HorizonteNovember 2005
Current transformer
Magnetising curve and point of remanence
B
I II III
H = i
m
⋅ w
I: closed iron core (TPX)
II: core with anti-remanence air-gaps (TPY)
III: Linearised core (TPZ)
up to about 80%
< 10%
negligible