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Steady-state and transient response model of multi-winding layers transformer due to lightning surge
STEADY-STATE AND TRANSIENT RESPONSE MODEL OF MULTI-WINDING LAYERS TRANSFORMER DUE TO LIGHTNING SURGE
K. Denno New Jersey Institute of Technology Newark, New Jersey 07102 U.S.A.
Abstract. Modeling of multi-layers winding transformer has been developed by this author in three dimensional system which took into consideration space variations of electromagnetic components along a transverse axis with respect to the two dimensional plane used before as the approximate model for the transformer in calculating the operational performance parameters.
.
SOLUTION OF THE INDUCED VOLTAGE SURGE ----.
-
The following 4th order differential
’
EC
(X,S)
= A2
(4)
The total solution for E (X, S):
represents theproper vehicle tosolve for the E(X.S) = A2 + A3 IlcV(X.S) induced voltage surge:
A_& a2.+ywA__= ax2
at*
kc
2x_ at*
ax2at2
(1) - A2 6(t)
=(X.t)
where,
+ A3 (AC+
[
t
-
“(t-t+]
where,
Y = transformer inter-turn capacitance/unit length
f(X)
(5)
f(X) is the associative space function of v(X.t)
c - transformer external capacitance to ground A,, each represent the conductive and the
Acs
I = transformer self-inductance/unit,length . W = one complete turn length of conductor.
convective amplitude of the lightning current
e = the induced voltage surge.
storke
v = the inducing incident voltage surge.
t
is the time delay for the singularity function 1 included for v(X.t)
Taking the Laplace transform of equation (1) wit
To find A2 and A3, the followingsinitial &-
fc S2E + ,Qlf*S* a2E_
I= 52 "
(2)
ax*
ax*
and boundary conditions are e(X,t) +OasX
where,
+m (6) (7)
e(X.t) + E. at t - 0, X - 0
E is a function of S. X V is a function of S. X
E. is the nominal transformer terminal voltage.
S is Laplace Transform Variable
Values of A2 and A3 are expressed below:
The complementary solution of eq. (2): gc (X.S) = A2e
-aX
A2
-
E
6(t)
-lll(t-tl)
+
t
(8)
and
where
go (9) A3' A,-AV where 6(t) is the impulse Delta Dirac function Ul(t-tl) is the delayed step function by tl
(3) a=
af +t
6(t)-U,(C-t,)-U,$)
0
z&
However since fc <
'0 478
TRANSIENT RRSPCIRSE MODEL
expressed
in equation
i(x.t)
=
1 -r
I
+
CONCLUSIONS
(81, the induced current
surge could be obtained fromthe
479
following: Mathematical models in the form of closed form
edt
.-
A2IJ(t) + A3'A,-Av'
+t' - t IJ(t-t1) I
Jc(t)-J,(t)
-$&
s&,tio"s have bee" secured Successfully resulting from the incidence of propagating inducing Voltage produced by lightning conductive return stroke and the convective time varying bound charges in the ground terrain.
] +’
( E?-$?
The f01lOwi"g are secured:
I 1.
I
"+%
,"-k e-jnX dX+k
-2 I" R
n_l
-jnX
." 3
2.
J'2"-l (n-l)! 3fl
J
where,
x 2 2
(10)
R - Rc for
conductive state
- R, for convective
t,o
t > tl
(11)
Denno, Khalil. "Dynamic Modeling for the Process of Inducing and Indiced Voltage Surges Due to Lightning", 3. of Electrostatics, No. 13. pp. 55-69. 1982.
1 (12)
g(X,t) - e(x.t)!i(x,t)
Denno, KhaliJ, "Mathematical Modelling of PruP+g ati"g Inducing and Induced Power Due to a" Act ual Pulse Shape of Lightning Surge", J of Electrostatics, No. 15, pp. 43-51, 1984.
DISTRIBUTION OF THE SURGE IMPEDANCE --
Denno, KhLhalil."Three Dimensional Electromagnetic (13)
Field Model of Power Transformer with MultiLayers Winding", will appear soon in the
from equations 8 and 11
wx,tj
Att-0 g
Time pattern of the surge impedance imposed 0" the transformer StTUCture. It is established that at the instant of voltage surge incidence, the surge impedance is a" impulse, while the steady-state value approaches the total nom inal self-inductance of the transformer. REFERENCES
- Av(t-tl) Ul(t-tl). for t > tl
:.
3.
State
Jc - Act , 0 < t < t1 - Ac(t-t1) IJ1(t-t1), for J" = A"
Distribution of the induced current transm itted surge.
Xn-2 e-jnX dx +
j"
x"-l
Distribution of the induced voltage surge taking into account, effectsof the transformer self inductance and the external capacitance to ground.
A26(t)+A3(Ac-A,)[t-U(t-tl)]
Partial Differential Equat Garabedian, P. R. ions, John Wiley, 1984.
A2U(t)+A3(Ac-A&t2-tU(t-tl)]
Harrington,
= L
(14)
+
Time-Harmonic Electromagne Roger, tic Fields, McGraw-Hill. N.Y., 1961.
Rudenberg,
+ L6(t)
(15)
Wood, Allen, and at t + 2 +.
l
(16)
.'. Distribution of imposed surge impedance as incidence on a
transformer. could be represented
as show" in the
figure
“f
Electric ihocks in Power Syst
.
and Wollenberg, b.f., power Gene
ratio", Operation and Sons. 1984.
zero
a result of lightning voltage
P.,
ems, Harvard Press. 1968.
Lb(t)
Surge Impedance Distribution
Control, John Wiley 6
K. Denno New Jersey Institute of Technology Newark, New Jersey 07102 U.S.A.
Abstract. Modeling of multi-layers winding transformer has been developed by this author in three dimensional system which took into consideration space variations of electromagnetic components along a transverse axis with respect to the two dimensional plane used before as the approximate model for the transformer in calculating the operational performance parameters.
.
SOLUTION OF THE INDUCED VOLTAGE SURGE ----.
-
The following 4th order differential
’
EC
(X,S)
= A2
(4)
The total solution for E (X, S):
represents theproper vehicle tosolve for the E(X.S) = A2 + A3 IlcV(X.S) induced voltage surge:
A_& a2.+ywA__= ax2
at*
kc
2x_ at*
ax2at2
(1) - A2 6(t)
=(X.t)
where,
+ A3 (AC+
[
t
-
“(t-t+]
where,
Y = transformer inter-turn capacitance/unit length
f(X)
(5)
f(X) is the associative space function of v(X.t)
c - transformer external capacitance to ground A,, each represent the conductive and the
Acs
I = transformer self-inductance/unit,length . W = one complete turn length of conductor.
convective amplitude of the lightning current
e = the induced voltage surge.
storke
v = the inducing incident voltage surge.
t
is the time delay for the singularity function 1 included for v(X.t)
Taking the Laplace transform of equation (1) wit
To find A2 and A3, the followingsinitial &-
fc S2E + ,Qlf*S* a2E_
I= 52 "
(2)
ax*
ax*
and boundary conditions are e(X,t) +OasX
where,
+m (6) (7)
e(X.t) + E. at t - 0, X - 0
E is a function of S. X V is a function of S. X
E. is the nominal transformer terminal voltage.
S is Laplace Transform Variable
Values of A2 and A3 are expressed below:
The complementary solution of eq. (2): gc (X.S) = A2e
-aX
A2
-
E
6(t)
-lll(t-tl)
+
t
(8)
and
where
go (9) A3' A,-AV where 6(t) is the impulse Delta Dirac function Ul(t-tl) is the delayed step function by tl
(3) a=
af +t
6(t)-U,(C-t,)-U,$)
0
z&
However since fc <
'0 478
TRANSIENT RRSPCIRSE MODEL
expressed
in equation
i(x.t)
=
1 -r
I
+
CONCLUSIONS
(81, the induced current
surge could be obtained fromthe
479
following: Mathematical models in the form of closed form
edt
.-
A2IJ(t) + A3'A,-Av'
+t' - t IJ(t-t1) I
Jc(t)-J,(t)
-$&
s&,tio"s have bee" secured Successfully resulting from the incidence of propagating inducing Voltage produced by lightning conductive return stroke and the convective time varying bound charges in the ground terrain.
] +’
( E?-$?
The f01lOwi"g are secured:
I 1.
I
"+%
,"-k e-jnX dX+k
-2 I" R
n_l
-jnX
." 3
2.
J'2"-l (n-l)! 3fl
J
where,
x 2 2
(10)
R - Rc for
conductive state
- R, for convective
t,o
t > tl
(11)
Denno, Khalil. "Dynamic Modeling for the Process of Inducing and Indiced Voltage Surges Due to Lightning", 3. of Electrostatics, No. 13. pp. 55-69. 1982.
1 (12)
g(X,t) - e(x.t)!i(x,t)
Denno, KhaliJ, "Mathematical Modelling of PruP+g ati"g Inducing and Induced Power Due to a" Act ual Pulse Shape of Lightning Surge", J of Electrostatics, No. 15, pp. 43-51, 1984.
DISTRIBUTION OF THE SURGE IMPEDANCE --
Denno, KhLhalil."Three Dimensional Electromagnetic (13)
Field Model of Power Transformer with MultiLayers Winding", will appear soon in the
from equations 8 and 11
wx,tj
Att-0 g
Time pattern of the surge impedance imposed 0" the transformer StTUCture. It is established that at the instant of voltage surge incidence, the surge impedance is a" impulse, while the steady-state value approaches the total nom inal self-inductance of the transformer. REFERENCES
- Av(t-tl) Ul(t-tl). for t > tl
:.
3.
State
Jc - Act , 0 < t < t1 - Ac(t-t1) IJ1(t-t1), for J" = A"
Distribution of the induced current transm itted surge.
Xn-2 e-jnX dx +
j"
x"-l
Distribution of the induced voltage surge taking into account, effectsof the transformer self inductance and the external capacitance to ground.
A26(t)+A3(Ac-A,)[t-U(t-tl)]
Partial Differential Equat Garabedian, P. R. ions, John Wiley, 1984.
A2U(t)+A3(Ac-A&t2-tU(t-tl)]
Harrington,
= L
(14)
+
Time-Harmonic Electromagne Roger, tic Fields, McGraw-Hill. N.Y., 1961.
Rudenberg,
+ L6(t)
(15)
Wood, Allen, and at t + 2 +.
l
(16)
.'. Distribution of imposed surge impedance as incidence on a
transformer. could be represented
as show" in the
figure
“f
Electric ihocks in Power Syst
.
and Wollenberg, b.f., power Gene
ratio", Operation and Sons. 1984.
zero
a result of lightning voltage
P.,
ems, Harvard Press. 1968.
Lb(t)
Surge Impedance Distribution
Control, John Wiley 6