ELEKTROTEHNIŠKI VESTNIK 78(3): 106-111, 2011 
ENGLISH EDITION 
Surge wave distribution over the power transformer continuous 
disc winding 
Mislav Trbušić
1
, Marko Čepin
2
 
1
Kolektor Etra,Šlandrova ul. 10, 1000 Ljubljana, Slovenija  
2
 Fakulteta za elektrotehniko, Univerza v Ljubljani, Tržaška 25, 1000 Ljubljana, Slovenija 
E-mail: mislav.trbusic@etra33.si 
 
Abstract. The paper presents a method enabling calculation of the radial and serial capacitance, inductance and 
voltage between adjacent coils of the power transformer continuous disc winding. Definition of the equivalent 
electric circuit is based on the assumption of a short circuit and grounded winding next to the iron core. This way 
all radial capacitances are at the same time also ground capacitances. Calculation of the serial capacitance is 
based on the assumption of linear voltage distribution along the coil. The inductance is obtained from the stray 
magnetic field in the transformer window calculated in the FEMM 4.2 computer code. The equivalent circuit with 
respect to the 1st and 2nd Kirchhoff’s law gives two systems of ordinary differential equations for the voltage and 
current that are solved using MATLAB calculation. Results are compared with measurement results obtained on a 
dry type transformer TBS (22/6.3 kV, 800 kVA, uk= 5.5 %). A comparison shows that differences between the 
calculated and measured values are acceptable. 
 
Keywords: Transformer, continuous disc winding, surge wave, capacitance, inductance 
 
 
1 PREFACE 
Overvoltages occurring in the power transformer 
windings, are caused by lightning or switching 
overvoltages. The distribution of voltages along the 
windings is heavily affected by the forehead and back 
shape of the high voltage pulses.  As these events are 
inevitable during the transformer operation, winding 
insulation must be carefully planned to withstand them. 
 The IEC 60076 standard specifies shapes and levels 
of insulation at which the transformer is tested [6]. The 
surge wave that simulates the atmospheric discharge 
takes the form of 1.2/50 and it is the standard test that 
every power transformer must pass. 
 Interleaved and continuous disc type of the coil is 
commonly used for the power transformer high voltage 
windings. Due to technological and time reasons, the 
use of interleaved windings is usually avoided. On the 
other hand, the interleaved winding has 3-6 times lower 
initial voltage distribution than the continuous disc 
winding, which is vital for voltage fluctuations in the 
winding. 
 Because of the large volume of work and congestion 
of winding machines in the production cycle, the need 
for continuous disc winding is growing. In order to 
adapt to new requirements, it is essential to know the 
voltage conditions inside the winding during the surge 
wave test as this is the basis for assessing the winding 
suitability.  
The paper presents a method enabling calculation of the 
capacitance, inductance and voltage distribution over 
the transformer continuous disc winding. By comparing  
the obtained measurements and results, the applicability 
of the method in everyday engineering practice is 
proven. 
 
2 SURGE WAVE CALCULATION 
Surge wave distribution over the continuous disc 
winding of power transformer is calculated based on the 
equivalent electric circuit, which is represented by lump 
parameters L, C, K (Figure 1). Mutual inductances M 
are also included, but are not plotted for better clarity. 
The equivalent circuit elements are made of two coils, 
so the voltage along each element represents the voltage 
in the gap between the adjacent coils. The winding end 
is grounded. 
 
 
Figure 1. Lumped parameter model  of the transformer 
continuous disc winding. 
 
Received 6 April 2011 
Accepted 23 May 2011 

SURGE WAVE DISTRIBUTION OVER THE POWER T
 
2.1 Capacitance calculation 
 To determine the capacitance, the 
represented as a capacitive circuit that includes both
shunt and serial capacitance. Figure 2
made of six coils and associated capacitive circuit.
 
Figure 2. Equivalent capacitive circuit of the tr
continuous disc winding. 
 
 Letters C and K mark the shunt and serial capacitance 
of the winding element, respectively
between the serial capacitance of coil
capacitance of equivalent element 
equation (1) [1]. 
 
2
sv
K
K =  
 
2.1.1 Serial capacitance calculation 
The serial capacitance refers to the capacitance between 
the turns and adjacent coils of the test 
sufficiently good approximation 
capacitance can be obtained by term (2
 
1 3
4
−
+ ⋅ =
m
ov
C
d
C
sv
K 
 
The end coil serial capacitance with
can be calculated from equation (3) [1
 
ION OVER THE POWER TRANSFORMER CONTINUOUS DISC WINDING 
the winding can be 
that includes both the 
2 shows a winding 
and associated capacitive circuit. 
 
Equivalent capacitive circuit of the transformer 
shunt and serial capacitance 
, respectively The relationship 
serial capacitance of coil KSV and the serial 
 K is defined by 
(1) 
calculation  
the capacitance between 
the test winding. With a 
sufficiently good approximation the coil serial 
2) [1], [2]. 
(2) 
with no potential ring 
1]. 
 
1 3
2
−
+ ⋅ =
m
ov
C
d
C
sv
K 
 
m …number of turns in the coil
 
C
d
 is the capacitance in the radial gap between adjacent 
coils (Figure 3) calculated using
 
( 2
8 , 27







−
+
+ ⋅ +
⋅ ⋅ =
Oil TB
k
h
pap
pap
pap
k
h
ti
a
i
d
d
C
ε ε
 
h
k 
… height of the radial gap  
a
ti 
…width of winding 
pap …double side thickness of paper insulation
d
i
 …mean diameter of the winding
β' …spacing coverage factor of the 
ε
pap
 …relative dielectric constant of paper
ε
TB-Oil
 …relative dielectric constant
between the adjacent coils 
 
Figure 3. Capacitance in the radial gap between 
coils 
 
C
d
' is the capacitance in the radial gap between the 
adjacent turns of the adjacent coils
 
m
d
C
d
C =
'
 
 
The capacitance between the adjacent turns (
determined by equation (6) [3]. 
2
0
8 , 27
+
⋅ ⋅ ⋅ =



pap
b
i
d
pap ov
C ε
107  
(3) 
coil 
apacitance in the radial gap between adjacent 
using equation (4) [3]. 
15
10
)
−
⋅







Oil
pap
[F]  (4) 
 [mm] 
[mm] 
…double side thickness of paper insulation [mm] 
…mean diameter of the winding [mm] 
of the winding  
relative dielectric constant of paper insulation 
relative dielectric constant for radial gap   
 
radial gap between the adjacent 
apacitance in the radial gap between the 
adjacent coils (5). 
(5) 
adjacent turns (Figure 4) is 
 
15
10
2
−
⋅
⋅



pap
pap
 [F] (6) 

TRBUŠIĆ, ČEPIN 
 
b
0 
…height of the bare wire or transpose cable
d
i 
…mean diameter of the winding  
pap …double side thickness of paper insulation
ε
pap
 …relative dielectric constant of paper
 
Figure 4. Capacitance between the turns. 
 
2.1.2 Shunt capacitance calculation
The shunt capacitances are capacitance
winding at the adjacent windings and grounded parts of 
the transformer (yoke, tank). During the
adjacent windings are short-circuited and grounded, 
thus all shunt capacitance become ground
The capacitance is determined depend
position of the test winding. Figure 
capacitance of the middle phase of the 
 
Figure 5. Shunt (ground) capacitances for the middle phase of 
the three phase transformer. 
 
Among the windings of the same phase (
are capacitances C
f
, which are calculated by equation
(7) [3]. 
 
15
10
) 2 ( 8 , 27
−
⋅
⋅ −
⋅ + ⋅ ⋅
=
deb fak d
d
g
b
sr
d
f
C [F]
  
d
sr 
…mean diameter of the gap between the windings
b
g
 …height of the shorter winding [mm
d …double width of the gap between windings
deb …thickness of solid insulation between 
windings [mm] 
bare wire or transpose cable [mm] 
[mm] 
paper insulation  [mm] 
relative dielectric constant of paper insulation 
 
calculation 
capacitances of the test 
cent windings and grounded parts of 
, tank). During the test, all the 
ed and grounded, 
become ground capacitance. 
depending on the 
 5 shows the earth 
of the transformer. 
 
Shunt (ground) capacitances for the middle phase of 
windings of the same phase (Figure 6) there 
calculated by equation 
[F] (7) 
between the windings[mm] 
mm] 
…double width of the gap between windings [mm] 
…thickness of solid insulation between the the 
ε
TB
 …relative dielectric constant of solid insulation 
between the windings 
ε
Oil
 …relative dielectric constant of oil
fak …(ε
TB-
 ε
Oil
)/ε
TB 
 
 
Figure 6. Capacitance between the two windings of the same 
phase. 
 
Equation (8) [3] holds for the capacitance between the 
outer windings of the adjacent phases (
 
1
2
ln
8 , 27












−








+
⋅
=
s
R
n
x
s
R
n
x
g
b
mf
C
 
TB
deb
Oil
deb
s
R
n
x
s
R
n
x
ε ε
κ
+
− −
−
= 
 
x
n
 … half the distance between the limb center
R
s
 …outer diameter of the most outer winding
Other marks have the same mean
 
Figure 7 : Geometry used in determin
capacitances between the most outer windings 
phases and between the most outer winding and 
 
108 
…relative dielectric constant of solid insulation 
…relative dielectric constant of oil 
 
Capacitance between the two windings of the same 
the capacitance between the 
adjacent phases (Figure 7).  
15
10
−
⋅ ⋅κ [F] (8) 
distance between the limb center [mm] 
most outer winding  [mm] 
meaning as above. 
 
determination of the shunt 
most outer windings of the adjacent 
phases and between the most outer winding and the tank. 

SURGE WAVE DISTRIBUTION OVER THE POWER T
Equation (8) can be used to determine the capacitance 
between the external winding and 
Instead of half the distance between the limb center 
the minimum distance between the axe of the external 
winding and the tank is inserted
expression multiplied by 2 (9)[3]. 
 
mf
C
k
C ⋅ = 2 
 
The total ground capacitance of winding 
of all partial ground capacitances 
particular case. 
 Capacitance C of the equivalent 
determined by dividing the total ground capacitance
the number of elements in the equivalent winding circuit
(10). 
 
el
g
N
C
C = ,  
 
N
el
 … number of elements in the equivalent 
circuit 
 
2.2 Inductance calculation 
 Inductances in the equivalent electric circuit
determined based on the assumption 
winding next to the iron core. This assumption allows us 
to neglect the influence of the iron core in the 
inductance system [1].  
 
 
Figure 8. Elements layout and calculation 
computer program FEMM 4.2. 
 
 Due to a good magnetic coupling
windings and coils, self and mutual inductance
determined from the stray magne
transformer window (11) [1]. 
 Figures 8 and 9 show a layout of
directions of the currents in the inductance 
using the FEMM 4.2 computer program
 
2
2
I
W
L
mag
⋅
=
σ
 (I = 1 A, DC) 
ION OVER THE POWER TRANSFORMER CONTINUOUS DISC WINDING 
) can be used to determine the capacitance 
and tank (C
NN
, C
VN
). 
distance between the limb center x
n
, 
axe of the external 
is inserted, and the entire 
(9) 
of winding C
g
 is the sum 
s appearing in a 
 winding circuit is 
ground capacitance by 
the equivalent winding circuit 
(10) 
equivalent electric 
equivalent electric circuit are 
n the assumption (of a short circuit 
. This assumption allows us 
of the iron core in the 
 
 
KS i
L
− σ
 using the 
coupling between the 
and mutual inductances can be 
magnetic field in the 
layout of the elements and 
inductance calculation 
computer program [4].  
(11) 
 
W
mag
 …magnetostatic energy in the 
 
Figure 9. Elements layout and calculation 
FEMM 4.2 computer program. 
 
 Self inductance of the winding
by multiplying the leakage inductance between the 
element and the short-circuit winding reduced to one 
turn and multiplied by the square of the number of turns 
in the element (12) [1]. 
 
KS i i i
L w L
−
⋅ =
σ
2
  
 
Mutual inductance between the eleme
by equation (13) [1]. 
 
j i KS j KS i
j i
w
L L L
M ⋅
− +
=
− − −
−
2
σ σ σ
 
w
i
 …number of turns in i-th element
w
j
 … number of turns in j-th element
L
σi-KS
 …leakage inductance between 
and short circuit winding reduced 
L
σj-KS
 … leakage inductance between 
and short circuit winding reduced to
L
σi-j
 … leakage inductance between 
element of the winding reduced to
 
2.3 Equations 
 
  
Figure 10. Equivalent electric circuit of the winding build up 
of three elements. 
 
109  
the transformer window 
 
Elements layout and calculation 
j i
L
− σ
 using the 
of the winding element is calculated 
by multiplying the leakage inductance between the 
circuit winding reduced to one 
by the square of the number of turns 
(12) 
Mutual inductance between the elements is determined 
j i
w w ⋅ (13) 
th element 
th element 
leakage inductance between the i-th element 
and short circuit winding reduced to one turn. 
leakage inductance between the j-th element 
uit winding reduced to one turn. 
leakage inductance between the i-th and j-th 
reduced to one turn. 
Equivalent electric circuit of the winding build up 

TRBUŠIĆ, ČEPIN 110 
 Figure 10 shows the equivalent electric circuit of the 
winding. First Kirchoff's Law holds with respect to the 
currents in the nodes. 
 
0 ) ( ) (
1
1 2 1 2 1 0 1 2 1
= ⋅ − − ⋅ − − ⋅ + −
dt
du
C u u
dt
d
K u u
dt
d
K i i 
0 ) ( ) (
2
2 3 2 3 2 1 2 3 2
= ⋅ − − ⋅ − − ⋅ + −
dt
du
C u u
dt
d
K u u
dt
d
K i i 
 
Rearrangement and translation to the matrix notation 
gives the following equation. 
 
dt
du K
C K K K
K C K K
i
i
i
C K K K
K C K K
u
u
dt
d
0 1
1
2 3 2 2
2 1 2 1
3
2
1
1
2 3 2 2
2 1 2 1
2
1
0
1 1 0
0 1 1
⋅






⋅






+ + −
− + +
+










⋅






−
−
⋅






+ + −
− + +
=






−
−
 
 
Or in a shorter form: 
 
[ ] [ ] [ ] [] [ ] [ ]
dt
du
EK CK i D CK u
dt
d
0
1 1
⋅ ⋅ + ⋅ ⋅ =
− −
 (14) 
 
 The second Kirchoff's Law holds for the voltages 
along the closed loops. 
dt
di
M
dt
di
M
dt
di
L u u
3
3 1
2
2 1
1
1 1 0
⋅ + ⋅ + ⋅ = −
− −
dt
di
M
dt
di
L
dt
di
M u u
3
3 2
2
2
1
2 1 2 1
⋅ + ⋅ + ⋅ = −
− −
dt
di
L
dt
di
M
dt
di
M u
3
3
2
3 2
1
3 1 2
⋅ + ⋅ + ⋅ =
− −
 
 
In the matrix notation:  
 
0
1
3 3 2 3 1
3 2 2 2 1
3 1 2 1 1
2
1
1
3 3 2 3 1
3 2 2 2 1
3 1 2 1 1
3
2
1
0
1
1 0
1 1
0 1
u
L M M
M L M
M M L
u
u
L M M
M L M
M M L
i
i
i
dt
d
T
D
⋅






⋅










+






⋅










−
−
⋅










=










−
− −
− −
− −








−
−
− −
− −
− −
43 42 1
 
 
In a shorter form: 
 
[] [ ] [ ] [ ] [ ] [ ]
0
1 1
u F L u D L i
dt
d T
⋅ ⋅ + ⋅ ⋅ − =
− −
 (15) 
 
Putting terms (14) and (15) together gives us a linear 
system of ordinary differential equations (16) solved by 
MATLAB.  
 








⋅
⋅
+






⋅






=










dt
du
u
u
i
dt
du
dt
di
0
0
0
0
δ
γ
β
α
 (16) 
 
T
D L ] [ ] [
1
⋅ − =
−
α 
] [ ] [
1
D CK ⋅ =
−
β 
] [ ] [
1
F L ⋅ =
−
γ 
] [ ] [
1
EK CK ⋅ =
−
δ 
 
[u
0
] …surge wave 
[u] …vector with unknown nodal potentials 
[i] …vector with unknown branch currents 
[CK] …matrix with capacitances 
[EK] …vector with capacitive connections between 
surge wave u
0
 and other nodes  
[D] …first difference matrix 
[L] …matrix with inductances 
[F] …vector with direct connections between surge 
wave u
0
 and other nodes 
 
3 RESULTS 
 Results of our calculation of the surge wave in the 
power transformer winding are gradients between the 
adjacent coils in the winding and are determined by 
equation (17). They are compared with measurements 
obtained on a dry type transformer TBS (22/6.3 kV, 800 
kVA, uk=5.5%) found in reference [1]. 
 
) max(
1 1 + +
− = Δ
i i i
u u u    i=0,1,2…N
el
 (17) 
 
u
i
 …potential of the i-th node in the equivalent electrical 
circuit  
N
el
 …number of elements of the winding 
 
 The surge wave used in our calculation and 
measurements is 1.2/50 μs with the peak value 95 kV. 
The surge wave is described by equation (18). 
 
) ( 95
2 1
0 0
t p t p
e e U u
⋅ − ⋅ −
− ⋅ ⋅ = [kV] (18) 
 
U
0
=1.043325 
p
1
=14732 s
-1 
p
2
=2080313 s
-1
 
 
 Figure 11shows the calculated and measured values 
of gradients in the radial gaps between the equivalent 
circuit elements of the winding, which correspond to the 
gradient between the adjacent coils. The test voltage is 

SURGE WAVE DISTRIBUTION OVER THE POWER T
selected so that the gradients between the
correspond to the percentage values of 
wave. 
 
Figure 11. Gradients in the radial gaps between 
coils, calculated (*) and measured values (o).
 
Figure 12 shows relative deviations of the measured 
values from the calculated values. 
 
Figure 12. Relative deviations of the measured values 
calculated values. 
 
4 CONCLUSION
The paper presents a method to be used in
capacitances, inductances and voltage
transformer continuous disc winding. 
computer code was developed to sup
made according to the presented method
results were compared with measurement
available in the literature.  
 As seen from the obtained results, the
the measured and calculated values 
which is acceptable for the engineering 
It can be concluded that the calculation of volt
distribution along the power transformer continuous 
disc winding approximates the actual voltage conditions 
inside the winding well enough and can therefore be 
used to design winding insulation. 
 
ION OVER THE POWER TRANSFORMER CONTINUOUS DISC WINDING 
s between the coils 
es of the applied surge 
 
radial gaps between the adjacent 
(o). 
viations of the measured 
 
elative deviations of the measured values from the 
ONCLUSION 
to be used in calculating 
and voltages along the power 
continuous disc winding. A special 
support calculations 
method. The calculated 
results were compared with measurement results 
the obtained results, the deviations of 
values are within 20%, 
engineering practice [1], [5]. 
that the calculation of voltage 
he power transformer continuous 
the actual voltage conditions 
and can therefore be 
5 REFERENCES
[1] B. Ćućić: Raspodjela udarnog napona po preloženom namotu 
transformatora, Magistrsko delo, Fakultet elektrotehnike i 
računarstva, Zagreb, 2001. 
[2]  B. Heller, A. Veverka, Surge Phenomena in Electrical Machines, 
Publishing House of the Czechoslovak Acade
Prague, 1968. 
[3]  K. Lenasi, Program ZVRU – koda, Tovarna transformatorjev, 
Ljubljana, 1991. 
[4]  http://www.femm.info/wiki/HomePage
[5] K. Lenasi, Udarne prenapeto
transformatorjev z vpeljavo dušenja induktivnih tokov, doktorska 
disertacija, Ljubljana 1975. 
[6]  IEC 60076-1, Power Transformers, Part 1: General, IEC 
Standard, 2000. 
[7]  H. M. Ryan, High Voltage Engineering and Testing, Institut
Electrical Engineers, 2001. 
 
Mislav Trbušić received his B.Sc.
engineering from  the University of Ljubljana, Slovenia in 
2011. He works at Kolektor Etra as
power transformers. 
 
Marko Čepin received his B.Sc., M.Sc. and Ph.D. degrees 
from the University of Ljubljana, Slovenia, in 1992, 1995 and 
1999, respectively. In the period from 1992
employed with the Jožef Stefan Institute. 
invited to Polytechnic University of Valencia
of the Nuclear Society of Slovenia (2010 
he is associate professor at the Faculty of Electrical 
Engineering, University of Ljubljana. His r
include power engineering, power plants and 
reliability. 
 
 
111  
EFERENCES 
: Raspodjela udarnog napona po preloženom namotu 
agistrsko delo, Fakultet elektrotehnike i 
B. Heller, A. Veverka, Surge Phenomena in Electrical Machines, 
Publishing House of the Czechoslovak Academy of Sciences, 
koda, Tovarna transformatorjev, 
http://www.femm.info/wiki/HomePage (last viewed 27.1.2011). 
K. Lenasi, Udarne prenapetosti v navitjih močnostnih 
transformatorjev z vpeljavo dušenja induktivnih tokov, doktorska 
1, Power Transformers, Part 1: General, IEC 
H. M. Ryan, High Voltage Engineering and Testing, Institution of 
B.Sc. degree in electrical 
the University of Ljubljana, Slovenia in 
Etra as a design engineer for 
c., M.Sc. and Ph.D. degrees 
from the University of Ljubljana, Slovenia, in 1992, 1995 and 
In the period from 1992-2009 he was 
Jožef Stefan Institute. In 2001, he was 
invited to Polytechnic University of Valencia. He is president 
ociety of Slovenia (2010 - 2014). Currently, 
professor at the Faculty of Electrical 
Engineering, University of Ljubljana. His research interests 
engineering, power plants and power system