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8/9/2019 High Frequency Model and Transient Response
1/6
High Frequency Model and Transient Response
of
Transformer W indings
Y osh i kdzu S h i buya .
and
Shigeto Fuj ita
Ab,wocr-Thr high frequenc y niudel of translnrmer winding
is indispcnsahle in analyzing the transients particularly
cause
by the very last transients which
uccur
at the lime of
disconnecting switch rrperations. The bansients in transfnrmer
have heen analyzed using a circuit ur interlinked inductances and
capacitances whnse values have lo he pniperly determined. 'Ihe
circuit constants have heen hitherto calculated in a niudel laking
each
cui1
sectinn pair as a huilding hlnck. T h e present paper
proposes the niethnd thal enahles further suhdividing the unit.
lh e circuit Iramients are computed utilizing the FFT technique.
T h e voltage mcillatiuns n l the winding subjected tn an inipulse
voltage are calculated. The co mp cm de nc e with the experimental
resulls is satisfactory. The response to a chopped impulse shows
this method's applicahility tu high frequenc y analyses. Since the
cnnstants are calculated direclly Item the design parameters
of
transformer winding, this technique is particularly useful in
developing and designing transfurm en.
Index
T e m v - Transformers, Transformer winding,
Inductance, Capacitance, Transient analysis, Transient response,
Disconnecting switches
I
INTRODUCTION
complex voltage oscillation can be excited in
the
A ransformer subjected to a lightning surge or switching
surge. Since it may cause
the
dielectric breakdown, the
analysis
of
voltage oscillation has
k e n
attempted for
ii
loug
time [l], 2].
The
transformer winding is
usirally
described by
a
circuit of
interlinked inductances and capacitances.
It
has been the
comm on practice
that
the circuit constants are evaluated for an
abbreviated equivalent circuit taking each coil section pair as
the
building
block
131,
141.
n e ast transients with high frequency com ponents of
MHz
order generated
in
GIs
by disconnecting switch operation may
cause
a
high frequcucy oscillation in
the.
directly rounccted
GIs-transformer system
[5).
To
analyze this phenomcnon,
the
conventional equivalent circuit is no t precise enough because
the
coil Ienglh in a section pair exceeds the spatial wavelenglh
at such high frequencies.
The
present paper proposes a model
in which each scclion pair is subdivided into groups of smaller
number of toms.
l'he
constants are lo
be
evaluated
in turn-tm
turn basis.
Calculated rcsiilts are shown for the experimental model
winding with which voltage oscillations are observed applying
an impulse. Transients generated by a chopped inipulsc
demonstrate
the
applicability of ibis method.
11. W I N D I N G
ONSTAN-I
A. Shupe of Uirirrliiig
Fig.
1
shows the winding of corc-type transformer. The
high voltage (HV) winding with electrostatic plates (SP)
surrounds the low voltage (LV) winding.
A
high frequency
surge is assumed to arrive at the I.IV tcrniiual. The voltages of
LV winding and core are assumed to he zero in this paper. In
the case of high frequency transients, the mapetic field or
flux in those areas are estimated to be relatively sm all because
of the eddy currents.
A number of disk coils are connected
in
scries iu I.IV
winding. Usually, turns in the roil are interleaved within a pair
of coils as shown in Fig. 1
to
improve the initial voltage
distribution. A set
of
these tw o coils
or
section pair has
been
so far used as the building block
of
HV winding in considering
the equivalent inductances and capacitances. In below, the
constants are examined io turn -to-turn basis.
B.
Lr i lucfarrces
The self and mutual inductances in
all
the turns
are
expressed b y the inductance matrix [ L ] .The size of matrix is
N,xN,,
where N , i s
the total
number of
turns.
In thr first approximation, thc effects
of
LV winding and
I
Hvwinding .
0-7803-7525-4/021~17.00
0
2002
IEEE.
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@) elf-mduclauce @)Mutual
inductance
Rg.2.
Assumptioiis inolculsting
inductances.
core are neglected, i.e. the fluxes of H V winding are to exiend
freely in those spaces. The self-inductance of one turn having
the length can bc evalu ated from the inductance of straight
conductor of the same length and cross-section as shown in
Fig.2a. The following expression is obtained [ 6 ] :
2 e
L , ;
-%[ In - 1
Where,
R
is the geometrical mean diameter (GMD) of c ross-
sectional area where current flows. Because high frequency
currents mainly distribute in the surface region, R can be
calculated
as
GMD of the conductor peripheral (the thick
rectangular lines in the figure). The mutual inductance
between tums I and j in
Fig.2b
is obtained from that of the two
ring wires whose positions are sp ecified by
r z , ,.
z
[7]:
Where,
k - 4 ~ , j / { ( ~ ~ + , , ) * + ~ } ,
nd K ( k ) , E ( k ) are the
complete elliptic integrals.
The next approximation assumes that the flux
is
exc l uded
from a cylindrical region r
c ro
as the result
of
eddy-currents
in the LV winding and core. This situation is simulated
considering an opposing current at position
i
against the
current at turn i as sho wn in Fig.2b. Th en, the s e If and mutual
inductances are approximated, respectively, by
using the inductances by (1) and (2).
C . Capacitances
The capacitance matrix
[ C ]
is defined reg arding all the
turns and SPs as so many isolated conductors. The matrix size
is (N,t2)X(N,t2). Most
of
elements will be close to zero
except those of two turns si tuated fac e to face.
As
an approximation, let only the capacitances shown in
Fig.3 count. They are:
(a)
Conductor-ground capacitances
C e , :
etween turn and ground (or LV windin r)
Fig.3. Capacitances evaluated
hy
paralle~plale pproximation
ri insulation thickness,
W
conducto r width, E oue turn length.
Other
capacitances are obtained likewise, in which different
relative dielectric constants may
be
introduced for the
insulation be.tween sections E.) and HV-LV space
E~).
The capacitance matrix [C ] can be composed arranging
above capacitances in the following manner. The capacitance
C g j
is
taken as a diagonal element
C i j . As
for K i j the
corresponding element Ci j
is
set as
- K j j
and, at the same time,
the diagonal eleme.nt
on
the same row
C i i
s increased by
tKij.
These are
to
be repeated for the tnm s
si, j
, and SPs.
111.
REDUCTION OF
DMENSION
A.
Circuit Eq uat ion
The equivalent circuit
of
transformer winding at high
frequencies is shown in Fig.4. Sinusoidal voltage
Eo
(angular
frequency
ID
represents the oncoming high frequency surge..
The voltage and current at each turn are expressed by vectors
(V) and (I), espectively. The circuit equations are described
in the form
[SI:
( A V ) = - [ Z ] . ( I ) ,
( A I ) =
- [ Y ] . V ) 5 )
Hen, AV) and ( A I ) mean the
veclors
composed of Vt-K.1
and
li i.,.
respectively. The impedance and admittance
matrices [ Z ] , [yl n (5) are related to the matrices
[ L ] ,
C ] .
?be followin gs are obtained, if the Joule loss and the dielectric
loss are taken into account 191.
[ z I = ( ~ ~ + J G G X ) [ L I6 )
{ [Y]
( j m + o t a n 6 ) [ ~ ]
Where,
a
d, and
tan6
are the conductor conductivity, typical
insulation thickness, and
loss
tangent of insulation,
p
being
the permeabil ity of vacuum.
- -
C Cge: etween SP and ground (or LV winding)
(b) Couductor-conductor capacitances
K , : between turns (within a co il or in separate coils)
K O ,K.: between turn and SP
These values can he estimated by the parallel plane
obtainable fro m the formula:
Ki j - E ~ E ~
..------
_ _
:> L 1
G--P
y
.-+
approximation. Fo r examp le, the interlum capacitance i s y ;ft
_ _
(4)
w.e
d
Where,
E~ ~ d :
ielectric constant
of
interturn insulation, Fig.4.
Equivalent circuit oftransformerwinding
at
high frequencies.
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The dimension of (6) is normally too large for numerical
calculation for a conventional computer as long as the
constants are given turn-by-turn.
B.
Constants
of
Grouped-Turn Model
Using the section pair as the building block in the circuit
model has been common in calculating the transients
following a lightning or switching surge [4]. Usage of a
smaller unit is preferable for higher frequency simulations.
Fig.5 shows the case that the section pair
is
further subdivided
into groups of smaller number
of
turns. Let the function n , ( i )
indicate the group number the turn
i
belongs. This redu ces the
dimension to the total number of groups N .
Iectmn Po S p .
tw7gtoup
Rg.5. Formarion
of
urn groups by
dividing
section
pair.
Assume the tum-base inductance
L j j is
known. Then, the
reduced inductance L' j j for the grouped turn system can be
calculated iterating the following proc edure
* A d d t h e v a l n e L j j t o L 'ng( j )ns ( j )
( l s i , j s N , )
As for the capacitance C X j including CEO,C8=), the
grouped turn base quantity C' , j can be obtained by the
iteration:
Add the value C S j o
As
or the capacitance Kjj (including
K O ,
K e ) , he grouped
Add the following
AK:;
equivalent to K j j ) o
K;, j )n, j )
(19 i
6N,
turn base quantity
can be obtained by the iteration:
15
i $ N , )
(7)
The equivalent capacitance is the concept postulating the
transformer action in the groups turns
i
and j belong 121,
[lo].
Here, Nk enotes the nominal tnm number of group k. In the
case that the denominator becomeszero (i.e. n E ( i )
=
n g ( l )
,
e i t h e r n J i ) or ngU as to be deliberately shifted by 1.
The capacitance matrix [
C']
of reduced dimension can
be
constructed using the CAscaand K L 8 ( + ( j l , just as in the same
way as [C] is composed from
C E j
nd
K j j
n section
ILC.
C .
Frequency and Time Doniairi Analys es
The circuit equations of grouped-turn model are obtained
using the constants
[ L ' ] , C']
in
(5)
and
(6)
or
in
Fig.4. If the
terminal voltage Eo is assumed to be a sinusoidal voltage, the
solution is obtainable by a matrix manipulation such
as
given
by Wilcox
[SI.
Then, the voltage or current of any point is
numerically calculable for a given frequency.
In the case the terminal voltage is given in the form of time
dependent waveform, it is possiblr lo calculate the voltage or
current waveform of any
p i n t
using the fast Fourier transfon
FIT)
echnique.
In
the analysis, the sinusoidal response has
to
be
calculated for
t i p
frequencies:
f
=o, fo ,
Z f o , . - . * f m ( = l i P f 0 )
From this result, the transfer function is obtained in FIT form.
Multiplying the above transfer function by the Tof input
voltage, the wanted voltage
(or
current) is obtained in the form
of
FFT.
Taking the inverse T gives the timc response which
is given at 2np sampling times:
t = o , t o , 2
i o ,
. T
( = 2 n p t o )
The following relations exist between frequency and time
domain sampling parameters.
1, = 11 2 f.). T 11 fo
8)
Iv. MODELWINDING A N D CONSTANTS
A. Model Winding
The model winding shown in Fig.
6
is used in the
experiment to observe th e voltage oscillation. The interleaved
HV winding has 76 coil sections and 960 tums in all. Major
dimensions are given in the figure. The winding consists of
three regions of slightly different specifications. Their
dimensions are listed in Table
1.
The exp eriment is conducted
in absence of the
LV
winding and core in the air. In
substitution, a cylindrical aluminum plate is inseded in the
center. In this condition, the relative dielectric constants are
estimated as sd=3.0, ,=1.77,
,=1.0.
A low impulse voltage of 12/50 11s) shape is applied at
the upper terminal of HV winding. The voltage w aveforms are
observed at the outermost turns
of
selected sections using an
oscilloscope with voltage probes.
TABLE
1
DIMENSIONS
OFMODBL
INDING
reaion
I A . C
number
of
sections
14
8
number
of
urns per section
12
13
conductorcross-section
Imm)
I 3.2x9 .5
3 . 2 x 9 . 5
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A. Coll.sfurlfs o f w; ~ t r f i r g
The constants of the model winding are calculated for the
four cases shown in l a h l e 2. The uumhcr of divisions refers
thc subdivided number per section pair disciissrd in section
1II.B.
Whether the zero-llux region is
assumcd
or not refers
to
TXBE 2
zero-flux region
4 divisions / section pair
I
divisions / section pair
V .
VOLTAGE
OSCILIATIONS
A.
VolfagrDisfribufioir by Siritisoirlal iripril
The solution for a constant sinusoidal voltage input gives a
voltage distribution if the frequency is given as mentioned in
Section
1II.C.
Fig.9 shows calculated results in three
cases
0
0
for the frequencies:
0.5
and
2
MHz. Here, and here
after, the loss related constants of 0 =5
X 10' Sim.
tand=O.04
are used.
In Fiy.9a
(0.5
MHz), there are no siguificant differences
among three cases. However, in Fig.9b
(2
MHz), differences
appear: the voltage distributious in
the
cases of subdivided
section pair a, ) become bumpy compnred
to
case
0.
Fig.7. Di&hutiom
of
iod c1rncr~~lculaledo r inodel winding.
(a)
4 div I
sectionpairfa@))
@)
I di /section p a i r m a )
Fig.8.
Diatributioin
of apaci lancewlculated fur mudcl
winding.
the two ways of approximations in the ev;iluation of
inductances mentioned in section
11.B.
In the case of assuming
zero-flux region, thr inductances are evaluated hy ( 3 ) ,with ro
set to the position of duminum plate.
Fig.7 shows cxamplcs of the distributions of calculated
inductance
L ' j j
(varyiug
j
for fixed
i
for the four cases.
Fluctuations seen in the case of subdivided section pair
(Fig.7a) are duc to
the
effects of interleaved winding. The
assumption of zrro-flux leads
to
decreased iuductauces
as
seen
k t w e e n
cases
a
nd
@
(or
cases0
nd
a).
Fig.8 shows graphs of
] C j j ] .
hey cover only a small part
of w hole distributions.
All
the elements except the diagonal
3
lines are zero
in
the case the section pair is
used
as division
unit (Fig.8b). However, in the case of subdivided section piir
(Fig.Ra), the ele men ts som e distance awa y from the diagonal
are
not
zero. This is also
due to
the interleaving.
@) 2 MHr
Fig.9.
Vollafe
dislrihutioii
wlculrlrd
forniodel winding
Comparing cases 0 nd 8 he two ways of assump tions in
the flux region do not lead to much differences.
E.
Frequency C haracteristics
In the frequenc y dom ain calculation, voltages are calculated
for
a
rangr of frequency. Fig.10 shows the frequency
characteristics of induced voltages at the
turn
numbers:
4 9 (5), 1 6 9 (15), 481 (39). 793 (63)
These four points arc the outemiost tiims of selected coil
sections. The num bers in parcnthcs rs are the section
numbers.
As expectcd from the last section's result, there arc
resonance p eaks in the characteristics obtained for the
condition of
4
divisions per sectioii (i.e. Fig.lOa). In contrast,
there are
no such
peaks appear in the case of 1 divisioii pcr
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I 5
4
1.
section
pair
(Fig.lOb). This suggests that the conventiouvl
method adopting the latter is not suitable to analyze
the
high
frequency
resonance
phenomena.
C. Volruge Oscrllatioris
by Itnpulse
T h e
f u l l lines in Fig.11 are the experimrntally obtained
voltage oscillations at the
four
lums including the input
impulse.
The
corresponding
broken
lines are calculated using
the frequency domain data b y case0 n the time domain. The
.
tol lowing
FIT
parameters
are
used:
fo=1.22
kIIz,
f m = 5 MHz,
r,=0.1
ps,
T=820
ps
IIp.2048
' h e experimental and calculated results
agree except
at
the wavetail area where
some extra oscillations
are
present in
the
experimental.
The
calculation using the constauts
ot
case 0 gives similar
results
showing
uo
significant differences between
in
calculation cases @ and
Q.
experimenlol-
VI. DISCUSSIONS
According
lo the
present method,
our
cau detemiiue the
iuductmcc and
capacitance
matrices of the
niodel
iu thc
condition
that
the scctiou pair
is
divided into a mimber of
groups.
Since these constauts
can
be zalciilatcd from the
groinrtry
o f
the winding,
Ihe
whole winding charactcristics
iiicluding t'requeucy and
time
domains
are
to
be
analyzed
using the des@ paramcters. These
are
the salieut features of
the method.
As
seen in the frcquency charactrnslics
of
Fig.10,
resonance frequencies appear in lhe region 2-3 MHz in
the
case
the
section pair is subdivided, but
no resonance
in the
rase the section pair is used as unit. The latter corresponds to
the common practice of modeling transformer used hitherto.
The
model with subdivided
section
pair constructed by the
present method seems
to
comprise high
natural
frequencies,
which have not been included
i n
tbe conventional model.
However, the two cases give almost the same time-responses
for
an
impulse input as mentioned in the. pr ev ia~ ~scctiou. This
is probably due to the fact that the full standard impulse
scarcely has frequency compo nents highcr thau I MHz where
the resonauce might have occurred.
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Fig.12 shows the computed responses of the model winding
subjected to a chopped impulse voltage.
As
seen in Fig.lZb,
the voltage at turn
169
changes abruptly at the time of
chopping, and an oscillation of about
3 MHz
follows in case
0 f 4 division per section pair, whereas no such oscillations
occur in case
Q.
One conseque nce of the voltage oscillation is
the increase
of
interlum voltage. Fig.12c sho ws the estimated
voltages to be induced in the first interrum (i.e. between tums
1 and 13) for the cases 0 nd 0 he maximum induced
voltage level calculated in case
0 s
twice larger than that
in
Q. It is expected that the present method of dividing the
section pair provides more accurate evaluation of voltage
oscillation in the case very fast transients are involved.
The present method
of
determining the constants c an be
applied not only to the interleaved winding but also to any
type of windings iucluding the simple continuous disk type
winding. Specifyin g the connection can be done simply
defining the function
r tg ( t )
adequately.
The multiconductor transmission line model is proposed to
analyze the fast transients in the shell -type transform er
winding 1111, which is difficult
to
apply in the interleaved
core-type winding. There is an attempt to introduce resistors
(to represent the transmission line model's characteristic
impedance) in the conventional model
141,
but the rational
determination of resistance va lues seem s difficult.
Time domain calculations in the present paper are
conducted utilizing the FFT technique. To use a time-domain
software such a s EM is more straightfornard. This will be
possibly pun ued in future.
VII.
CONCLUSIONS
The
following points are clarified in this paper.
l )A method is proposed to calculate the constan ts in the high
frequency circuit of HV winding in which the subdivided
p u p s of turns in the section pair are taken
as
the building
blocks.
2)The
voltage oscillations of transformer windings are
calculated using the
FI T
technique.
Those
calculated for a
model winding subjected to an impulse voltage correspond
with the experimental.
3)The applicability to high frequency transient calculation is
demonstrated in the analysis of response to a chopped
impulse revealing an oscillation of about 3 m z .
Another feature of the method is that the high frequency
characteristics can be evaluated once the geometry of
transformer winding
or
the design paramrters are given. 7his
kind of analysis will be needed more frequently with
increasing number
of
transformers directly connec ted to GIS.
VIII. REFERENCES
[ I ]
121
131
P. A. Abetti. Bibliography
OD
the
surge
performance
of transformers
and rotarisg machines . Trans.
AIEE vol.
77, pp. 1150-1168.1958.
A.
G r e e n w o o d
Eleciricol rronsirnrs in
power
rysremr. New York
John
Wiky, 1991,p.322.
W. cNutt
T.
J.
Blalock
and R.
A
Hinloa
Response
of
transformer
windings to system transieot votiges , IEEE Tronr. Power
Apparnrur
ondSysrems.
vol .
PA S93 (2) pp. 175-467.1974.
S . Okabc. M. Koto. 1. Teradnirbi. M. Irhikawa, T. Kobayarhi. and T.
Saida, An electricmodel of gasinsulatcd shunt reactor and analysis
of
wigni l ion
surge
voltagef.
IEEE Trans. Power D c l i w v , vol. 14, pp.
378-386, Feb. 1999.
C E R E WG 33/19-03. Very fast transient phenomena associated with
gas insulated substationP .CIGRE Report, 33-13.1988.
S. Takeyama, Theory of declroma gnelism phenomena (idapaoere ).
Tokyo: Maruzen, 1939, p.386.
K.
Okuyama, A
numerical
analysis of impulse voltage distribution in
lransfurmer windings (in Japanese):
Trans .
IEE Japan
~01.87-I ,
pp.181-189, Jan. 1967.
D.
. Wilcax. W. .
Hurley. T.
P. M ch l e . and M.Conlon 'Aool iat ion
[ J ]
[SI
IS]
171
I81
.
..
of modified md a l theory io the modell ing of practical transformers,
Proc.
IEE,val .
139 PI. C (6). pp. 513-52 0,19 92.
K.
Curnick,
B.
Filliat. C. Kieny. and W. MCller. Distribution of very
fast
transient
overvol tages
in
transformer
windings . CIGR E Report, 12
204.1992.
191
.~
[IO] Y. Kawaychi. Calculation of the circuit
constants
or the compulation
of
internal oscillating
vol tage
in transformer winding* Trans. I E E
Jopon.vol.89-3, 88.537-546, Mar. 1969.
Ill]
Y. Shibuya, S . Fujita andE. Tamaki, Aaalysisof V ery Fast Transienb
in
Transformers, IEE Proc. Gmer. Tranrm. Distrib.. ~01.148 p.377-
383. Sepl.2001.
Yoshihzo Shibuya received B.S. and M.S. in
electrical engineeriog from Kyoto U niversity, Japan,
in 1964 and 1966, respenively. He eceived PhD in
dec1ric.l
enginesing in 1976
from
University of
Salford, UK.
Sincc he joined Mitsubirhi Elenric Corporation
in 1966. he has been engaged in researches
on
lightning
a m s t e r s ,
GIS and transformer iorulalion
problems. Since 1999. he is with Department of
Electrical Engineering Shibaura lnslimte of
Technology. He is ioteresled io fast m n s i en t s
power system. particularly their influence to
transformers Profesxrr Shibuya is a Fellow of IEE and a member of IEE
Japan.
Shieto Fojila received
B.S.
in ph yj i a in 1983
from Saitama University. Japan.He received PhD
in electrical engioeering io Z o l f r o m University of
Tokyo.
Since be joined Mitsubirhi ElectricCorporation
in
1983.
he has been engaged in research
and
development of GIS and other power equipmnt
including iransformers. His cumnt interest is
the
overvol tages
in power systems and the insulation
mordinatioo of power equipment. Dr Fujita i s a
member of Pbysical Society of Japan aod IEE of
Japan.