High Frequency Model and Transient Response

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.

I839


2/6

@) 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.

1840


3/6

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

1841


4/6

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

1842


5/6

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.

1843


6/6

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.

Documents

High Frequency Model and Transient Response