approximate expressions of parasitic capacitances for ...

APPROXIMATE EXPRESSIONS OF PARASITIC CAPACITANCES FOR ISOLATION TRANSFORMER

DOINA

MARIN1, CONSTANTIN VIOREL MARIN2. VASILE TRU$CA2

Key words: Parasitic capacitances, Isolation transformer, Equivalent circuit. This paper presents a method for modelling the isolation transformer like an equivalent electric circuit in order to calculate the total parasitic capacitances. The parasitic capacitances

of the

isolation

transformer

cannot

be

neglected

with

increasing

frequency. These capacitances are important for a good prediction of the response of

the isolation transformer.

I. INTRODUCTION

An imp01iant step in the design of electric circuits is represented by device modeling. The model structure is usually known and its parameters have to be identified. The equivalent electric circuit of the transformer for high frequencies has to take into account the parasitic capacitances [1 ]. The calculation of parasitic capacitances of the isolation transformer is very important in order to simulate the transformer's behavior at different frequencies. The calculation proved to be a difficult task because parasitic capacitances are distributed parameters. Attempts for equivalent transformer electric circuit considering parasitic capacitances were made [2, 3] for impulse tension stability calculation. A simplified method for calculating parasitic capacitances of inductors made of a ferromagnetic core and multiple-layer windings is presented in [4]. A method for calculating parasitic capacitances of single-layer solenoid air-core inductors is presented in [5]. In this paper a new calculation method for the total parasitic capacitances of the

isolation

transformer

is

proposed.

This

method

relies

on

approximate

expressions of parasitic capacitances in terms of geometric transformer parameters [8, 9]. 1

Electroaparataj, Bucharest.

2

"Politehnica'· University of Bucharest.

Rev. Roum. Sci. Techn. -Electrotechn. et Energ., 51, 3, p. 191-198, Bucarcst, 2006

192

Daina Marin, Constatin Viorel Marin. Vasile Tru�ca

Section

2

2 presents the approximate expressions of parasitic capacitances. The

calculation of the parasitic capacitances for an isolation transformer taking into considerations the shields is presented in Section

3.

2. APPROXIMATE EXPRESSIONS OF PARASITIC CAPACITANCES Equations for parasitic capacitances were determined for a transformer with windings consisting of several layers of turns that are close to one another. The equation for the capacitance of two strait parallel conductors infinitely long of circular cross section placed into a homogeneous medium is [6]:

C = c

nd In

2r d

-

,

(1)

�d 2 -4r2

where d is the distance between the center of two adjacent turns, radius, l is the length of conductors and

r

is the wire

is the permittivity of the homogeneous

E

medium. The effect of the conductor's insulation is determined with the equation for the cylindrical capacitor [6]:

2nd CI = R '

(2)

ln-

where

E

,

is the relative permittivity, R

=

r r +a

is the insulation radius and

a

is the

radial thickness of the conductor insulation. In the hypothesis that the turn curvature is neglected, the capacitance between two adjacent turns Cu is the equivalent capacitance of three capacitances in series connection: two related to the conductor's insulation (relation

2

is used) and the

third related to the gap of air between the two conductors (relation I is used). After simplifications the equation for the capacitance between two adjacent turns C,,

(turn-to-tum capacitance) [8] is:

cu

=

TrE0f

--"--

-----

In

----

I (1 +a/r)� d/2 d/2 -1 a+r a+r

--�r=======

( )2

(3)

3

Parasitic capacitances for isolation transformers

193

The turn-to-shield capacitance is calculated [4, 8] with equation (4): (4) The equivalent circuit for a coil with two layers is represented in Fig. 1.

Fig. I

For

-

Equivalent circuit for a coil with two layers.

10 turns the equivalent capacitance between terminals may be

n >

calculated [4, 8] with

CAB

=

1.366 X C11,

(5)

where C11 is C represented in Fig. 1. The equivalent circuit for a coil with one layer and a shield is represented in Fig. 2.

Fig. 2

For

n

>

-

The equivalent circuit for a coil with one layer and a shield.

10 turns the equivalent capacitance between terminals may be

calculated. [8, 9] with equation (6):

CAB= 2.302

x

Ctr,

(6)

where C,, is represented in Fig. 2 as C. The leakage capacitance is the equivalent capacitance of two capacitors arised between the extremity turns of the primary and secondary in parallel connection. This kind of capacitance is approximated with a capacitance arised between two rectangular conductive and equal plates.The capacitance is calculated [7] with equation (7):

194

Doina Marin. Constatin Viorel Marin, Vasile Tru�dt

CI l

=

2:

4

( n;}

In 1 +

(7)

where a is the thickness of coil, l is the lenght of coil and D is the distance between coils.

3. EXAMPLE The expressions for parasitic capacitances were approximated for an isolation transformer with the following data: S11 f

=

=

3 .5 kVA, U1n

=

220 V, U211

=

220 V,

50 Hz. The coils have 2 layers with 50 turns/layer. The electrical scheme is

represented in Fig. 3. PRIMARY A ' '

U1

22£1V

5 ..

> > > >

c

�

>

[ S1

)I

expressions

c

\_.

i" i">

c

CORE

� > > > :>

c

'

0

The

\_. >

c

c

> > > > >

< • •

-

.. . ... .

rt 82 rt-

• •

Fig. 3

L

e1

22DY

U2 • SECONDARY

The electrical scheme for the isolation transformer.

for parasitic capacitances were approximated

different cases: - the isolation transformer with decoupled shields; - the isolation transformer with coupled shields.

111

two

5

195


The equivalent circuit for the transformer with decoupled shields is presented in Fig.

4.

I Ctsl

�

c4i

core

E1

Fig. 4

-

_L

S1

The equivalent circuit of isolation transformer with decoupled shields.

The primary coil is connected between B and C terminals and the secondary

coil between D and F. E1

E2 is the shield between S1 is the shield for the primary coil and S2 is the

is the shield for the core,

primary and secondary coils,

shield for the secondary coil. The transformer has accessible shields and in this case the shields are not coupled. The primary-ground capacitauce is the equivalent

capacitance CAc of the series connection of capacitances CAB and CBc· CAB is approximated with an equivalent capacitance of number of turns/layer). CAB calculated with equation

n

turn-to-shield capacitances

is determined with equation (6), where

ell

(n is

(3): C AB

=

0.209nF.

CBc is calculated with equation (4):

CBC

=

0.565nF.

CAc (primary-ground capacitance) is the equivalent capacitance of CAH and Csc in series connection:

196

Doina Marin, Constatin Viorel Marin, Vasile Tru�ca

C rmnary-grounJ

=

6

0.152 nF.

The primary-secondary capacitance is the equivalent capacitance of turn-to shield capacitances calculated with equation ( 4), in parallel connection:

C rrimary-secondmy

=

0.442 nF.

The secondary-ground capacitance is the equivalent capacitance primary ground and secondary-ground with

Cm: in

series connection (similar calculus with

Csc): CDF

=

0.628nF'

csecondary-i;rotmd

=

0.096 nF .

The calculus was performed for a single column. The total capacitance is

obtained by considering the partial capacitances of both columns in parallel connection:

C rnmary-ground

=

0.304 nF,

C rnman-seco11Jmy

Csecondary-growu1

=

=

0.884 nF,

0.192 nF

(8)

The capacitances (8) were obtained for the isolation transformer with decoupled shields which can be assimilated with an usual transformer .

The equivalent circuit for the isolation transformer with coupled shields 1s

presented in F ig. 5.

core

S1

El \

c

/

''---1 �/

Fig. 5 -The equivalent circuit of isolation transformer with coupled shields.

7

197


The primary-ground

capacitance is the equivalent capacitance of two

equivalent capacitances in parallel connection (one is Cr2

capacitance in series connection, the other is

Cp1

and a turn-to-shield

and a turn-to-shield capacitance in

series connection). The calculated capacitances are: C pl

=

1.75 nF,

C r2

=

2.17 nF and

C p mary-ground n

=

1.27 nF.

The primary-secondary capacitance is the equivalent capacitance of two capacitances arised between the extremity turns of the primary and secondary in series connection: C

rnmary-secondary

=

0.02685 nF.

The secondary-ground capacitance is an equivalent capacitance of C., and

Cm

in series connection . C

The

total

secondmy-ground

capacitance

is

=

obtained

0.487 nF.

by

multiplying

with

2

the

partial

capacitances calculated for a single column, considering the partial capacitances in parallel connection: C

rnmwy-ground

=

2.54 nF,

Csecondarv-i;round

C primary-secondary

=

=

0.974 nF'

53 7 PF·

(9)

·

The values obtained for the capacitances in both cases are presented in table 1. Table 1

Cases

Cp-s

Cp-g

c,_g

Decoupled shields

0.884 nF

0.304 nF

0.192 nF

Coupled shields

53.7 pF

2.54 nF

0.974 nF

The shield between primary and secondary coils reduces the capacitance between primary and secondary from 0.884 nF to 53.7 pF.

4. CONCLUSIONS

The paper presentes a method for calculus of the parasitic capacitances in order to obtain a prediction referring to the frequency response of the transformer. The shield between primary and secondary reduces the capacitance between primary and secondary for almost sixteen times, from 0.884 nF to 53.7 pF.

198

Daina Marin, Constatin Viorel Marin, Vasile Tru�ca

8

The paper proposes an approach for the study of the isolation transformer using the equivalent electric circuits. Starting with the equivalent electric circuit, the total parasitic capacitances are calculated in different cases. This approach allows the using of a soft for circuits, simulating the behaviour of the transformer in order to predict the attenuation characteristic. Further research will use the obtained values for simulating frequency behavior of the isolation transformer with multiple shields to the common-mode and transverse mode noises. Received on August 2(), 2()05

REFERENCES 1. R. Morrison, Grounding and Shielding Techniques. Fourth Edition. John Wiley & Sons, Inc. 1998. 2. R. Richter. Ma:;ini electrice, vol. III ('Transformatorul"). Edit. Tehnica. Bucharest. 1966. 3.

E.

Jezierski.

Z.Gogolevski.

Transformatoare electrice.

I Szmit.

Construcfie �i proiecwre.

Bucure�ti, Edit. Tehnica, 1966. 4. A. Massarini, M. K. Kazimierczuk, G. Grandi, Lumped Parameter Models for Single-and Jhtlt1jJ/e

Layer Inductors, Proc. PESC' 96. Baveno, Italy. June 1996, pp. 295-30 I. 5. G. Grandi, M. K. Kazimierczuk, A. Massarini, U. Reggiani, Stray Capacitances of Single-Layer

Solenoid Air-Core Inductors, IEEET Transaction on Industry Applications. 35, 5. September October 1999, pp. 1162-1167. 6. R. Radulet, Bazele Electrotehnicii. Probleme, vol. I, Edit. Didactica �i Pedagogica. Bucharest, 1970. 7. A. Timotin, V. Hortopan, S. Mastera, A. Ifrim, Lec{ii de Ba::i'le Electrotehnicii, vol. II, Edit. Didactica �i Pedagogica, Bucharest. 1964. 8. D. Marin, Distributed Stray Capacitances Models for the Isolation Transformers with .1'1ult1jJle

Shields, Simpozionul National de Electrotehnica Teoretica SNET'2003. Bucharest. June 18, 2003, pp. 35-38. 9. D.

Marin,

Theoretical

and

Experimental Contributions to

Transformers with Multiple Shields

-

Dissertation, 2003.

the

Structure

of

the

Isolarion