Leakage field of a transformer under conventional and superconducting condition

Journal of Materials Processing Technology 108 (2001) 246±252

Leakage ®eld of a transformer under conventional and superconducting condition P. Raitsios* Construction and Operations Department, Distribution Engineering, Public Power Corporation of Greece, Chalcocondyli 22, 104 32 Athens, Greece

Abstract The purpose of this work is to determine the leakage ®eld in a transformer and the current density distribution in its windings. These are of concrete height and concrete thickness and are assumed not to be in proximity to the yokes. Also the conductivity of the conductive material is taken into account. The primary winding is formed by many layers of round conductors with negligible cross-section. The secondary winding consists of two layers of rectangular conductors. Using Maxwell's differential equations and introducing the vector potential, general equations result for the components of the magnetic induction in two-dimensional ®eld space. The formulae of these components are converted into Fourier series, so that boundary conditions can be applied for determining the integrating constants of the general solutions. The leakage inductance results from the calculation of the magnetic energy. Since its expression depends on the conductivity, so depends on the temperature too. In the superconducting condition in the region of the windings, the leakage inductance becomes equal to 0. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Leakage ®eld; Transformer; Leakage inductance; Conductivity

1. Introduction Some of the most important problems in the design of a transformer is the predetermination of the reactive voltage drop at the secondary terminals when the transformer is loaded and the electromagnetic forces developing when the transformer is short-circuited. The exact calculation is a matter of considerable dif®culty and designers usually employ approximate formulae for the computation of the leakage reactance corrected by some suitable factor which experience and the results of tests have shown to be necessary. For avoiding empirical factors, the exact knowledge of the leakage ®eld is of great importance. Rogowski treatment, i.e. in power up today, refers, as is well known, to windings being in proximity to the core yokes and having constant current density. It does not take into account the conductivity of the conductive material. In this research work, a basic arrangement of a two windings transformer has been investigated for calculating the leakage ®eld from a new aspect, i.e. taking into consideration the conductivity of the conductive material and the concrete height and concrete thickness of the windings. The secondary winding consists of two layers of rectangular conductors and eddy currents are taken into account. The *

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primary winding is formed by many layers (or by individual coils) of round wires of negligible cross-section, so that eddy currents can be omitted. The insulation between conductors is assumed negligible too. The secondary winding lies next to the core limb, outside the primary winding. The windings are cylindrical and coaxial. The core limbs are assumed to be cylindrical too. Fig. 1 represents a vertical section through the window of the transformer, so that the problem can be solved in two dimensions. The electromagnetic ®eld in the air and in the space of conductive material has been investigated using Maxwell's differential equations and introducing the vector potential. In the air space, Laplace differential equation must be satis®ed. In the conductive material, the use of Poisson's differential equation is required for constant current density, or the diffusion differential equation, when eddy currents are taken into account. The current density in the primary winding may be expressed by a Fourier series. For the vector potential a suitable formula, ful®lling Poisson's differential equation, can be selected in this case. For the boundary problem at the surfaces of the secondary winding, the diffusion differential equation is applied and the vector potential is assumed to consist of two vector potentials. The ®rst one is chosen in such a way that the horizontal induction component, resulting at the horizontal boundary surfaces of the winding,

0924-0136/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 7 6 4 - 0

P. Raitsios / Journal of Materials Processing Technology 108 (2001) 246±252

Nomenclature A a1, a2 B d, b, s, c g h 1, h2 I1 J1 J2 N1 N2 Q, K x, z

vector potential winding thicknesses magnetic induction horizontal spacings limb height winding heights primary current of a conductor primary current density secondary current density number of primary turns number of secondary turns integrating constants horizontal, vertical axes

Greek symbols d, x, c distances m magnetic permeability of the conductive material s conductivity Subscripts k, n

integers

becomes equal to 0. This is realized when the separating constant for the solution of the diffusion differential equation of the vector potential is inversely proportional to half of the winding's height. The second vector potential is chosen in such a way that the vertical induction component at the vertical boundary surfaces of the winding becomes equal to 0. This is realized when the separating constant of this vector potential is inversely proportional to the winding width. The formulae derived for the whole vector potential in the winding space are ®rstly converted into Fourier series over the interval between the yokes and over the winding width, subsequently they are transformed boundary functions. The corresponding formulae for the vector potentials in the air spaces outside the windings are also converted into Fourier series over the interval between the yokes. Applying to them the boundary conditions at the winding surfaces, the

247

integrating constants can be estimated. The current density distribution in the secondary winding, where eddy currents are taken into account, is also derived from the vector potential. The arrangement is considered symmetrical to the horizontal axis. 2. Determination of the magnetic ®eld The investigating arrangement consisting of N1 primary turns (n1 turns per layer and n2 layers) and N2 secondary turns in two layers of 12N2 turns per layer is shown in Fig. 1. The magnetic ®eld can be determined by the following equations: In the air spaces, the Laplace differential equation must be satis®ed, @ 2 AI @ 2 AI ‡ 2 ˆ0 @x2 @z

(1)

where AI is the vector potential in the air spaces. In the conductive material, two cases are distinguished: In the primary winding, where the current density is assumed to be constant, Poisson's differential equation holds, i.e. @AII1 @ 2 AII1 ‡ ˆ ÿmJ1 @x2 @z2

(2)

where J1 ˆ

I1 N1 ˆ constant h1 a1

(3)

is the current density of the primary winding. In the secondary winding, the diffusion differential equation may be applied, @ 2 AII2 @ 2 AII2 ‡ ˆ jp2 AII2 @x2 @z2

(4)

where AII is the vector potential of the conductive material, p2 ˆ mos;, m is the magnetic permeability of the conductive material, s the conductivity and o the cyclic frequency. 2.1. Air spaces (equations) Since the same current ¯ows in all turns of the secondary winding, the contribution of each layer can be expressed by the same integrating constant. Hence, under the consideration that the separating constant (m) takes all real values from 0 to 1 for the solution of Eq. (1), the general solution for the vector potential can be written as follows [1,2]: Air space I1 between the two windings: AI1 ˆ ‰Q1 eÿm…b‡x† ‡ Q2 …1 ‡ eÿm…a2 ‡s† † emx Š cos mz

(5)

Air space I2 between the two layers of the secondary winding: Fig. 1. Window of a two windings transformer.

AI2 ˆ ‰Q1 eÿm…b‡x† ‡ Q2 …eÿmx ‡ eÿm…a2 ‡1ÿx† †Š cos mz

(6)

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P. Raitsios / Journal of Materials Processing Technology 108 (2001) 246±252

Air space I3: AI3 ˆ …M1 e

mx

‡ M2 e

ÿmx

† cos mz

(7)

AI4 ˆ …M3 emx ‡ M4 eÿmx † cos mz

(8)

Air space I4: The magnetic induction is given by the well-known relation BˆDA

(9)

from which the components of the magnetic induction (along the axes x, z) can be determined. At the boundary surface  12 g …mr ! 1†, the horizontal induction component equals 0. Therefore mg kp sin ˆ 0 ! mk ˆ 2 g=2 (10) with k ˆ 0; 1; 2; 3; . . . ; 1 Consequently there is an in®nity of solutions. Eqs. (5)(8) are now expressed as sums, 1 X ‰Q1k eÿmk …b‡x† ‡ Q2k …1 ‡ eÿmk …a2 ‡s† † emk x Š cos mk z

AI1 ˆ

kˆ0

(11) 1 X BI1x ˆ ‰Q1k eÿmk …b‡x† ‡Q2k …1 ‡ eÿmk …a2 ‡s† † emk x Šmk sin mk z kˆ0

(12) BI1z ˆ

1 X

‰ÿQ1k eÿmk …b‡x† (13)

2.2. Conductive material spaces 2.2.1. Primary winding The constant current density may be expressed by Fourier's series [4] in the interval ÿ 12 g  z  12 g, ak cos mk z

kˆ0

1 ak ˆ g=2

Z

h1 =2

I1 N1 cos mk z dz ÿh1 =2 h1 a1

(14)

mI1 N1 …b ‡ a1 ÿ x†2 ga1 1  X ÿ Q5k emk x ‡ Q6k eÿmk x ‡

AII1 ˆ ÿ

kˆ1

2I1 N1 sin mk h1 =2 …g=2†h1 a1 mk

 cos mk z



(19)

2.2.2. Secondary winding For satisfying the boundary conditions at the windings surfaces, the vector potential is assumed to consist of two vector potentials which certainly ful®l Eq. (4). 2.2.2.1. Space of the ®rst layer. AII2 ˆ AII21 ‡ AII22 AII21 ˆ

1 X

(20)

…K1k eq1k x ‡ K2k eÿq1k x † cos mk z

(21)

K3k cos m2k x cosh q2k z

(22)

kˆ0 1 X

where m1k ˆ

kp ; h2 =2

m2k ˆ

kp ; a2

1 2 I1 N1 2 I1 N1 X sin mk h1 =2 ‡ cos mk z g a1 g=2 h1 a1 kˆ1 mk

q2k ˆ

q jp2 ‡ m21k

q jp2 ‡ m22k

1 X

(23) (24)

…K1k eq1k x ‡ K2k eÿq1k x †m1k sin m1k z

kˆ1 1 X

(15)

K3k q2k sinh q2k z cos m2k x

ÿ

(25)

kˆ1

(16)

BII2z ˆ

1 X …K1k eq1k x ÿ K2k eÿq1k x †q1k sin m1k z kˆ0 1 X

ÿ

K3k m2k cosh q2k z sin m2k x

(26)

kˆ1

2

@ AII1 @ AII1 2mI1 N1 2mI1 N1 ‡ ˆÿ ÿ @x2 @z2 ga1 …g=2†h1 a1 1 X sin mk h1 =2  cos mk z mk kˆ1

q1k ˆ

From the vector potential AII1, according to (9), the induction components are: BII2x ˆ

Eq. (2) then becomes 2

(18)

which results in the following general solution:

Eq. (3) may be written as J1 ˆ

1 X mI1 N1 …b ‡ a1 ÿ x†2 ÿ Y…x† cos mk z ga1 kˆ1

kˆ0

‡ Q2k …1 ‡ eÿmk …a2 ‡s† † emk x Šmk cos mk z

J1 ˆ

AII1 ˆ ÿ

AII22 ˆ

kˆ0

1 X

Since Eq. (17) contains a constant term, AII1 must involve a term of second degree in x. Selecting the suitable expression for the vector potential

(17)

2.2.2.2. Space of the second layer. Following the same method as for the space of the first layer, a similar formula for the vector potential can be written.

P. Raitsios / Journal of Materials Processing Technology 108 (2001) 246±252

3. Boundary conditions

Correspondingly for BII1z,

At the vertical windings surfaces, the following boundary conditions can be applied:

BII2z ˆ

For x ˆ 0 : BIIx ˆ BII2x ; BIIz ˆ BII2z For x ˆ a2 : BII2x ˆ BI2x ; BII2z ˆ BI2z

where

(27)

For applying these boundary conditions (27), formulae (12), (13), (25) and (26) are converted into Fourier series. For the horizontal components: BIIx ˆ

1 X

a1n sin mn z

(28)

c2n ˆ

c3n

Z

1 g=2 X

ÿg=2 kˆ1

‰Q1k eÿmk …b‡x†

‡ Q2k …1 ‡ eÿmk …a2 ‡x† † emk x Šmk sin mk z sin mn z dz (29) Z g=2 1 sin mk z sin mn z dz a1 …k; n† ˆ g=2 ÿg=2   1 sin…mk ÿ mn †g=2 sin…mk ‡ mn †g=2 ÿ ˆ g=2 mk ÿ mn mk ‡ mn (30) BII2x ˆ

1 X …a2n ‡ a3n † sin mn z

(31)

nˆ1

with 1 g=2

a2n ˆ

a3n

Z

1 h2 =2 X

ÿh2 =2 kˆ1

‰K1k eq1k x ‡ K2k eÿq1k x Šm1k sin m1k z

(32)  sin mn z dz Z h2 =2 X 1 1 ˆ K3k q2k cos m2k x sinh q2k z sin mn z dz g=2 ÿh2 =2 kˆ1 (33)

For the vertical components: BIIz ˆ

1 X

1 g=2

Z

1 h2 =2 X

ÿh2 =2 kˆ0

…K1k eq1k x ÿ K2k eÿq1k x Šq1k cos mk z

(38)  cos mn z dz Z h2 =2 X 1 1 ˆ K3k m2k sin m2k x cosh q2k cos mn z dz g=2 ÿh2 =2 kˆ0 (39)

where a1n

c1n cos mn z

Looking at the formulae for a1(k, n) and c1(k, n), Eqs. (30) and (38), these formulae become equal to 1 for values k ˆ n, while they become equal to 0 for all values k 6ˆ n. That is a1 …k; n† ˆ a1 …k† and c1 …k; n† ˆ c1 …k†. Integrating constant K3k can be estimated as follows: the horizontal induction component equation (25), resulting from the vector potential assumed in the plate's region, must have the same value in this region after converting it into a Fourier series (Eq. (31)) over the interval ÿ 12 g  z  12 g (regions II ‡ III). Hence, both equations must have the same value at the horizontal boundary plate's surfaces, too. Subsequently they are expressed as boundary functions and after converting again in Fourier series over the interval z ˆ  12 h, 0  x  a and equating them, integrating constant K3k can be estimated. Thus, for the horizontal induction component of the plate's region (Eq. (25)), at these boundary surfaces, 1 X BIIx ˆ ÿ K3k q2k sinh…12 q2k h† cos m2k x

(40)

kˆ1

Converting it into Fourier series as follows: BIIx ˆ

1 X

a4l cos m2l x

(41)

lˆ1

where (34)

nˆ0

a4l

2 ˆ a

Z aX 1 0 kˆ1

ÿ K3kl q2k sinh …12 q2k h† cos m2k x cos m2l x dx (42)

where c1n ˆ

(37)

nˆ0

nˆ1

1 ˆ g=2

1 X …c2n ‡ c3n † cos mn z

249

1 g=2

Z

1 g=2 X

ÿg=2 kˆ0

‡ Q2k …1 ‡ eÿmk …a2 ‡s† † emk x Šmk cos mk z cos mn z c1 …k; n† ˆ

1 g=2

Z

Setting

‰ÿQ1k eÿmk …b‡x†

g=2

ÿg=2

(35)

a4 …l; k† ˆ

2 a

Z 0

a

cos m2k x cos m2l x dx

(43)

a4 …l; k† ˆ 1 for l ˆ k, i.e. a4 …l; k† ˆ a4 …k† ˆ 1, for all other values becomes equal to 0. Hence, BIIx …x; z ˆ 12 h† can be written as follows:

cos mk z cos mn z dz

  1 sin…mk ÿ mn †g=2 sin…mk ‡ mn †g=2 ‡ ˆ g=2 mk ÿ mn mk ‡ mn (36)

BIIx …x; z ˆ 12 h† ˆ ÿ

1 X kˆ1

K3k q2k sinh …12 q2k h† cos m2k x

(44)

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P. Raitsios / Journal of Materials Processing Technology 108 (2001) 246±252

4. Calculation of the secondary current

Correspondingly for Eq. (37) at this boundary surface BIIx ˆ

1 X

‰…K1k eq1k x ‡ K2k eÿq1k x †m1k a2 …k†

kˆ1

ÿ K3k q2k a3 …k† cos m2k xŠ sin …12 mk h†

(45)

Converting this equation also in a Fourier series, then for lˆk BIIx ˆ

1 X

For validation of the results of this method the secondary current is calculated. Assuming, for simpli®cation, that the secondary winding of the transformer consists of only one layer, then its width can be set a2, instead of 2a2 ‡ s. This is given by the well-known expression, Z I2 ˆ

‰…K1k a5 …k† ‡ K2k a6 …6††m1k a2 …k†

xˆ0

kˆ1

ÿ K3k q2k a3 …k†Š sin…12 mk h† cos m2k x where 2 a5 …k† ˆ a 2 a6 …k† ˆ a

Z

a

0

Z

0

a

e

q1k x

e

ÿq1k x

cos m2k x dx cos m2k x dx

(46)

(47) (48)

Setting expression (44) equal to (46) gives the following expression results for K3k: ÿ K3k q2k …sinh…12 q2k h† ÿ a3 …k† sin…12 mk h†† ˆ …K1k a5 …k† ‡

K2k a6 …k††m1k a2 …k† sin…12 mk h†

(49)

The same expression (49) for K3k results by applying the ampere-turns law around the boundary line of the plate, on one hand from the assumed induction components (25) and (26), for the plate's region, on the other hand from the induction components resulting from the Fourier series (31) and (34), equating the two expressions. Integrating constant M2k (7) can be estimated by applying boundary conditions at the vertical iron surface x ˆ 2a2 ‡ s ‡ c. Hence, follows a system of ®ve equations for determining the ®ve integrating constants, Q2k, K1k, K2k, K3k, and M1k, from which the induction components can be estimated as a function of the known integrating constant Q1k. For d ˆ 0, the integrating constant Q1k becomes [5] Q1k ˆ

ÿmk a1

m0 I1 N1 1 ÿ e ga1 h1 =2 m3k

sin…12 mk h1 †

(50)

For d 6ˆ 0, i.e. for a primary winding divided in two coils, Q1k becomes: Q1k

m 0 I1 N 1 1 ÿ eÿmk a1 ˆ …sin …12 mk h1 † ÿ sin …mk d†† ga1 …h1 =2 ÿ d† m3k (51)

From the ®nal formulae (44)±(47), the induction components in the ®rst layer of the secondary winding and in the winding space between primary and secondary windings can now be calculated. Following the same method, the ®eld distribution in the second layer can be determined. Applying boundary conditions at the primary winding, the ®eld distribution in this winding is derived.

xˆa2 Z zˆh2 =2 zˆÿh2 =2

J2 dx dz

(52)

where J2 ˆ ÿj

p2 AII1 m

(53)

According to (4), J2 is the current density and AII2 the vector potential from Eq. (20). Then the following expression results for the secondary current:  1  p2 X eÿq1k a2 ÿ 1 eq1k a2 ÿ 1 ÿ K2k K1k I2 ˆ ÿj m kˆ0 q1k q1k 2 sin …12 m1k h2 † sinh …12 q2k h2 † sin m2k a2 ÿ 2K3k  m1k q2k m2k

 (54)

Examining this expression for k ˆ 0, I2 ˆ ÿI1 N1

(55)

independently from the dimension b between the windings. For all other values k 6ˆ 0, I2 becomes equal to 0. This means that in any transformer the primary and the secondary ampere-turns are equal in short-circuited condition and in phase opposition. 5. Calculation of the leakage inductance The leakage inductance according to Rogowski's treatment is given, as it is well known, by the following expression: Ls ˆ

mr m0 2 0 N DU g 1

(56)

where g is the distance between the yokes, N1 the turns of the primary winding, D0 the equivalent windings' width, U the middle peripheral of the windings, D0 ˆ …13 a1 ‡ b ‡ 13 a2 †KR

(57)

where a1 is the width of the primary winding, a2 the width of the secondary winding, b the distance between the two windings, and a (58) KR ˆ 1 ÿ …1 ÿ eÿpg=a † pg is the Rogowski factor.

P. Raitsios / Journal of Materials Processing Technology 108 (2001) 246±252

251

According to this work, for the leakage inductance results [3]: Ls ˆ

mr m0 2 N RK U g 1

(59)

where for rough approximation, following expression for RK results: RK ˆ

1 ÿ…1=q†…1 ÿ e4qa1 † ÿ 4a1 e2qa1 ‡b 2 …1 ÿ e2qa1 †2 ‡

1 ÿ…1=q†…1 ÿ e4qa2 † ÿ 4a2 e2qa2 2 …1 ÿ e2qa2 †2

Factor RK is given in cm. For this factor following expressions are valid: q kp ; q ˆ jp2 ‡ m2k ; p2 ˆ mos mk ˆ g=2

(60)

Fig. 2. Magnitude of the vertical and horizontal induction component versus distance z for d ˆ 0 m.

(61)

where a1 is the width of the primary winding, a2 the width of the secondary winding, g the distance between the yokes, U the middle peripheral of the windings, m0 the magnetic permeability of the air, k the integer, s the conductivity of the windings material, o the cyclic frequency. Factor RK (60) is different from the equivalent width D0 of the leakage ®eld given by Kapp±Rogowski (57). If now the conductivity s tends to in®nity, i.e. the superconductive condition, then for RK results: RK ˆ b

(62)

i.e. a magnetic ®eld remains only in the region between the two windings, while in the regions inside the windings disappears. Hence, in this work the Meissner±Ochsenfeld phenomenon is predicted. 6. Example By an arrangement of a two windings transformer, the primary consisting of N1 ˆ 10 turns, the secondary of two plates connected in series, the induction components have been calculated. The following data have been considered: g ˆ 0:22 m; h1 ˆ 0:2 m; a1 ˆ 0:03 m; h2 ˆ 0:2 m or 0:1 m; a2 ˆ 0:007 m; s ˆ 0:001 m; b ˆ 0:01 m; I1 ˆ 1 A, s ˆ 3:6  107 Oÿ1 mÿ1 ; o ˆ 314 sÿ1 ; m0 ˆ 4p10ÿ7 V s=A m; c ˆ 1; d ˆ 1. In Fig. 2, the magnitudes of vertical and horizontal induction components are shown versus distance z for three different heights of the secondary winding, if the primary winding consisting of one coil (d ˆ 0 m) [6]. In Fig. 3, the magnitudes of vertical and horizontal induction components are shown versus distance z, the primary winding consisting of two coils (d ˆ 0:008 m). In Fig. 4, the magnitude of vertical induction component all over the transformer window is shown versus distance x, the primary winding consisting of one coil (d ˆ 0 m).

Fig. 3. Magnitude of the vertical and horizontal induction component versus distance z for d ˆ 0:008 m.

Fig. 4. Magnitude of the vertical induction component versus distance x.

252

P. Raitsios / Journal of Materials Processing Technology 108 (2001) 246±252

ones obtained by Kapp's and Rogowski's treatments. Furthermore, taking into account the conductivity of the windings material, different values follow also for the leakage reactance with different conductivities (copper, aluminium). This occurs under load, in short-circuit condition, and ®nally, in the superconducting condition. Especially in the superconducting condition, the ®eld inside the windings disappears, predicting the Meissner± Ochsenfeld phenomenon. The leakage inductance becomes smaller, than in conventional condition by similar dimensions of the windings. Fig. 5. Magnitude of the horizontal induction component versus distance x.

In Fig. 5, the magnitude of horizontal induction component all over the transformer window is shown versus distance x, the primary winding consisting of one coil (d ˆ 0 m). 7. Conclusions This method can be applied to transformers with a primary winding wound from a wire of negligible crosssection and with a secondary of rectangular conductors, for the determination of the leakage ®eld, from which the shortcircuit voltage and the short-circuit forces are derived. If the real height of the windings is considered, and the windings are not assumed to be in proximity to the core yokes, then more accurate values follow for these magnitudes than the

References [1] P. Raitsios, Distribution of current and magnetic ®eld density of many in series connected plates and many parallel conductors, Elect. Eng. 80 (1997) 17±20 (in German). [2] P. Raitsios, A. Safacas, Distribution of current and magnetic ®eld density in ¯at conductive plates in an array composed from plates and many parallel conductors, Elect. Eng. 75 (1992) 411±417 (in German). [3] P. Raitsios, Distribution of magnetic ®eld and current density and leakage inductance estimation in a ¯at conductive plate, in: Proceedings of the English Universities Power Engineering Conference, UPEC'96, Vol. 2, 1996, pp. 473±476. [4] B. Hague, The Principles of Electromagnetism Applied to Electrical Machines, Dover, New York, 1962. [5] J.A. Tegopoulos, M. Papadopoulos, Eddy current distribution and losses in solid plates due to a parallel current ®lament, Part I and II, IEEE Trans. Pas. 74 (1974). [6] Working Group 12-04 of Study Committee No. 12, Calculation of short circuit forces in transformers, ELECTRA-GIGRE, No. 67, December 1979.