1

Determination of the air-core reactance of transformers by analytical formulae for different topological configurations and its comparison with

an electromagnetic 3D approach.

Michel Rioual, IEEE Senior M., Electricité De France – R & D Division

1, avenue du Général de Gaulle 92141 Clamart (France)

Yves Guillot Electricité De France – R & D Division

1, avenue du Général de Gaulle 92141 Clamart (France)

Cyrille Crepy Supélec

3, rue Joliot Curie 91192 Gif-Sur-Yvette

Abstract – This document describes the determination of the saturated inductance of a transformer, which is the slope of the saturation curve Φ(I) under highly saturated conditions. This parameter, which has a strong impact on the overvoltages when energizing a transformer, has been determined from analytic formulae for different transformer technologies. A comparison with the values derived from an electromagnetic 3D calculation is also given in this paper. Keywords – Energization, power transformer, inrush currents, saturated inductance, air-core reactance, air-core inductance, core-type, shell-type.

Nomenclature Lsat : saturated inductance of a transformer

under highly saturated conditions (H) Laircore : air-core inductance of a transformer (H) Lleakage : leakage inductance of a transformer

winding (H) Zr : reference impedance (Ω) Sr : rated power (VA) Ur : rated voltage (V) Φ(I) : saturation curve of a transformer (Wb) N : number of turns of the winding Lt,i : self-inductance of the turn i (H) Mt,ij : mutual inductance between the two turns

i and j (H) For a core-type transformer with layer-type winding: Ll,i : self-inductance of the layer i (H) Ml,ij : mutual inductance between the two

layers i and j (H) C : diameter of a limb of the magnetic core

(m) E : width of a window of the magnetic core

(m) F : length of a window of the magnetic core

(m) D : average diameter of the HV-MV

windings (m) H : average longitudinal height of the HV-

MV windings (m)

T : radial thickness of the HV-MV windings

(m) Ni : number of turns of the layer i Nl : number of layers of the winding ri : radius of the layer i (m) h : average longitudinal height of a layer (m) For a shell-type transformer with pancake-type winding: Ms,km : mutual inductance between the two sides

k and m (H) Ms,km(L,d): mutual inductance between the two

equal sides k and m, which length is L and separated by the distance d (H)

a : width of a turn (m) b : length of a turn (m) α, β : dimensions of the section of a conductor

(m) dij : orthogonal distance between the

pancakes i and j (m) hi : radial thickness of the pancake i (m) ti : longitudinal thickness of the pancake i

(m) Ni : number of turns of the pancake i 1. INTRODUCTION The energization of power transformers may create the saturation of the magnetic core and lead to high overvoltages and inrush currents. The magnitude of those stresses depends on the following different parameters: - closing times of the circuit-breaker poles, - residual fluxes in the core, - transformer parameters as the winding connections, the hysteretic curve of the magnetic core and the value of its air-core inductance. This last parameter, which characterizes the slope of the saturation curve Φ(I) under highly saturated conditions, has been determined in this paper for different transformers (rated power, technology). Firstly, this paper describes the guiding principles of the analytical formulae giving the air-core reactances values, based on the determination of the sum of self and mutual inductances. Secondly, analytical formulae have been determined, for two usual technologies: a core-type

978-1-4244-4241-6/09/$25.00 ©2009 IEEE

2

transformer with layer-type windings and a shell-type transformer with pancake-type windings. Thirdly, the saturated inductance has been determined and applied to a 600 MVA, then a 96 MVA transformer, and compared to simulations made with a 3D electromagnetic field program. 2. GENERAL PRINCIPLES OF THE ANALYTICAL CALCULATION OF THE AIR-CORE REACTANCE The calculation of the air-core reactance of a transformer is mainly based on the calculation of sums of self and mutual inductances, derived from assumptions enabling those calculations. From a theoretical point of view, it can be said that the self-inductance of a winding, constituted by two elements A and B in serie may be determined by making the addition of the self inductances of A and B, and also the double of the mutual inductance between A and B; extending those assumptions to a winding made of N turns, the air-core inductance may be derived as follows:

∑∑∑≠===

+=N

jj

ijt

N

i

N

i

itaircore MLL

11

,

11

, (1)

This is also described by the fig. 1 below:

Fig. 1: Interaction between the two turns i and j, for a core-type transformer. The exact theoretical calculation of the mutual between two circular turns has been described by Maxwell [1], [2], and is based on elliptical integrals derived from the Neumann’s formula. Those approaches are at the origin of those generally used by manufacturers for the calculation of air-core inductances. 3. DETERMINATION OF THE AIR-CORE INDUCTANCE FROM ANALYTICAL FORMULAE FOR A CORE-TYPE TRANSFORMER 3.1 Approach by the Kalantarov’s and

Tseitlin’s formula 3.1.1 Principle and formula

This approach describes the determination of the air-core inductance based on the simplified geometry of the winding, as shown in fig. 2 below, in order to be able to have a quick estimation of the air-core inductance; it is applied for transformers having circular turns, and is used by several manufacturers.

Fig. 2: Geometry considered for the Kalantarov’s and Tseitlin’s formula This approach is based on the fact that the winding is formed by a uniform disposition of the turns in the longitudinal and radial axes; it is especially applicable for thin and long windings, where the dimensions may satisfy:

65.0≥D

H et 8.0≥D

T (2)

The air-core inductance is then given by the Kalantarov’s and Tseitlin’s formula:

72 10 −⋅⋅⋅= KDNL aircore (3) In that case, the parameter K can be read on abacuses, and takes into account the ratios H/D and T/D. In must be noted that, if we consider a real transformer, as for instance the 600 MVA autotransformer (from Manufacturer 1) described in 3.1.2, the windings are divided into layers which characteristics (thickness, height, space between two layers) may vary from one layer to another one. The insulation between conductors has not been taken into account in this paper. 3.1.2 Application to a 600 MVA core-type

transformer The transformer described in the fig. 3 is a 5 limbs core-type transformer; the coupling between phases is wye for the HVand MV windings and delta for the LV winding. The winding consists of a concentric arrangement of layers made of joined circular turns rolled around an insulating cylinder. The iron core and the disposal of the HV (400 kV), MV (225 kV) and LV (21 kV) windings are given by fig. 3 below:

3

Fig. 3: Description of the 600 MVA autotransformer: 5 limbs magnetic core (a), zoom on the disposal of the layers in a window of the magnetic core (b). For the determination of the air-core inductance, the HV winding is usually the only one to be taken into account, as in the case of a no-load energization made from the HV side, others windings being left open. However it must be noticed that in the specific case of an autotransformer, the serial connection of the HV and MV windings implies that both have to be considered. The LV winding is neglected, as the induced current circulating because of the delta coupling has an insignificant impact on the flux distribution. The reference impedance Zr is required to express the Laircore parameter in p.u. (per unit), and given by:

r

rr S

UZ

2= (4)

r

aircoreaircore Z

LupL

⋅=ω

.).( (5)

The geometrical parameters have been determined from the plans given by the manufacturer. The assumption made is that the thickness T, as seen globally, is thus given by (Dext+Dint)/2 (see Fig. 3). In that case the Kalantarov’s and Tseitlin’s formula (3) takes the following parameters as data inputs and leads to the following value of Laircore :

1336=N , 526.1=D

H , 287.0=D

T

..23.107.1 upHLaircore ==⇒ In fact, the parameter which is important for transient programs is Lsat, given by:

leakageaircoresat LLL −= (6)

where the Lleakage inductance represents the magnetic losses in the considered winding. In this paper, only the determination of Laircore is adressed,

without any assumptions on the value of Llosses, this last value being derived either from the on-site short-circuit tests, or from a 3D electromagnetic field program. 3.2 Description of the multi-layer approach 3.2.1 Principle In paragraph 3.1, a simplified geometry has only been considered; in order to take into account the real geometry of a transformer with layer-type windings, a more precise analytical approach has to be defined, which is given in the following paragraphs. This approach consists of dividing the winding in appropriate elements, the layers in this case, for which the self and mutual inductances can be calculated by precise formulae, being validated experimentally for simple geometries. The self and mutual inductances are then summed-up in order to obtain the total air-core inductance value. 3.2.2 Choice of the formulation In that case, the self and mutual inductances are determined from analytical formulae from two reference books of GROVER, the first one having been written in 1911 [3], and the second one being a revised version written in 1973 [4]. The calculations described in those books are originated from the publication of the “National Bureau of Standards”, and give hundreds of formulae of self and mutual inductances established by different authors for very different simple geometries. Specific formulae have been chosen:

- The Nagaoka’s formulation, which is the most efficient in order to determine the self-inductance of a layer having circular turns.

- The Havelock’s formulation, which can be applied for the calculation of mutual inductances between layers, when the layers are not joined and have the same axis and close longitudinal heights.

- An alternative method for the calculation of mutual inductances is the Cohen’s formula, which is more complex than the Havelock’s one, and provides almost identical results, except in the case of joined layers, or layers of very different longitudinal heights, for which the accuracy is slightly better.

The self-inductance of the layer i in the case of the Nagaoka’s formula is given by:

( )h

rNKL ii

il

72

,104 −⋅⋅⋅⋅⋅

=π (7)

4

where the coefficient K is developed in Appendix 1; Appendix 1 gives also the calculation of the mutual inductance Ml,ij in the Havelock’s formula; at a final stage, the aircore reactance is given by:

∑∑∑≠===

+=lll N

jj

ijl

N

i

N

i

ilaircore MLL

11

,

11

, (8)

3.2.3 Application in the case of a 600 MVA autotransformer; comparison with the value given by manufacturers.

The Table 1 gives the air-core reactance value, derived from two approaches, with respectively a simplified representation of the winding or a detailed representation of the layers:

Envelope of the winding

Real geometry (multi-layers)

Kalantarov & Tseitlin

Laircore

= 1.07 H (1.23p.u.)

Nagoaka & Havelock

Laircore = 0.97 H

(1.114 p.u.)

Nagaoka & Cohen

Laircore = 0.967 H (1.110 p.u.)

Table 1: Comparison between both approaches, by a simplified representation of the winding or a detailed representation of the layers. The “Nagaoka+Cohen” and “Nagaoka+Havelock” formulae lead to almost identical values (variation lower than 0.5%), which are 10% lower than the value given by the Kalantarov’s and Tseitlin’s formula. A comparison with the results obtained with an electromagnetic 3D field program is also adressed in chapter 5, confirming the assumptions and calculations performed.

4. DETERMINATION OF THE AIR-CORE REACTANCE FROM ANALYTICAL FORMULAE FOR A SHELL-TYPE TRANSFORMER 4.1 Description of the approach The shell-type technology makes the calculation of air-core inductance from analytical formulae more complex, because of the rectangular form of the turns, as described on fig. 4 below.

Fig. 4: Magnetic circuit for a shell-type transformer

Fig. 5: View of a shell-type transformer with windings in pancakes The proposed analytical approach consists in subdivising the HV winding in well chosen parts so that the self and mutual inductances could be calculated with accurate formulae: - The self-inductances of a pancake is obtained by

calculating, for each turn, the self-inductance and the mutual inductances with the other turns, and then by summing all those values. The mutual inductance between two turns is calculated from the sum of the different sides of both rectangles.

- The mutual inductance between two pancakes is obtained by summing all the mutual inductances corresponding to each pair of turns belonging to two different pancakes.

- The global self-induction is the sum of self and mutual inductances of every pair of pancakes.

The insulation and the rounded edges of the pancakes are not taken into account in this approach. The self-inductance Lt,i of a single turn i is given by the following formula [4]:

( ) ( )

( ) ( ) )9(447.022

log

log2log410 9,

⎟⎠

⎞++++−+⋅−

⎜⎜⎝

⎛+⋅−⎟⎟

⎠

⎞⎜⎜⎝

⎛

+⋅⋅+⋅⋅= −

βα

βα

dba

dbb

daaba

baL it

with 22 bad += .

Fig. 6: Geometrical dimensions of a turn The mutual inductance between two orthogonal sides is equal to zero. The mutual inductance Ms,nq between two parallel sides separated by a distance d (see parameters on fig. 7) is deduced from mutual inductances, noted Ms,n’q’(L,d), between two parallel equal sides of appropriate length L and separated by

Pancake

Magnetic circuit

5

the distance d. The relationship is given by the formulae (10) and (11), [4].

Fig. 7: Parameters of the mutual inductance in the following cases: two parallel equal sides (a), two parallel unequal sides (b). The mutual inductance between two parallel equal sides (case (a) in the fig. 7) n and q of length L is given by the following formula:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛++−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛++⋅⋅= −

L

d

L

d

d

L

d

LLdLM nqs 2

2

2

29

, 11ln210),(

(10)

The mutual inductance between two parallel unequal sides (case (b) in the fig. 7) n and q can be deduced from the previous formula as follows:

),(),( '''','',, dpMdpmMM qnsqnsnqs −+= (11)

Considering geometrical symmetries, the mutual inductance between two turns can be calculated from the mutual inductances between the parallel sides of the two rectangles [4], as follows:

( )28,26,17,15,, 2 sssskmt MMMMM −+−= (12)

Fig. 8: Parameters of inductive coupling between: two turns (a), two pancakes (b). 4.2 Application to a 96 MVA auxiliary

transformer The previous formulae have been put together in an algorithm which data are the geometrical characteristics of each pancake. This analytical approach was applied to a 96 MVA auxiliary transformer (from Manufacturer 2) for a thermal power plant. It consists of a 400 kV/6.8 kV shell-type transformer, with pancakes winding made of 960 turns.

In that case, the calculated value of the air-core inductance is 0.22 p.u., 17% under the value of 0.26 p.u. given by the manufacturer (see Table 2). A comparison with an electromagnetic 3D software is given in chapter 5.

Analytical approach for shell-type transformers

Manufacturer value

Laircore = 1.157 H (0.22 p.u.)

Laircore = 1.4 H (0.26 p.u.)

Table 2: Comparison between the analytical approach developed and the value given by the manufacturer, for the 96 MVA transformer. The mutual inductances between pancakes represent around 85% of the global air-core inductance. 5. VALIDATION OF THE ANALYTICAL APPROACHES BY A FLUX3D CALCULATION (SINGLE-PHASE AND THREE-PHASE SIMULATIONS) 5.1 Description of the 3D approach The results of the analytical approaches have been compared to the simulations performed with the electromagnetic 3D software “FLUX3D” [5]. It enables, in particular, the calculation of the 3D magnetic field developed in a transformer and requires a 3D geometrical meshed model of the transformer. The FLUX3D approach was applied to the core-type transformer and the shell-type transformer previously mentionned. The relative permeability μr

is set to 1 in order to meet very high saturation requirements faced during the energization. The simulations were performed in steady state conditions, first in a single-phase mode, secondly in a three-phase mode, in order to evaluate the mutual inductances between phases. The transformers were supposed to be no-loaded, which is the case during their energization. Considering the results given by FLUX3D, the air-core inductance can thus be determined from three equivalent ways: the calculation of the energy, the determination of the impedance or the calculation of the magnetic fluxes. 5.2 Case of a single-phase simulation The FLUX3D single-phase approach was applied to the determination of the air-core reactance for the core-type 600 MVA autotransformer. Its high

6

voltage and middle voltage windings are connected in serie. The three-phase coupling is a wye one for MV-HV windings, a delta connection for the LV winding. In the simulation, a single MV-HV phase was energized, no matter which phase it was because of the symmetry, and other windings were all left open.

Fig. 9: 3D model of the core-type 600 MVA transformer

Fig. 10: Description of the FLUX3D circuit involving the 600 MVA autotransformer

From the results given by FLUX3D, the three methods (either energy, impedance, or flux) give 1.11 p.u. as value of the air-core inductance for the 600 MVA transformer (Manufacturer 1). 5.3 Influence of the mutual inductances:

results of three-phase simulations for two technologies

As the energization of a transformer is a three-phase operation, the inductive coupling between phases slightly modifies the slope of the saturation curve Φ(i), compared to the single-phase case. The coupling phenomena can be represented by mutual inductances between phases.

Fig. 11: Magnetic field lines for a saturated core-type 600 MVA transformer calculated by FLUX3D.

To take into account the mutual-induction between phases and to estimate its influence on Laircore, three-phase simulations have been performed on the core-type 600 MVA autotransformer and on the shell-type 96 MVA transformer, in steady state conditions, with no-load conditions at the secondary side and in the following conditions: - three-phase power conditions, - presence of the low voltage required, as its delta

coupling enables induced currents. In the case of the 600 MVA transformer, the method of the impedance has led to the results displayed on table 3. The slight increase of Lsat, 1 to 2% compared to the single-phase case, is caused by the mutual inductances between phases. The LV windings have almost no influence, with induced currents which are 50 times lower than the MV-HV currents.

Analytical approach FLUX3D

Single-phase

Three-phase

Laircore = 0.967 H (1.11 p.u.)

Laircore = 0.965 H (1.11 p.u.)

Phases 1 & 3

Laircore = 0.976 H (1.12 p.u.)

Phase 2

Laircore = 0.988 H

(1.135 p.u.)

Table 3: Comparison between the values of the air-core inductance calculated by FLUX3D and by the analytical approach, for the 600 MVA tranformer. The air-core inductance for the middle phase is the highest one, due to the geometrical proximity with both windings for phases 1 and 3, which leads to higher mutual induction coefficients. In the case of the shell-type 96 MVA transformer, the FLUX3D approach has led to a value of 1.219 H (0.23 p.u.) for phase 1. This value is 5% higher than the value of 1.157 H (0.22 p.u.) calculated by the analytical approach for shell-type transformers presented in chapter 4. The value provided by the manufacturer is equal to 0.26 p.u. It must be noticed that the determination of the air-core reactance by the FLUX3D approach requires

HV_1 MV_1 LV_1

HV_2 MV_2 LV_2

HV_3 MV_3 LV_3

7

several hours for the modeling of the transformer and simulations to be performed. With the analytical approaches developed, the calculation time essentially depends on the number of turns and the transformer technology. It remains less than: - 1 s for a 1340 turns (per HV-MV phase) layer-

type winding of a core-type autotransformer; - 5 mn for a 960 turns (per HV phase) pancake-

type winding of a shell-type transformer (simulations being performed with a ~2 GHz processor).

6. DISCUSSION The air-core inductance is not, in most cases, a key parameter for manufacturers, which are more focused on short-circuit issues from the specifications required by utilities. Yet they often provide a value for the air-core reactance, generally calculated by approximate formulae using a global geometry of the transformer (envelope of the winding), with a 10 to 20% accuracy. For utilities, the Lsat value is highly critical for certain studies, because a 10% surestimation of the Lsat can lead to underestimate stresses during the transformer energization by a factor of 30% [6]. The analytical approaches developed in this paper enable an improvement in the Lsat determination, with a better accurary than those given by more simple approaches. 7. CONCLUSION This document describes the determination of the saturated inductance of a transformer, which is the slope of the saturation curve Φ(I) under highly saturated conditions. This parameter, which has a strong impact on the overvoltages when energizing a transformer, has been determined from analytic formulae for different transformers technologies. More powerful algorithms have been proposed, adapted to the geometry of the windings (layers, pancakes, form of the turns). A comparison with the values derived from an electromagnetic 3D calculation is also given in this paper. Applied to a core-type 600 MVA transformer and a shell-type 96 MVA transformer, a good agreement between both approaches has been obtained. APPENDIX 1: CALCULATION OF THE AIR-CORE

INDUCTANCE FOR A CORE-TYPE TRANSFORMER

WITH A LAYER-TYPE WINDING The self-inductance of the cylindric layer i can be expressed by the Nagaoka’s formula, as follows:

( )h

rNKL ii

il

72

,104 −⋅⋅⋅⋅⋅

=π (13)

The coefficicent K can be calculated with an excellent accuracy from the following formula:

5432 26011644122'3

41 qqqqq

k

kK +++++−=

π

76

3

3760576 qq ++

(14)

with: 95

215

22

2⎟⎠

⎞⎜⎝

⎛+⎟⎠

⎞⎜⎝

⎛+= LLLq ;

'1

'1

k

kL

+−= ;

22

2

4

4

hr

rk

i

i

+= ; 22

2

4'

hr

hk

i += (15)

Most of the time, considering that the self-inductances of the different layers only weight around 10% in the global air-core inductance, the expression of K can be simplified by the following formula, with no difference on the final result:

⎟⎠

⎞⎜⎝

⎛ +=

118

100

2

h

rK

π

(16)

Fig. 12: Parameters of Nagaoka’s and Havelock’s formulae

For the calculation of the mutual inductance between layers, the Havelock’s formula is given by the following expression:

( )βπμ ⋅−⋅⋅⋅⋅= jjiiijl rhnnrM 220, (17)

with the coefficient β approximated by the following formula:

( )18661256

5

3132

11

16

1

4

1

32768

35

2048

5

128

1

16

1

2

1

7

6

6

4

4

2

2

5

4

4

2

23

2

2

8

8

6

6

4

4

2

2

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛++++

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛++−⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛++

−−−−−=

h

r

r

r

r

r

r

r

h

r

r

r

r

r

h

r

r

r

h

r

r

r

r

r

r

r

r

r

j

j

i

j

i

j

i

j

j

i

j

ij

j

i

j

j

i

j

i

j

i

j

iβ

Despite its complexity, the Cohen’s formula gives a slightly more accurate value for windings either being very compact or having layers of very different longitudinal heights. The Cohen’s formula consists in complex combinations of the elliptic integrals defined in Maxwell’s theory:

8

( )( )

dxxk

kF ∫ −=

θθ

0 22 sin1

1, (19)

( ) ( ) dxxkkE ∫ −=θ

θ0

22 sin1, (20)

(for further details about Cohen’s formula, see Grover’s book [3]).

REFERENCES [1] J. C. Maxwell “Treatise on Electricity and

Magnetism”, Editions Jacques Gabay, 1887, Paris.

[2] P. L. Kalantarov, L. A. Tseitlin, “Calculation of Inductance, Handbook”, 3rd ed., Leningrad, Energoatomisdat, 1986, 488 p. [in Russian].

[3] E. B. Rosa, F. W. Grover, “Formulations and tables for the calculation of mutual and self-induction [Revised]”, Washington Government Printing Office, 1911, from Bulletin of the Bureau of Standards Vol. 8 N°1.

[4] F. W. Grover, “Inductance Calculations: Working Formulas and Tables”, 1946 & 1973, Dover Phoenix Edition, 2004.

[5] J. Coulomb, Y. Du Terrail, G. Meunier, “Flux3D, a finite element package for magnetic computation”, IEEE Transactions on Magnetics, vol. 21, issue 6, pp. 2499-2502, Nov. 1985.

[6] M. Rioual, C. Sicre, “Energization of a no-load transformer for power restoration purposes: sensitivity to parameters” PES Summer Meeting, 2000, IEEE Volume 2, July 16-20, pages 892-895.

Michel Rioual was born in Toulon (France) on May 25th, 1959. He received the Engineering Diploma of the “Ecole Supérieure d'Electricité” (Gif sur Yvette, France) in 1983. He joined the EDF company (R&D Division) in 1984, working on electromagnetic transients in networks until 1991. In 1992, he joined the Wound Equipment Group as Project Manager on rotating machines. In 1997, he joined the Transformer Group, as Project Manager on the transformers for nuclear plants. He is a Senior of IEEE, belongs to CIGRE and to the SEE (Society of Electrical and Electronics Engineers in France). Yves Guillot was born in Paris, France, on April 1st, 1967. He received his Electrical Engineering diploma of the “Ecole Supérieure d'Electricité” (Supélec) in 1990. Then he joined the research center of EDF as a research engineer mainly involved in modeling power transformers : high

frequency modeling, diagnosis methods and electromagnetic field calculation. Cyrille Crépy was born in Paris on June 30th, 1986. He received the Engineering Diploma of the “Ecole Supérieure d'Electricité” (Gif sur Yvette, France) in 2008. He gathered his first work experience during interships at the nuclear power plant of St-Laurent-des-Eaux, and MBDA. He completed his studies by a 6 months internship at EDF R&D where he worked on stresses during transformers energization.