Electrical Engineering 83 (2001) 41±46 Ó Springer-Verlag 2001

Calculation of self and mutual inductances and 3-D magnetic fields of chokes with air gaps in core B. Tomczuk, K. Babczyk

Contents This work deals with the analysis of 3-D magnetic ®eld in electric choke (reactor) with two windings. The analysis is related to the solutions of Laplace's and Poisson's equation by the Finite Element Method. The package OPERA-3d has been used. The self and mutual inductances of windings, which are connected in series aiding as well as in series opposing, have been determined. The calculation results of magnetic ¯ux density components and the inductances are compared with the measured values.

1 Introduction Electric reactors, which are called chokes, are magnetic energy ``accumulators''. Their sizes are mainly de®ned by the term LI 2 [7]. The ferromagnetic parts of them obviously depend on current frequency. Thus, it is necessary to distinguish between the alternating current (AC) and direct current (DC) chokes. The ®rst ones carry (in windings) pure alternating current while the second ones. additionally carry the premagnetizing current component. The inductance of the chokes may differ considerably in cases of the identical ones but applied in direct current or AC circuits. It must be taken into account especially when direct DC chokes are adjusted to the AC mains. Since the chokes produce leakage ®elds [4, 7], the stray losses must occur. The negative effect of the additional losses is observed. The magnetic ®eld in the air-gaps may also produce additional losses. The ¯ux density components, parallel to the winding axes, produce the stray losses in the windings and core. However, transversal stray ®elds may also occur. To reduce the negative effects of the ®elds, it is advisable to provide the individual air-gaps in the core as small as possible [3]. 2 Basic equations for the magnetic field The total ®eld H can be expressed as the sum of the ®eld HS obtained from the sources and the ®eld HM arisen from the induced magnetisation [2]. Since the ®rst Maxwell equation (for time-invariant situation) Received: 14 August 2000

B. Tomczuk (&), K. Babczyk Dept. of Electrical Eng. & Automatic Control, Technical University of Opole, Luboszycka 7, 45-036 Opole, Poland

rHˆJ

…1†

must be hold, it follows that HM is irrotational. The line integral of the vector HM around the closed contour is identically zero, therefore the vector ®eld can be written as the gradient of some scalar u which is called the reduced scalar potential

HM ˆ

ru :

…2†

Regarding the above, the divergence of B in the air can be expressed as:

r
…B† ˆ l0 r
fl…HS

ru †g ;

…3†

where l0 is the vacuum magnetic permeability. Since the divergence must be equal to zero, the nonlinear partial differential equation (PDE) of the Poisson type governs the ®eld

r
…lru † ˆ r
…lHS † :

…4†

The reduced potential formulation is not acceptable for all magnetic ®eld problems because of errors in the total ®eld calculation. The space variations of HS and HM are quite different due to various methods of evaluating them [2]. Moreover, the magnetic ®eld intensities strongly cancel themselves in some volumes of the analysed region, especially in magnetic materials. The above dif®culty can be avoided when the eddy currents (in the magnetic materials) are omitted. If the regions carrying currents are excluded, we can introduce for the rest regions the scalar potential w called the total potential

Hˆ

rw :

…5†

Taking into account the r
B ˆ 0, a non-linear PDE of the Laplace's type is given by:

r
…lrw† ˆ 0 :

…6†

The combination of the two potentials consists of using the reduced potential u inside the regions where the currents ¯ow and the total potential w everywhere else. At the interface between the regions, the continuity conditions are obviously included.

3 The calculation of inductances The energy-storage method for ®nding the inductances of the reactor windings is used in this work. The inductance, which is usually de®ned as the ¯ux linkage of a coil per ampere of its currents,

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Electrical Engineering 83 (2001)

Lij ˆ

Ni /i Ij

…7†

can also be calculated from the magnetic energy of the analysed system. For a magnetic system the magnetic energy can be calculated from integral over all the analysed area

Z Z

Wˆ

 H dB dV :

…8†

V

42

When the current RMS values of I in the reactor winding (which are ``introduced'' in the OPERA-3d package), and the magnetic energy W in the analysed region are known, we can obtain the total inductance of the reactor winding

Lt ˆ 2W=I 2

…9†

When currents ¯ow through the reactor windings they establish a resultant mutual ¯ux /m that is con®ned to the magnetic core. However, a small amount of ¯ux, known as the leakage ¯ux /L , links only one winding and does not link the other ones. The leakage path is primarily in air [4, 5]. The effects of the ¯ux can be accounted for by the leakage inductance. From the value of the magnetic energy we can obtain the total inductance of the reactor windings. To evaluate the leakage inductance we must calculate the Fig. 1. Outline of the reactor with main dimensions (in mm) self- and mutual inductances. The object of consideration is presented in Fig. 1. The numbers given along the co-ordinate axes denote distances from the reactor to the boundaries of the analysed region. The 6-mm long air gaps, in each limb of the reactor core are modelled. The physical model with the dimensions given in Fig. 1, and 292-turn winding (wound on each limb) has been examined. Different connections of the windings for the singlephase reactor have been considered. They are series opposing connection (anti-parallel coupling) ± case 1, and series aiding (push±pull or tandem) connection ± case 2. In Fig. 1, two directions of the winding currents are depicted. In case 1 the direction of the currents I1 create an opposite system of the two windings. The direction of the currents I2 deals with the push±pull connection of the windings in case 2. The former case deals with the leakage ¯ux /r and inductance Lr [4]. The ¯ux /r passes through air. When the coils are connected in series (case 2) almost all ¯ux /s , produced with one coil, is linked with the second one and mostly depends on the length of air gaps in the magnetic core. Thus, the value of the inductance Ls is much greater than the Lr . Taking into account the same values of the currents in the two windings, the inductances for cases 1 and 2 can be expressed respectively

Lr ˆ L1 ‡ L2 2M Ls ˆ L1 ‡ L2 ‡ 2M

…10†

Knowing the Lr and Ls values, for the identical windings, Fig. 2a, b. Main component of the ¯ux density in the air gaps. a Opposite system of coils (case 1), b tandem connections of coils which are located on each reactor column, we can calculate (case 2) the leakage inductances LrR (for case 1) and LsR (for case 2)

LrR ˆ L ‡ M LsR ˆ L

M

…11†

For suf®ciently small individual air-gaps the inductance of a choke winding can be estimated via the approximate formula [3]

B. Tomczuk, K. Babczyk: Calculation of self and mutual inductances and 3-D magnetic ®elds of chokes with air gaps in core

  1=2 
1=2 L ˆ l0
N
SFe =Rd 1 ‡ 2 d=SFe

case 1) passes mostly through the air outside the core, the values in Fig. 2a) are ten times lower than those in Fig. 2b), (case 2). They do not exceed 11 mT, which is not where d is individual air gap length; Rd, the total incor- visible in the ®gure because the graphs (for comparison) porated air-gap; SFe , the active cross section of core; N, the are made intentionally in the same scale. number of wires in the winding. We calculated the B values in the core as well. Selected distributions of the By and Bz components in the core 4 keeper (z ˆ 51:1 cm) for opposite- (Fig. 3a, b) and tandem The sample results of calculations and measurements (Fig. 3c, d) connections of the windings are presented. The The opposite connection of the windings (case 1) deals By values for case 1 are very small. They do not exceed with an open boundary problem [5, 6]. There are tech0.005 T in almost all presented regions (Fig. 3a). niques that accurately model the in®nite domain [6, 7]. The Bz values for case 1 (Fig. 3) are between 0.004 and One of them is the combination of the FEM and integral 0.014 T, which is not visible because of the same scale as method. As the coupled techniques in three dimensions for case 2. For the tandem connection (case 2) of the are relatively expensive, the approximate method of windings, the Bz modulus does not exceed 0.085 T extending the ®nite element mesh has been used in the (Fig. 3d) even at the points near the core edges. The By work. By trial and error in the FEM analysis we ®xed a direction (in the core keeper) is collinear with the direcsuf®ciently distance from the core surface to the tion of the main ¯ux. Thus, the By values attain 0.15 T boundaries of the calculated region. The dimensions of (Fig. 3c). the region (boundary box) for case 1 were greater The B components have been veri®ed experimentally. than those in Fig. 1. For case 2 the dimensions can be They were measured at many points in the parts of air lower because the analysed system is without large region. The Bz and By values are extreme at the air-gap leakage [8]. region, especially near the core and keeper edges. Thus, The values of the magnetic ¯ux density have been cal- the Bz distributions (for cases 1 and 2) are presented in culated in- and round the core. In Fig. 2 the distribution of Fig. 4. the Bz component (in the air gaps of the core) for the The quantities, which we are interested in, are the current excitation of 1.5 A is presented. The distribution stored magnetic energy and winding inductances. of the Bz values (Fig. 2) refers to the points with the co- The total- and self-inductances have been veri®ed exordinate z ˆ 47:8 cm (see Fig. 1). As the main ¯ux (for perimentally. The inductances have been measured for 2

…12†

Fig. 3a±d. Flux density components in the middle of the core keeper (z ˆ 52:1 cm). a, b Opposite system of the windings

(case 1), c, d tandem connections of the windings (case 2)

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Electrical Engineering 83 (2001)

Table 1. Comparison of calculated and measured inductances Number Inductances in [mH] of air-gaps Connections Total For Measured Self Mutual of windings inductance (12) (L) (M) 1

Push±pull (Lr) 63.95 In series (Ls) 446

± 72 634 430

127 96

5

Push±pull (Lr) 59.63 In series (Ls) 323

± ± 536 ±

95.5 65.9

44 lower if one gap is divided into several smaller ones (Table 1). The over-current in the reactor windings is likely to effect the local saturation of the core. The effect for oneand ®ve air gaps in the two considered cases is presented in Fig. 5. The excitation current is 22 times greater than the rated one (which refers to the previous ®gures). For such a current (in case 1) the permeability of the core slightly decrease in the regions of yokes where the ¯ux comes in. The noticeable decreasing of the lr is observed in the neighbourhood of the air gaps, especially in the corners of the ferromagnetic parts (Fig. 5a, b). Figure 5a, b, d, e deal with air-gap of 6 mm long. For case 2 (tandem connection of windings) the overexcitation current (438  22 ampere-turns) saturates nearly all core region. Thus, the relative permeability of the two columns is of the order of 100 and increases in the core corners. The decreasing of the lr is observed in the neighbourhood of the air gaps (Fig. 5d, e). In the vicinity of the gaps the permeability value is almost equal to the vacuum permeability. This phenomenon practically causes the increase in the air gap length. Figure 5c and f concern the rated excitation ampereturns H ˆ 438 A but for a relatively small air-gap (d ˆ 2:5 mm). We assumed the gap length intentionally. The small length of the gap ensures that the work point of the reactor is on the B/H line bent. Fig. 4a, b. Comparison of the Bz values in the air gaps. a Opposite 5 connections of the windings (case 1), b tandem connections Conclusions of the windings (case 2) The application of the ®nite element method (FEM) to the

two connections of the reactor windings. The magnetic core with one gap in each column has been examined. In Table 1 the values of inductances have been given for the two cases of the windings' connections. By the magnetic energy determination we have calculated the total- and self L- and mutual M inductances as well. In fact, we have found the total inductances from computer simulations. We have considered one and ®ve gaps in each column (leg) of the core. The total length of the ®ve air gaps in each leg was equal to the length of the single gap (for the variant with one gap in each leg). For the push± pull connection of the windings (in the reactor with small air gap), the division of the gap into several smaller ones almost does not change the reactor inductance Lr . For the connection in series (of the windings) the inductance Ls is

calculations of the inductances of the reactor windings has been demonstrated. The practical problems for connections of windings have been discussed. The main obstacles to the discretization of the analysed region for the computational model exist in restriction of hardware (PC) and software (OPERA-3d) [2]. It is obvious that the inductance obtained in case 2 (tandem connection of the windings) as observed is much larger than for case 1. Still another conclusion from these calculations is that the geometry investigated in case 1 has to be modelled in its entirety. For the opposite connection of the windings the natural boundary of the reactor is essentially at in®nity. It is also necessary to determine the computed ®eld at a suf®ciently large distance. The distance was ®xed by trial and error in numerical experiments.

B. Tomczuk, K. Babczyk: Calculation of self and mutual inductances and 3-D magnetic ®elds of chokes with air gaps in core

45

Fig. 5a±f. Distribution of the relative magnetic permeability lr in the reactor core. a±c Opposite connection of the windings

(case 1), d±f push±pull connections of the windings (case 2)

The calculated values of the total inductances are lower than the measured ones for the opposite connection of the windings (case 1) and greater for the tandem connection of the windings (case 2). It results from measurement and calculation errors. The former concern inaccuracies in making the physical model and location of the measuring probe.

For the linear core (lr ˆ 104 and more), the inductance value is nearly the same as for non-linear provided that the excitation current does not exceed ten times the nominal current. The use of ®nite elements package OPERA-3d has been successful especially in determining of the ®eld for push± pull connection of the windings. However, for very small

Electrical Engineering 83 (2001)

air gaps the calculations require mesh re®nement, and much calculation effort.

References

46

1. Lavers JD (1993) Electromagnetic ®eld computation in power engineering. IEEE Transactions on Magnetics, Vol. 29, No. 6: 2347±2352 2. OPERA-3d (1999) Reference Manual. Vector Fields Limited, Oxford, England 3. Standard Programme (1994) BLUM GmbH, Postfach, Germany 4. Tomczuk B (1994) Analysis of 3-D magnetic ®elds in high leakage reactance transformers. IEEE Transactions on Magnetics, Vol. 30, No. 5: 2734±2738 5. Tomczuk B (1994) Three-dimensional Modeling of Open Magnetic Fields in Transformers and Reactors by Means of

Integral Methods. Habilitated thesis published by The Technical University of Opole, Studies and Monographs, No. 77, Opole (in Polish) 6. Tomczuk B (1992) Three-dimensional leakage reactance calculation and magnetic ®eld analysis for unbounded problems. IEEE Transactions on Magnetics, Vol. 28, No. 4: 1935±1940 7. Zakrzewski K, Tomczuk B (1998) On the use of the FEM and BIE methods in the 3-D magnetic ®eld computation of the reactors with air gaps, COMPEL ± The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, London, Vol. 17, No. 1±3: 116±120 8. Zakrzewski K, Tomczuk B (1996) Magnetic Field Analysis and Leakage Inductance Calculation in Current Transformers by Means of 3-D Integral Methods. IEEE Transactions on Magnetics, Vol. 32, No. 3: 1637±1640