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Electric Power Systems Research 101 (2013) 63– 70

Contents lists available at SciVerse ScienceDirect

Electric Power Systems Research

jou rn al hom e page: www.elsev ier .com/ locate /epsr

ptimum design of core blocks and analyzing the fringing effect inhunt reactors with distributed gapped-core

bbas Lotfia, Ebrahim Rahimpourb,∗

Electric Power Engineering Department, Norwegian Uni. of Sci. and Tech. (NTNU), N-7491 Trondheim, NorwayABB AG, Power Products Division, Transformers, Lohfelderstrasse 19-21, D-53604 Bad Honnef, Germany

a r t i c l e i n f o

rticle history:eceived 29 August 2012eceived in revised form 4 March 2013ccepted 18 March 2013

eywords:

a b s t r a c t

Shunt reactors are designed in the form of cores with air gaps which are distributed along the legs. Itis crucial to determine the height of blocks based on its impact on the magnitude of the fringing flux,the number of blocks and other size-related parameters. This study has applied three-dimensional finiteelement method to a single-phase reactor which is calculated by well-known equations of magneticcircuit theory. This is done to analyze the effects of the fringing flux on the inductance.

ore blocksistributed gapped-core, Fringing effect,hunt reactor, Optimum design

In this work, air gaps are distributed along the legs in order to determine the effects of the numberand height of core blocks on the inductance. The analytical equations, which are used to calculate theoptimum heights of core blocks based on the Schwarz-Christoffel transformation, are developed.

These equations are derived from the size-related parameters of the reactor and their validity has beenfully confirmed by the outcomes of the finite element analysis. There is a good agreement between theheight calculated by empirical relations and that estimated using the proposed method in this paper.

. Introduction

Shunt reactors are usually installed to overcome power systemroblems caused by the capacitive reactive power. These include:ver voltages that occur during low load periods of long transmis-ion lines as a result of the inherent capacitance of the line; leadingower factors at generating plants resulting in lower transientnd steady-state stability limits; and open-circuit line chargingVA requirements in extra-high-voltage systems that exceed thevailable generation capabilities [1]. Nowadays, shunt reactors areesigned in the form of cores with air gaps. To reduce the fring-

ng effects of the flux, air gaps are distributed along the leg; as aesult, the legs of the reactor are divided into several laminatedagnetic steel disks called core blocks, which are separated by air-

ap wedges [2]. Fig. 1 shows this structure. The fringing effect isodeled as a permeance parallel to the main air gap, which leads

o a higher inductance in the winding than was originally calculatedrom the length of the main air gaps. Further, flux leakage aroundhe air gap results in an increase in the Eddy current losses in the

inding, core blocks and other iron components [3,4].

For designing the reactor, the volume of the air gap is deter-ined by taking the following into consideration: (a) the required

∗ Corresponding author. Tel.: +49 2224 14200; fax: +49 2224 14398.E-mail addresses: abbas.lotfi@ntnu.no (A. Lotfi),

brahim.rahimpour@de.abb.com (E. Rahimpour).

378-7796/$ – see front matter © 2013 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.epsr.2013.03.006

© 2013 Elsevier B.V. All rights reserved.

reactive power (Q) of the reactor, (b) the maximum magnetic fluxdensity (Bm in the core, (c) frequency (f), (d) the amount of increasein the inductance caused by the fringing effect, and (e) the energystored in the space between the core leg and the winding. Theincrease rate of the inductance caused by the fringing flux can becontrolled by the number of core blocks in each leg. Therefore, it isessential to pay attention to the correlation between the inductanceof the windings and the number of core blocks at every step of thedesign. In addition, it is crucial to determine the height of the blocksbased on (a) its impact on the magnitude of the fringing flux, (b)the number of blocks, and (c) other size-related parameters of thereactor. It is worth mentioning that the diameter of the core blockswill be equal to the diameter of the air gap, which in turn will bedetermined by the specified volume and financial considerations[5]. Regarding this subject, there are no remarkable works aboutthe above mentioned problem and most of the research is relatedto the interaction between shunt reactors and the power system indifferent transient conditions [6,7], the nonlinear dynamic model-ing [8,9], and the manufacturing of core blocks and the materials[10].

Firstly, this study will apply three-dimensional finite elementmethod (FEM) to a single-phase reactor in order to analyze theeffects of the fringing flux on the inductance. After that, the air gaps

will be distributed along the legs of the reactor to determine theeffects of the number and height of core blocks on the inductance.Then, the analytical equations used in calculating the inductanceand the optimum height of core blocks will be presented. These

64 A. Lotfi, E. Rahimpour / Electric Power Sy

ett

2

2

rott

Q

V

I

A

welfpvfetb

P

wbiti

L

fcubltc

(Hw) and the width (Bw) of the winding are 146.13 mm and 70 mm.Fig. 2 shows the calculated reactor. The main gap can be seen alongthe middle leg.

Table 1Specifications of the modeled reactor.

Parameters Symbol Value

Reactive power Q [kVAR] 100Voltage Ve [kV] 10Current Ie [A] 10Outer diameter of leg Doc [m] 0.1774Length of air gap lg [m] 0.0225

Fig. 1. Gapped core shunt reactor.

quations are derived from the size-related parameters of the reac-or and their validity has been fully confirmed by the outcomes ofhe FEM.

. The case study and FEM

.1. The design and structure of the modeled reactor

For the purposes of the present study, a single phase shell-typeeactor of 10 kV, 100 kVAR is designed using the equations basedn magnetic circuit theory. Neglecting the reluctance of iron andhe copper losses, the equations below determine the volume ofhe air gaps.

= Ve · Ie (1)

e =(

2�√2

)N · Bm · Ag · f (2)

e =(

1√2

)(Bm · lg�0 · N

)(3)

By manipulating the above equations:

g · lg = Q(�

�0· B2

m · f) (4)

here Q, Ve, Ie, lg, Ag„ f, Bm and �0 represent the reactive power, theffective nominal voltage, the effective nominal current, the totalength of the air gap, the area of the air gap, number of turns, therequency, maximum flux density in the air gap, and the free spaceermeability, respectively. The left-hand side of Eq. (4) is the gapolume and will be constant if the reactive power, flux density andrequency are kept constant. Also, it assumes that all the magneticnergy is stored in the main air gap. Consequently, by disregardinghe fringing flux, the inductance of the winding can be obtained aselow:

g = �0

(Ag

lg

)→ L = N2 · Pg (5)

here Pg, N and L represent the permeance of the air gap, the num-er of turns, and the inductance of the winding, respectively. The

nductance of the winding, moreover, can be obtained by means ofhe reactive power and the nominal voltage, both of which are thenputs of the design problem, as follows:

= V2e

ω · Q(6)

Since the proportion of Ag/lg affects the size of the reactor, dif-erent values for this ratio lead to different design in terms of theonsumed and losses of copper and iron [5]. Ultimately, these val-es are decided according to economical considerations. As it can

e seen, the stored energy in the spaces between the winding and

eg for inductance calculation is not considered. As a matter of fact,he method of this paper is to make a comparison of the inductanceorresponding to the main air gap with inductance corresponding

stems Research 101 (2013) 63– 70

to overall energy stored in all the regions. Then the best value for theheight of blocks is found by minimizing the ratio of stored energyin the main gap to all the regions in terms of corresponding induc-tances. Thus, for analytically calculated reactor in this section, it isnot necessary to consider the energy stored in other regions. How-ever, economically determining of the above mentioned designparameters is not significant for the purpose of this paper, and sothe Ag/lg ratio has been presumed to be 1.1. The validity of the equa-tions in various designs can be confirmed by changing the valuesonce the equations have been sufficiently developed. Therefore, thenumber of turns can be achieved by:

N =√√√√ L

�0

(Ag

lg

) (7)

The cross-sectional area of the winding can be reached throughthe current density equation as below:

Ku · Aw = N · Acu → Aw = N

ku

(IeJ

)(8)

where Aw, Ku, Acu and J represent the area of the winding, thefilling factor, the cross-sectional area of the wire, and the currentdensity, respectively.

Incorporating Eq. (7) into Eq. (8) provides the following:

Aw = IeKu · J

√√√√ L

�0

(Ag

lg

) (9)

After Aw has been determined, the height of the winding is calcu-lable by obtaining the Hw/Bw with a view to technical and financialconsiderations [5]. In this paper, the above mentioned ratio regard-less of these considerations is selected to be 2.1. It is necessaryto mention that the wire insulations as well as number of coolingchannels between winding discs for thermal considerations havea direct impact on the value of the filling factor. In the optimiza-tion process the dimensions of the winding will be changed. So, thepower loss of winding and finally number of cooling channels willbe changed. As a result, the filling factor is not a constant. For thesake of simplicity, Ku is considered constant. Table 1 displays thespecifications of the reactor derived from the equations mentionedabove, the inputs of the problem, and values of 0.6, 1.1 and 3 for Ku,Ag/lg, and J, respectively.

The distance from the winding to the middle and side legs aswell as to the top and bottom yokes (Bi, Bs, Hu and Hd accordingto Fig. 1) is 20 mm, and is finally decided by the insulation design.Further, the yoke is rectangular with a width (dy) equal to the outerdiameter of the leg (Doc) and a thickness (Hy) of 70 mm, the height

Inductance L [H] 3.19Number of turns N [t] 1519Width of winding Bw [m] 0.07Height of winding Hw [m] 0.14613

A. Lotfi, E. Rahimpour / Electric Power Systems Research 101 (2013) 63– 70 65

2

2

d

1

2

ict

∇

wp

vtts

w

W

L

wos

Fig. 3. Meshing of the model.

Table 2Result values from the FEM without air-gap distribution.

Energy densityof the air gap

Inductance of themain air gap

Totalinductance

Fig. 2. The modeled single phase reactor.

.2. Modeling of the reactor

.2.1. Modeling methodTo model the described reactor in Section 2.1 using three-

imensional FEM, following simplifications have been applied:

) Because of the symmetry, only one half of the reactor is modeledin FEM software, and

) Since this study does not investigate the eddy losses, these lossesare not considered. Consequently, magneto static formulation isused for the inductance calculation.

The past research has proven the validity and accuracy of FEMn reactor modeling [11–13]. Thus, a three-dimensional model isreated. Meshing the space of the problem and presuming linearity,he Poisson vector equation (Eq. (10)) is solved [14].

×(

1�0�r

∇ × �A)

= �J (10)

here A is the magnetic potential vector and �r is the relativeermeability.

Once the problem has been solved and the magnetic potentialector has been obtained, other parameters such as flux density andhe magnetic field strength will be easily calculable. Subsequently,he equations below will provide the density of the energy in thepace of the problem, and ultimately the inductance of the winding.

m = 12

�H · �B (11)

m =∫ ∫ ∫

Wmdv (12)

= 2Wm

Im(13)

here Wm, wm, �B and �H represent the magnetic energy, the densityf the energy per volume, flux density vector, and magnetic fieldtrength, respectively.

Fig. 4. FEM results, (A) Distribution of magnetic

FEM Results 571400 [J/m3] 3.18 [H] 4.6 [H]

During the FEM, the maximum current is used, and in order toachieve the desirable accuracy, smaller meshes are used in each runof the program. The process of decreasing the size of meshes con-tinues until the energy error is below 0.1 in two consecutive runs.Once this is achieved, the program will stop, and the calculated val-ues, i.e. inductance, flux distribution and so on, will be consideredas final results. Fig. 3 presents the model with the mesh plotted.

2.2.2. FEM analysis results without distributing air gapThis section will present the results of the application of the FEM

to the modeled reactor, where the air gap was not distributed andinstead placed in the center of the middle leg. Additionally, Fig. 4displays the distribution of the density of the energy, and of themagnetic flux density. As the flux is noticeably scattered in andaround the air gap, it is anticipated to have a considerable effecton the inductance magnitude of the winding. Table 2 provides thevalues obtained from Eq. (12) and Eq. (13) for the energy of themain air gap, its inductance, and the total inductance.

Fig. 5 illustrates a plot of energy variation along the line at themiddle of air gap. The plotted energy density along the mentionedline regards half of the model, and then the value of the energy den-sity has to be multiplied by two for the overall density of energy. Thefringing effect around the gap can be observed in two-dimensionalplot of the magnetic flux lines. Fig. 6 shows this effect.

To achieve the inductance of the main air gap, the density ofthe energy is multiplied by its volume. Despite the fact that theobtained value is favorably close to the expected value in Section 2.1(Table 1), the total inductance is much larger. This is to be expectedbecause of the presence of considerable fringing flux. In addition,some energy is stored in the space between the winding and the

core that is not considered in analytical formulation in Section 2.1.The fringing flux path is generally modeled as a permeance parallelto the main air gap, resulting in an increase in the total permeanceand finally a rise in the inductance. Also, the fringing flux finds its

energy, (B) Distribution of magnetic flux.

66 A. Lotfi, E. Rahimpour / Electric Power Systems Research 101 (2013) 63– 70

g the

wli

2t

gebaaryh

aosmcfattcovborow

Fig. 5. Energy variation alon

ay to the leg through the sides, and owing to the fact that theayers of the core are grain-oriented, these fluxes lead to excessiveron losses [3,4].

.2.3. Investigating the effects of distributing the air gap alonghe leg

As mentioned before, the air gap is divided into several smalleraps along the leg to reduce the fringing effect by decreasing theffective length of the air gap. Consequently, the effects of the num-er of core blocks and their height on decreasing the fringing fluxnd the inductance of the winding are studied at different steps ofpplying the FEM. To this end, 10 different states of the modeledeactor with different numbers of core blocks are used. In this anal-sis, m and lC will represent the number of core blocks and theeight of each block, respectively.

The total length of the air gap has been kept constant. Besides,s Fig. 7 illustrates, for different values of m, the total inductancef the winding is obtained according to the increase in lc. Fig. 8hows the magnetic flux lines for three different values of lC, where

= 3. As demonstrated in Fig. 7, for all values of m, an increase in lCauses a decrease in the inductance of the winding which followsrom the decrease in the fringing effect in the air gaps; however,fter reaching a specific amount, it starts to cause an increase inhe inductance of the winding. The findings indicate that to reducehe fringing effect, there is an optimum value for the height of theore blocks in the leg. The changes in the inductance can also bebserved in the magnetic flux lines drawn in Fig. 8. Table 3 gives thealues of lC corresponding to the minimum inductance (indicatedy lC,p) and the total inductance of the winding for different values

f m. In this table Lm, L0, and Hw represent the inductance of theeactor corresponding to m number of core blocks, the inductancef the reactor without distributing the air gap, and the height of theinding, respectively.

Fig. 6. Magnetic flux line.

line at the middle of air gap.

According to Table 3, increasing the value of m up to 6 bringsa decrease in the inductance of the winding; however, for valueslarger than 6, the magnitude of the inductance remains unchanged.The air gap length between two consecutive core blocks for m = 10is equal to 22.5/(10 + 1) = 2.045 mm. At this point, the difference inmagnitude between the inductance of the winding and the induct-ance of the main air gap is mostly due to the energy stored in thespace between the winding and the core leg; the fringing flux exertsno significant influence. It is worth noting that the obtained val-ues indicate that the proportion of the height of the winding (Hw)to the number of air gaps (m + 1) closely approximates the valueof lC,p. Of course, this equation is not generalizable, and may notwork in other designs. The validity of this equation can be fur-ther explored by using various values in Ag/lg to design a varietyof reactors. The appendix includes the optimum values of lC forAg/lg = 2 and Ag/lg = 0.5, as well as the proportion of the height ofthe winding to the number air gaps, which proves that the equationis satisfactorily accurate.

In addition to the FEM-based equation used for calculating theoptimum lC, a new equation is developed in the next section. To doso, an equation is derived from the Schwarz-Christoffel transforma-tion and used to model the fringing flux. Besides the optimum lC, theinductance value is also determined. It is necessary to mention thatthis equation confirms the results obtained from the FEM-basedcalculations.

3. Analytical modeling of the fringing flux path

3.1. Analytical method for considering the fringing effect

There are a number of analytical methods to measure the effectof the fringing flux, most of which incorporates an equivalent per-meance in the mentioned flux path [15–18]. The equation below

Table 3Results of the FEM with distributed air gaps.

m (number of blocks) lC,p[mm] Lm [H] Lm/L0 Hw/(m + 1) [mm]

1 72 4.1 0.89 73.12 49 3.98 0.865 48.73 38 3.9 0.847 36.534 30 3.86 0.84 29.235 25.5 3.82 0.83 24.366 22 3.8 0.826 20.877 19.5 3.79 0.82 18.38 17.3 3.78 0.82 16.249 15.5 3.77 0.819 14.61

10 14 3.76 0.817 13.3

A. Lotfi, E. Rahimpour / Electric Power Systems Research 101 (2013) 63– 70 67

us the

wr

P

wopcdg

tbt

P

Fig. 7. The changes in the inductance vers

hich is derived from the Schwarz-Christoffel transformation rep-esents this [15]:

fring = �0D

(1 + Ln

�l

2lg

)(14)

here, as shown in Fig. 9, D, l, lg and Pfring represent the diameterf the core, the length of the leg, the length of the air gap, and theermeance of the fringing flux, respectively. Eq. (14) can be used toalculate the permeance of the fringing flux when the air gap is notistributed. This equation helps to correct the inductance equationiven in Eq. (5).

Thus, all that needs to be done is to add the above permeance,he permeance of the fringing flux, and the permeance for the space

etween the winding and the limb to Eq. (5), which will result inhe following:

tot = Pg + Pfring + Pw (15)

Fig. 8. The changes in the flux path caused by increasing the lC for m = 3.

height of blocks for different values of m.

where Ptot, Pfring, Pg and Pw are the permeance of the total flux pathof the winding, the permeance of the fringing flux path, the perme-ance of the main air gap, and the permeance for the energy storedin the space between the winding and the core leg, respectively. Inthis equation, Pfring is obtained from Eq. (14) and Pg from Eq. (5).However, to determine Pw, the equivalent surface should be calcu-lated by considering the shape of the fringing flux. To this end, ifthe permeance of the fringing flux is geometrically assumed to bean empty cylinder, then the outer diameter equaling the flux arccan be obtained as follows:

Pfring = �

4([Do

fring]2 − D2oc) (16)

Dofring =

√4�

Pfring + D2oc (17)

Considering Eq. (14) will result in:√ ( ( ))

Do

fring = 4�

Doc · lg 1 + Ln�lC2lg

+ D2oc (18)

Fig. 9. The fringing flux path considered in calculating the permeance.

68 A. Lotfi, E. Rahimpour / Electric Power Sy

Table 4The values of the permeances and inductance obtained from the analytical equationsfor the modeled reactor.

Parameters Symbol Value

Gap permeance Pg 1.384 × 10−6

Fringing permeance Pfring 0.61 × 10−6

Permeance of space between core and winding Pw 0.0126 × 10−6

−6

wof

P

wtFa

L

om

roma

3

rtFbfitlcbfbl

P

well. The amount of error in either case is acceptable. In case thenumber of blocks increases to the point where the fringing effect forthe space between the blocks reaches zero, then the only perme-ance will be that of the air gaps and the space between the winding

Total permeance of winding Ptot 2.007 × 10Inductance L 4.63Error % (analytical and FEM) Error −0.65%

here Doc and Dofring

represent the outer diameter of the leg and theuter diameter for the fringing flux, respectively. This leads to theollowing:

w = �0

�4

[(D + 2Bi)

2 − (Dofring

)2]

Hf(19)

here Bi and Hf represent the distance between the winding andhe core block, and the height of the core window, respectively (seeig. 1). Therefore, the inductance of the winding can be calculateds follows:

= N2 · Ptot (20)

Table 4 lists the values of the permeances and inductancebtained by employing these equations and the dimensions of theodeled reactor.The error percentage, concluded by comparing the inductances

esulted from the above-mentioned equations with the valuesbtained from the FEM, confirms that the Schwarz-Christoffelethod is able to estimate the permeance of the fringing flux path

ccurately.

.2. Analytical equation for the height of the core block

The equation obtained in the previous section as well as theesults achieved from the FEM will be taken into considerationo develop an equation for the suitable height of the core blocks.ig. 8 illustrates that the higher lC gets, the less scattered the fluxecomes. For smaller values of lC, most of the fringing flux is blockedrom the upper leg to the lower leg. This means that its magnitudes insignificant and thus the core blocks cannot affect it much. Ashe height of the blocks (lC) increases, the fringing flux of the uppereg and lower leg, as well as the fringing flux between the non-onsecutive blocks disappears, and only that of the consecutivelocks remains. At this point, it appears that there is the lowestringing effect. In order to check this hypothesis, the permeanceetween the upper leg and the second block corresponding to each

eg will be formulated (see Fig. 10) below:

fring,u2 = �0 · D

(1 + Ln

(�(2l′)

2l′g

))(21)

Fig. 10. The fringing flux path used in calculating lC .

stems Research 101 (2013) 63– 70

l′ = 12

(Hf − mlC − lg − lC ) (22)

l′g = lC + 2m + 1

lg (23)

where l′ is the length of the core leg from which the fringing fluxexists, and l′g is the intended air gap (see Fig. 10).

To calculate lC,p, the permeance calculated by Eq. (21) should beneglected, which will lead to the equation below:

�l′

l′g= 1

e→ lC,p =

(�e)2 Hf −

((�e)

2 + 2m+1

)lg

1 + (�e)2 (m + 1)

(24)

Fig. 11 shows the values calculated by Eq. (24) and the FEM.This figure demonstrates the remarkable accuracy of the proposedanalytical equation in predicting the height of core blocks leading tothe lowest inductance. The appendix includes the values obtainedfrom Eq. (24) for different designs of the modeled reactor.

3.3. Calculating the inductance of the reactor with distributed airgap

The inductance value of the reactor can be accurately estimatedby using the calculated height of the core blocks and incorporatingEq. (15). Subsequently, the permeance for each distributed air gapcan be obtained from the equation below:

Pfring = �0D

(1 + Ln

�lC2lg

)(25)

To determine the inductance, it is necessary for the perme-ance of each air gap to be connected in series with the fringingpermeance, and for this series-connected set to be parallel to thepermeance for the space between the winding and the core. There-fore:

Ptot = 1m + 1

(Pg + Pfring) + Pw (26)

where Pw is obtained through Eq. (16), Eq. (17), Eq. (18) and Eq. (19).Once Ptot is obtained from Eq. (20), the inductance of the reactor canbe calculated. Table 5 displays the predicted values of the modeledreactor for different values of m. Evidently, the results attained fromthe analytical equations and the FEM are favorably similar.

Table 5 indicates that as the number of core blocks rises and thefringing effect declines, the value of the inductance decreases as

Fig. 11. Comparing the values of lC obtained from the analytical method with thosefrom FEM.

A. Lotfi, E. Rahimpour / Electric Power Systems Research 101 (2013) 63– 70 69

Table 5The inductance calculated by the analytical equations and the FEM.

Number of core blocks LFEM [H] LCal [H] Error %

m = 1 4.13 4.124 −0.48m = 2 3.98 3.92 1.5m = 3 3.91 3.8 2.8m = 4 3.86 3.73 3.4m = 5 3.83 3.68 3.9m = 6 3.81 3.64 4.46m = 7 3.79 3.62 4.48m = 8 3.78 3.6 4.76m = 9 3.77 3.58 5m = 10 3.76 3.57 5.05

Table 6Validation by the built and tested shunt reactor.

m = 13, Hf = 1621 mm, Hw = 1360 mm, lg = 300 mm

Parameters 3D-FEM[12] Eq. (24) Empirical Hw/(m + 1)

aitwao

4

tioido3ommi

oair

5

sCdsitFbaoufte

Fig. A1. Height of core block versus number of blocks for Aglg

= 2.

Height of the core blocks 92 [mm] 92.1 [mm] 90 [mm] 97 [mm]

nd the core. However, according to Eq. (20), this will lead to annductance of 3.43 H. In this state, the error percentage in relatedo the inductance obtained from the FEM for m = 10 will be 8.77%,hich is acceptable. Obviously, this error will be lower and more

cceptable if the inductance is calculated by FEM for a large valuef m.

. Validation by a manufactured and tested reactor

In order to verify the proposed equation, a manufactured andested three-phase shunt reactor of 50 MVAR, 400 kV is stud-ed. In previous works [5,12], different values for the heightf blocks and the Ag/lg ratio were considered to calculate thenductances in a different number of core blocks using three-imensional FEM. This shunt reactor has 13 core blocks. The heightf the core window and overall length of air gap are 1621 and00 mm, respectively. The height of blocks at the minimum amountf fringing effect is calculated by empirical relations from theanufacturer. It is compared with the three-dimensional FEModeling [12] and the analytical equation presented in this paper

n Table 6.According to Table 6, there is a good agreement in the results

btained from 3-FEM, the empirical relations and Eq. (24). Thisssures that the presented equation can be used for estimat-ng the best dimensions for core blocks in the design of shunteactors.

. Conclusion

Reduction of the fringing flux and stray losses are the main rea-ons for distributing the air gap along the leg in shunt reactors.onsequently, it is vital to have an analytical equation to enable theesigners to calculate the best value for the core height at differentelections of core block numbers. This paper incorporates a numer-cal and an analytical method to investigate the fringing effect andhe inductance of shunt reactors with distributed air gaps. First, theEM analysis indicates that there is an optimum height for each corelock to minimize the fringing effect. Second, an empirical relations a function of winding height and the number of core blocks isbtained by the FEM. Then, a new analytical equation is derived

sing fringing flux permeance based on Schwarz-Christoffel trans-ormation. There is a good agreement in the results obtained fromhree-dimensional FEM, empirical values and proposed analyticalquation.

Fig. A2. Height of core block versus number of blocks for Aglg

= 0.5.

Appendix

Validity of the developed equations on the different designs ofthe modeled reactor is demonstrated in Figs. A1 and A2.

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