A study on stray capacitance modeling of inductors by using the finite element method

88 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 43, NO. 1, FEBRUARY 2001

coaxial extensions of the balanced ports explain the origin of the pos-tulated series inductor. The unshielded line is excited in the differen-tial mode by the LPDA, which allows the 50- load presented insidethe shielded box at the balanced terminals of the bifilar winding to beviewed as two 25- resistors with a virtual ground between them. Eachof the 25- resistors is transformed to a frequency dependent seriesimpedance on the unshielded line side of the associated 50- coaxialextension. The two transformed 25- resistors appear in series withthe inductance presented by the short-circuited air-line formed by theouter cylindrical screens of the coaxial extensions and the outside ofthe balun box.

The total series inductance obtained from the transmission linemodel for the coaxial extensions of the balun ports is in good agree-ment with the values obtained previously by analyzing the shift inthe measured frequencies of maximum coupling: 145 nH versus 140nH for the maximum at 203 MHz, and 154 nH versus 174 nH formaximum at 337 MHz.

We incorporated this high frequency equivalent circuit of the baluninto the numerical model as a final test of our explanation for the posi-tions of the maxima. The predicted and measured maxima of the “NECwith balun model” in Fig. 2 were now found to differ by less than 4MHz and 1 dB. The out-of-band computational artifact at 140 MHz isnot pertinent to this discussion.

VII. CONCLUSION

The lesson is simple but important. Numerical codes, like NEC, arecapable of yielding acceptable coupling data for a simple antenna andtransmission line system like ours,onlyprovided sufficient care is takento make the numerical model and the physical system consistent. (Theneed for compliance with the various intrinsic restrictions on numericalmodels, [2]–[4], is taken as obvious.)

In our case, a prerequisite for such consistency was the use of twobaluns, one at the antenna feed point, and the second at the junctionbetween the unshielded transmission line system and the coaxial cableto the measuring instrument. Until the baluns were working properlythere was no meaningful agreement between the measurements andnumerical predictions.

After the modifications, and careful characterization of the outputbalun, the simulated coupling displayed the essential features of themeasured response over the common operating band of both balunsand the LPDA. Below and above this band the agreement is poor. Thisis expected, since the numerical model cannot take into account theeffects of common mode currents flowing on the cables due to balunmalfunction.

This study also highlights a modeling dilemma which often arisesin engineering practice. The engineering purpose of a numerical simu-lation is to predict adequately the properties of a physical system. Wewere only able to simulate the behavior of our rather simple experimentafter it had been modified to effectively eliminate the effects of the longcoaxial cables and their immediate environment. When dealing withthe much more complex systems of practical EMC engineering it mayoften be unfeasible to change the system to suit the modeling code.

ACKNOWLEDGMENT

The authors thank their colleague Prof. D. Davidson for acquiring,via the U.S. Department of the Army, a copy of NEC-4 for use in theirresearch and for advising them on subtle aspects of NEC models. Theythank T. Stuart for his perceptive editorial advice.

REFERENCES

[1] F. M. Tesche, M. V. Ianoz, and T. Karlsson,EMC Analysis Methods andComputational Models. New York: Wiley, 1997.

[2] G. J. Burke and A. J. Poggio,Numerical Electromagnetics Code(NEC)—Method of Moments. Part III: User’s Guide. Livermore, CA:Lawrence Livermore Nat. Lab., Jan. 1981.

[3] G. J. Burke,Numerical Electromagnetics Code—NEC-4 Method of Mo-ments. Part I: User’s Manual (NEC 4.1). Livermore, CA: LawrenceLivermore Nat. Lab., Jan. 1992.

[4] J. Peng, C. A. Balanis, and G. C. Barker, “NEC and ESP codes: Guide-lines, limitations, and EMC applications,”IEEE Trans. Electromag.Compat., vol. 35, pp. 124–133, May 1993.

[5] D. A. Frickey, “Conversions betweenS; Z; Y; h; ABCD, andT pa-rameters which are valid for complex source and load impedances,”IEEE Trans. Microwave Theory Tech., vol. 42, pp. 205–211, Feb. 1994.

[6] T. A. Milligan, Modern Antenna Design. New York: McGraw-Hill,1985.

[7] L. H. Becker, “An investigation of radiated EMI induced in wire-con-nected electronic systems,” M. Eng. thesis, Dept. Elect. Electron. Eng.,Univ. Stellenbosch, Stellenbosch, South Africa, Nov. 1996.

[8] C. L. Ruthroff, “Some broad-band transformers,”Proc. IRE, vol. 47, pp.1337–1342, Aug. 1959.

A Study on Stray Capacitance Modeling of Inductors byUsing the Finite Element Method

Qin Yu and Thomas W. Holmes

Abstract—Stray capacitance modeling of an inductor is essential forits RF equivalent circuit modeling and inductor design. Stray capacitancedetermines an inductor’s performance and upper frequency limit. In thispaper, a method has been proposed for modeling the distributed straycapacitance of inductors by the finite element method and a node-to-nodelumped capacitance network. The effects of wire insulation layer, ferritecore, number of segments used to model the circumference of a wire crosssection, pitch and coil-to-core distances, and the capacitance betweennonadjacent turns, etc., on an inductors’ self-capacitance and calculationaccuracy, have all been considered. The calculated equivalent lumpedstray capacitance for a rod inductor with ferrite core is compared to thatestimated from measurement. Good agreement between them has beenobserved.

Index Terms—Inductors, self-capacitance modeling.

I. INTRODUCTION

The stray capacitance of an inductor (or choke), widely used forsuppressing radio frequency (RF) noise, plays an important role in af-fecting the frequency characteristic and performance of the inductor.The stray capacitance modeling is, therefore, essential for accurate in-ductor RF equivalent circuit modeling. It helps identify the key factorsthat affect the stray capacitance of an inductor. It is very useful forimproving the inductor design which requires certain desired self-res-onant frequency and insertion loss.

Manuscript received March 9, 1998; revised August 11, 2000.Q. Yu was with ITT Automotive, Inc. (now Valeo, Inc.), Dayton, OH 45459

USA. He is now with Lucent Technologies, Inc., Columbus, OH 43054 USA(e-mail: [email protected]).

T. W. Holmes was with ITT Automotive, Inc. (now Valeo, Inc.), Dayton, OH45459 USA. He is now with Hewlett-Packard Company, Dayton, OH 45459USA (e-mail: [email protected]).

Publisher Item Identifier S 0018-9375(01)01727-6.

0018–9375/01$10.00 © 2001 IEEE

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 43, NO. 1, FEBRUARY 2001 89

Fig. 1. Equivalent circuit model of an inductor.

The bulk rod inductor used for noise suppression typically consists ofasingle-layerwinding which is made of a round copper wire wound onaslug-type ferrite core. The parasitic capacitance between the windingturns appears in shunt with the inductor and results in the occurrence ofresonance at some frequency [1]. Above this self-resonant frequency,the impedance of the inductor becomes capacitive.

Massarini and Kazimierczuk [2] derived expressions for calculatingthe self-capacitance of single and multiple-layer inductors with andwithout a conductive core, including an analytical equation for calcu-lating the capacitance between twoadjacent air-coreturns. Their equa-tions ignore small variations in turn-to-turn capacitance caused by theexistence of nearby turns, that is, the turn-to-turn capacitance calcu-lated is independent of the relative positions of turns in the inductor.Coils with dielectric cores were also not considered. Because of thedifficulty in accurately estimating the actual paths of electric field linesand the surface area, the analytical expressions of the capacitance givenin [2] might underestimate the self-capacitance by 10% to 40%.

At RF, the direct measurement of the stray capacitance of the in-ductor is difficult. In [3], a technique was developed for estimating theshunt self-capacitance of a ferrite-core inductor, as well as its otherequivalent circuit parameters.

In this paper, a study on the stray capacitance modeling of a single-layer slug-type inductor with a ferrite core has been conducted by usingthe finite element method (FEM) and a lumped node-to-node capaci-tance network method. The lumped stray capacitance calculated by theproposed method is compared with that estimated by a measurementmethod given in [3]. The correlation between two results is good.

II. I NDUCTOR SELF-CAPACITANCE MODELING

In [1], it states that an inductor can be modeled by an equivalentcircuit at RF as shown in Fig. 1, whereRL, LL, andCS represent, re-spectively, the equivalent resistance, inductance, and capacitance of theinductor.RL is mainly caused by winding and core losses, andCS rep-resents the distributed turn-to-turn parasitic capacitance effects of thewinding. However, in Fig. 1, the effects of the ground on the inductor’sstray capacitance are ignored. As a result, this model is appropriateonly for the cases where there are no grounded conductors nearby, orthe coil-to-ground stray capacitance is negligible. Otherwise, a� cir-cuit model for the stray capacitance (see Section II-C) has to be used.At higher frequencies, some other models have to be used, such as thetransmission line model, for predicting the high-frequency behavior ofinductors with multiple resonant modes [4].

The stray capacitance of a coil consists of coil turn-to-turn capaci-tance, turn-to-ground capacitance between coil turns and ground con-ductors, and turn-to-core capacitance between turns and core if the corematerial is conductive.

In the following, the methods for calculating the turn-to-turn andturn-to-ground capacitance, modeling the distributed stray capacitanceand obtaining the lumped stray capacitance of the inductor are de-scribed in details.

Fig. 2. The 2-D axisymmetric finite element model of anN -turn inductor.

Fig. 3. Node-to-node lumped capacitance network.

A. Calculation of Stray Capacitance of the Inductor by FEM

The turn-to-turn and turn-to-ground capacitances of a single-layerferrite-core rod inductor are calculated by a 2-D electrostatic axisym-metric finite element model. In the 2-D axisymmetric FEM, a helicalcoil is modeled by a number of coaxial planar loops as shown in Fig. 2.The number of the loops is equal to the number of turns and the dis-tance between the centers of two loops is equal to the pitch of thecoil. The left edge of the problem outer region, which passes throughthe axis of the inductor, is treated as an axisymmetric boundary. Theother three edges of the outer region are assigned as balloon-type openboundaries which simulate the infinite large boundaries where eitherthe potential or charge equals zero. Other interfaces between objectsare treated as natural boundaries. Triangular cells are used to mesh thewhole problem region.

Maxwell2-D and 3-D field simulators are FEM packages from An-soft Corporation [5], [6]. InMaxwellelectrostatic simulators, which arethe solvers here, the capacitance between a conductor and other con-ductive structures or conductors is calculated by

1) applying 1 V to the conductor and 0 V to the other conductivestructures;

2) calculating the scalar electric potential� by using the FEM;3) computing the electric field strength and the electric flux density

from �;4) obtaining the capacitance values by computing the energy stored

in the field.

B. Distributed Stray Capacitance Modeling

The distributed capacitance of an inductor is modeled by a networkof lumped node-to-node capacitance elements as given in Fig. 3, whereeach node represents a turn.

In Fig. 3, the symbolCij represents the capacitance between theturns i and j, andCio gives the total capacitance of the turni toground. The relationship among the node voltages, node currents


and node-to-node capacitance can be represented by anN � Nadmittance matrix [see (1)], where the turn-to-core capacitance andthe capacitance between the coil and other conductors are ignored. In(1), Ii, Vj , andYij are current, voltage, and admittance, respectively,wherei; j = 1 . . .N . Equation (1) gives a general expression aboutthe capacitive couplings among conductors. In practice, for the turnk,only its coupling with the turnk � 2, k � 1, k + 1, andk + 2 need tobe taken into account and others can be ignored. In other words, in theadmittance matrix,Yk;j = 0 wherej 6= k, k � 2, k � 1, k + 1, andk+ 2. If other nearby conductors including the core exist, more nodesneed to be added to the network shown in Fig. 3

I1I2......IN

=

Y11 Y12 . . . . . . Y1NY21 Y22 . . . . . . Y2N

...... . . . . . .

......

... . . . . . ....

YN1 YN2 . . . . . . YNN

V1V2......

VN

: (1)

For a coil turn, the existence of nearby conductors orits own coilturns will affect its charge distribution. Therefore, the assumptionCi; i+k = Ci+j; i+j+k (i; j = 1 . . .N andk = 0 . . .N � 1),generally, is not valid. If an inductor and its surrounding objects aresymmetric about its horizontal center plane (perpendicular to theinductor’s axis), the calculations for its turn-to-turn capacitance canbe reduced half.

The calculation of the turn-to-turn capacitance by a 2-D electro-static axisymmetric FEM is very fast.In addition, inductors used in RFEMI noise filtering circuits generally have small number of turns. Foran inductor with 11 turns, the real time for calculating the turn-to-turncapacitance is less than 30 s and the CPU time is less than 10 s with anHP Apollo series 700 workstation.

C. Calculation of Lumped Equivalent Stray Capacitance

The equivalent stray capacitance of an inductor between its two ter-minals can be obtained by eliminating all intermediate nodes. Here, thereduction of the nodes is done by the appropriate matrix operation. Itcan also be done by�=Y transformation which eliminates one internalnode at a time.

Equation (1) is rearranged as follows:

I1IN

I2...

IN�1

=

Y11 Y1N Y12 . . . Y1;N�1YN1 YNN YN2 . . . YN;N�1

Y21 Y2N Y22 . . . Y2;N�1...

......

......

YN�1; 1 YN�1;N YN�1;2 . . . YN�1;N�1

�

V1VN

V2...

VN�1

: (2)

Then, (2) becomes

IxIy

=Yxx YxyYyx Yyy

VxVy

(3)

Fig. 4. Equivalent circuit of the inductor between two terminals.

where the corresponding vector and matrix partitions can be readilyidentified

[Ix] =I1IN

; [Vx] =V1VN

[Iy] =

I2...

IN�1

; [Vy] =

V2...

VN�1

[Yxx] =Y11 Y1NYN1 YNN

; [Yxy] =Y12 . . . Y1N�1YN2 . . . YNN�1

[Yyx] =

Y21 Y2N...

...YN�11 YN�1N

and

[Yyy] =

Y22 . . . Y2N�1...

......

YN�12 . . . YN�1N�1

:

WhenIy = 0, it can be derived from (3) that

Vy = �Y �1yy YyxVx: (4)

Substituting (4) into (3) yields that

Ix = (Yxx � YxyY�1

yy Yyx)Vx = YxVx (5)

whereYx is a2 � 2 matrix and has the form

Yx = Yxx � YxyY�1

yy Yyx:

After eliminating all intermediate nodes, the equivalent circuit of theinductor between its two terminals becomes Fig. 4.

The nondiagonal admittance inYx matrix is equal to�j!C1N .Hence, the equivalent lumped stray capacitance between the inductortwo terminals can be obtained easily.

III. CALCULATION RESULTS

The accuracy of calculating the turn-to-turn capacitance by using theelectrostatic 2-D axisymmetric FEM has been evaluated by comparingthe calculation results with those of an electrostatic 3-D model. Theeffects of conductor insulation layer, ferrite core, modeling methods,number of segments for modeling the circumference of a circular wirecross section, types of open boundary condition used, presence ofnearby conductors, and capacitance between nonadjacent turns, ontthe stray capacitance and calculation accuracy have also been studied.

A. Sample Inductors

Four single-layer rod sample inductors were used in this study. Thewindings of all sample inductors are made of round copper wires with adiameter of 1.59 mm and have the same mean radius of the conductor


TABLE ITURN-TO-TURN CAPACITANCE OF COIL1 VERSUS THEVARIOUS NUMBER OF

SEGMENTS PERCONDUCTORCIRCUMFERENCE BYUSING 2-D AXISYMMETRIC

FINITE ELEMENT MODEL

loop, which is 3.4 mm and the same pitch distance, i.e., the distancebetween the centers of two adjacent turns, which is equal to 1.73 mm.The sample inductors used include

• COIL1: anair-core coil which consists of two coaxial toroidalloops made of bare wires.

• COIL2: a two-turnair-corehelical coil made of bare conductors.• COIL3: an 11-turnair-core helical inductor. The dielectric con-

stant of the conductor insulation is 4. The thickness of the con-ductor insulation is 0.07 mm.

• COIL4: an 11-turnferrite-corehelical inductor. Its core is madeof Fair-Rite #43 soft ferrite material with an RF dielectric con-stant"r of 14 and conductivity about10�3( �m)�1. The lengthof the core is 21.44 mm and the radius of the core is 2.54 mm.Its winding has the same properties and geometry as that of theCOIL3.

An outer region of 70 mm�100 mm (see Fig. 2) was applied toall following 2-D studies. The open boundary conditions were imple-mented by setting the potential or charge to zero along the assignedopen boundaries which are 256 times the size of the above user definedouter region.

B. Parametric Analysis

1) Number of Segments for Modeling a Round Conductor:Asmentioned before,Maxwell 2-D electrostatic field solver was usedto calculate the inductor stray capacitance, such as the turn-to-turnand turn-to-ground capacitance. In theMaxwell2-D field simulator, acircle is modeled by a polygon. The number of sides of the polygonaffects the size of the wire meshes. Therefore, the number of segmentsper circumference of the conductor cross section is important forobtaining accurate turn-to-turn capacitance. It is obvious thattheoverall distance between turns decreases as the number of sides ofthe polygon increases. The stray capacitance thus increases with theincrease in number of segments used.

For example, for COIL3, the capacitance between its turn 1 and turn2 increases 60% when the number of segments per conductor circum-ference is increased from 8 to 32.

In order to identify the appropriate number of segments for mod-eling a round conductor in capacitance calculation, a test case was runfor COIL1 with various number of segments, where the charge-typeballoon open boundary condition was used. The results are tabulatedin Table I. The geometry of COIL1 is rotationally symmetric about itscenter axis. The 2-D axisymmetric finite element model thus shouldbe able to deliver accurate results if the mesh size is appropriate. FromTable I, it can be seen that when the segment number is changed from 16to 32, the turn-to-turn capacitance increases 5.3%. However, when thenumber of segments is further increased from 32 to 64, the turn-to-turncapacitance changes only 1.44%. There is always a tradeoff betweenthe number of segments per wire circumference and computation time.

Fig. 5. The 3-D finite element model of COIL2.

Hence,the appropriate number of segments per circumference of around conductor is around 32.

2) 2-D Modeling Versus 3-D Modeling:Theoretically speaking,the 3-D modeling for a device is more accurate than its 2-D modeling.The geometry of a device can be modeled more accurately in its3-D modeling. However, the accuracy of 3-D modeling is limited byavailable computer resources and numerical techniques.

In order to compare the computation accuracy, the turn-to-turn ca-pacitance of COIL2 was calculated by a 3-D finite element model withMaxwell3-D electrostatic field solver. In the 3-D model, a helical coilis artificially broken into a number of helical turns as shown in Fig. 5.This can be done by artificially reducing the length of one turn or bothturns slightly, e.g., 1.0% for one turn or 0.5% for both turns. The short-ening of the coil turns will give a gap of about 3.6� between two adja-cent turns. The cross section of the wire is modeled by a polygon, likein a 2-D model. Each helical turn is modeled by a number of straightsegments (see Fig. 5). A cylinder which surrounds the object was cre-ated and used as the problem boundary. The surfaces of the problemouter region are treated as Neumann boundaries, that is, the electricfield is tangential to those surfaces and the charge on those surfaces iszero. The results are given in Table II, where the height and radius ofthe background cylinder that forms the boundary are 24 mm and 13mm, respectively.

For COIL2, its 2-D axisymmetric finite element model is same asthat of COIL1. Its turn-to-turn capacitance is given in Table II.

There is a difference in the mean circumference of a planar ring wireand a helical wire. However, for a tightly wound coil, such differenceis small, and can be neglected.

By comparing the results given in Tables I and II, it can be seen thatthe error between the results of the 2-D and 3-D modeling methods is2.3% with 16 segments per wire circumference. Therefore, it can beconcluded that the 2-D axisymmetric model is reasonably accurate forcalculating the turn-to-turn capacitance of a helical coil.

3) Effects of Conductor Insulation Layer:Including the wire insu-lation layer into the model is crucial for obtaining accurate capacitance,especially for tightly wound coils. For a tightly wound winding, theonly space between two adjacent conductors is their insulation layer.Therefore, the insulation layer is the major medium for forming theturn-to-turn stray capacitance between the two adjacent turns.

For example, for COIL3, if its conductor insulation layer is not con-sidered in its 2-D axisymmetric model, its turn-to-turn stray capaci-tanceC12 is 38% lower than that if the insulation layer is taken intoconsideration. In both cases, eight segments were used for modelingthe conductor circumference.

4) Effects of Core Materials:For an inductor with a core whose rel-ative permittivity is not one, the dielectric core should be included in themodel. The existence of the dielectric core will increase the couplingbetween turns. Hence, the stray capacitance between turns, especiallyfor nonadjacent turns, will increase.

The only difference between COIL3 and COIL4 is that COIL4 has adielectric core and COIL3 does not. A capacitance calculation was thus


TABLE IITURN-TO-TURN CAPACITANCE OF COIL2 VERSUS THEVARIOUS NUMBER OF

SEGMENTS PERWIRE CIRCUMFERENCE ANDSEGMENTS PERTURN,RESPECTIVELY, BY USING 3-D FINITE ELEMENT MODEL

conducted for COIL3 and COIL4, respectively, by using the 2-D mod-eling method. In both cases, 32 segments per wire circumference wereused. For COIL3, the calculated turn-to-turn capacitanceC12 = 2:57pF andC13 = 0:046 pF and the equivalent lumped stray capaci-tance is 0.439 pF. For COIL4, the calculated turn-to-turn capacitanceC12 = 2:95 pF andC13 = 0:19 pF, and the equivalent lumped straycapacitance of COIL4 is 0.563 pF. Therefore, for inductors with softferrite cores, e.g., nickel zinc ferrites whose RF relative permittivity is14, the core must be simulated in the model. Otherwise, the stray ca-pacitance might be underestimated.

The conductivity of the core material will also affect the coil straycapacitance. Generally, the conductivity of nickel zinc soft ferrites isvery low, around10�7 to 10�3(�m)�1. For a manganese zinc ormanganese soft ferrite, its conductivity usually is in the range of 0.5to 2 (�m)�1 [7]. For other types of core materials, their conductivitymight be higher. Just for verification purposes, let the core conductivityof COIL4 be changed from10�3(�m)�1 to 104(�m)�1. Then, theturn-to-turn capacitanceC12 is changed from 2.95 pF to 2.49 pF, andC13 is changed from 0.19 pF to 3.44� 10�2 pF, with 32 segments perconductor circumference used. It is obvious that with the increase inconductivity of the core material, the turn-to-turn capacitance will bereduced slightly. In addition, the turn-to-core capacitance will increasesignificantly [2].

5) Effects of the Capacitance Between Nonadjacent Turns:Thenonadjacent turn-to-turn capacitance and turn-to-ground capacitanceshould be considered in the calculation of the total lumped capaci-tance, if their values are comparable with their adjacent turn-to-turncapacitance values. In the literature related to the coil self-capacitancecalculations, many authors assume that the stray capacitance betweennonadjacent turns can be ignored in the total equivalent capacitancecalculation [2]. This might be true only for a loosely wound air-corecoil. Otherwise, the equivalent lumped stray capacitance will beunderestimated.

A test was run for calculating the equivalent lumped stray capaci-tance of both COIL3 and COIL4, with and without the nonadjacentcapacitance taken into account. In the turn-to-turn stray capacitancecalculation, 2-D axisymmetric modeling method was used with 32segments per conductor circumference. For COIL3, whose maximumcapacitance between nonadjacent turns is two orders lower than thatbetween adjacent turns, the calculated equivalent stray capacitance is0.439 pF when the nonadjacent turn-to-turn capacitance is includedin the coil node-to-node capacitance network model. However, ifall nonadjacent turn-to-turn capacitance is ignored, the calculatedequivalent capacitance becomes 0.253 pF. Therefore, ignoring thenonadjacent turn-to-turn capacitance for COIL3 gives a calculationerror of 42.4%. For COIL4, its nonadjacent turn-to-turn capacitanceis higher than that of COIL3 due to the existence of the ferrite coreand the maximum value is one order lower than the adjacent turncapacitance. If only considering adjacent turn-to-turn capacitance in

Fig. 6. Geometry of COIL4 surrounded by a grounded metal box.

TABLE IIICALCULATED TURN-TO-TURN STRAY CAPACITANCE C OF COIL3

VERSUS THEPITCH DISTANCE OF THECOIL

the model, the calculated equivalent stray capacitance is 0.286 pF,which is much lower than the total lumped capacitance of 0.563 pFwith nonadjacent turn-to-turn capacitance included in the model. ForCOIL4, this gives a calculation error of 49.2%.

6) Effects of Nearby Conductors:Due to proximity effects, the ex-istence of nearby conductors affects the distributed stray capacitance.This reduces the lumped series stray capacitance of the inductor, butincreases the coil-to-ground capacitance. Therefore, they must be in-cluded in the model.

Giving an example, if COIL4 is surrounded by a grounded cylin-drical metal box as shown in Fig. 6, it can be seen from the calculationthat the stray capacitance of COIL4 will be changed, as expected. Theturn-to-turn capacitance changes slightly, but the turn-to-ground capac-itance increases significantly, where in the calculation the number ofsegments per wire circumference equals 32. Referred to Fig. 4, theequivalent series stray capacitanceC1; 11 reduces from 0.563 pF to0.314 pF and the equivalentC1o andC11; o increase to 0.679 pF and0.502 pF, respectively.

7) Effects of Pitch and Coil-to-Core Distance:Both the pitch dis-tance and the distance between core and coil affect the stray capac-itance significantly, especially the former. The stray capacitance in-creases with the decrease of those distances.

A test case was run for COIL4 with various pitch distances by usingthe 2-D modeling method. The turn-to-turn capacitanceC12 for dif-ferent coil pitch distances is shown in Table III, where 32 segmentsper wire were used in the calculation. It can be seen that when the coilpitch distance is increased, the space between turns is increased and,thus, the stray capacitance is reduced accordingly.

For COIL4, if the diameter of the core is reduced slightly, e.g., 0.005mm, then a very small air gap is inserted between the core and thewinding. Then the turn-to-turn capacitanceC12 is changed from 2.98pF to 2.95 pF andC13 is changed from 0.196 pF to 0.194 pF, andthe total stray capacitance is reduced slightly to 0.560 pF from 0.563pF, where the number of segments per conductor circumference equals32. This indicates that the self-capacitance can be reduced slightly byincreasing “air” space between the coil and core.


Fig. 7. Measurement setup diagram.

8) Effects of the Type of Open Boundary Condition:There are twodifferent types of open boundary conditions. One is charge-type openboundary condition, where the total charge on the open boundary iszero. The other type is the voltage-type open boundary condition,where the potential on the open boundary is zero. The application ofdifferent types of open boundary conditions in the model will givedifferent turn-to-ground capacitance and slightly different turn-to-turncapacitance. With the charge-type open boundary condition, theturn-to-ground capacitance calculated usually is very small and canbe ignored, that is, in Fig. 3,Cio

�= 0 (i = 1; . . . ; N). With thevoltage-type open boundary condition, the turn-to-ground capacitancecalculated usually can not be ignored in the node-to-node capacitancenetwork.

Which type of open boundary condition should be used dependson the problem’s assumptions. Generally, for the applications wherethere are no grounded conductors in the nearby area of the inductor,the charge-type open boundary condition is appropriate. Otherwise, thevoltage-type open boundary condition should be used.

C. Comparison with measured results

As mentioned above, the equivalent lumped series stray capacitanceof COIL4 calculated by the 2-D modeling method presented in thispaper is 0.563 pF, where there are no other grounded conductors nearby.

The equivalent lumped stray capacitance of two sample inductors ofCOIL4 was also estimated by using the measurement method describedin [3].

The stray capacitance of an inductor can be estimated by measuringthe resonant frequencies of the inductor circuit with external capacitorsin parallel with the inductor, as shown in Fig. 7, whereCext is thecapacitance of the external capacitor. The lumped stray capacitance canbe estimated by the following equation:

CS = C1 �

C2

C1

�

f2

f1

2

� 1

1�f2

f1

2(6)

whereC1 andC2 capacitance of two external capacitors;

f1 resonant frequency of the circuit withC1 in parallelwith the inductor and similarly;

f2 resonant frequency of the circuit withC2 in parallelwith the inductor.

The two external capacitors used in the measurement are 3.31 pFand 5.53 pF, measured by HP 4276 LCZ meter at 20 kHz. HP 8753B

TABLE IVCOMPARISONBETWEEN THECALCULATED AND ESTIMATED EQUIVALENT

LUMPED STRAY CAPACITANCE OF COIL4

network analyzer was used to measure the resonant frequency of theinductor circuit. The measured and calculated equivalent lumped straycapacitance of COIL4 are tabulated Table IV. Good agreement betweenthe calculated and measured results has been observed. The discrep-ancy between them is about 1.7% to 4.5%.

IV. CONCLUSION

This paper proposes a method for modeling the self-capacitanceof rod inductors. The 2-D axisymmetric finite element method wasused to calculate the stray capacitance of the inductor, and the lumpednode-to-node capacitance network was used to model the distributedcapacitance of the inductor. Four samples inductors were used in thestudies. The calculation results are compared with the estimated onesfrom measurement. Good agreement has been observed. In addition,a thorough parametric analysis has been conducted, which includesstudying the effects of number of segments for modeling a roundconductor, insulation layer, core material, pitch distance, coil-to-coredistance, nonadjacent turn-to-turn capacitance, etc., on the straycapacitance and its calculation.

REFERENCES

[1] H. W. Ott, Noise Reduction Techniques in Electronic Systems, 2nded. New York: Wiley, 1988.

[2] A. Massarini and M. K. Kazimierczuk, “Self-capacitance of inductors,”IEEE Trans. Power Electron., vol. 12, no. 4, pp. 671–676, 1997.

[3] Q. Yu, T. W. Holmes, and K. Naishadham, “Radio frequency characteri-zation of ferrite materials,” inIEEE AP-S Int. Symp. URSI North Amer-ican Radio Science Meeting, vol. 106.03, 1997.

[4] R. W. Rhea, “A multimode high-frequency inductor model,”Appl. Mi-crowave Wireless, vol. 9, no. 6, pp. 70–80, 1997.

[5] Maxwell 2D Simulator User’s Reference: Ansoft Corp., 1994.[6] Maxwell 3D Simulator User’s Reference: Ansoft Corp., 1993.[7] Fair-Rite Soft Ferrites, 13th ed: Fair-Rite Products Corp., 1996.

A study on stray capacitance modeling of inductors by using the finite element method

Documents

Transcript of A study on stray capacitance modeling of inductors by using the finite element method