270 IEEE TRANSACTIONS ON MAGNETICS, VOL. 26, NO. 1, JANUARY 1990

Effect of Finite Thickness of Magnetic Substrate on Planar Inductors

Abstmct-The effect of finite thickness of magnetic substrate or film on the inductance of planar inductors is studied using the method of current images. The results obtained here reduce to previous results [l] in the limit of infinite thickness. It is found that the thickness and permeability play complementary roles in the enhancement of induc- tance, with the normalized inductance determined roughly by the product of permeability and thickness in the most interesting region of parameter space. The results derived here allow one to determine the required minimum thickness of the substrate which will produce max- imum inductance enhancement for a given permeability of the sub- strate material. The same results can also be used to determine the permeability of a magnetic film or substrate quickly and are directly applicable to the design of planar inductors in co-fireable ceramic, hy- brid, and thin film technologies.

I. INTRODUCTION LANAR INDUCTORS have been studied for some P time [ 11-[7]. Most of the current interest in planarized

inductive devices results from push towards higher and higher operating frequencies in dc-to-dc converters in or- der to obtain smaller sizes and higher power densities [2]- [5]. Planar inductive devices (inductors and transformers) have many advantages over their conventional counter- parts which includes better thermal management, lower high-frequency losses in magnetic material and in electri- cal conductors, low profile, and higher power densities.

Recently we derived a simple result which describes the inductance enhancement of planar inductors due to a thick magnetic substrate of permeability p [ 11. We used method of current images to show that a maximum of 100-percent enhancement over the corresponding air-core value can be obtained. Theoretical result was found to be in good agreement with the experimental data for thick substrates. These results were an extension of the air-core results of Rodriquez el al. [2].

In our previous calculations [l], we assumed an infi- nitely thick substrate. However, in many practical situa- tions the substrates may not be sufficiently thick to vali- date the assumption of infinitely thick magnetic substrate. In such cases, large error could result from using the re- sults of [ 13. Examples of technologies where the magnetic layers may have small thickness are conventional thick film technology as well as now emerging low temperature

Manuscript received June 19, 1989; revised August 18, 1989. The author is with ITT Powersystems Corporation, Technical Center,

IEEE Log Number 8931785. Tucson, AZ 85706.

co-fireable ceramics technology [8], [9]. Furthermore, as will be seen in Section 111, the effect of finite magnetic layer thickness is more pronounced if the permeability of the magnetic material is low ( p < = 100). Typical high- frequency magnetic materials, NiZn ferrites , have rela- tively low permeabilities [lo]. Thus there is a need to generalize the result derived in [ l ] to case of finite thick- ness of magnetic layer in order to determine the thickness of magnetic substrate of a given permeability which would be sufficient to give maximum inductance enhancement.

In this paper we generalize the results of [ 11 to include the variation of inductance of planar inductors with the thickness w of the magnetic substrate (layer). Other basic assumptions are the same as in [ 11 : It is assumed that the distance of the planar inductor from the surface of the substrate is small compared to the typical dimension of the inductor. Circular spirals are considered explicitly but the qualitative conclusions should be valid for other planar geometries as well.

As in [ l ] , we use current images to derive a formula for the inductance of circular spiral on magnetic substrate of arbitrary thickness w and permeability p. In the limit of large w, the result reduces to that obtained in [l]. It is found that permeability and thickness complement each other in enhancing inductance with large permeability compensating for smaller thickness and vice versa in al- most a linear fashion in the region where the inductance is changing the most. We show that for low values of permeabilities ( p < = 100) there is a minimum substrate thickness which will give maximum inductance enhance- ment for a given permeability.

11. THEORY

Consider an N-turn circular spiral carrying a current I , placed parallel to a magnetic substrate (film) of permea- bility p and thickness w as shown in Fig. 1. The top sur- face of the substrate is at z = 0 with the spiral located at a small distance zo( >0) above it. As shown in [ l ] and [2] the spiral can be modeled quite accurately by N con- centric loops each carrying a current Z as shown in Fig. 2. For latter convenience the cross-sectional view of the model in Fig. 2 is shown in Fig. 3.

In the next step of building our model we will use method of current images [11]-[13]. This is a powerful method in many magnetostatic problems with the images

0018-9464/90/0100-0270$01 .OO 0 1990 IEEE

ROSHEN: FINITE THICKNESS OF MAGNETIC SUBSTRATE IN PLANAR INDUCTORS 27 1

t -I-- - - - - - -

+O - - - d - - -

- @ ( l d ) I - - - - - - -(2w+z0)

-a3(1-a2)1 - - - - - - -( 4 W + Z O ) Fig. 1 . A circular spiral carrying a current Z near the surface of a magnetic

substrate of permeability p and thickness w .

-a5(1-a2)1 - - - - - - -( Gw+z")

Fig. 4. Magnetic substrate replaced with an infinite series of image loops. The distance of each set of images from the substrate surface and current carried by a single loop in each set is also shown. CY = ( p - 1) / ( + 1 ). Shown at the very top are the source current loops. Each of the source loops carries a current I and is located at z = ZO.

Fig, 2 , Concentric circulars loops as a model of circular spiral, Each loop carries a current I .

- - - - - - are appropriate for the calculation of the field in the z > 0 region, which is the region of interest to us. The fields in this region are obtained by the superposition of the fields of the source and the images. In the following we obtain the total vector potential in the z > 0 region by

Fig. 3. Cross-sectional view of concentric circular loops model of Fig. 2.

positions and strengths ensuring proper boundary condi- tions at the interfaces of different media. The method al-

using the supe~osition principle. Although the spiral shown in Fig. 1 does not have,

lows one to calculate the fields in all regions except inside the current-carrying conductors as if one has a homage- neOuS medium having no interfaces. We will use this method to replace the magnetic substrate (film) with a se-

'peaking, cy1indrica1 symmetry, Our Of

Figs* 2-4 have cy1indrica1 symmetry about an axis pass- ing through the center Of the loops and perpendicular to the planes Of the loops. This cy1indrica1 symmetry is the

ries of images which will allow us to calculate the fields in the > region. Calculation of inductance requires knowledge of the fields in the > region We note

electrostatic problem considered by Silverster [ 141. The

result of approximating the spiral with concentric rings. Thus cylindrical coordinates with the z axis passing through the center of the loops is the preferred coordinate

currents have Only component9 and thus the that the problem at hand is completely analogous to the 'Ystem for Our In this ''Ordinate 'Ystem the

have Only this component* Fur- the physical quan-

magnetic vector potential in our case and the scalar elec- trostatic potential satisfy the Same form of Laplace q u a -

vector potential thermore, because Of this symmetry

tion and the boundary conditions are similar if dielectric tities are independent of p. In particular, the vector po- constant E is replaced with p - l [11]-[13]. his work tential is independent of p, i.e., A ( r , p, z ) = A ( r , z > - Silvester considered line electrostatic charges near a di- electric sheet of finite thickness and obtained equivalent

Next we various parts Of the vector potential' There are three types Of contributions to the vector poten-

image representations for all regions of interest. By ex-

of images for our problem as shown in Fig. 4.

strate (film) surface and each loop of this set carries a current aZ where a = ( p - 1 ) / ( p + 1). This first set

was the only one considered in [ 11. The remaining images result from multiple reflections [ 141 from the two surfaces

rection to the inductance due to the finite thickness of the magnetic substrate. Each set of these additional images is placed 2w apart and each of the loop carries a current

1 - a2)Z. Here w is the thickness of the substrate and 1 is an integer greater than one. All of these images

tial. The first contribution comes directly from the source

rent gives rise to a vector potential A;( r, z ) given by plaiting the analogy, we easily obtain the equivalent set current located at z = Z O . The nth loop Of the Source cur-

The first set of images are a distance zO from the sub- &(I-, z ) = A p , z - zo) (1)

where A;( r , z ) is the vector potential of a current loop of

given by [11]-[13] of images corresponds to the infinitely thick substrate and radius rn, carrying a "ITent and located at z = O . It is

of the magnetic substrate and are responsible for the cor- A g r , z ) = - (2)

Here the function \k (x) is given by

*(X) = (1 - "?') [ K ( x ) - E ( x ) ] (3)

212 IEEE TRANSACTIONS ON MAGNETICS, VOL. 26, NO. 1, JANUARY 1990

and ko is given by

47rrrn

( r + rn)' + (2)'. k i = (4)

K ( x ) and E ( x ) are complete elliptic integrals of the first and second kind, respectively. The second part of the vec- tor potential comes from the first set of images located at z = -zo which correspond to the infinitely thick sub- strate. Each of the loops in this set of images carries a current a I and the nth ring of this first set of images gives the following contribution to the vector potential:

AY(r, z ) = d $ ( r , z + zo) ( 5 )

= ( P - + 1 ) . (6 )

where a is given by

The third part of the vector potential comes from the re- maining sets of images located at z = - (20 + (21 - 2 ) w) ( 1 > 1 ) which are due to the finite thickness of the substrate. All of the nth loops of these remaining sets of images give a net negative contribution to the vector po- tential and hence will tend to reduce the inductance. It is given by

m a21 - 3 A ' t ( r , z ) = - ( 1 - a2) c

1=2

A$(r , z + z,, + 2(1 - 1 ) w ) . (7)

As is clear from the above, in the limit of large perme- ability A; -, 0, thus the correction to these additional im- ages vanishes in spite of the finite thickness of the mag- netic substrate. This is due to the fact that if the permeability is very large, even a very thin layer of mag- netic substrate would be able to confine the magnetic field within itself and hence there will not be any reflections from the bottom surface of the substrate.

The net vector potential is obtained by superposition of ( l ) , (6), and (7) and summing over all the N loops

A ( r , z ) = c [AE(r, z ) + A;(r , z ) + A:(r , z ) ] . N

( 8 )

From the net vector potential (8), the total inductance of the spiral can easily be obtained by integrating A ( r , z ) around each of the source loops and dividing the result by the current.

L = -

n = l

N l N 1 ds, [AE(r, z) + AY(r, z) + A f ( r , z ) ] . I * = I n = l

(9)

For m # n terms, r = r, and z = zo along the path of integration. Form = n terms, the contours of integrations have r = ( r n - t ) and z = zo, where 2t is the width of conductor. For flat narrow conductors, such as screened- printed conductors, 2t may be taken to be actual width of the conductor trace. The above includes the mutual as well

as self-inductances of the source loops. The terms with m # n give the mutual inductances while the terms with m = n give the self-inductances of the source loops. The above result neglects the internal self-inductances of the loops which are due to fields that exist inside the conduc- tors. Their net contribution to the total inductance L is usually negligible.

To clearly identify various effects, we write (9) as the sum of two terms

L = L, + L, (10) where

. N r N

is the inductance of the spiral for an infinitely thick sub- strate. The integrations in (1 1) were performed in [ 13 with the following result:

L, = ( a + 1 ) b = *b. P + l

It was assumed in [ 11 that zo, t << r,. Lo is the inductance of the spiral in absence of the magnetic substrate. The reader is referred to [ 11, [2] for further details regarding

L, is the correction to (12) due to the finite thickness of Lo *

the magnetic substrate and is given by . N r N

(13) L, = - 1 ] ds, Af(r, 2 ) .

I m = l n = l

In the Appendix we perform the integrations in (13) to obtain the following result:

m

L, = - ( 1 - a*) 1 = 2 c a21-3

N

J ( 2 r n - t)' + ((21 - 2)wf 9'(kkn)

+ m + n 5 d ( r n + r,)' + ((21 - 2)wf " ( k h n ) ]

where

and

(16) 4rn(rn - t )

(kk,)' = (2rn - t)' + ((21 - 2 ) w f '

In arriving at the above results (14)-(16) we have as- sumed that conductor width 2t is small compared to radii r, and the substrate thickness w.

ROSHEN: FINITE THICKNESS OF MAGNETIC SUBSTRATE IN PLANAR INDUCTORS

111. DISCUSSION

Equation (14) is our central result which gives the cor- rection to the inductance of a planar inductor due to the finite thickness of the magnetic substrate. The inductance for infinitely thick substrate is given by (12) and thus the total inductance for finite substrate thickness is given by the sum of (12) and (14), as expressed by (10). A number of interesting points emerge from the study of these re- sults as noted below.

a) The correction L, is always negative and therefore always reduces the inductance. Thus the limit of 100-per- cent inductance enhancement due to magnetic substrate obtained by us previously [l] remains valid. This is an expected result.

b) In the limit of large permeability, a -+ 1 and the correction L, due to finite thickness vanishes irrespective of the thickness w of the magnetic substrate. This shows that for large permeabilities even a relatively thin sub- strate is able to provide maximum inductance enhance- ment of 100 percent. The reason for this is that for large permeabilities the substrate is able to confine all the field within itself and there are no reflections from the bottom surface of the substrate. Since the bottom surface does not contribute, the substrate of finite thickness behaves as if it were infinitely thick.

c) For a fixed permeability p , larger magnetic substrate thickness w has the effect of reducing the magnitude of L, and increasing the total inductance L. This is due to the fact that for large w, only a small number of higher order ( I > 2 ) images are able to contribute significantly to L, as their distance from the source loops increases rapidly. In fact, it can be shown [15] that the relative contribution of the Ith image falls off as d 3 , where d = 2 (1 - 1 ) w + zo is the distance of the image from the substrate top sur- face. Of course, in the limit w + 03, contributions of all the higher order images approach zero, L = L,, and thus the results derived in [ l ] for infinitely thick substrate are recovered.

d) As will be seen clearly in item e), in the region of parameter space ( p , w ) where the inductance is changing the most, the permeability p and the thickness plays com- plementary roles in reducing the magnitude of L, and in- creasing the total inductance L. In other words, a decrease in the value of p can be compensated by an increase in the value of the thickness w so as to obtain the same value of the inductance L. Furthermore, in this region the in- ductance depends roughly only on the product of perme- ability and thickness. Thus a decrease of p by a factor of two with a simultaneous increase of w by a factor of two will roughly leave the value of L unchanged.

e) All of the features discussed above, especially in items d) and e), can be seen more clearly in Figs. 5 and 6. In Fig. 5 , we have used the theoretical results (lo), (12), and (14) to plot normalized inductance L/Lo for a four-turn spiral as a function of the permeability p with the thickness w varied as a parameter. The geometrical

P e r m e a D l l i t y

Fig. 5 . Normalized inductance LILO as a function of permeability p of the substrate/film for various values of thickness w (m). Lo is the inductance f o r p = 1.

le-I .. . Thickness (m)

Fig. 6. Normalized inductance LILO as a function of thickness w ( m ) , with permeability p varied as a parameter. LO is the inductance for p = 1.

parameters for this spiral are as follows: inside radius = 0.5 cm, radius increment = 0.125 cm, and conductor width = 0.05 cm. These curves generally follow an S-shape, with each curve starting from air-core value ( L o ) for L at p = 1 and approaching the maximum possible value ( L = 2L0) for large p . The knee ( L = 1.9L0) of each of the curves occurs at a value of p which depends on the value of the parameter w. For larger values of w, the knee occurs at lower values of permeabilities, while for smaller values of w one needs to go to much higher values of p to obtain the same inductance enhancement. Thus w and p have complementary roles as mentioned in item d). As a more specific example, L 1.8Lo for p = 300 and w = 0.01 cm. Roughly the same value of L can be obtained by decreasing p to 30 and simultaneously in- creasing w to 0.1 cm. It is interesting to note that a very similar “inverse linear” relationship between p and w also occurs in the inverse problem of magnetic shielding [ 1 I].

214 IEEE TRANSACTIONS ON MAGNETICS, VOL. 26, NO. I , JANUARY 1990

Fig. 6 shows another interesting feature of results (lo), (12), and (14). In this figure we have plotted normalized inductance as a function of the thickness w with the permeability varied as a parameter. All the curves start from the same value' L = L,,, as those shown in Fig. 6 for small values of w. However, in contrast to the curves shown in Fig. 5, curves shown in Fig. 6 approach differ- ent limiting values as the thickness w increases. The lim- iting value of L depends on the permeability p , with larger values of p giving larger limiting values of L. This effect is seen to be specially strong for low permeabilities ( p = < 100). Thus for such low permeabilities, for the sub- strate material it may not be possible to obtain 100-per- cent inductance enhancement, no matter how thick the substrate is made. These curves also show, for such low values of permeability, the inductance does not increase as the thickness of the film is increased beyond a certain value. This value is then the required minimum thickness for a given permeability which gives the maximum pos- sible inductance. There is no advantage in going to thicker films. On the other hand, thinner films would not give maximum allowable inductance enhancement for a given permeability. Thus the required minimum thickness is the optimum thickness for a fixed permeability.

These results have direct application to planar magnet- ics. Planarized inductors can be fabricated using thin [6]- [8] and thick film technologies as well as the new emerg- ing low temperature co-fireable ceramic technology [8], [9], [ 161. This latter technology is especially promising for realizing planar inductors with high values of induc- tances for two reasons: 1) the substrate or film thickness could be substantially larger than those that can be achieved through the other two technologies. 2) The lam- ination process allows for higher densification of mag- netic material and hence should yield high permeabilities. Furthermore, this technology allows for the integration of various passive components in a single monolithic planar structure by building various passive components with substantial power-handling capability as planar structures on different layers. This integration would be very advan- tageous in achieving higher power densities, better high- frequency performance, and lower profiles for power sup- plies.

IV. CONCLUSIONS To summarize, we have extended the use of the method

of current images to obtain new results which describe the effect of finite thickness of the magnetic substrate on the inductance of planar inductors. These results can be used in design of planar inductors on a magnetic film or sub- strate. In particular, an optimum thickness of a magnetic substrate which will provide maximum possible induc- tance enhancement can be determined. The results also allow one to easily determine the permeability of the sub- strate material if the thickness, inductance, and the cor-

'Although this cannot be seen clearly for all the curves shown in Fig. 6, numerical results show that all curves start at L = Lo.

responding air-core value of the inductance are mea- sured.

The losses in magnetic substrate (film) can be estimated by using the current images method used here. This method allows one to calculate the magnetic field distri- bution inside the film by replacing the source distribution with another set of image currents [ 113-[ 131. This can be then combined with the specific loss data for a particular material to estimate the core losses. Further extension of the present work employing the image method to planar magnetics include inductors with top and bottom mag- netic films as well as to other magnetic devices such as planar transformers [ 171.

Finally, we point out that some additional information regarding the effect of finite thickness of magnetic sub- strate on planar inductors can be found in [ 181.

APPENDIX In this Appendix derivation of (14) is given by perform-

ing the integrations in (13). If we separate m = n and m # n terms and use (7) we can write (13) as

n # m " 1 01 N [ z, Z i n + c Zkrn . - 1

L, = - (1 - CY2) c Z 1 = 2

(A1 ) The integrals I,,, and I,,, are given by

If, = ds,A;(r, z + zo + 2(1 - 2)w) (A2)

Z f , = 1 ds,A;(r, z + zo + 2(1 - 1)w). (A3)

The integration in (A2) is to be performed along the inner radius of the nth source loop. Thus z = zo and r = r, - t, where r, is the radius (mean) of the loop and 2t is the trace width. Note that inclusion of the finite width effect is essential in order to avoid divergent result. With these values of r and z (A2) becomes

Z k n = 27r(rn - t)A&(r,, - t , 2z0 + 2(1 - 1)w) . (A4)

If we now assume that the distance zo of the source current from the surface of the magnetic substrate is much smaller than the width w of the substrate, (A4) can be simplified to

ZL,, = 27r(rn - t ) A k ( r n - t, 2(1 - 1)w) . (A5)

Next we use (2) for A; and (4) for k i in (A5) to obtain the final result for Z!,,,

s

Z k n = J ( 2 r n - t f + (2(1 - 1)wf * ( k L n ) (A6) where

and function !P ( x ) is defined by (3). The integration in (A2) is performed in a similar man-

ner. In this case, the integration is to be performed along

ROSHEN: FINITE THICKNESS OF MAGNETIC SUBSTRATE IN PLANAR INDUCTORS 275

the mth source loop and thus r = rm and z = zo. For these values of r and z (A3) becomes

ZLm = 27rrmA;Z((rm, 2z0 + 2(1 - 1 ) ~ ) . (A8)

Once again if zo << w, (A8) simplifies to

zkm = 2 7 r r m ~ ~ ( r m , 2(1 - 1 ) ~ ) . (A91 Use of (2) and (4) in (A9) finally gives

Z:m = d ( r m + rn? + ( (21 - 2 ) w f \k (kkm) (A10) where

[7] N. Saleh and A. H. Qureshi, Electron. Lett., vol. 6, pp. 850-852, 1970.

[8] F. R. Gleason, “Thin film microelectronic inductor,” in Proc. Nut. Electron. Conf., vol. 20, pp. 197-198, 1971.

[9] A. L. Eustice, S . J . Horowitz, J . J . Stewart, A. R. Travis, and H. T . Sawhill, Hybrid Circuit Technol., pp. 9-14, June 1987 and pp. 15- 20, July 1987.

[lo] C. Snelling, So3 Ferrites. [ l l ] J . D. Jackson, Classical Electrodynamics, 1st ed.

Wiley, 1962, ch. 5. [I21 J . D. Stratton, Electromagnetic Theory.

Hill, 1941, ch. 4. [13] W. R. Smythe, Static and Dynamic Electricity.

McGraw-Hill, 1950, ch. 7. [14] P. Silvester, Proc. Inst. Elec. Eng., vol. 115, pp. 43-48, 1968. [15] A. Zisserman, “Fresh approaches to magnetostatic field problems with

emphasis on analytic techniques,” Ph.D. dissertation, Sunderland Polytechnic, Tyne and Wear, England, 1984.

[16] D. Zwemer, W. Roshen, and J. Woodard, U.S. Patents pending, 1989.

London, UK: Iliffe, 1969, ch. 3. New York, NY:

New York, NY: McGraw-

New York, NY:

Substituting (A6) and (A10) in (Al) one immediately ob- [l71 W. A. Roshen, to be published. [18] V. Sementov and C. Afanasyev, Radiorechnika, no. 3, 1966 (in Rus- tains the result (14). Note that (15) and (16) are the same cirrn) I_-..

as (A7) and (All) , respectively.

REFERENCES [ l ] W. A. Roshen and D. E. Turcotte, “Planar inductors on magnetic

substrates,” IEEE Trans. Magn., vol. 24, pp. 3213-3216, 1988. [2] R. Rodriquez, J . M. Dishman, F. T. Dickens, and E. W. Whelan,

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[4] E. Rodriquez and A. Estrov, U.S. Patent 4 622 627, Nov. 1 1 , 1986.

[6] R. F. Sahoo, IEEE Trans. Magn., vol. MAG-15, pp. 1803-1805,

535-541, 1980.

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[SI A. Estrov, PCIM, pp. 14-24, Aug. 1986.

1979.

Waseem A. Roshen (PM’87) received the Ph.D. degree in physics from the Ohio State University, Columbus, in 1982.

From 1982 to 1984, he was a postdoctoral Research Associate at the University of Virginia, Blacksburg. He taught Physics at Ohio State Uni- versity from September 1984 to April 1986. Since April of 1986, he has been employed at ITT Powersystems Corporation, Tucson, AZ, as a Se- nior Scientist. His research interests are in the areas of magnetism and magnetic devices, superconductivity, and electronic components used in power supplies.