Analysis of planar sandwich inductors by current images

2880 IEEE TRANSACTIONS ON MAGNETICS. VOL 26. NO 5. SEPTEMBER 1990

Analysis of Planar Sandwich Inductors by Current Images

WASEEM A. ROSHEN, MEMBER, 1EEE

Abstract-Using the method of current images, inductors with spiral windings sandwiched between top and bottom plane thick magnetic plates are investigated theoretically. Expressions are obtained for the inductances and magnetic fields of these planar structures assuming infinite plate thicknesses. The dependence of the inductance on the permeability and the distances between the spiral windings and the top and bottom plates is studied by numerically evaluating the inductance expressions. I t is shown that, except for some limiting cases, the inductance depends only on the net distance between the top and bottom magnetic plates and is independent of position of the spiral windings within the gap. Furthermore, it is found that, in contrast to the open planar structures studied earlier, the sandwich inductor can yield large values of inductances with the enhancement over air-core values limited only by the permeability of the magnetic plates. Numerical evaluation of the formulas for magnetic fields both inside the magnetic plates and in the gap are used to calculate energy density variation in the device, to estimate lateral dimensions of these sandwich inductors, to discuss construction of multiple devices in a single sandwich structure, and to discuss calculations of losses in the sandwich inductors. Planar magnetics investigated in this work can be realized through low- temperature multilayer co-fired ceramics technology, screen printing on ferrite slabs, as well as conventional thick- and thin-film technologies.

I. INTRODUCTION

NTEREST in planarized magnetics has been active for I some time [1]-[9]. Earlier studies on planar inductors focused on thin-film Permalloy types [7]-[9]. Recent in- terest in planar inductive devices is a result of pushing up the operating frequencies of dc-to-dc converters in the megahertz range to obtain smaller sizes and higher power densities [ 11-[6]. Planar inductive devices (inductors and transformers) have many advantages over their conventional counterparts; these include better thermal manage- ment, lower high-frequency losses in magnetic material and in electrical conductors, low profile, and higher power densities.

Recently, we investigated planar inductors on magnetic substrates [ 11, [ 2 ] by generalizing air-core results of Rod- riquez et al. [ 3 ] for spiral inductors. By deriving a simple result we showed that a 100% inductance enhancement over the air-core value is the maximum enhancement one can achieve by the use of a single magnetic plate. There-

Manuscript received November 7 , 1989; revised April 9 , 1990. The author was with ITT Power Systems Corporation, Tucson, AZ. He

is now with General Electric Corporate Research and Development, K1- 3C40, Schenectady, NY 12301.

IEEE Log Number 9036778.

fore, the inductance values one can achieve either by using an air-core spiral or a planar inductor on a magnetic substrate are limited.

For many applications larger values of inductances are required. In this paper we study a simple planar structure that could yield substantially higher inductance values. The sandwich inductor we study in the present paper consists of a spiral winding between a top and a bottom flat magnetic plate. This simple structure has the additional advantage of being easy to manufacture and is particularly suited for the now emerging low-temperature co-fireable ceramics technology [ lo] , [ l 11. If the magnetic plates have large enough surface resistivity [ 121, then the spiral windings can be put directly on one of the plates but USU- ally one would need an insulator sheet for putting down the conductors for the windings. Other technologies through which such structures can be realized are the conventional thick-film hybrid and thin-film technologies as well as screen printing on ferrite slabs [4].

In this work current images are used to account for the magnetic plates. To simplify the analysis, we have assumed that these layers are infinitely thick. Thus the inductance values as well as other results derived here give the limiting values. Although only circular spiral windings are considered explicitly, qualitative conclusions should be valid for other planar windings as well.

By using current images, we obtain expressions for the inductance and magnetic fields in sandwich inductors. Numerical evaluation of these formulas are used to study the effect of permeability of the magnetic plates, the variation of the distances of the spiral from the two plates and the distance between the two plates on the inductance. Numerical results for the flux density are employed to estimate the lateral dimensions of the sandwich inductor, discussion of multiple inductors in the same device, and the placement of a secondary coil for maximum coupling. It is pointed out that field distribution can also be used for calculating core losses as well as estimating conductor losses.

The remainder of the paper is organized as follows. In Section I1 we describe out model and the current images and derive expressions for the vector potential A . In Sec- tion I11 we derive expressions for the inductance and study its implications by numerical evaluation of the expressions. In Section IV we obtain expressions for the flux density by differentiating the vector potential and investigate the numerical results for various consequences. Fi-

0018-9464/90/0900-2880$01 .OO 0 1990 IEEE

ROSHEN: ANALYSIS OF SANDWICH INDUCTORS BY CURRENT IMAGES 2881

nally , in Section V , we summarize some of the important conclusions and point out some possible extensions of the present work.

11. IMAGES AND VECTOR POTENTIAL We consider an N-turn circular spiral carrying a current

I , sandwiched by a top and a bottom thick magnetic plate of permeability p . The spiral is located at z = 0. The top and bottom magnetic plates are located at z = z1 and z = -z2, respectively. As shown before [1]-[3], the spiral can be modeled quite accurately by concentric loops. Thus we replace the spiral with concentric loops as a first step in our modeling. The cross-section view of this model is shown in Fig. 1.

We will use the method of current images [13]-[15] to model the effect of the magnetic plates. For this purpose we will assume that the magnetic plates are infinitely thick. Although this is not an essential assumption, it sim- plifies the analysis considerably. The simplification allows us to concentrate on the fundamental properties of the sandwich structure without unduly complicating the problem due to the finite thickness effects studied in [2]. The current images method is a powerful one in many magnetostatic problems with the image positions and strengths ensuring proper boundary conditions at the interfaces of different media. The method allows one to calculate the fields in all regions except inside the current- carrying conductors as if one has a homogeneous medium having no interfaces.

We are interested in obtaining images that will properly describe the effect of magnetic plates for calculating vector potential and magnetic field in two regions: region I ( -22 < z < z l ) and region I1 ( z > z l ) . We will not consider images that are appropriate for region I11 ( z < -z2 ), explicitly since the vector potential and magnetic fields in this region should be similar to those in region 11. To proceed further, we note that the electrostatic problem considered by Silvester [ 161 is analogous to the problem we are considering here. The scalar electrostatic potential and the magnetic vector potential satisfy the same form the Laplace equation and the boundary conditions are similar if the dielectric constant E is replaced with p- I [ 131-[ 151. Silvester obtained an image representation for line electrostatic charges near a dielectric surface. By ex- ploiting the analogy and straightforward generalization of his results, we obtain the equivalent set of images for our problem as shown in Figs. 2 and 3.

The images shown in Fig. 2 are appropriate for calculating vector potential and magnetic fields in region I ( -z2 < z < z l ) . The group of images shown in Fig. 2(a) results from starting the reflection from the bottom plate. Each set of these images are located at z = -2 ( lzl + ( 1 - 1 122) and 21 ( z l + z2) alternatively. 1 is an integer equal to or greater than one. Each loop of a given set of images carries a current a2'- 'I and ( ~ ~ ' 1 , respectively. Here CY = ( p - 1 )/( p + 1 ). Similarly, the group of images shown in Fig. 2(b) are obtained by starting the reflections from the top plate. Each set of images in this group are located

Fig. 1 . Cross-sectional view of sandwich inductor model canying current I .

21(z1+z2) - - - -- - 2' I

21(z1+z2)-2z2 - - - $-11

3 I

---

- - -

4(Z1+Z2) - - -

2 I - - -

Fig. 2. Images for region I obtained by starting reflections at the bottom plate. (a) First group. Left: positions of each set of images. Right: currents carried by each loop in given set. 01 = ( p - l ) / ( ~ + l ) . (b) Second group. Left: positions of each set of images. Right: currents carried by each loop in given set.

2882 IEEE TRANSACTIONS ON MAGNETICS, VOL 16. NO 5, SEPTEMBER 1990

at z = 2(1z2 + ( 1 - l ) z l ) and z = -21(z1 + z 2 ) alternatively. Each loop in a given set carries a current Q 21p ' I and a2'I, respectively. All of these images contribute to magnetic fields in region I. The fields in this region are obtained by the superposition of the fields of these images and the source current located at z = 0.

Fig. 3 shows the images appropriate for calculating the vector potential and the magnetic field in the top magnetic plate, i.e., in region I1 ( z > z l ) . The images shown in Fig. 3(a) result from starting the reflections from the bottom plate surface. Each set of images in this group is located at z = -2 ( lzl + ( 1 - 1 ) z 2 ) with the corresponding image strength given by ( 1 + a ) a 2 ' - ' I . The images shown in Fig. 3(b) are obtained by starting the reflections at the top magnetic plate. Each set of these images is located at z = -2 ( 1 - 1 ) ( z I + z 2 ) with the image strength given by ( 1 + Q ) Q ~ ' - ~ Z . The fields in region I1 are then obtained by the superposition of the field of all these images. Source current does not contribute directly to the fields in this region. This statement can be made more clearly by considering the analogous situation in optics. From above the surface of water, only the images of the objects below the surface can be seen. The objects them- selves are not directly observables from above the surface of water.

Next we use this image representation to derive vector potentials for regions I and 11. We start by noting that the sandwich inductor considered here does not posses cylindrical symmetry in a strict sense. However, our models of Figs. 1-3 have cylindrical symmetry about an axis passing through the center of the loops. This cylindrical symmetry is the result of approximating the spiral with concentric loops. Thus cylindrical coordinates with the z axis passing through the center of the loops is the con- venient coordinate system for our model. This coordinate system has the advantage that all the currents have only one component, namely the azimuthal component, thus ensuring that the vector potential will also have only this component. Furthermore, because of cylindrical symmetry all the physical quantities are independent of $. Thus the vector potential is independent of $, i .e. , A ( r , 4, z ) = A ( r , z ) 4 .

Vector potential A , ( r , z ) for region I can now be calculated. It consists of three contributions. The first contribution comes from the source current located at z = 0.

The nth loop of the source current gives rise to a vector potential A { ( r , z ) = A"( r , z ) , where A" ( r , z ) is the vector potential of a current loop of radius r,, carrying a current I and located at z = 0. It is given by [12]-[14]

22 -21(z1+2 ) --- - - - (1+a$1-11 2 .

-2(z1+z2) - - -

4 z 1 + z 2 ) ---

-2(l-l)(z +a ) - - - --- ( 1 + a) P 2 1 1 2

(b)

Fig. 3. Images for region I1 obtained by starting reflections at bottom plate. (a) First group. Left: positions of each set of images. Right: currents carried by each loop in given set. (b) Second group. Left: positions of each set of images, Right: currents carried by each loop in given set.

and k is given by

. . . 4 rr,, k2' (1. + l")* + ( z ) ? (3 )

K ( k ) and E ( k ) are complete integrals of the first and second kind, respectively. The second part of the vector potential comes from the group of images shown in Fig. 2(a). The nth of this group of images give a net contribution Ayl ( r , 2):

+ a2 'An(r , z - 2 f ( z 1 + z 2 ) ) ] ( 4 1 where Q is given by

A " ( r , z ) = \ k ( k ) . (1) r k

Q = ( P - 1 ) A P + 1 ) . ( 5 )

The third contribution to vector potential in region I is due

Here the function \k ( k ) is given by

[ K ( k ) - . . . ( 2 ) to the images shown in Fig. 2(b). The nth loops of this

ROSHEN: ANALYSlS OF SANDWICH INDUCTORS BY CURRENT IMAGES 2883

group of images give a net contribution to AY2 ( r , z ) given by

m

~ : 2 ( r , z ) = C [CF'A'(~, z - 21(z1 + z 2 ) + 2z2) 1 = 1

+ a2'An(r , z + 21(z1 + z 2 ) ) ] . ( 6 ) Finally, the net vector potential in region I A I ( r , z ) is

now obtained by adding A",r, z) , (4), and (6) and summing over all the N loops:

N

A1(r, z ) = c [A;; (r , z ) + A:l(r, z ) + A:z(r, z ) ] . n = I

(7) Vector potential A, ( r , z ) in region I1 is similarly ob-

tained. In this case the source current does not contribute. Only the images shown in Fig. 3(a) and (b) contribute to A2 ( r , z). The net contribution Ay3 ( r , z ) of all the nth loops in the group of images shown in Fig. 3(a) is given by

m

& ( I - , z ) = ( 1 + a ) c &IA"(r, z I = I

+ 21(z1 + z2) - 24). ( 8 ) In a similar manner all the nth loops of the group of images shown in Fig. 3(b) give rise to a total vector potential 4 4 ( r , z ) :

m

~ : ~ ( r , z ) = ( I + a ) C c ~ ~ ' - * ~ ~ ( r , z I = I

+ 2 ( 1 - l)(Zl + 4). (9) Thus the net vector potential A2 ( r , z ) in region I1 is obtained by adding results (8) and (9) and summing over all the N loops:

N

A2(r , z ) = C [Ay4(r, z ) + & ( r , z ) ] . (10)

This completes our derivation of vector potentials for the regions. In the following section we employ the vector potential A I ( r , z ) for region I to obtain an expression for the inductance of the sandwich inductor, while in Section IV we use both A I ( r , z ) and A2 (r, z ) as given by (7) and (10) to derive formulas for the magnetic fields in regions I and 11, respectively.

n = I

111. INDUCTANCE From the vector potential A I ( r , z ) as given by (7), the

inductance of the sandwich inductor can easily be calculated by integrated A I (r, z ) around each of the source loops and dividing the result by the current I:

N

+ AT'(r, z ) ] . (11) For m # n terms, r = rm and z = zo along the path of integration. For m = n terms, the contours of integration

have r = ( r , - t ) and z = z,,, where 2t is the width of conductor. For flat narrow conductors such as screened printed conductors, 2t may be taken to be the actual width of the conductor trace. The foregoing 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 foregoing result neglects the internal self-inductances of the loops due to fields that exist inside the conductors. Their net contribution to the total inductance L is usually negligible.

It is now instructive to write (1 1) as a sum of two terms, separating the two types of contributions:

L = L ( ) + L c (12) where

. N P N = A 9 ds, c A;f ( r , z )

n = l I m = l

is the inductance of the spiral in the absence of,magnetic plates. The integrations in (13) were performed in [ 2 ] , [3], and we refer the readers to these two references for further details regarding the air-core value Lo of the inductance.

Lc is the inductance enhancement over the air-core value Lo due to the magnetic plates. It is given by

N

where Ayi( r , z ) and A$( r . z ) are given by (4) and (6) . The integrations in (14) can be performed as in [ 2 , ap- pendix]. These integrations are fairly straightforward and give the following result for L,:

m N

Lc = C c [ 2 a " 4 2 r n - t ) 2 + (21(21 + Z 2 ) ) * / = I n = I

+ C C [r,, + r,, r,? - t -+ r,,] / = I m f n

where the function Kp ( x l , x2 , x 3 ) is defined as

and + ( k ) is defined by (2 ) . Equation (15) is our central result for the inductance

and gives the enhancement of the inductance of a spiral due to sandwiching the spiral between two magnetic

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16

12

8 -

plates. The total inductance of the sandwich spiral is given by the sum of (15) and air-core inductance given by (13). As mentioned in the introduction, in our analysis we assumed infinitely thick magnetic plates, therefore this result represents the maximum enhancement in inductances one can obtain with this type of structure.

We have studied numerically the result (1 5 ) along with (13) for the inductance of the sandwich inductor as a function of z I , z2 , the distances of the spiral from the top and bottom plates, and permeability 1.1. The spiral used for this study has the following geometrical parameters: inside radius = 0.5 cm, radius increment = 0.125 cm, conductor width = 0.05 cm, and number of turns = 4. Some of the results are shown graphically in Figs. 4 and 5. In Fig. 4 the normalized inductance is plotted as a function of permeability 1.1 of the magnetic plates. Gap length = z , + z2 is varied as a parameter. In Fig. 5, the normalized inductance L I L O , used to define an effective permeability prff, is plotted as a function of the gap length z, + z2 for different values of 1.1. The number of terms kept in evaluating (15) and (13) was determined by error tolerance, which was set at less than 5 %. Some of the salient points of these results are as follows.

1) Except for some limiting cases (see items 4 and 5 , to follow), the inductance does not depend on individual values of distances z I and z2 but depends only on the net gap length z I + zz. In other words, the inductance of the sandwich inductor depends only on the net distance between the top and bottom plates but is independent of where the spiral is placed in the gap.

2) As I, + zz --f 0 , L -+ pL, . This is the case of embed- ded windings and the result says that the inductance in this case is enhanced over the air-core value by a factor equal to the relative permeability of the plate material. This is an expected result and demonstrates that in contrast to the open structures considered in [ 11, [2], the sandwich structure can yield large inductance enhancement over the corresponding air-core values. The enhancement is limited only by the permeability of the material of the magnetic plates.

3) A given value of gap length ( z I + z? ) causes larger reduction in the inductance in case of larger permeability than in the case of smaller permeability of plates material. This is especially significant for small values of gap length. The effect is analogous to the effect seen in more conventional magnetics where the air-gap effect is determined by the ratio of the permeability to the gap length.

4) For large values of the gap length, all the curves shown in Fig. 5 approach each other. Thus indicating that for large values of z I + zz inductance becomes somewhat independent of the permeability of the magnetic plates. In particular, if z I -+ 03, z2 -+ CO, then the inductance Lc goes to zero and one obtains L = Lo.

5 ) I f z l + ~ ~ a n d z , - + O , t h e n L - + ( a ! + I)Lo.Same result is obtained in case of z1 + 0 and z2 -+ CO. This is the case when either the top or the bottom plate is re- moved from the structure and hence the structure is re- duced to that of a planar inductor on an infinitely thick

- Gap = 0.0 mm

0.2 mm 0.4 mm 0.8 mm

-

-

_ - _ _ - - - -- .' , ,, -/5- __- -

I I I I I

IEEE TRANSACTIONS ON MAGNETICS. VOL. 26. NO. 5. SEPTEMBER 1990

+ao . ,.a

Permeability

Fig. 4. Normalized inductance L I L O as function of permeability j .~ of magnetic plates for various values of gap length z , + z2 . Lo is inductance for fi = 1.

20.0 r-----7 16 0

$ 12.0

1 1

2" 8.0

4.0

0 2 0 4 0 6 0.8 1.0

Gap (mm)

Fig. 5 . Normalized inductance L I L O as a function of gap length, with permeability varied as parameter. Lo is inductance for p = 1.

magnetic substrate. The result obtained here is consistent with that in [ 11.

IV. MAGNETIC FIELDS

The magnetic field B in general can be obtained from the vector potential A by using the relation

B = V x A . (17) The vector potentials for the sandwich inductor in two regions were derived in Section 11. Since the vector potentials derived there for region I and I1 have only the azimuthal components A ( r , z ) , (17) shows that there are


only two components of the magnetic field: the radial component B, and the z-component B,.

The magnetic field in region I can now be obtained by using the vector potential A I ( r , z ) as given by (7), along with (4) and (5) as well as (1)-(3) in (17) and carrying out the differentiations. The results are

B&-, z) = 5 ( b ; ( r , z) + I = 5 I [a%;(r , z n = I

- 21(z1 + z2))

+ a 2 @ ( r , z + 2qz1 + 2 2 3 )

+ a 2 / - 1 n b , (r , z + 21(z1 + z2) - 2z1)

+ a2'- 'b:(r , z - 21(ZI + z2) + 2z2)])

(18)

B,(r , z) = 5 (b ; ( r , z) + / = I 5 [a2%;(r, z n = l

- 2% + z2))

+ a 2 @ ( r , z, + 2qz1 + z2))

+ a 2 1 - I n b&, z + 21(ZI + z2) - 221)

+ a2'-'b;(r, z - 21(z1 + z2) + 2z2)])

(19 ) where b:(r , z ) and b;(r , z ) are the components of the magnetic field of a circular current loop of radius r, and located at z = 0. These components are given by

. . . POI Z b;(r, z) =

r [ ( r n + r)' + z2]I '2

X [ - K ( k ) + r 2 + rf + z2

. . . cl01 1 1 /2

b;(r , z) = - 27r [ ( r n + r ) 2 + 221

r2 - rt - z2 X [ - K ( k ) +

(rn - r)' + 22

where k is defined by (3). The magnetic field in region I1 is similarly obtained by

using (10) for vector potential A2 ( r , z ) along with (9) and (8), in (17):

N m

Br(r , z) = ( 1 + a ) c c [a2'-lb:(r, z n = l I = l

N m

Bz(r , z) = ( 1 + a ) C C [a2'-lbZ(r, z n = l / = I

+ 21(ZI + 2 2 ) - 221)

+ a2'-2b:(r , z + 2(1 - l ) ( z , + 4 1 . (23 )

This completes our derivation of magnetic fields for the two regions. Equations (18) and (19) give the magnetic fields for region I ( -z2 < z < z l ) while (22) and (23) are our results for magnetic field in region I1 ( z > ZI 1. We have verified that these fields satisfy the boundary conditions: a) B, continuous and b) B r / p continuous at the interface at z = z l .

To investigate the implications of these results, we have obtained numerical results for the magnetic fields in the two regions for the sandwich spiral whose inductance was calculated in Section I11 of this paper. We remind the reader that geometrical parameters for this sandwich inductor are inner radius = 0.5 cm, radius increment = 0.125 cm, conductor width = 0.05 cm, and number of turns = N = 4. Numerical results are shown in graphical form in Figs. 6-8. A current I = 1 A is assumed.

In Fig. 6 we show the variation of B,, B,, and the mag- nitude B of the magnetic field as a function of z, the ver- tical distance from the spiral for a fixed value of r (=0.045 cm). This value of r places us just inside the inner radius of the spiral. Values of other parameters used are z l = z2 = 0.01 cm and p = 20.0 for the plate material. As seen from these graphs, B, and B both decrease fairly rapidly with increasing values of z . More interestingly B and B, show a maximum near the interface in region IT. This maximum is quite a general feature which we have seen for all other values of the parameters we have investigated. Furthermore, both B, and B shows a jump at the interface z = z I . This discontinuity occurs due to boundary condition that H , = B r / p be continuous. The occur- rence of a maximum near the interface suggests that most of the energy is stored in the magnetic material is near the interface. Since the losses in a magnetic material vary as B P (2 .0 < 0 < 3.0), most of the core losses in the sandwich inductor will also occur in the region near the interface.

In Fig. 7 the normalized flux density B / B o is plotted as a function of normalized radius r / r o for radius r greater than outside radius ro of the spiral. The curves are shown for only two values of z ( z = 0.1 mm and z = 0.2 mm), though we also obtained numerical results for other values of z, including z = 0.0. The results for other values of z , especially z = 0, are very similar. Values of other parameters for these curves are the same as those for Fig. 6. For both values of z , the normalized flux density falls very rapidly with the average exponent of r o / r greater than 3.7 in for ro < r < 2r0. This is an important piece of infor- mation if one is interested in building more than one independent device in the same sandwich structure, since it tells us that the field of this first device does not extend

2886

10.0

8.0

F

$ 6.0

n

G 4.0

- v) c a

3

2.0

I-

0.4 0.8 1.2 1.6 2.0

(mm)

Fig. 6. Variation of magnetic field with z for the sandwich spiral studied in the text. r is slightly smaller than inner radius of the spiral and z , =

z2 = 0.01 cm.

L I I I I I I 1.0 1.2 1.4 1.6 1 8 2.0

rlr,

Fig. 7. Normalized flux density as function of normalized radius. E , is the flux density at r = r, where r, is slightly larger than the outside radius of the spiral. z = 0.1 mm and z = 0.2 mm for the top and bottom curves, respectively. See the text for more details regarding other parameters.

significantly beyond r = 2r0. Thus the second independent device can be built such that its outer dimensions fall at r 5 2r0 with no significant coupling between the two devices. Additionally, this field distribution can be used to set the lateral dimensions of the magnetic plates for the single device. For this reason in Fig. 8 we have used this field distribution to plot the normalized “energy density for a given radius ( 5 rB2),” to show how the energy density falls with the radius. Once again the results are shown for two values of z . As can be seen this quantity falls even faster than the magnetic field B. For the particular structure studied, the energy stored beyond r = l.5ro is negligible and hence suggest that the top and bottom magnetic plates need not extend beyond r = 1.5r0. This has been verified qualitatively using low-temperature multi-

IEEE TRANSACTIONS ON MAGNETICS, VOL 26. NO 5 . SEPTEMBER 1990

I 1 I I I 1 I I

0.8 2 = 0 1 mm-

0 2 rnrn----- I \ \

---+---+----e

1.1 1 2 1 3 1.4 1.5 1.6 1 7 1 8 rlr,

Fig. 8. Energy density rB’ as a function of r . Energy density and radius have been normalized by their values at r = ro. z = 0.1 mm and : =

0.2 mm for top and bottom curves, respectively. See the text for details regarding other parameters.

layer ceramic technology. Some idea of the required thickness of the plates can be obtained from reading [2], where we show the required minimum thickness depends on the permeability in an inverse manner.

Since our analysis is magnetostatic, our results for magnetic fields cannot be used in a straightforward manner to calculate high-frequency conductor losses. However, one can adopt a simple model for the skin effect in which the fields decay exponentially along the width of the flat conductors starting from the two edges and then use our magnetostatic fields to define the starting fields at the two edges of the conductor. This, combined with the con- straint that the total current through each conductor be equal to I , should be sufficient to estimate to first-order accuracy the current distribution and hence the high-frequency ac conductor losses in the sandwich inductor. As mentioned previously, if the magnetic plates material is fully characterized for high-frequency losses, then the magnetic field results obtained in this paper can be used to calculate the core losses as well. Thus from the magnetic field results one should be able to obtain a good estimate of the net high-frequency losses in the sandwich inductor.

V. CONCLUSION

To summarize, we have analyzed a planar inductor formed by sandwiching a spiral winding between two infinitely thick magnetic regions. Expressions are obtained for the inductance and magnetic fields for this structure. It is shown that the inductance of this type of structure depends only on the net distance between the top and bottom plate and is independent of the position of the spiral placed within the gap formed between these two plates. The effect of gap length on the inductance has been studied. Numerical results for the magnetic fields are used to discuss the setting of the lateral dimensions of the device, construction of multiple devices in the same sandwich structure and placement of secondary coil for maximum coupling. Fields distribution can also be used to estimate


the device losses if the loss characteristics of magnetic material of the top and bottom plates are known.

A straightforward extension of the present work includes series and parallel connected spirals sandwich by the same magnetic plates. Parallel connected coils might be used for reducing the conductor losses, while the series connected coils will yield larger values of inductance. Another application of the present approach would be to coupled inductors or transformers.

In the present work the thicknesses of the top and bottom magnetic plates are assumed to be infinite. Thus this work neglects the effect of finite thickness [2] of the magnetic plates and hence only limiting values of inductance are obtained. The present work might be extended to include the finite thickness effect, though the large number of image sets required may make this a formidable task. It is possible that, if the plates are thin enough, there would be some dependence of the inductance on the position of spiral in the gap.

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141 P. M. Gradzki, in Power Elecironics Seminar, Virginia Power Elec-

[5] E. Rodriquez and A. Estrov, U.S. patent 4 622 627, Nov. 11, 1986. [6] A. Estrov, in Proc. Power Conversion and Intelligent Motion, Aug.

1986, pp. 14-24. [7] R. F. Sahoo, IEEE Trans. Magn., vol. MAG-15, pp. 1803-1805,

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Waseem A. Roshen (M’87) received the Ph.D. degree in physics from Ohio State University, Columbus, in 1982. From 1982 to 1984, he was a postdoctoral Research Associate at the

University of Virginia. He taught physics at Ohio State University from September 1984 to April 1986. He then joined ITT Power Systems Cor- poration, Tucson, AZ, as a Senior Scientist. He is currently with General Electric Corporate Research and Development, Schenectady, NY. His research interests are in the areas of magnetism and magnetic devices, su- perconductivity, and electronic components used in power supplies.

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Analysis of planar sandwich inductors by current images