IEEE TRANSACTIONS ON MAGNETICS, VOL. 21, NO. 4, JULY 1991 375 1

/// /// //

/

‘p = -v / POLE / REGION A

c / 2g

An Analysis of the Magnetic Field of a Ring Head with a Highly Permeable Underlayer

\

\ POLE

\ cp =. +v 3\

\

David T. Wilton

Abstract-An idealized mathematical model of the magnetic UNDER LAY ER field due to a ring head with a highly permeable underlayer is solved using Fourier analysis. Accurate Fourier coefficients are derived for a range of gap dimensions. The leading term of the resulting exact series is shown to be equivalent to a previously published approximation, but it is demonstrated that this term alone is not normally sufficient to predict the field accurately. REGION C

INTRODUCTION HE magnetic field due to an idealized semi-infinite T two-dimensional ring head with a finite gap has been

found by a variety of methods, most notably Fourier anal- ysis [ 11, conformal mapping techniques [2]-[4] , and finite difference and finite element methods [5], [6], each ap- proach having its advantages and limitations.

Following the suggestion by Iwasaki [7] of adding a highly permeable underlayer, or “keeper” layer, to im- prove the performance of single-pole heads for perpen- dicular recording, interest has been shown in using a ring head with such an underlayer [8]-[ 111 for perpendicular recordifig . Published theoretical solutions to this problem include a conformal mapping solution based on a pre- scribed approximation for the potential in the gap [2], a general conformal mapping solution [8], a Fourier inte- gral solution based on a linear gap potential approxima- tion [9], and a finite difference solution [6]. In this paper a Fourier analysis will be presented for which the approx- imation in [ 2 ] and [9] is seen to be the first term of the resulting exact infinite series obtained here. Also it will be shown that this leading term alone is not sufficient to predict accurately the field in the region close to the ring pole heads.

THE MAGNETIC FIELD Consider the mathematical model of a ring head with

an underlayer shown in Fig. 1 with infinitely permeable pole pieces at potentials + V and - V separated by a semi- infinite gap of width G = 2 g at a distance t from an in- finite plane at zero potential. The magnetic potential sat- isfies Laplace’s equation in the “T-shaped” region exte- rior to the poles and the infinitely permeable underlayer.

Manuscript received October 5 , 1990; revised December 21, 1990. The author is with the Department of Mathematics and Statistics, Poly-

technic South West, Drake Circus, Plymouth, Devon, PL4 8AA, UK. IEEE Log Number 9143166.

I 4 Y

Fig. 1. Ring head with an underlayer.

It is convenient to split this region into two parts A and C , where A is the gap region between the poles -g < x < g, y < 0 and C i s the region -CO < x < a, 0 < y < t in which the recording medium lies.

The Fourier analysis presented here follows that of Fan [ l ] in his study of a ring head without a keeper layer and consequently much of the algebraic detail will be omitted. The general solution of Laplace’s equation in region A satisfying the given boundary conditions at x = f g and which has the correct behavior as y + --09 is the series

while in the region C the corresponding solution is ex- pressed in integral form as

where the harmonic coefficients A, and the function C(k) are to be determined.

Matching the potential cpc (x, y) with the potential pA(x, y) and the pole potentials f V at y = 0, and taking a Fourier sine transform permits C(k) to be expressed in terms of the A, coefficients, so that ( 2 ) becomes

(3) 2 v -

cpc(x, y) = - Io + A,2n(- 1)” I,, , a n = l

0018-9464/91/0700-3751$01.~ 0 1991 IEEE

3752 IEEE TRANSACTIONS ON MAGNETICS, VOL. 21, NO. 4, JULY 1991

with

sin k sin k x / g sinh k(t - y ) / g dk . (4)

The coefficients A, may then be determined by match- ing the normal derivatives acp,(x, y ) / a y and acpc(x, y ) / a y along the line - g < x < g , y = 0 and using the orthog- onality properties of the functions sin m7rx/g, where m is an integer, to give the infinite set of linear algebraic equa- tions

I n = jm o k 2 - (n7r)' sinh k t / g

m

lr - A m + C (-l)m+nn.rrl,nAn = (-1)""VZm0, 4 n = I

where

k sin2 k coth k t / g dk . (6)

= .!om [k2 - [k2 - (n7r)'I

The normalized coefficients An/V, n = 1 , 2, 3, seen to depend only on the ratio g / t .

field in region C in particular are now

- . are

The horizontal and vertical components of the magnetic

m 2 v 2n 7rg

= -- Jo - C A n - ( - l )n J , (7) n = l g

with

k sin k cos k x / g sinh k(t - y ) / g dk (8) sinh k t / g

and

with

k sin k sin kx/g cosh k(t - y ) / g dk , (10) o k 2 - (n7r)2 , sinh k t / g Kn = im

respectively.

without an underlayer [ 11, [ 121 are The corresponding well-known results for a ring head

and

with

(13) sin k sin k x / g - k Y l g dk

!om k 2 - e z; =

where the coefficients A;, n = 1 , 2, 3, - - equations,

satisfy the

m

with

k sin2 k dk . (15) '" = !om [k2 - (ma)2] [k2 - (nn)*]

The magnetic field components in region C follow from (12).

It is clear from physical considerations that these two solutions should be similar in the region close to the ring head and within the gap as the keeper layer moves farther away from the poles, i.e., as g / t -+ 0. This correspon- dence will be discussed further in the following section.

RESULTS AND DISCUSSION The infinite system of linear algebraic equations (5 ) ,

(6) for the coefficients A,, may be solved approximately by restricting the system to some finite size N . For a ring without a keeper layer the integrals Z:,, are expressible in closed form [ 11. Unfortunately here the integrals Zmn have resisted attempts to find closed form analytic expressions, and relatively expensive numerical integration [ 131 has been used to evaluate them. This places a restriction on the size of N it has been possible to consider. Table I shows the first six normalized coefficients A n / V for a range of ratios g / t computed with N = 40.

As g / t + 0 it is observed, and will be confirmed be- low, that the coefficients A, do approach the correspond- ing coefficients A; for a ring head with no underlayer. This fact may be used to assess the accuracy of the coef- ficients computed here. Based on solving systems of or- ders up to N = 640 together with an extrapolation tech- nique, it is possible, for a ring head without an underlayer, to determine the coefficients A: very accurately [12] as AT/V = -0.086157, A f / V = 0.029150, A f / V = -0.015254, A f / V = 0.009593, A t / V = -0.006680, and A t / V = 0.004965.

A comparison with the coefficients for a ring head with- out an underlayer obtained from an N = 40 system, also shown in Table I, reveals that these latter values are ac- curate to within about one figure in the fourth decimal place. Consequently, this is the expected accuracy of all of the coefficients shown in Table I.

Once the Fourier coefficients are known, the magnetic field components may be found from (7)-(10). While the integrals corresponding to (8) and (10) may be evaluated in closed form for a ring head with no underlayer [14], this has not yet proved possible here for all values of x and y and it is again necessary to use numerical integra- tion [13].

Representative horizontal and vertical field components for the case g / t = 0.5 are shown in Figs. 2 and 3 where the magnetic field has been normalized by the deep gap

WILTON: MAGNETIC FIELD OF RING HEAD 3753

TABLE I COEFFICIENTS A, / V COMPUTED FROM A N N = 40 SYSTEM

Ring Head g / t 1 .o 0.75 0.5 0.25 0.1 0.025 AII

A l / V -0.1322 -0.1144 -0.0996 -0.0896 -0.0866 -0.0861 -0.0861

A 3 / V -0.0209 -0.0187 -0.0169 -0.0156 -0.0153 -0.0152 -0.0152

A S / V -0.0089 -0.0080 -0.0073 -0.0068 -0.0067 -0.0066 -0.0066

A , / V 0.0412 0.0366 0.0327 0.0300 0.0292 0.0291 0.0291

A , / V 0.0129 0.0116 0.0105 0.0098 0.0096 0.0095 0.0095

A , / V 0.0066 0.0059 0.0054 0.0051 0.0049 0.0049 0.0049

Legend a & = A 3

yLG yo.?

v/c. p:9

./G

Fig. 2. Horizontal magnetic field of a ring head with an underlayer, g / f = 0.5.

11 I

I I I I I

-2 -1.5 - 1 -0.5 0 0.5 1 1.5 2

x/G

Fig. 3. Vertical magnetic field of a ring head with an .underlayer, g / t = 0.5.

field Hg = V / g and all distances have been normalized by the full gap width G = 2s. Note that the negative -H, /Hg of the horizontal field has been plotted for con- sistent visual comparison with other work. There seems to be some confusion in the literature over the correct sign of HI, even allowing for the reversed polarity of the ring poles in some cases. The horizontal field decays to zero as x / G -+ ko3 while the normalized vertical field tends

Figs. 4 and 5 show the corresponding solutions for a ring without an underlayer. The horizontal component of

to k g / t .

Legend

&=A3

v/c=.l r/c= 9.9

yLG ~ 0 . 2

-2 -1.5 - 1 -0.5 0 0.5 1 1.5 2

./G

Fig. 4. Horizontal magnetic field of a ring head without an underlayer.

'1 0.5 -

I b 2 (U z 0-

U

(U

0 m I

+

-0.5 -

I I I

-2 -1.5 - 1 -0.5 0 0.5 1 1.5 2

x/G

Legend

&&=A3

e . 7 v/c= 0:9

yLG EO.?

Fig. 5 . Vertical magnetic field of a ring head without an underlayer

the field is generally lower in amplitude with an under- layer than without but may have slightly steeper gra- dients, although very close to the ring poles higher am- plitudes are achieved with an underlayer. The vertical field component has increased amplitude in the presence of an underlayer and increased gradients underneath the gap but decreased gradients below the pole pieces. These results are consistent with previous observations [6], [8], [9] al- though only [6] considered the region y / G < 0.5 close to the poles with a numerical finite difference solution. All results were obtained by truncating the infinite series (7) and (9) each to six terms only. For y / G < 0.1 more terms would be required.

IEEE TRANSACTIONS ON MAGNETICS, VOL. 27, NO. 4, JULY 1991 3754

It is instructive to compare the analytical results for the ring head both with and without an underlayer in greater detail. For k t / g >> 1, coth k t / g = 1 and some analysis shows that as g / t + 0, I,,,, -+ I:,, leading to A, -+ A: as expected and as already observed in Table I, and cpA(x, y ) --t cp,”(x, y ) . Also, the integral (4) may be written in the equivalent form

sin k sin k x / g e-kT/g - e - k r l ~ sinh /cy/g] dk.

(16)

sinh k t / g In = !ow k 2 - (nr)* r As g / t + 0 and for y / t << 1 this tends to the integral I f (13) and so cpc(x, y ) -+ cp;(x, y ) . The familiar solution for a ring head without an underlayer is therefore seen to be a limiting case of the more general solution derived here as g / t -+ 0 and for y / t << 1 , i.e., everywhere in the gap region A and in region C close to the ring poles.

The first term in (16) is that for a ring with no under- layer while the second term may be considered a “cor- rection” term due to the presence of an underlayer. Of course, except for the leading n = 0 terms, the coeffi- cients of these integrals will be different in the two series (3) and (12).

The exact infinite series solution derived here may also be compared with previous analytical results for this prob- lem [2], [9]. Karlqvist [2] used conformal mapping tech- niques to obtain the potential function as an integral along the line y = 0 based on a prescribed potential variation along this line. Assuming a linear variation in the gap he obtained an expression ([2], equation (23)) for the vertical component of the magnetic field along the line y = t , the surface of the underlayer.

Bloomberg [9] generalized the results of Karlqvist based on a Fourier integral approach but assumed a linear vari- ation in potential across the gap ( - g < x < g, y = 0). He obtained approximations for the magnetic field com- ponents as, in the notation and style of this paper,

2 V jw sin k cos h x / g

g r 0 k M x , Y ) = --

e -krlK sinh ky /g] sinh k t / g

dk (17)

and

e - k t / g cosh k y / g ] e-kY18 + d k . (18)

sinh kt /g

These are seen to be precisely the first terms in the full expressions (7) and (9) for the horizontal and vertical components respectively if the alternative representation (16) for Io is taken. For y = t , H,(x, t ) = 0 and HY(x, t ) reduces to the expression given by Karlqvist [2].

These leading terms for a ring head with an underlayer are equivalent to the popular Karlqvist arctangent and log-

1

0.8 rn $

1 0.6

a,

U f 0.4

C

9 L

0, Legend

I & o w 5 1

LA)

r / O l O . l c t )

0-2 LG=o’m

d?;O B PI 0

-2 -1.5 - 1 -0.5 0 0.5 1 1.5 2

x/G

Fig. 6. Comparison of the exact horizontal magnetic field of a ring head with an underlayer (E-curves) with that predicted by the leading term alone (A-CUNeS), g/ t = 0.5.

arithmic approximations to the field components of a ring head with no underlayer. They correspond, as mentioned above and as in the case of no underlayer, to a linear vari- ation in potential across the gap, as may be easily verified by direct evaluation of the leading term in (3) for y = 0 for which I,, = I f .

It is well-known that in the case of a ring without an underlayer the Karlqvist approximations differ signifi- cantly from the true fields in the region close to the poles [ 143-[ 161, these works naturally concentrating on the hor- izontal field for longitudinal recording. Similarly in the case of a ring with an underlayer, the leading terms (17) and ( 1 8) are not sufficient to predict accurately the true fields. The differences are most significant in the gap re- gion and up to just past the pole edges and increase as y / G gets smaller. Figs. 6 and 7 show for the case of g / t = 0.5 considered previously this effect at y / G = 0.1, 0.3, 0.5 where six additional terms of the infinite series have been taken for comparison. For y / G < 0.1 the dif- ferences are even more pronounced but also more terms of the infinite series would be needed to give the exact solution to acceptable accuracy. Only for y / G 2 0.5 could these leading terms be considered reasonable ap- proximations for all x / G.

There are three main approaches in use today for cal- culating head fields: conformal mapping techniques, fi- nite difference/element methods, and Fourier analysis. While the first two methods may have advantages in terms of speed and the ability to handle complex geometries re- spectively, neither gives a convenient representation of the solution for further calculation or for parameter in- vestigation. Fourier analysis does provide such a repre- sentation albeit in the form of an infinite series. It is some- times implied that accurate calculation of the Fourier coefficients or evaluation of the field integrals is a prob- lem. However, reliable automatic integration routines and linear equation solvers are widely available today on both mainframe computers and PC’s, e.g., in the NAG [17] and IMSL (181 subroutine libraries, and these computa- tional “difficulties” may be easily overcome. The possi-

WILTON: MAGNETIC FIELD OF RING HEAD 3155

l1 I

I I I

- 2 -1.5 - 1 -0.5 0 0.5 1 1.5 2

x/G

Fig. 7. Comparison of the exact vertical magnetic field of a ring head with an underlayer (E-curves) with that predicted by the leading term alone (A-curves), g / t = 0.5.

ble disadvantage of a Fourier series solution, apart from its limitation to relatively simple geometries, is the slow convergence of the series in certain regions when a large number of terms, perhaps combined with an extrapolation technique, may be necessary to achieve the desired ac- curacy. However, as the cost of computing decreases, this presents less of a problem.

CONCLUSION An exact analytical solution that is amenable to com-

putation for a model of the magnetic field of a ring head with an underlayer has been derived. It has been shown that the presence of an underlayer can significantly influ- ence the magnetic field and that commonly used approx- imations are not sufficiently accurate in the region close to the ring head.

REFERENCES G. J. Y. Fan, “A study of the playback process of a magnetic ring head,” IBMJ. Res. Devel . , vol. 5 , pp. 321-325, Oct. 1961. 0. Karlqvist, “Calculation of the magnetic field in the ferromagnetic layer of a magnetic drum,” Trans. Roy. Inst. Technol. Stockholm,

N. Curland and J. H. Judy, “Calculation of exact ring head fields using conformal mapping,” IEEE Trans. Magn. , vol. MAG-22, no.

vol. 86, pp. 1-27, 1954.

6 , pp. 1901-1903, NOV. 1986.

J. S. Yang and H. L. Huang, “Calculation of exact head and image fields of recording heads by conformal mapping,” IEEE Trans. Magn., vol. 25, no. 3, pp. 2761-2768, May 1989. T. J. Szczech, D. M. Perry, and K. E. Palmquist, “Improved field equations for ring heads,” IEEE Trans. Magn. , vol. MAG-19, no. 5, pp. 1740-1744, Sept. 1983. T. J. Szczech and K. E. Palmquist, “A 3-D comparison of the fields from six basic head configurations,” in Int. Con$ Video and Data Recording, IERE Conf. Proc . , no. 59, pp. 17-22, Apr. 1984. S . Iwasaki, “Perpendicular magnetic recording,” IEEE Trans. Magn., vol. MAG-16, no. 1 , pp. 71-76, Jan. 1980. 0. Lopez, W. P. Wood, N. H. Yeh, and M. Jursich, “Interaction of a ring head and double layer media-field calculations,” IEEE Trans. Magn. , vol. MAG-18, no. 6, pp. 1179-1181, Nov. 1982. D. S . Bloomberg, “Spectral response from perpendicular media with gapped head and underlayer,” IEEE Trans. Magn., vol. MAG-19, no. 4 , pp. 1493-1502, Jul. 1983. 0. Kitakami, Y. Ogawa, S . Yagamata, and H. Fujiwara, “Improve- ment of the reproduced output of CO-Cr thin film media by insertion of very thin soft magnetic underlayer,” IEEE Trans. Magn. , vol. 25, no. 5, pp. 4177-4179, Sept. 1989. S. Iwasaki and K. Ouchi, “On design of CO-Cr perpendicular re- cording media for usage of a ring-type head,” IEEE Trans. Magn. , vol. 26, no. 1 , pp. 97-99, Jan. 1990. D. T. Wilton, “A comparison of ring and pole head magnetic fields,” IEEE Trans. M u m . . vol. 26. no. 3. DD. 1229-1232. Mav 1990. ” , I ‘ . I ,

[13] R. Piessens, E. deDoncker-Kapenga. C. Uberhuber, and D. Kahaner, QUADPACK: A Subroutine Package for Automatic Integration. New York: Springer-Verlag, 1983.

[14] A. W. Baird, “An evaluation and approximation of the Fan equations describing magnetic fields near recording heads, ” IEEE Trans. Magn., vol. MAG-16, no. 5, pp. 1350-1352, Sept. 1980.

[15] B. K. Middleton and A. V. Davies, “Gap qffects in head field distri- butions and the replay process in longitudinal recording,” in Int. Conf. Video and Data Recording, IERE Conf. Proc. , no. 59, pp. 27-36, Apr. 1984.

[16] H . L. Huang and H. Y. Deng, “Comparison of ring head and SPT head write fields,” IEEE Trans. Magn. , vol. MAG-22, no. 5 , pp. 1305-1309, Sept. 1986.

[17] NAG, Numerical Algorithms Group Ltd, Oxford, UK. [ 181 IMSL, International Mathematical and Statistical Library, Houston,

TX .

David T. Wilton received the B.A. degree in mathematics from the Uni- versity of York, UK, in 1969 and the D.Phil. in numerical analysis from the University of Oxford, UK, in 1974.

He spent three years at the University of Dundee, Scotland, and then three years with the Ministry of Defence working on dynamic fluid-struc- ture interaction problems in underwater acoustics. Since 1978 he has been lecturing mathematics at Polytechnic South West, Plymouth, UK. During 1987-1989 he spent two years at the City Polytechnic of Hong Kong. His main research interests are in numerical analysis and applied mathematics in the areas of acoustics and electromagnetics.