Chapter 2. Inductive Head Field Functions and Their Fourier Transforms Introduction

Transcription

1 Chapter Inductive Head Field Functions and Their Fourier Transforms.. Introduction In the writin process, the manetic core in an inductive head serves to enerate a concentrated field in the ap reion. This field is limited by the saturation manetisation of the core material. Therefore core M s should be lare. However, when the head is not enerised when switchin from one current sense to the other, the field is required to be zero to avoid unwanted writin. This implies low remanance. Also to avoid hysteresis losses the coercivity should be very small. Such low coercivity materials are said to be manetically soft. The core material should also have hih permeability to provide an easy path for the coil flu. Since the data transfer rate is important, these heads must maintain their properties to very hih frequencies ( MHz). The eometry of the transducers should also be desined so that they are efficient in terms of the translation of input current to recordin head field at all operatin frequencies. For inductive heads, the most widely used class of materials was ferrites in particular NiZn and MnZn. With their hih resistivity, ferrite heads are limited to frequency dependent hysteresis power loss. Two disadvantaes of ferrite heads have led to the development of new materials. A low saturation flu density limits their use to recordin on media with coercivities no reater than Oe. Furthermore, the permeability falls off rapidly with increased frequency due to domain wall motion loss. Metallic film heads, such as nickel-iron, offer hiher saturation manetisation than 4

2 Chapter ferrites makin them attractive for writin on hih coercivity media. Metallic alloys have very hih permeabilities and low resistivities compared to ferrites. They are thus used in the form of thin-sheets or laminations to minimise eddy current losses (time varyin manetic flu density in ferrites induces currents in the core which accordin to Lenz s law, tends to oppose the chane in the field inducin it; eddy currents result in joule heatin and thus enery loss in the head core (Kraus, 99)). Reduction of head dimensions was limited by restrictions imposed by the fabrication process of ferrite and metal heads. The thin-film techniques developed by the electronics industry offer ecellent control over critical head dimensions. Common materials used in thin-film heads are sendust (AlFeSi) which are often used as pole tips because of their mechanical hardness. Permalloy is another class of materials used in thin-film heads. Permalloys are made of alloys of Fe and Ni, which due to their composition have low coercivity and hih permeability. Their low resistivity, however, requires laminated structure to minimise eddy current loss. They are also mechanically soft for which they are limited to non-contact applications i.e. flyin heads. The write head eometry and characteristics shape the written manetisation transition in the medium. To analyse the recordin process correctly, the head field must be described accurately. In the replay process, the output voltae is effectively iven by the reciprocity formula correlatin the head sensitivity function with the diverence of the recorded manetisation. The enerated head fields also influence the shape of the medium noise spectrum. Thus in order to fully understand and characterise these processes, the head fields must be interated in the analysis. This chapter will attempt to review and, where appropriate, derive closed form epressions for the two dimensional fields from semi-infinite apped heads and thin-film heads. In semiinfinite apped heads, the poles are assumed to be infinitely lon and infinitely deep with finite ap lenth. In addition to the finite ap lenth, thin-film heads are characterised by their finite pole lenths and finite head depths (throat heihts). The derivations will be based upon the assumption of hih permeability cores. This allows the fields to be determined everywhere above the head from the scalar manetic potential at the head surface. 5

3 Chapter The chapter will start by illustratin the parameters that limit the head efficiency in inductive heads and derive an epression for the field in the ap reion. This will be followed by analyses of the simple epression iven by Karlqvist (954) for the semiinfinite apped head. This approimation will then be compared with more accurate field calculations of Fan (96; 96) and Ruirok (99) for the same head eometry. Thin-film heads will then be discussed as a eneralisation of inductive heads. The validity of the simplified assumptions used in derivin the approimate field equations will be investiated by consideration of the lon wavelenth response of inductive heads. In all inductive heads, current is applied to the coil turns that results in flu circulatin the structure. In the head core the flu density, B, is predominately due to the manetisation since the internal fields are small. In the tip reion the flu density is larest because of the decrease in cross-sectional area, A, ( B = φ / A, φ is the flu) and saturation enerally beins in this reion. A relatively lare, steady current throuh the coil causes pole tip saturation, which chanes the head permeability and thus alters the surface potential distribution, and in turn modifies the head fields. To prevent appreciable manetic saturation in the pole tips, it is necessary that H µ.6bs / (Mallinson, 996) where H is the deep ap field and B s is the saturation flu density of the poles. In the ap reion, the flu density is due solely to the manetic field i.e. B = µ oh. For hih density diital recordin, the field in the ap must be H πh c (Mallinson, 996; Charap, 997) where H c is the storae medium coercivity. Flu that frines into the reion above the ap where the medium passes provides the recordin field. o In the forthcomin analysis effects such as contoured poles (rounded tip corners), pole tip saturation, finite head coil dimensions and coil fields will be inored for simplicity. The head permeability will be considered fied at different write frequencies to maintain the constant manetic scalar potential at the head surface. 6

4 Chapter.. Head Efficiency The head efficiency is the same for both readin and writin (Mallinson, 993). In this analysis, the epression for the head efficiency is derived from the write process. An inductive head can be represented usin an equivalent manetic circuit as shown in Fiure.. A c R c / I N, A NI R " (a) R c / (b) Fiure. Diaram showin (a) an inductive head with (b) its equivalent manetic circuit. In Fiure., N is the number of coil turns, I is the applied current, " is the lenth of the flu path in the core, is the ap lenth and A c and A are the core and ap crosssectional areas respectively. The equivalent circuit is driven by a manetomotive force (mmf) of manitude NI. In electrical circuits, resistance is iven by: " R = (.) σ A where σ is the conductivity. Similarly, for a manetic circuit the reluctance is iven as: where µ (Hm - ) is the permeability and is defined as permeability (dimensionless) and " R = (.) µ A o 7 µ = µ r µ o where µ r is the relative µ = 4π Hm is the permeability of free space. It can be seen that (.) and (.) have the similar form with the conductivity, σ, in (.) replaced by the permeability, µ, in (.). Thus in Fiure., R c and R are the head core and ap reluctances respectively. The frinin field reluctances above 7

5 Chapter and below the head ap are drawn in the diaram as dashed lines in parallel with the ap reluctance. The frine field reluctances are normally hih compared to the ap reluctance and are thus inored. The inductive head is required to transfer the mmf enerated in the head windins efficiently to the ap reion where the recordin medium passes. A measure of this ability is defined as: mmf across the ap E fficiency( η ) = (.3) mmf of the coil The flu around the equivalent circuit usin Kirchhoff s law (Fiure.) is: NI R + R c and thus the mmf across the ap reluctance is the efficiency is therefore: R c NI + R R. Substitutin in (.3), NI R Rc + R R η = = (.4) NI R + R Writin the efficiency in terms of the head reluctances defined in (.) ives: η = A " + Ac µ r c (.5) where µ r is the relative permeability of the core. It can be seen from (.5) that the head efficiency can be improved by increasin the ap lenth, maintainin hih core permeability and optimisin the head eometry usin a lare A c /A ratio. Ampere s circuital law states that the mmf around a closed path equals to the total current enclosed. Thus the total mmf around the core and ap is iven by the closed interal: H d " = NI (.6) which can also be written for the head structure of Fiure. as: Hcore " + H = NI 8

6 Chapter where H is the ap mmf and H core " is the mmf around the core. Substitutin into H (.3) for neliible H core, the efficiency can now be written as: η =. Solvin for NI H yields the deep ap field: H NIη = (.7) It is assumed that the ap depth is much larer than the ap lenth, in which case the potential deep in the ap varies approimately linearly across the ap, yieldin a constant deep-ap field, H (Bertram, 994)..3. The Semi-Infinite Gapped Head Structure Fiure. shows a semi-infinite head structure where the pole pieces and the head depth tend to infinity. The centre of the coordinate system is the mid-point of the ap lenth. y, y` µ =, ` -V +V µ = Fiure. Semi-infinite apped head structure and coordinate system. In order to derive closed form epressions for the fields beyond the head surface and into the reion of interest, simplifyin assumptions have to be made. With the hih scalar permeability of the core material (assumin no losses at hih write frequencies), the pole faces are considered at constant manetic potential, V = NI / and + V = NI /, as shown in Fiure.. As a result, the flow of flu is from the riht pole to the left pole in approimately a circular contour implyin a neative horizontal field component, H, and a positive vertical field component, H y, directed upwards from the riht hand pole and downwards at the left hand pole. This notation for the 9

7 Chapter flow of flu will be maintained throuhout this thesis. The track width and head depth dimensions are taken to be lare compared to the ap lenth dimension, which reduces the field problem to the two co-ordinate aes; and y. When the vector head fields are determined from the scalar surface potentials, scalin by the head efficiency is necessary. Due to the very hih permeability, the head efficiency factor will be omitted in the discussion of manetic potentials ( η when µ r = ) but, however, is implied whenever the vector field equations are derived or implemented..3.. Karlqvist (954) Linear Gap Potential Approimation Usin the above simplifications and assumin a linear variation of the manetic potential between the ap corners, Karlqvist (954) derived an approimate closed form epression for the horizontal and vertical components of the field produced by a semi-infinite apped head. For unit permeability of the manetic storae medium, the potential distribution alon the head surface (y = ) was written as: V, < - V ϕ(,) =, - < < (.8) ( / ) + V, > Equation (.8) is illustrated diarammatically in Fiure.3. ϕ(,) V -V Fiure. 3 Surface potential distribution for a semi-infinite apped head (Karlqvist, 954). 3

8 Chapter For infinite head-to-medium separation, the potential everywhere beyond the head surface boundary is iven by the two dimensional Green s function (Karlqvist, 954; White, 984): ϕ(`,) `= ( `) + y ϕ(, y) = d` (.9) π y where (`,) denotes the head surface and (,y) defines the location of the observation point. Substitutin (.8) into (.9), and interatin alon the surface boundary notin that V = NI / = H / ives the scalar manetic potential as: / + / ( / + ) tan ( / ) tan H y y ϕ(, y) = (.) π y ( / + ) + y ln ( / ) + y The vector field, H, above the head surface can be obtained from a scalar potential usin: H = - ϕ (.) Thus the surface field for a Karlqvist type head is constant in the ap reion and vanishes at the pole faces (rectanular function). On substitution from (.) and evaluation of the derivatives, the horizontal field component is iven by: ϕ(, y) H / + / H (, y) = = tan + tan (.) π y y and the vertical component is: ϕ(, y) H ( / + ) + y H (, y) = = ln (.3) y π ( / ) + y Karlqvist has found that equations (.) and (.3) provide a ood approimation for the actual fields from a semi-infinite apped head for values of y > / 4. For very lare distances beyond the head surface, the semi-infinite apped head can be approimated by a semi-infinite block with a vanishinly thin ap, i.e.. Thus the manetic potential alon the pole faces in this approimation reduces to: V, - < < ϕ(,) = + V, < < 3

9 Chapter Evaluation of the potential usin the two dimensional Green s function and differentiatin with respect to the co-ordinate aes ives the field components for an infinitely narrow ap head (known also as the far field approimation) in the form: H H y H y (, y) = (.4) π ( + y ) H (, y) = (.5) π ( + y ) Fiure.4 compares the vector fields produced by the Karlqvist and narrow ap head equations for y / =. 5. In Fiure.4, the lonitudinal field component is symmetric and bell shaped while the vertical field component ehibits odd symmetry and vanishes at the ap centre. To quantify the error in the approimations, the percentae root mean square error estimate is used; it is defined as (Wells, 985): ε RMS N qˆ i qi = (.6) N i= qi ε RMS is a measure of the oodness of fit between the approimated quantity qˆ and its accurate value q. It was estimated, usin (.6), that the narrow ap approimation differs by 8% from the more accurate Karlqvist field for y / =. 5. The percentae error increases to 36% for y / =.. At lare distances from the head surface, on the other hand, for eample y / =, the difference reduces to 3% as epected since the Karlqvist approimation approaches the far field limit. 3

10 Chapter.4. Karlqvist Narrow ap head. H (,y)/h / (a) Karlqvist Narrow ap head H y (,y)/h (b) Fiure. 4 (a) Lonitudinal field and (b) vertical field components of a semi-infinite head evaluated usin the Karlqvist and narrow ap head approimations for y/=.5. / 33

11 Chapter It must be noted that the Karlqvist and narrow ap field approimations only apply for values of y. Since these field equations were derived based on the potential distribution on the head surface, they yield, incorrectly, twice the value of the deep ap field for values of y <. Alternativly, by derivin the field equations arisin from equal and opposite sheets of manetic chares on the inner ap surfaces, half the Karlqvist field is obtained producin the correct deep ap field for values of y < (Hoaland and Monson, 99). In this case, a correction factor of must be accounted for when evaluatin the fields beyond the head surface..3.. Fourier Transform of Laplace s Equation The Fourier transform pair of a spatial function f() are defined by: = jk d F (k) = f () e (.7) = f () = F(k) e jk dk (.8) π where k = π / λ is the wavenumber, λ is wavelenth and j =. Reardless of the head eometry or the manetisation pattern, all potentials obey Laplace s equation in free space, which in two dimensions is written as: ϕ(, y) ϕ(, y) ϕ = + = (.9) y The Fourier transform of Laplace s equation with respect to ives a differential equation involvin y only: ϕ y whose eneral solution is iven by: ( k, y) + k ϕ(k, y) = ky ky ϕ (k, y) = A(k)e + B(k)e (.) If the reion of interest is completely to one side of the field sources, such as the reion above a head with no permeable material present, then the boundary conditions include that the potential vanishes at y = and that at the source plane, y =, the 34

12 Chapter Fourier transform of the potential is ϕ(k,). Thus the solution to Laplace s equation for the potential in (.) is (Mallinson, 974): k y ϕ (k, y) = ϕ(k,) e (.) Usin (.), the vector fields follow a similar relationship since the Fourier transform of the radient operator is: =, jk, y y where the two-way arrow indicates Fourier transformation. The Fourier transform of the vector field is thus: (- jkϕ(k, y), k ϕ(k, y) ) H (k, y) = (.) and hence the field components are (Mallinson, 974; Bertram, 994): H H y k y (k, y) = jkϕ(k, y) = H (k,) e (.3) (k, y) = k ϕ(k, y) = ± jsn(k) H (k, y) (.4) where sn(k) denotes the Sinum function and represents the sin of k and vanishes at k =. Consequently, the spectrum of the frinin field of a head (or medium) can be determined completely by the field or the potential spectrum on the top surface of the head (or medium). Equations (.3) and (.4) are lobal relations and apply to fields in free space where the field source is only on one side of the reion of interest. It can also be observed that the Fourier transforms of the field components are identical in manitude and shifted in phase by ±π/ (the j factor in (.4)) dependin on the sin of k. Thus if the horizontal field component is known, the correspondin vertical component can be determined. This is translated into the spatial domain usin the Hilbert transform, h(), which for a function f() is iven by the formula (Mallinson, 973): ( r= f (r) h () = dr (.5) π r) where r is a dummy variable. The Fourier transform of (.5) is effectively iven by equation (.4). Thus, all two dimensional field components in free space, on one side of a source, are Hilbert transforms of each other. This technique has been used in conjunction with microloop measurement of the vertical field from a thin-film head to 35

13 Chapter evaluate the horizontal field component (Weismehl et al, 988). All heads also ehibit the same k y e 974; White, 984). type of spacin loss behaviour (Wallace, 95; Mee, 964; Mallinson, Since the surface field of the Karlqvist field is the rectanular function, then its Fourier transform is simply the sinc function (Briham, 988). Consequently, accordin to (.3), the -component of the Fourier transform of the Karlqvist field is iven by: H sin(k / ) k y (k, y) = NIη e (.6) (k / ) where NIη = H. The spectrum of the Karlqvist field is thus the product of the sinc function, which is the surface field transform at y =, and the spacin loss term. The spectrum of the surface field represented by the sinc function is an oscillatin function with decreasin amplitude as shown in Fiure.5. The surface field spectrum also vanishes at ap-nulls that are submultiples of the ap lenth. The first null occurs eactly at λ =.. H (k,)/niη.8.4 Karlqvist Lindholm Narrow ap head Fiure. 5 Gap loss function spectra usin Karquist and narrow ap approimations compared with Lindholm s accurate spectrum. /λ 36

14 Chapter In real heads, however, the effective ap is approimately % loner than the mechanical ap lenth (Westmijze, 953). Lindholm (975) ave an accurate closed form epression for the spectrum of a semi-infinite apped head obtained by curve fittin to tabulated values of the eact field transform produced by Westmijze (953). The fitted surface field wavelenth response, denoted by S, depends only on the ratio /λ and is iven in terms of the dummy variable f as: sin(.πf ), (.πf ) S ( f ) = NIη.36 sin -/3 f sin( π( f + / 6) ) ( π( f / 6) ), +.55 f 4 / 3 f <.5 (.7) f.5 Equation (.6) is also plotted in Fiure.5 with f = /λ. In (.7) and from Fiure.5, the first ap-null occurs at / λ = in areement with Westmijze s tabulated values. Hence the first ap null occurs in reality at a wavelenth λ.36. Multiplyin the ap lenth in the arument of the sinc function in (.6) by a factor of.36 ives the Karlqvist field spectrum accurately up to the first ap null (Bertram, 994): sin(.36k / ) k y π H (k, y) NIη e, k < (.8) (.36k / ).36 Generally, for wavelenths shorter than.36, accurate results can be obtained usin equation (.7). The wavelenth response of the narrow ap head field is simply found by settin = in (.6). In the limit, the surface spectrum is equal to the constant NIη illustrated by the dotted horizontal line normalised to unity in Fiure.5; hence the field transform becomes: H k y (k, y) = NIη e (.9) The Karlqvist field equations and their spectra alon with the narrow ap simplified formulae are used etensively in the recordin and replay studies. This is mainly due to the ease in which such equations can be utilised to yield closed form epressions that ive insiht into the eneral properties of these processes. To investiate the 37

15 Chapter validity of the simplified and widely used Karlqvist epressions, a more riorous study of the semi-infinite apped head structure that does not assume a linear variation of potential in the ap reion must be performed. This will allow the simplified Karlqvist equations to be compared with more accurate field calculations and indicate the conditions in which the simple Karlqvist epressions can be considered as close approimations to real fields Fan (96) Fourier Analysis In his study, Fan (96; 96) has derived an eact solution for the semi-infinite apped head potential problem in the form of the Karlqvist approimations plus an infinite series of correction terms. This involved solvin the two dimensional Laplace s equation of the potential for an infinitely permeable head with the confiuration iven in Fiure., and a medium with unit permeability subject to boundary conditions. The satisfyin conditions were that the two pole pieces are at equipotential (and hence by symmetry, ϕ = at = ). Moreover, the potential vanishes at lare distances from the head surface i.e. in the direction of increasin y, and that the potential between the infinite poles varies linearly deep in the ap (Mee, 964). The solution of Laplace s equation for the potential and the derivations of Fan s equations are iven in detail in Appendi I. The manetic potential in the ap reion, denoted by ϕ A, is iven by: ϕ π < = + y V n / A (, y) A n sin e, (.3) ( / ) n = y while the potential above the pole faces, ϕ C, is: α= 38 nπ αy < < ϕc (, y) = sin( α) C( α)e dα, (.3) y > In (.3), α is an arbitrary constant satisfyin the boundary conditions, V = NI / and C(α) is determined from matchin the potentials above and in the ap reion at the head surface (y = ). The coefficients A n in (.3), normalised to V, are determined from the solution of an infinite series of linear alebraic equations and are independent of the ap lenth (Appendi I). Wilton (99) ave the first si coefficients correct to

16 Chapter 6 decimal places usin a truncated series of 64 terms with etrapolation. Huan and Den (986) and Baird (98) have also provided the first si coefficients but with a truncated series of only 6 terms and are thus less accurate than the coefficients obtained by Wilton. The first coefficients were computed usin LU decomposition (Press et al, 994) with N N set of linear alebraic equations where N = and are in areement to 6 decimal places with Wilton s (99) calculations. Until recently, the solution of a lare number of linear alebraic equations was the only means by which the Fan coefficients could be calculated accurately. Furthermore, there is no other method or tabulated results to check the residual error in the evaluated coefficients. By epandin a conformal mappin solution for the potential of a Karlqvist type head eometry and matchin to Fan s (96) solution for the potential in the ap reion, it is now possible to determine the Fourier coefficients eactly (Wilton, Middleton and Aziz, 999). Usin Schwarz-Christoffel transformation, Westmijze (953) has derived an implicit epression for the semi-infinite apped head field in the comple z-plane. However, the iven function cannot be inverted to yield the field components as a function of position and thus numerical inversion has to be performed. Nevertheless, at the boundary =, the transformation function reduces to the simple form: where U = H (, y) H U y = ln (.3) π U + U. Equation (.3) is valid for all values of y (includin neative values) in the ap reion and above the head surface. Rewritin equation (.3) in the form: πy U = tanh U which for lare neative values of y can be epanded into the series (Dwiht, 96): n= n n / U nπy / U = + ( ) e e (.33) 39

17 Chapter Differentiatin the potential in the ap reion (.3) with respect to to obtain the field and usin the same notation in (.33), the lonitudinal component of the field in the ap reion at = usin Fan s Fourier series is: A n nπy / U = + nπe (.34) V n= The normalised coefficients A n /V can then be evaluated by matchin the coefficients of powers n y / e π in the two series (.33) and (.34). However, (.34) must first be substituted into the eponential term of equation (.33) since it is implicit in U. Before substitution, let A n n n y / ε = e π and βn = nπ for simplicity, then (.34) becomes: V substitution into (.33) yields: n= n U = + β ε (.35) n n= n + n βnε n n= n U = + ( ) e ε (.36) Dependin on the value of n, the eponential term in (.36) is epanded into a Taylor series and the coefficients of ε n are matched with (.34) to yield: n = : n = : β β = e 4 4 = 4e β + e = e n = 3: etc β = 4e β + 8e β e = 58e As n increases, the number of alebraic manipulations and eliminations per term becomes increasinly difficult. However, the computer alebra packae Mathematica (996) was used to evaluate further values of the normalised Fan s coefficients. Table lists the first eact normalised coefficients alon with their numerical equivalents with accuracy up to decimal places. 4

18 Chapter n A n /V A n /V correct to d.p. 3 e π π 4 5e 6 58e π e π 6934e π 388e π 9897e π 96785e π e π e π Table. Eact Fourier harmonic coefficients for Fan s equations. 4

19 Chapter n A n /V A n /V correct to d.p e 75π e 3π e 6875π e 6875π e 9555π e 583π e π e 9445π e π e 4547π (Continued) Table. Eact Fourier harmonic coefficients for Fan s field equations. 4

20 Chapter Havin obtained the Fourier harmonic coefficients, the potential in the ap reion at the head surface, ϕ A (,), can now be evaluated. Fiure.6 illustrates the normalised surface potential in the ap reion accordin to (.3) for n =, n = 6 and n = respectively...5 n = (Karlqvist) n = 6 n = ϕ(,)/v Fiure. 6 Surface potential calculated usin Fan s Fourier analysis for different number of harmonic coefficients. / In Fiure.6, the zeroth harmonic ( n = ) ives correctly the Karlqvist linear ap potential approimation where the summation in (.3) disappears. In the Fan calculations, the slope of the actual surface potential increases at the ap corners as shown in Fiure.6. Usin (.6) it was estimated that the actual surface potential for a rin type head, as measured usin Fan s analysis with n =, differs from the Karlqvist linear approimation by as much as 7% in areement with Lindholm (975). It can also be seen that evaluatin the surface potential with 6 correction terms is adequate to ive close areement to the eact solution (Huan and Den, 986; Mallinson, 99). The root mean square error percentae between the surface potentials with n = 6 and n = was estimated to be within % where the reatest disareement occurs at the ap corners. Since the harmonic coefficients were 43

21 Chapter determined eactly, terms will be used in the series evaluation of the subsequent field equations. The field components everywhere above the head surface for a semi-infinite head usin Fan s harmonic analysis are iven by (Appendi I): H H (, y) = π ( / ) tan / + + tan y n n( ) An n= α= / y αsin( α / ) cos( α)e [ α (nπ / ) ] αy dα (.37) H y H y (, y) = ln π y + ( / ) + ( / + ) + ( / ) n n( ) A n n= α= αsin( α / ) sin( α)e [ α (nπ / ) ] αy dα (.38) The first terms (or the zero harmonic terms) on the left hand side of H and H y in (.37) and (.38) are effectively the Karlqvist equations as a consequence of the linear potential obtained for n =. Added to the Karlqvist epressions is an infinite series of terms that can be considered as correction terms to the Karlqvist approimation. Closed form solutions for the interals in Fan s epressions were derived by Baird (98) in terms of the comple eponential interal (Appendi I) allowin the evaluation of the vector fields. The horizontal and vertical field components computed usin Fan s equations are demonstrated in Fiure.7 at different separations from the head surface. 44

22 Chapter..8 y/=.5 y/=. y/=.4 y/=.6 H (,y)/h / (a) H y (,y)/h y/=.5 y/=. y/=.4 y/= / (b) Fiure. 7 Plot of (a) lonitudional and (b) vertical field components as calculated usin Fan s equations for different spacin-to-ap lenth ratios with n=. 45

23 Chapter At lare distances from the head surface, equation (.37) effectively reduces to the first order Karlqvist approimation due to the eponential loss term. This can be seen clearly from Fiure.8 which compares the lonitudinal field component of Fan s equation with the Karlqvist field at different spacins from the head surface...8 y/ =.5, ε RMS = 9% y/ =., ε RMS = 5% Fan Karlqvist H (,y)/h.6.4 y/ =.4, ε RMS = 3% y/ =.6, ε RMS =.7% / Fiure. 8 Camparison between Fan and Karlqvist lonitudinal fields for different flyin heiht-to-ap lenth ratios. Also shown in Fiure.8 is the estimated percentae error of the difference between the accurate field iven by Fan and the Karlqvist approimation. It can be seen that the Karlqvist lonitudinal field approaches the actual field from a rin type head, evaluated usin Fan s Fourier series, for values of flyin heiht-to-ap lenth ratios reater than.4 as confirmed elsewhere (Huan and Den, 986; Baird, 98). The percentae difference decreases with the subsequent increase in the ratio y/. In equation (.3) it can observed that for y =, ϕ C (,) is the imainary part of the inverse Fourier transform of C(α) (since the surface potential is an odd function) with α as the Fourier operator. Hence C(α) is effectively the Fourier transform of Fan s surface potential as noted by Mallinson (99). Therefore, the Fourier transform of Fan s surface potential is : 46

24 Chapter ϕ (k,) = jπ C(k) (.39) C The Fourier transform of the field everywhere above the head, usin (.3), is thus: H k y (k, y) = kπ C(k) e (.4) Substitutin for C(k) from Appendi I, the wavelenth response of Fan s field is written as (Mee, 964): H n sin(k / ) k n( ) (An / V)sin(k / ) k y (k, y) = NIη + e (k / ) (.4) n= [(k / ) (nπ) ] Equation (.4) is in ecellent areement with the field transform iven by Lindholm (.6) and when plotted, the two spectra are indistinuishable. This is not surprisin since both spectra utilise the eact solution of the potential distribution for the semiinfinite apped head structure Ruirok (99) Thin-Gap Approimation The field due to a finite ap head that is infinitely thin (zero ap depth) and infinitely lon, known as a thin-ap head, was iven eplicitly by Westmijze (953) by conformal mappin. In the ap reion (-/ < < /) and for y =, the surface field is iven by: H thin H (,) = (.4) π ( ) Ruirok (99) has suested a more accurate approimation to the eact surface fields for a semi-infinite apped head with infinite depth in the ap reion can be found by superposin half the constant Karlqvist surface field and half the surface field of the thin-ap head (Ruirok, 99; Bertero et al, 993), i.e.: H H (,) + π ( ) (.43) The surface potential, ϕ(,), can easily be obtained by interatin (.43) with respect to. The normalised surface potential was found to be: 47

25 Chapter ϕ(,) V = + sin π (.44) The reduced surface potential of the positive half of equation (.44) is plotted in Fiure.9 alon with the constant slope potential due to Karlqvist and the potential from Fan (96) Fourier analysis...8 Ruirok Fan, n = Karlqvist ϕ(,)/v / Fiure. 9 Surface potential evaluated usin Ruirok thin-ap approimation compared with Fan and Karlqvist head surface potentials. In Fiure.9, the estimated difference error between Ruirok s approimation and Fan s surface potential was less than %. This illustrates the ood areement between the actual and approimated surface potential suested by Ruirok. The vector fields above the head surface can be obtained by differentiatin the two dimensional Green s function for the potential iven by (.9) with respect to coordinate aes in the ap reion since the fields vanish at the pole pieces. This produces the horizontal and vertical field components in terms of the surface fields for y > as: H `= ( `) + y (`,) H (, y) = d` (.45) π y 48

26 Chapter H (`,)( `= ( `) + `) H y (, y) = d` (.46) π y The interals in (.45) and (.46) were solved eactly by Bertero et al (993) for the surface field suested by Ruirok usin contour interation and the solution yields the horizontal field component as: H H (, y) = tan π / + + tan y / y H [ y ( / ) ] + 4 y + y + ( / ) π [ + y ( / ) ] + 4y ( / ) and the vertical field component as: H y H ( / + ) (, y) = ln 4π ( / ) H + sn() π [ y + y + y ( / ) [ + y ] + 4 ( / ) y ] + + 4y y ( / ) ( / ) / / (.47) (.48) Fiure.(a) compares the lonitudinal field computed usin (.47) with the correspondin fields from Fan (96) and Karlqvist for y/ =.5. It can be seen that Ruirok s simple approimation is in close areement with the accurate Fan field. The percentae root mean square error between Ruirok and Fan fields was % for y/ =.5 and reduces to.3% for y/ =.6. The vertical field component calculated usin (.48) is iven in Fiure.(b) alon with Fan and Karlqvist fields for comparison for y/ =.5. The percentae difference error between Ruirok s vertical field and that computed from Fan s analysis was also % and reduced to less than.6% for y/ =.6. Thus, it can be concluded that Ruirok s approimation provides an accurate means by which the fields from a semi-infinite head could be calculated from closed form epressions similar to that iven by Karlqvist. 49

27 Chapter..8 Ruirok Fan Karlquist H (,y)/h / (a).. Ruirok Fan Karlquist H y (,y)/h / Fiure. (a) Horizontal and (b) vertical field components from Ruirok s thin-ap approimation compared with correspondin Fan and Karlqvist fields for y/=.5. (b) 5

28 Chapter The ap-loss function of the thin-ap head field iven by equation (.4) was also derived by Westmijze and is iven by: H thin (k,) = NIη J (k / ) (.49) where J o (k/) is the Bessel function of the first kind. Combinin (.49) with half the Karlqvist ap-loss function yields the wavelenth response of Ruirok approimation iven by equation (.43) which for a distance y above the head is: NIη sin(k / ) k y H (k, y) = (k / ) e + J o (k / ) (.5) Equation (.5) is plotted is Fiure. and compared with Fan s eact ap loss function and Karlqvist approimation. It can be seen that Ruirok s thin ap approimation follows the eact Fan s surface spectrum closely. Furthermore, the same approimation was found to aree well in the position of the first ap nulls with Lindholm s semi-infinite apped fitted spectrum equation (Bertero et al, 993). o. Ruirok Fan, n = Karlqvist H (k,)/niη /λ Fiure. Gap loss functions for Ruirok, Fan and Karlqvist. 5

29 Chapter.4. Thin-Film Heads µ= y, y`, ` -V +V µ= P P L / L / Fiure. General structure of a thin-film head with boundary potential. A eneral structure of a non-symmetrical thin-film head with the oriin of the coordinate system bein at the ap centre is illustrated in Fiure.. In this structure, the poles are assumed to have infinite permeability. The track width dimension is assumed to be lare compared to the and y dimensions. The head is also assumed to be infinitely deep where the head coil dimensions and fields are nelected. In Fiure., is the ap lenth, = P and = P where P and P are the L + L + lenths of the left-hand and riht-hand poles respectively. In the previous section, fields were either obtained by assumin a particular approimation for potential in the ap reion or via the solution of Laplace s equation for the potential subject to boundary conditions. The epressions presented in the previous section for the ap potential and fields can, without loss of enerality, be applied directly to the ap reion of the thin-film head shown in Fiure.. The problem is now of findin an approimate epression for the surface potential, or surface field, beyond the pole ede reion. Once an accurate epression for this reion 5

30 Chapter is found, superposition can be utilised to obtain an epression for the surface potential, or field, for the entire head structure. Superposition is valid in the case of an infinitely deep head for different pole lenth-to-ap lenth ratios where the potential in the ap and beyond the poles are nearly independent of the pole lenths (Lindholm, 975)..4.. Potter (975) Head Field Function Motivated by remarks made by Mallinson (974) reardin the effects of finite head dimensions, Potter (975) has suested an inverse dependence of the scalar potential beyond the pole ede of a finite lenth head of the form: C ϕ(,) = Co +, P + / (.5) α with the approimation that α = where α in reality is reater than unity. The coefficients C o and C were determined by the continuity of ϕ and the condition that ϕ (,) NI / 4 as beyond the pole ede (see section.5). The pole pieces were assumed to be at constant manetic potential of ±NI/ with the surface potential varyin linearly across the ap reion, similar to the case of a Karlqvist head. Thus the potential for the thin-film head iven by Potter can be epressed as: NI L, 4 NI, NI ϕ(,) =, NI, NI L +, 4 < L L / < < / / > L / / L / / / (.5) 53

31 Chapter 54 NI/4 NI/ P +/ / ϕ(,) Fiure. 3 Potential distribution alon one half of a thin-film head for P = (potter, 975). The surface potential for a sinle pole of a symmetrical thin head about its centre is plotted in Fiure.3 usin (.5). The surface fields are obtained by differentiatin (.5) with respect to the co-ordinate aes where the fields at the pole surfaces vanish. The field components everywhere above the head are obtained usin the standard two dimensional Green s function for the surface fields. Potter (975) only provides the horizontal field component for a symmetrical thin-film head. The vector fields for a non-symmetrical thin-film head were reworked from (.5) and are iven by: π + + π π π = y ) / L ( ) / (L ln L y ) / L ( ) / (L ln L ) y ( y / L tan L y / L tan L ) y y( ) y ( ) y ( 4 8 y H y / tan y / tan H y) (, H (.53)

32 Chapter H ( / + ) + y H y (, y) = ln π ( / ) + y H 4 y 8π ( + y ) ( + y ) π L / + π L / L tan + L tan y y ( y ) (L / ) (L / ) + L ln L ln ( + y ) ( + L / ) + y ( L / ) + y (.54) For symmetrical poles (i.e. L = L ), (.53) reduces to the form iven by Potter (975). The first terms on the riht hand sides of (.53) and (.54) are effectively the Karlqvist field equations. This is the consequence of the linear potential in the ap reion. Equations (.53) and (.54) are plotted, normalised to H, in Fiure.4 for y/ =.5. The lonitudinal field component from a thin-film head is characterised mainly by the neative undulations at the pole corners whose manitude increases with reducin distance to the ap. The perpendicular field component also features two local etrema at the correspondin pole corners P = P = 3 H H(,y) / H. -. H y P P Fiure. 4 Lonitudinal and vertical field components of a non-symmetrical thin film head calculated usin Potter s surface potential distribution for y/=.5. / 55

33 Chapter.4.. Szczech et al (986) Head Field Function Szczech in 979 has presented similar epressions for the boundary potential beyond the poles to those iven by Potter (975) with adjustable parameters and also assumed initially that the potential varies linearly across the ap. The values of the coefficients, however, were determined by fittin to a lare-scale eperimental model of an asymmetrical thin-film head. His two dimensional approimations produced ood areement with the actual fields measured from the eperimental model and this areement improved with the increase in head spacin. Szczech s approach allows, via the adjustable coefficients, to take into account effects such as coil fields, dead layers, rounded corners, or other departures from the idealised case. It was also confirmed from his study that the effective ap lenth is larer than the mechanical ap lenth by a factor of. (i.e. %). Moreover, for different ap and pole lenths, Szczech (979) has found that the values of the coefficients chane only slihtly and this difference was attributed to the eperimental error. This, in fact, confirms that for heads with infinite depth, the scalar potentials in the ap and beyond the poles are almost independent of the pole dimensions. Szczech s boundary potential equations were later improved (Szczech and Iverson, 986; Szczech and Iverson, 987) providin reater accuracy at low flyin heihts and are applicable, with suitable values for the coefficients, to any head eometry. In the case of an asymmetrical thin-film head, the coefficients were determined by fittin to accurate finite difference field calculations for a small value of y correspondin to the practical limitation of an actual head-to-medium spacin, allowin the fitted coefficients to be more versatile for a wide rane of flyin heihts. Fiure.5 illustrates the horizontal and vertical field components as iven by Szczech et al (987) normalised to H s where H H (,). From the formula of the surface field s = in the ap reion iven by Szczech et al (see Appendi II), H s is effectively equal to the deep ap field H at =. This, however, is only applicable for the case of the Karlqvist field and thus proper scalin has to be applied at the ap centre when comparin with other field formulae. From Fan s Fourier analysis, the value of the surface field at the ap centre is.85h, and from Ruirok (99) thin-ap approimation it is.8h. The field equations and the values of the improved 56

34 Chapter coefficients as iven by Szczech et al (986; 987) are provided for reference in Appendi II. H (,y)/h s P = P = 3 y/=.65 y/=. y/=.4 y/= / (a) H y (,y)/h s y/=.65 y/=. y/=.4 y/=.6 P = P = / Fiure. 5 (a) Horizontal and (b) vertical field compoenents evaluated usin Szczech et al s fittin equations with the modified coefficients (987). (b) 57

35 Chapter.4.3. Lindholm (975) Surface Field Transform of a Thin-Film Head The wavelenth response of a thin-film head consists of a series of oscillations associated with the pole lenth P, and another series associated with the ap lenth. The ratio L/ (L=P +P +) or P/ has a stron influence on fluctuations about the ap nulls as a function of wavenumber and effectively determines the lon wavelenth limit of thin-film heads. In a comprehensive study by Lindholm (975), it was found that for a fied ap lenth and infinite depth of a finite pole lenth head, the potential in the ap reion chanes by less than 7% when reducin the pole dimension from infinity (semi-infinite apped head) to zero (parallel plate head). Furthermore, Lindholm (975) also observed, usin numerical computations of conformal mappins of the surface potential, that the maimum percentae difference between the potentials beyond the poles of a sinle pole head and a parallel plate head is also less than 7%. Thus it was concluded that the potentials in the ap and beyond the pole edes are insensitive to the pole lenths. This conclusion allowed the use of superposition of different head eometries to yield, to a ood approimation, the surface potential (or field) and hence the flu response of a thin-film head structure. The superposition process as described by Lindholm (975) will be analysed and the field spectra for the various head eometries iven by Lindholm (975; 976a) will be utilised. The wavelenth response of the thin-film head was obtained by superposition of the Fourier transform of the semi-infinite apped head iven by equation (.7) and the Fourier transform of the parallel plate head as shown in Fiure.6 (Lindholm, 975). In a parallel plate head, the poles are infinitely thin (i.e. P = P = ) and thus the ap lenth will equal the total head lenth i.e. = L. The surface potential beyond the thin poles of the parallel plate head will effectively be used to approimate the potential beyond the poles of an actual, finite pole lenth, thin-film head since the surface potential is insensitive to the pole lenth. The Fourier transform of the surface field of a parallel plate head, S, is iven in terms of the dummy variable f as (Lindholm, 975; 976a): ( S sin( πf ) e f ) = NIη Γ( f + ) ( πf ) f 58 f (.55)

36 Chapter where e =.78 and Γ is the Gamma function. The first ap null for the parallel plate head occurs eactly at / λ = L / λ =. Fiure.6 illustrates raphically the superposition process for the case of symmetrical poles, i.e. P = P = P. (a) (b) (c) (d) + _ = S (/λ) S (L/λ) S (L/λ) S(,L, ;/λ) P P L Fiure. 6 Superposition of semi-infinite and parallel plate heads to produce the wavelenth reponse of an infinitly deep thin-film head. Startin with the ap potential in (a) iven by S (/λ), a parallel plate head definin the head lenth L and providin a ood approimation for the potential beyond the poles is added in (b). After step (b), the potential is to too low by.5 for L / and too hih by.5 for L /. Subtractin the potential of a semi-infinite apped head of ap lenth L in (c) removes this ecessive potential and produces the required potential distribution of eometry (d). Thus the wavelenth response of the surface field for the symmetrical thin-head iven in Fiure.6, denoted by S(,L, ;/λ), is written as: { S ( / λ) + S (L / λ) S (L / )} S(,L, ;/ λ) = NIη λ (.56) The aruments in the surface spectrum S in (.56) indicate the three important dimensions of a thin-film head; namely the ap lenth, the head lenth L and the head depth where the infinity indicates an infinitely deep head. The last arument implies that the surface field spectrum is a function of inverse wavelenth. 59

37 Chapter. P = P =.8 S(,L, ;/λ)/niη /λ (a).5. P/= P/=4 P/= S(,L, ;/λ)/niη /λ (b) Fiure. 7 Frquency response of an infinitly deep thin-film head calculated usin superposition for (a) P/= and (b) for different values of P/ on a semi-lo scale. In Fiure.7(a), it can be seen that the wavelenth response of a thin-film head features ripples in the spectrum known as head bumps. The frequency and amplitude of these ripples are dependent on the ratio P/ as shown in Fiure.7(b). These 6

38 Chapter ripples interfere with the position of the ap nulls associated with the ap lenth. From the solution of equation (.56) for different values of pole lenth-to-ap lenth ratios, i.e. solvin S(,L, ;/λ) = for the first ap null, it was found that the position of the first ap null oscillates about the value / λ = (Lindholm, 975), i.e. the value for the etreme case of the semi-infinite poles, P /. Equation (.56) can easily be etended to accommodate the case of non-symmetrical poles. This is best illustrated in Fiure.8. (a) (b) + _ S (/λ) s s jπ s / λ S (L/ λ) e (c) (d) = S (L/ λ) e jπ / λ S(,L, ;/λ) s P P L Fiure. 8 Introducin a spatial shift in the superposition process to determine the wavelenth response of non-symmetrical thin-film heads. This is achieved by introducin a spatial shift, s, in the parallel plate and the semiinfinite apped heads in steps (b) and (c) that represents the required pole lenths. The shift distance s is calculated by solvin the two followin equations on either side of the head s line of symmetry, i.e.: solvin for s yields: L L + s s = P + = P + 6

39 Chapter P P s = (.57) This displacement is translated into the frequency domain usin the Fourier shift property (Briham, 988). Thus accordin to Fiure.8, equation (.56) becomes: jπ s / λ [ S ( / λ) + [S (L / λ) S (L / λ)] e ] S (, L, ;/ λ) = NIη (.58) It can be seen from (.58) that the wavelenth response is no loner real and an imainary term is introduced (the sine term in the comple eponential). This is epected since the pole lenths are not equal and hence the head field function is no loner even. For zero spatial shift, the eponential term is equal to unity and (.58) reduces to the form of equation (.56). The real and imainary parts of equation (.58) are plotted in Fiure.9 for a non-symmetrical thin-film head..5. Real part Imainary part S(,L, ;/λ)/niη /λ Fiure. 9 Plot of comple field spectrum for a thin-film head with P = and P = Bertero et al (993) Head Field Function For a head of finite lenth L with zero ap lenth and infinite depth, Westmijze (953) has approimated the surface field beyond the pole edes by a first order epansion of a conformal mappin solution iven by: 6

40 Chapter H H / 6 3 (,) L / (.59) / 3 5/3 / 3 / 3 L π ( L / ) Bertero et al (993) suested superposin the accurate surface field in the ap reion iven by Ruirok in equation (.43), with the surface field beyond the poles iven by (.59) to approimate the fields from a non-symmetrical thin-film head. Fiure. illustrates the surface field distributions described by Bertero et al compared with the surface fields of Potter (975) and Szczech et al (986) (after scalin by the factor.8). H (,)/H Bertero et al Szczech et al Potter P = P = / Fiure. Surface field distribution suested by Bertero et al (993). From Fiure., it can be seen that there is ood areement between the surface fields of Bertero et al (993) and that of Szczech et al (986) especially in the ap reion. Beyond the poles, the surface field due to Bertero et al decays slowly with increased distance as compared to that of Szczech et al (986). This can attributed to the first order approimation of the actual field beyond the pole as iven by Westmijze (953). To obtain a closer fit, hih order terms must be included in the surface field formula. Nevertheless, the first order term does provide a reasonable approimation for the fields beyond the head poles and allow the vector fields to be represented by simple closed form epressions. It can also be observed that the surface field due to 63