Designing a writer for magnetic recording

The model in Fig. 2 represents a typical ring structure used as a writer in longitudinal magnetic data storage systems. The ring is made of magnetic material e.g. Ni$_{45}$Fe$_{55}$ with a relative permeability $\mu_r \sim 3000$. We seek a solution to the value of the field in the gap, referred to as the deep gap field ($H_g$).

Figure 2: Schematic cross-section of a ring head. The dotted lines represent the closed loops C and D and are used to simultaneously solve Maxwell's equations.

Table 1 lists the variable names along with typical values. Without typical dimensions and material parameters, this design problem would continue to be an abstract mathematical exercise.

Table 1: Typical values for variables used.

Geometry		$\mu$m	Other
Yoke length	$l$	15	Relative permeability	$\mu_r$	3000
Yoke width	$w$	20	Conductivity	$\sigma$	5e6
Stack height	$h$	10	Magneto-motive force	$NI$	0.03
Gap	$g$	0.1	Skin depth	$\delta_f$
Pole thickness	$p$	3	Frequency	$\omega/(2\pi)$
Track width	$w_g$	0.25	Dissipation time	$\tau$

For starters, we assume that the material is isotropic and linear and separate the problem into two components

• Static case ...when a constant high current is applied to the coils and the writer erases the magnetic disk e.g. while formating a drive.
• Dynamic case ...when we rapidly the direction of current in the coils so as to write a pattern of 1's and 0's on the media.

The dynamic case, which requires a self-consistent solution to both Ampere's law around loop C and Faraday's law around loop D, is somewhat complicated. Hence it also becomes necessary to understand under what circumstances one can use the more convenient static approximation.

Static case

Referring to the closed loop C in Fig. 2, we write down Ampere's law (differential and integral forms respectively) as

$$\triangledown \times \vec{H} = \vec{J} \Rightarrow \oint_C \vec{H}.\vec{dl} = \int_A \vec{J}.\vec{dA} \hspace{0.3in}$$ (1)

The integral form is solved in terms of the fields at various points on the loop C. If we assume that there is no leakage of magnetic flux i.e. all the field is concentrated within the magnetic material. Following a flux line and expanding the above integrals yields

$$H(2l + 2h - g) + H_g g = N I$$ (2)

where, $NI$ is the total magneto-motive force (MMF) obtained by adding all the current traversing the area enclosed by C. (We currently ignore the change in width of the pole pieces and close the contour before the nose).

A second relation between $H_g$ and $H$ is obtained by solving the divergence equation $\triangledown \cdot \vec{B}$ across the gap. This merely yields a continuity equation (see Sec 5.4.2, Ref 2) of the form

$$(\vec{B}-\vec{B_g})\cdot \hat{n} = 0 \Rightarrow \mu_r H = H_g.$$ (3)

Combining (2) and (3) yields the deep gap field

$$H_g = \frac{\mu_{r} N I}{\mu_r g + 2(l+h) - g}.$$ (4)

Finally, the efficiency of the writer is defined to be

$$\eta \triangleq \frac{H_g}{NI} \approx \frac{\mu_r}{\mu_r g + 2(l+h)}.$$ (5)

We have thus illustrated the use of Ampere's law and an application of the divergence theorem $\int (\triangledown \cdot \vec{B}) dV = \oint \vec{B} \cdot \vec{dA}$.

Dynamic case

The reader is referred to Chap. 5 of Ref. 2 and the sub-sections below for a theoretical basis of the following discussion.

Consider the rectangular cross-section of the top pole denoted by the closed loop D in Fig. 2. Assuming a time variation of the form $e^{j\omega t}$, Faraday's law states that

$$\triangledown \times \vec{E} = \frac{-\partial B}{\partial t... ..._D \vec{E}\cdot\vec{dl} = -j\omega\int_A \vec{B}\cdot\vec{dA}.$$ (6)

Similarly, Ampere's law combined with $\vec{J}_e = \sigma \vec{E}$ becomes

$$\triangledown \times \vec{H} = \vec{J} + \frac{\partial D}{\... ...\mathrm{app} + (1 + j\omega\tau)\vec{J}_e\right)\cdot\vec{dA},$$ (7)

where $\vec{J}_\mathrm{app}$ is the applied current density (in the coils), $\vec{J}_e(x)$ is the spatial distribution of the eddy current density and $\tau = \epsilon/\sigma$ is the characteristic dissipation time for electric charge within the conductor. The three terms for current density integral arise from the external voltage applied to the writer coils, the electric field due to a changing magnetic field (eddy current) and the magnetic field due to the changing electric field (displacement current). For good conductors, $\omega \tau \ll 1$ and the last term is safely ignored.

Since the material used is a good conductor, we expect that all electric charges reside on the surface. Furthermore, eddy currents generated by the time varying magnetic field will flow in rectangular loops on the outer surface. The problem is to solve (6) and (7) simultaneously. We illustrate two possible approaches to the problem.

1. Assume pole pieces that are very wide ($w » t$) and the fields decay exponentially within the material. With a skin depth $\delta$ in the material we have an approximate form for the spatial distribution of the magnetic field in the pole piece,
   

   $$H(x) = H_0\left(\frac{e^{-\gamma y}+e^{\gamma (x-t)}}{1+e^{-... ...ech}\left(\gamma t/2\right)\cosh\left(\gamma(x - t/2)\right),$$ (8)

   
   
   where $\gamma = (1+j)/\delta$. In the low frequency limit, $\omega\rightarrow 0,\;\delta \rightarrow \infty$, $H(x) = H_0$ and is uniform through the material. Substitute $H(x)$ in (6) to estimate $J_e(x)$ and then integrate (7) around loop C. The resulting equation is
   

   $$[2(l+h)-g]H_{t/2} + g H_g = NI-[2(l+h)-g]\int^{t/2}_0 J_e(x)dy,$$ (9)

   $$\Rightarrow [2(l+h)+ - g]H_{t/2} + g H_g = NI - [2(l+h)-g](H_0 - H_{t/2}),$$ (10)

   
   
   where $H_{t/2}$ is the field at the center of the pole and we used the continuity conditions on the tangential component of $\vec{H}$ to evaluate the integral of $J_e(x)$. From the assumed spatial distribution of $H(x)$, we also get $H_{t/2} = H_0 \mathrm{sech}(\gamma t/2).$ Combining this with $H_g = \mu_r H\;\mathrm{and}\;\eta=\mathrm{Re}[H_g]/(NI)$ we finally obtain the dynamic writer efficiency
   

   $$\eta (\omega) \approx \mathrm{Re}\left[\frac{\mu_r \mathrm{s... ...gamma t/2)}{2(l+h)+g \mu_r \mathrm{sech}(\gamma t/2)}\right].$$ (11)

   
   
   It is instructive to note that we never quite used $J_e(x)$ in our calculation. However, the power loss due to the formation of eddy currents depends on the spatial distribution of eddy currents which are uniquely determined by $H(x)$ and (7). Furthermore, we could have derived $\eta(\omega)$, under the approximation $w » t$, from the static expression (5) by defining $\mu_r (\omega) \triangleq \mu_r \mathrm{sech}(\gamma t/2)$.
   
2. Use separation of variables to find a series expansion for $H(x,y)$ in a rectangular cross-section and use the lowest order terms to solve (1). Taking the origin at the center of the rectangle and from the symmetry of the expected solution, we assume a solution to Helmholtz's equation with the form
   

   $$H(x,z) = \sum_m\sum_n C_{mn} \cosh(\kappa_x x)\cosh(\kappa_z z),\;\;{\mathrm with} \kappa_x^2+\kappa_z^2 = \gamma^2.$$ (12)

   
   
   The coefficients $C_{mn}$ are to be determined by the boundary conditions $H(\pm w/2,z) = H(x,\pm t/2) = H_0$. Most text book problems take advantage of the fact that the field or potential is zero on one or more edges to determine the propagation constants. However this not being the case here and with $\gamma$ being complex, we do not have any obvious choices for $\kappa_x$ or $\kappa_z$.
   

Both the time and spatial scales play an important role in problems with harmonic excitations. Note that

• $H_g$ decreases as $\omega$ increases and it gets harder to design writers as we operate at higher frequencies.
• $H_g$ decreases as $\sigma$ increases since high resistivity materials restrict the flow of eddy currents.
• $H_g$ increases with $\mu_r$, though the increase is not quite as strong as in the static case.
• The effective permeability $\mu_r (\omega)$ decreases as the thickness $t$ of the pole pieces increases.

Characteristic dissipation time

Before we attack the problem of oscillating excitation currents in the writer coils, it is imperative that we understand when the proposed dynamic solutions are valid. The continuity equation for free charge is combined with Gauss' law to yield

$$\frac{-\partial \rho}{\partial t} = \triangledown \cdot \vec... ...iangledown\cdot(\sigma\vec{E}) = -\frac{\sigma}{\epsilon}\rho,$$ (13)

where we assumed that the medium was homogeneous. Further assuming a linear medium, we find that

$$\rho(t) = e^{-(\sigma/\epsilon)t}\rho(0).$$ (14)

Hence, we obtain a characteristic diffusion time $\tau = \epsilon/\sigma$ which is the time it takes for charge to flow out to the surface of the conductor. For our dynamic solution to be valid, one can adopt a rule of thumb that the write current must switch directions over a time scale at least 100 times larger than $\tau$, such that $\omega \tau \ll 1$. This would allow the formation of eddy current loops and permit an approximation of exponentially decaying fields in the interior of the material, while ignoring the displacement current.

Skin depth

For time harmonic (of the form $e^{j\omega t}$) excitations, combining the curl of the differential equation in (1) with (6) and Ohm's law yields Helmholtz's equation,

$$\triangledown^2\vec{H} - \gamma^2 \vec{H} = 0,\;\;\mathrm{where}\;\;\;\; \gamma^2 = j\omega\mu(\sigma + j\omega\epsilon).$$ (15)

Fields propagating along $\hat{y}$ have solutions to the above equation of the form $e^{-\gamma y}$. The complex propagation constant $\gamma$ has a real component that relates to the exponential decay of fields within matter, while the imaginary component represents the time harmonic solution. For good conductors,

$$\tau « 1/\omega \Rightarrow \sigma » \omega\epsilon$$ (16)

$$\therefore \gamma \triangleq \alpha + j\beta = \sqrt{\frac{\omega\mu\sigma}{2}}\left(1+j\right).$$ (17)

The skin depth $\delta_f \triangleq 1/\alpha$ is a measure of the distance it takes for the electromagnetic field to decay to $e^{-1}$ of its original value. The subscript is used explicitly to remind us that the skin depth depends on the frequency of the electromagnetic excitations. Given a field $H_0$ at the surface of the pole, we expect a solution that decays away from the surfaces of the of the material and has a form $H(x,t) = H_0 Re[e^{-\gamma y + j\omega t}] = H_0 e^{-x/\delta}\cos(x/\delta - \omega t)$.

Power loss