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Read Back Signals in Magnetic Recording - Research Group Fidler

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z<br />

Analytical Calculations<br />

Figure 3.9: a) Application of Ampere’s Law on the dashed path, b) application of Gauss’s<br />

Law on the dashed box.<br />

In addition we use Gauss’ Law for a box with <strong>in</strong>f<strong>in</strong>itesimal small height dz over the sens<strong>in</strong>g<br />

layer:<br />

x<br />

a) g g<br />

b)<br />

−μ ⋅μ ⋅H ( z) ⋅ t+ 2 ⋅μ ⋅H ( z) ⋅ dz+μ ⋅μ ⋅ H ( z+ dz) ⋅ t = 0.<br />

(3.25)<br />

r 0 sig 0 g r 0 sig<br />

Here μ r is the relative permeability of the sens<strong>in</strong>g layer. The last equation is equivalent to<br />

1 dH sig<br />

Hg( z) =− ⋅μr ⋅t⋅ (3.26)<br />

2 dz<br />

Comb<strong>in</strong><strong>in</strong>g (3.24) and (3.26) gives a differential equation.<br />

2<br />

dHsig<br />

2<br />

1<br />

Hsig ( z) = ⋅μr⋅g⋅t⋅ (3.27)<br />

2 dz<br />

Assum<strong>in</strong>g the boundary conditions<br />

ABS<br />

H (0) = H and H ( h)<br />

= 0<br />

(3.28)<br />

sig sig sig<br />

leads to the solution<br />

⎛h−z⎞ s<strong>in</strong>h ⎜ ⎟<br />

λ<br />

Hsig ( z) = Hsig⋅<br />

⎝ ⎠<br />

with λ=<br />

⎛h⎞ s<strong>in</strong>h ⎜ ⎟<br />

⎝λ⎠ ABS r<br />

μ ⋅g⋅t . (3.29)<br />

2<br />

Here λ is the flux propagation length. Now the effective field can be calculated under the<br />

terms of (3.22):<br />

Hsig<br />

t<br />

Hg Hg<br />

Hg<br />

dz<br />

z<br />

x<br />

Hsig<br />

t<br />

dz<br />

g g<br />

39

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