Read Back Signals in Magnetic Recording - Research Group Fidler

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out in in 2 2 1 ∂V Numerical Methods H − H = H (4.61) to guarantee (4.50) and (4.53). Again we have an equivalent problem as in Section 4.2, but for three dimensions. Analogously the field H 2 can be evaluated by ( − ′ ) 1 r r n H () r = ( ′ ) dA π � ∫ H r . (4.62) ′ 2 4 ∂V in 1 r−r 3 To save memory we only create an interaction matrix from boundary to boundary, and solve the Laplace equation (4.60) again to calculate H 2 . 4.4.6 Example: Long Wire To verify the program and the calculated fields, some relative simple models are simulated, which can also be calculated analytically. The first example is a long straight Cu-wire with length l = 200 mm and wire radius r = 2 mm . The conductivity of Cu is 7 -1 -1 σ= 5.961⋅10 Ω m . At least we have to specify the boundary conditions of the wire. At one side the electric potential was set to V 1 = 0 . The boundary condition of the opposite end was chosen in a way, that the total inflowing current has a value of I = 1 mA . With these parameters the total resistance of the Cu-wire can be calculated: R = 267 µ Ω . Using Ohm’s Law (2.1) we get for the electric potential V 2 = 0.267 µV . This analytically calculated value matches well with that of the FEM calculation 2 0.278 µV FEM FEM V = . The error ( 2 V is about 4.2% larger than V 2 ) is traced back to the mesh. The mesh has to approximate a cylindrical surface with triangles. So the cross sectional area of the mesh model is smaller than that of the realistic model. This leads to a larger resistance and further to a larger voltage. I 2 The current density of the realistic model is j = = 79.58 A/m . The magnetic field of an 2 r π infinite long straight wire with constant current can be easily calculated due to the symmetry in respect of rotations around the wire’s axis. The only non-zero field component is that in azimuthal direction and can be determined using Ampere’s Law (2.6). � ∫ H⋅ ds= ∫j× dA (4.63) ∂A A 54
Numerical Methods Integration of Ampere’s law gives for the radial field H ( ) ⎧ ρ ⎪ j : 0≤ρ< r ⎪ 2 . (4.64) ϕ ρ =⎨ 2 ⎪ r j : r ≤ρ ⎪⎩ 2ρ This function is plotted in Figure 4.3, and shows a linear increase, reaching its maximum at the wire’s surface and declines proportional to 1/ρ outside. H ϕ [A/m] 0.10 0.08 0.06 0.04 0.02 0.00 0 0 10 20 30 40 50 ρ [mm] Figure 4.3: Hϕ for an infinite long wire is plotted over the distance from the middle axis. The second curve shows the error of the numerical calculation. The second curve in Figure 4.3 shows the relative error of the FEM calculation compared to (4.64). The error is negative and increases according to amount with increasing ρ . The model with finite length used for the FEM calculation is the reason for that. The finite length has a greater effect for larger radial distances. The discontinuities of the error curve are due to the Adaptive Cross-Approximation Technique (see Section 4.3). For small ρ we have a very good match of the numerical and analytical calculated fields. 4.4.7 Example: Coil A coil as shown in Figure 4.4 is the second example. For the coil wire we have again the same 7 -1 -1 parameters as for the straight wire ( σ= 5.961⋅10 Ω m , I = 1 mA , r = 2 mm ). The radius of the coil is r′ = 20 mm , the length is l = 100 mm , and it has N = 20 windings. The model additionally has two straight leads with lengthl L = 80 mm . Similar to the straight wire we can now calculate the difference of the electric potential by analytical means: -10 -8 -6 -4 -2 Error [%] 55
Page 1 and 2:
DIPLOMARBEIT Read Back Signals in M
Page 3 and 4: Abstract This thesis concentrates o
Page 5 and 6: Contents 3.2.4 Signal of Written Bi
Page 7 and 8: Introduction 1 IntroductionEquation
Page 9 and 10: Introduction In 1955, before the RA
Page 11 and 12: 2 BasicsEquation Section (Next) 2.1
Page 13 and 14: Basics curl H = 0 . (2.15) So H is
Page 15 and 16: Basics This is justified, because t
Page 17 and 18: Basics Here J ij represents the exc
Page 19 and 20: s Basics ∂J α ∂J =−γ J× He
Page 21 and 22: Basics for spin down electrons. An
Page 23 and 24: Basics Figure 2.4: Schematic model
Page 25 and 26: Basics channels in the parallel and
Page 27 and 28: Basics Nowadays there are again eff
Page 29 and 30: Analytical Calculations 3 Analytica
Page 31 and 32: Analytical Calculations developing
Page 33 and 34: Analytical Calculations t C 8 π =
Page 35 and 36: µ 0 Hsig [T] 0.030 0.025 0.020 0.0
Page 37 and 38: Analytical Calculations n n ⎛HH,
Page 39 and 40: z Analytical Calculations Figure 3.
Page 41 and 42: 4 Numerical MethodsEquation Numeric
Page 43 and 44: Numerical Methods ϕ ( x ) =δ . (4
Page 45 and 46: Numerical Methods current flowing o
Page 47 and 48: Numerical Methods ∫ ∫ ∫ si =
Page 49 and 50: Numerical Methods Figure 4.2: The l
Page 51 and 52: Numerical Methods j =−σgrad Φ.
Page 53: Numerical Methods within the volume
Page 57 and 58: 4.5 Time integration Numerical Meth
Page 59 and 60: 5 Read Head DesignEquation 5.1 Bias
Page 61 and 62: Read Head Design The exchange bias
Page 63 and 64: Read Head Design free layer (see Fi
Page 65 and 66: FEM Simulations 6 FEM SimulationsEq
Page 67 and 68: FEM Simulations Magnetic Material J
Page 69 and 70: 6.2 Results FEM Simulations In the
Page 71 and 72: Output [V] 0.108 0.107 0.106 0.105
Page 73 and 74: FEM Simulations has a greater influ
Page 75 and 76: Output Voltage [V] 0.1074 0.1072 0.
Page 77 and 78: 7 OutlookEquation Section (Next) Ou
Page 79 and 80: Appendix A In case we have the func
Page 81 and 82: Appendix B Moreover the second inte
Page 83 and 84: List of Figures List of Figures Fig
Page 85 and 86: List of Figures Figure 6.1: The mag
Page 87 and 88: Bibliography [11] H. Katayama, M. H
Page 89 and 90: Bibliography [38] W. F. Brown Jr.,
Page 91: Lebenslauf Otmar Ertl Reinprechtsdo
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Read Back Signals in Magnetic Recording - Research Group Fidler

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