Read Back Signals in Magnetic Recording - Research Group Fidler

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V 2l + N2πr′ ≈ = 3.57 µV σrπ L 2 2 Numerical Methods (4.65) With the FEM we get V 2 = 3.81µV . The difference is again due to the smaller cross sectional area of the meshed model. H z (r=0) [A/m] 0.20 0.15 0.10 0.05 0.00 0 -100 -50 0 50 100 150 200 z [mm] Figure 4.4: The model of the coil (left). The color code shows the linear decrease of the electric potential inside the wire. The chart (right) shows the magnetic field of a solenoid in z-direction at the middle axis. The two dotted lines identify the position of the solenoid. The second graph shows the relative error of the coil’s magnetic field calculated with FEM calculation compared to the solenoid’s field. The magnetic field for a perfect solenoid (hollow cylinder with infinitesimal thin nappe) with length L, face current density J and radius r′ can be calculated exactly along the middle axis (z-axis): ⎛ ⎞ J L−z z Hz( z) = ⋅ ⎜ + ⎟. (4.66) 2 ⎜ 2 2 2 2 ( L− z) + r′ z + r′ ⎟ ⎝ ⎠ When we now approximate the coil by such a solenoid and set the face current density equal to NI J = , we have the possibility to compare the field of the solenoid H z with that of the L calculated field of our model. The run of the magnetic field along the middle axis of the solenoid is shown in Figure 4.4. In addition the relative error of the FEM calculated coil compared with the solenoid is given. So the quantitative values are again consistent. 10 8 6 4 2 Error [%] 56
4.5 Time integration Numerical Methods Using the LLG equation, the time evolution of the magnetization can be calculated. The discretization of realistic problems leads to an initial-value problem (4.2) with up to a million unknowns. To solve such problems the free solver CVODE is used [29] [30], which offers iterative and direct multistep methods. Multistep methods make use of the past values of the solution. For our purposes the iterative backward differentiation formula (BDF) seemed to fit best to solve our problem, in order to achieve fastest convergence times. The nonlinear system is solved using a Newton method, which requires only a few iterations. In order to improve the performance, CVODE allows preconditioning. By declaring an estimated Jacobian matrix the convergence time can be drastically decreased. CVODE is capable of choosing the time step size automatically. For rapidly changing processes a smaller time step is chosen to increase accuracy. Figure 4.5 schematically shows the complete simulation cycle. First the mesh model and the material parameters are read. Then the boundary matrices for the FEM/BEM methods for the stray field and the current field are set up. Afterwards the initial magnetization is read and set, and the time integration starts. CVODE solves the LLG equation, and requests the right hand side of the LLG equation for certain points of time. CVODE passes the actual time and the magnetic polarization J i of each node to the LLG calculation routine, which calculates the right hand side of the LLG equation. All magnetic fields which contribute to the total effective magnetic field H eff are calculated for each nodal point. Then the effective field is inserted in the right hand side of the LLG equation, which leads to new functions f i of time and magnetic polarization of the i-th node. These functions are transmitted back to the solver, which decides to modify the magnetic polarization. If the desired precision is not fulfilled the new magnetic polarizations are passed to the LLG calculation routine again, and the cycle is repeated. Otherwise the solver chooses the next time step and the system time is increased. The whole procedure is repeated until a user chosen final time is reached. 57
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 and 54: Numerical Methods within the volume
Page 55: Numerical Methods Integration of Am
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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