Continuous Time Considerations.md (3461B)
1 --- 2 author: Jatin Chowdhury 3 title: Continuous Time Equations for Analog Tape Modeling 4 date: 2/1/2019 5 --- 6 7 # Record head 8 9 For an instantaneous current I, the magnetic field output of the record head is given as a function of distance along the tape 'x', and depth into the tape 'y' (Karlqvist medium field approximation) [Bertram, page 60]: 10 11 $$ H_x(x,y) = \frac{1}{\pi} H_0 \Big(\tan^{-1} \Big(\frac{(g/2) + x}{y} \Big) + \tan^{-1} \Big(\frac{(g/2) - x}{y} \Big) \Big) $$ 12 $$ H_y(x,y) = \frac{1}{2 \pi} H_0 \ln \Big(\frac{((g/2) - x)^2 + y^2}{((g/2) + x)^2 + y^2} \Big) $$ 13 14 where $g$ = head gap, and $H_0$ = deep gap field, given by: 15 16 $$ H_0 = \frac{NIE}{g} $$ 17 18 where $N$ = number of turns coils of wire around the head, and $E$ = head efficiency given by: 19 20 $$ E = \frac{1}{1 + \frac{l A_g}{\mu_r g} \int_{core} \frac {d \vec{l}}{A(l)}} $$ 21 22 where $A_g$ is the gap area, $\mu_r$ is the core permeability relative to free space ($\mu_0$), $g$ if the gap width, and $A(l)$ is the cross-sectional area of the core as a function of length. 23 24 # Tape Magnetisation 25 26 ## Deadzone 27 28 For low current, the field is insufficient to create a change in magnetisation. For high current the field saturates. The effective field magnetising the tape $H_h$ can be described as follows: 29 30 $$ H_h = \begin{cases} 0 & H \le S^* H_c \\ 31 H & H > S^* H_c 32 \end{cases} $$ 33 34 where $S^*$ = hysteresis loop squareness, and $H_c$ = coercivity. 35 36 ## Hysteresis 37 38 The magnetostatic field recorded to magnetic tape can be described using a hysteresis loop. A circuit simulation of a hysteresis loop by Martin Holters and Udo Zolzer, using the Jiles-Atherton magnetisation model can be found at http://dafx16.vutbr.cz/dafxpapers/08-DAFx-16_paper_10-PN.pdf. They use the following differential equation to describe magnetisation 'M' as a function of magnetic field 'H': 39 40 $$ \frac{dM}{dH} = \frac{(1-c) \delta_M (M_{an} - M)}{(1-c) \delta k - \alpha (M_{an} - M)} + c \frac{dM_{an}}{dH} $$ 41 42 where $M_{an}$ is the anisotropic magnetisation given by: 43 44 $$ M_{an} = M_s L \Big( \frac{H + \alpha M}{a} \Big) $$ 45 46 where $M_s$ is the magnetisation saturation, and $L$ is the Langevin function. 47 48 # Playback head 49 50 ## Ideal playback voltage 51 52 The ideal playback voltage as a function of tape magnetisation is given by [Bertram, page 121]: 53 54 $$ V(x) = NWEv \mu_0 \int_{-\infty}^{\infty} dx' \int_{-\delta/2}^{\delta/2} dy' \vec{h}(x' + x, y') \cdot \frac{\vec{M}(x', y')}{dx} $$ 55 56 where $N$ = number of turns of wire, $W$ = with of the playhead, $E$ = playhead efficiency, $v$ = tape speed. Note that $V(x) = V(vt)$ for constant $v$. $\vec{h}(x, y)$ is defined as: 57 58 $$ \vec{h} (x, y) \equiv \frac{\vec{H} (x, y)}{NIE} $$ 59 60 where $\vec{H} (x, y)$ is the same as for the record head. 61 62 ## Loss effects 63 64 There are several frequency-dependent loss effects associated with playback, described as follows [Kadis, page 126]: 65 66 $$ V(x) = V_0(x) [e^{-kd}] \Big[\frac{1 - e^{-k \delta}}{k \delta} \Big] \Big[\frac{\sin (kg /2)}{kg/2} \Big] $$ 67 68 where $k$ = wave number. 69 70 ### Spacing Loss 71 72 $$ g_s = e^{-kd} $$ 73 74 where $d$ is the distance between the tape and the playhead. 75 76 ### Gap Loss 77 78 $$ g_g = \frac{\sin (kg /2)}{kg/2} $$ 79 80 where $g$ is the gap with of the play head. 81 82 ### Thickness Loss 83 84 $$ g_t = \frac{1 - e^{-k \delta}}{k \delta} $$ 85 86 where $\delta$ is the thickness of the tape. 87 88 # Conclusion 89 If each of these components is digitized, a digital physical model of the analog tape machine can be constructed.