commit 5bed50f025048349b15687e54ec0a13e2763237e
parent 54f4f0161af99a7e057e6385981e5010b4da58c2
Author: jatinchowdhury18 <jatinchowdhury18@users.noreply.github.com>
Date: Thu, 7 Feb 2019 15:47:00 -0800
Paper updates and hysteresis parameters
Diffstat:
4 files changed, 103 insertions(+), 41 deletions(-)
diff --git a/Paper/420_paper.pdf b/Paper/420_paper.pdf
Binary files differ.
diff --git a/Paper/420_paper.tex b/Paper/420_paper.tex
@@ -33,7 +33,7 @@
%------------------------------------------------------------------------------------------
% ! ! ! ! ! ! ! ! ! ! ! ! user defined variables ! ! ! ! ! ! ! ! ! ! ! ! ! !
% Please use these commands to define title and author(s) of the paper:
-\def\papertitle{Real-time Physical Model for Sony TC-260 Tape Machine}
+\def\papertitle{Real-time Physical Model for Analog Tape Machines}
\def\paperauthorA{Jatin Chowdhury}
% Authors' affiliations have to be set below
@@ -115,12 +115,14 @@ DAT tape, and then by DAWs \cite{Kadis}. Despite being replaced
by higher quality technology,
many have sought to recreate a "tape" sound through digital effects, despite
the distortion, tape "hiss", and other oddities analog tape produced.
-To quote Brian Eno, "Whatever you now find weird, ugly, uncomfortable, and nasty
+The following describes
+a physical modelling process for an analog tape machine,
+starting from basic physical principles.
+\newline\newline
+"Whatever you now find weird, ugly, uncomfortable, and nasty
about a new medium will surely become its signature. CD distortion, the jitteriness
of digital video, the crap sound of 8-bit - all of these will be cherished
-and emulated as soon as they can be avoided." \cite{Eno}. The following describes
-a physical model of a Sony TC-260 tape machine, starting from basic physical
-principles.
+and emulated as soon as they can be avoided." -Brian Eno \cite{Eno}.
\section{Continuous Time System}
Audio recorded to and played back from a tape machine can be thought of as going
@@ -149,7 +151,7 @@ where $g$ = head gap, and $H_0$ = deep gap field, given by:
\end{equation}
%
where $N$ = number of turns coils of wire around the head, and $E$ = head
-efficiency given by:
+efficiency which can be calculated by:
\begin{equation}
E = \frac{1}{1 + \frac{l A_g}{\mu_r g} \int_{core} \frac {d \vec{l}}{A(l)}}
@@ -236,9 +238,10 @@ described as follows \cite{Kadis}:
\begin{equation}
V(x) = V_0(x) [e^{-kd}] \Big[\frac{1 - e^{-k \delta}}{k \delta} \Big] \Big[\frac{\sin (kg /2)}{kg/2} \Big]
+ \label{eq:lossEffects}
\end{equation}
%
-where $k$ = wave number, $d$ is the distance between the tape and the playhead,
+for sinusoidal input, where $k$ = wave number, $d$ is the distance between the tape and the playhead,
$g$ is the gap width of the play head, and $\delta$ is the thickness of the tape.
\section{Digitizing the System}
@@ -305,7 +308,7 @@ We use linear interpolation to find the half-sample values used to calculate $k_
\subsubsection{Numerical Considerations}
To account for rounding errors in the Langevin function for values close to
-zero, we use the following for numerical evaluation, as in \cite{Hysteresis}:
+zero, we use the following approximation about zero, as in \cite{Hysteresis}:
\begin{equation}
L(x) = \begin{cases}
\coth(x) - \frac{1}{x} & \text{for $|x| > 10^{-4}$} \\
@@ -329,7 +332,7 @@ meter, the following plot shows the Magnetisation output.
\begin{figure}[ht]
\center
\includegraphics[width=3in]{../Simulations/Sim2-M_H.png}
- \caption{\label{flowchart}{\it Digitized Hysteresis Loop}}
+ \caption{\label{flowchart}{\it Digitized Hysteresis Loop Simulation}}
\end{figure}
%
\subsection{Play Head}
@@ -343,6 +346,58 @@ or,
\hat{V}(n) = NWEv \mu_0 g \hat{M}(t)
\end{equation}
%
+The loss effects described in \cref{eq:lossEffects}
+can be modelled as a series of digital filters.
+
+\subsection{Oversampling}
+To reduce frequency aliasing caused by the hysteresis non-linearity,
+we recommend oversampling by at least 4x.
+%
+%\section{Determining System Parameters}
+%\subsection {Tape Machine Parameters}
+%%@TODO: Do I need this section?
+%\subsubsection {Record Head}
+%The input current to the record head is approximately 0.1 mA peak-to-peak
+%\cite{RefManual}. Plugging into equation (16)
+%
+%g = 20 microns, N = 100, E = 0.1
+%
+%gives a peak-to-peak field amplitude of 5E5 A/m
+%
+%\subsubsection{Play Head}
+%N = 100, W = 0.25in = 6.3e-3m, v = 7 ips = 0.18 m/s,
+%g = 20 microns, mu = 12.57e-7
+%
+%Gain of play head = 2.85e-14
+\section{System Parameters}
+\subsection {Tape Parameters}
+For the tape hysteresis \cref{eq:dmdt}, following parameters are necessary:
+\begin{itemize}
+\item Magnetic Saturation ($M_s$): For Ferric Oxide ($\gamma F_2O_3$)
+magnetic tape as would typically be used for a reel-to-reel machine,
+the magnetic saturation is $3.5e5$ (A/m) \cite{jilesBook}
+\item Hysteresis Loop Width ($k$): For soft materials, $k$ can be approximated
+as the coercivity, $H_c$ \cite{Jiles1992}. For Ferric Oxide, $H_c$ is approximately
+$27$ kA/m \cite{jilesBook}.
+\item a: Knowing the coercivity and remnance magnetism of Ferric Oxide
+\cite{jilesBook}, we can calculate $a$ = 22 kA/m by the method described includegraphics
+\cite{Jiles1992}
+\item c: From \cite{Jiles1992}, $c$ = 1.7e-1.
+\item alpha: From \cite{Jiles1992}, alpha = 1.6e-3
+\end{itemize}
+
+\subsection{Tape Bias}
+A typical analog tape recorder modulates the input signal by a
+"bias" signal to avoid the "deadzone" effect whenever the input
+signal crosses zero \cite{Camras:1987:MRH:27189}. Thus, the
+input signal to the record head can be described by:
+\begin{equation}
+ \hat{V}_{head}(n) = \hat{V}_{in}(n) \cos(2 \pi f_{bias} n T)
+\end{equation}
+%
+An output signal in the audio frequency range can be recovered
+by a simple low-pass filter \cite{Camras:1987:MRH:27189}.
+
%\newpage
\nocite{*}
diff --git a/Paper/references.bib b/Paper/references.bib
@@ -1,26 +1,3 @@
-@book{Zolzer:2011:DDA:2028616,
- author = {Zolzer, Udo},
- title = {DAFX: Digital Audio Effects},
- year = {2011},
- isbn = {0470665998, 9780470665992},
- edition = {2nd},
- publisher = {Wiley Publishing},
-}
-
-@Misc{Carl,
- title={Cassette Tape Physical Modeling and Analysis},
- author={Richard Carl, Daniel Fine, Benjamin Schmitz},
- school={University of Rochester, Department of Audio and Music Engineering},
- howpublished={\url{http://www2.ece.rochester.edu/~zduan/teaching/ece472/projects/2018/RichardCarl_DanielFine_BenjaminSchmitz_Cassette_FinalReport.pdf}}
-}
-
-@Misc{Howell,
- title={A Comparison of Analog vs Digital Tape Echo},
- author={Benjamin Howell},
- school={Birmingham City University},
- howpublished={\url{https://www.academia.edu/10437785/Benijah_HowellBSc_-_A_Comparison_of_analog_VS_Digital_Tape_Echo}}
-}
-
@BOOK{1994tmr..book.....B,
author = {{Bertram}, H.~N.},
title = "{Theory of Magnetic Recording}",
@@ -91,3 +68,32 @@ booktitle = {Theory of Magnetic Recording, by H.~Neal Bertram, pp.~372.~ISBN 052
adsurl = {http://adsabs.harvard.edu/abs/1986JMMM...61...48J},
adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}
+
+@ARTICLE{Jiles1992,
+ author = {{Jiles}, D.~C. and {Thoelke}, J.~B. and {Devine}, M.~K.},
+ title = "{Numerical determination of hysteresis parameters for the modeling of magnetic properties using the theory of ferromagnetic hysteresis}",
+ journal = {IEEE Transactions on Magnetics},
+ year = 1992,
+ month = jan,
+ volume = 28,
+ pages = {27-35},
+ doi = {10.1109/20.119813},
+ adsurl = {http://adsabs.harvard.edu/abs/1992ITM....28...27J},
+ adsnote = {Provided by the SAO/NASA Astrophysics Data System}
+}
+
+@book{jilesBook,
+ title={Introduction to Magnetism and Magnetic Materials},
+ author={Jiles, D.},
+ isbn={9781482238884},
+ url={https://books.google.com/books?id=2diYCgAAQBAJ},
+ year={2015},
+ publisher={CRC Press}
+}
+
+@Misc{RefManual,
+ title={Sony TC-260 Service Manual},
+ author={Sony},
+ year={1965},
+ url={https://www.vintageshifi.com/repertoire-pdf/pdf/telecharge.php?pdf=Sony-TC-260-Service-Manual.pdf}
+}
diff --git a/Simulations/hystersis.py b/Simulations/hystersis.py
@@ -5,10 +5,11 @@ from matplotlib import animation
fs = 48000
T = 1/fs # sample interval
M_s = 350000 # Jiles book
-a = 1.1e3 # adjustable parameter
+a = 2.2e4 #adjustable parameter
alpha = 1.6e-3
-k = 4.0e2
-c = 1.7e-1
+k = 27.0e3 #Coercivity
+c = 1.7e-1 #1.867e-1
+#M_r = 0.5e6 (A/m)
#H_c (coercivity) = 25000 - 30000 A/m (Jiles book)
# Langevin function
@@ -52,13 +53,13 @@ def M_n (M_n1, k1, k2, k3, k4):
#input signal
t = np.linspace (0, 1, fs * 50)
-#H_in = 100 * np.sin (2 * np.pi * 100 * t)
+#H_in = (5e5) * np.sin (2 * np.pi * 100 * t)
freq = 2000
-H_in = np.concatenate ((2000 * np.sin (2 * np.pi * freq * t[0:fs*5]), 1800 * np.sin (2 * np.pi * freq * t[fs*5:fs*10]), \
- 1600 * np.sin (2 * np.pi * freq * t[fs*10:fs*15]), 1500 * np.sin (2 * np.pi * freq * t[fs*15:fs*20]), \
- 1400 * np.sin (2 * np.pi * freq * t[fs*20:fs*25]), 1300 * np.sin (2 * np.pi * freq * t[fs*25:fs*30]), \
- 1200 * np.sin (2 * np.pi * freq * t[fs*30:fs*35]), 1100 * np.sin (2 * np.pi * freq * t[fs*35:fs*40]), \
- 1000 * np.sin (2 * np.pi * freq * t[fs*40:fs*45]), 800 * np.sin (2 * np.pi * freq * t[fs*45:fs*50])))
+H_in = np.concatenate ((5e2 * np.sin (2 * np.pi * freq * t[0:fs*5]), 1e3 * np.sin (2 * np.pi * freq * t[fs*5:fs*10]), \
+ 3e3 * np.sin (2 * np.pi * freq * t[fs*10:fs*15]), 5e3 * np.sin (2 * np.pi * freq * t[fs*15:fs*20]), \
+ 1e4 * np.sin (2 * np.pi * freq * t[fs*20:fs*25]), 3e4 * np.sin (2 * np.pi * freq * t[fs*25:fs*30]), \
+ 5e4 * np.sin (2 * np.pi * freq * t[fs*30:fs*35]), 1e5 * np.sin (2 * np.pi * freq * t[fs*35:fs*40]), \
+ 3e5 * np.sin (2 * np.pi * freq * t[fs*40:fs*45]), 5e5 * np.sin (2 * np.pi * freq * t[fs*45:fs*50])))
# plt.plot (t, H_in)
# plt.show()
