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Martin Kennedy 2025-05-05 10:27:34 -04:00
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Final.tex
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@ -2,6 +2,34 @@
% \IEEEoverridecommandlockouts
% The preceding line is only needed to identify funding in the first footnote. If that is unneeded, please comment it out.
\usepackage{cite}
\usepackage{xcolor}
\definecolor{codegreen}{rgb}{0,0.6,0}
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\definecolor{codepurple}{rgb}{0.58,0,0.82}
\definecolor{backcolour}{rgb}{0.95,0.95,0.92}
\usepackage{listings}
\lstdefinestyle{mystyle}{
backgroundcolor=\color{backcolour},
commentstyle=\color{codegreen},
keywordstyle=\color{magenta},
numberstyle=\tiny\color{codegray},
stringstyle=\color{codepurple},
basicstyle=\ttfamily\footnotesize,
breakatwhitespace=false,
breaklines=true,
captionpos=b,
keepspaces=true,
numbers=left,
numbersep=5pt,
showspaces=false,
showstringspaces=false,
showtabs=false,
tabsize=2
}
\lstset{style=mystyle}
\usepackage{amsmath,amssymb,amsfonts}
\usepackage{algorithmic}
\usepackage{graphicx}
@ -10,7 +38,7 @@
\usepackage{framed}
\usepackage{longtable,booktabs,array}
\usepackage{caption}
\usepackage{xcolor}
\usepackage{siunitx}
\usepackage{circuitikz}
\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
@ -54,7 +82,7 @@ the frequency response of the open-loop gain.
One of the most well-known op-amps is the LM741. In no small part to
demonstrate one weakness of the LM741, a cutoff frequency of
$\SI{40}{kHz}$ is selected for the high-pass filter design. To compare,
$\SI{400}{kHz}$ is selected for the high-pass filter design. To compare,
three op-amps are selected beyond the LM741:
\begin{enumerate}
@ -67,10 +95,11 @@ three op-amps are selected beyond the LM741:
\end{enumerate}
To focus on the real-world impact of the variations in performance of
these op-amps: imagine that the application for our high-pass filter
is as a pre-amplifier for a hobbyist kit, to be sold in the United
Kingdom, which can use the MSF signal ($\SI{60}{\kHz}$) \cite{b4} to
tell the current time.
these op-amps at higher frequencies: imagine that the application for
our high-pass filter is as a pre-amplifier for a hobbyist AM receiver
kit to receive emergency weather and traffic alerts in the Northeast
US. These alerts are broadcast between $\SI{530}{\kHz}$ AM and
$\SI{1650}{\kHz}$ \cite{b4}.
\section{Design of a High-pass Active Filter}
@ -117,8 +146,8 @@ divider, and so
This high-pass filter design is a common one, with a well-known cutoff
frequency $\omega_c = \frac{1}{R_1 C}$ and gain $K = \frac{R_2}{R_1}$
\cite{b3}. To acquire a cutoff frequency of
$\omega_c = \SI{40}{\kHz} \approx \SI{2.51E5}{}$ rad/s,
$R_1 = \SI{100}{\ohm}$ and $C_1 = \SI{39}{\nano\farad}$ will
$\omega_c = \SI{400}{\kHz} \approx \SI{2.51E6}{}$ rad/s,
$R_1 = \SI{100}{\ohm}$ and $C_1 = \SI{3.9}{\nano\farad}$ will
suffice. Only $R_2$ remains to be selected to determine the limit on
gain; selecting $R_2 = \SI{100}{\kohm}$ yields
$K = 1000 = \SI{60}{dB}$.
@ -227,8 +256,8 @@ Combining equations \ref{eqn:kcl} and \ref{eqn:known},
\begin{align}
\begin{split}
0 &= \frac{v_{in}-v_n}{R_1+\frac{1}{s C_1}} - \frac{v_n - A(-v_n)}{R_2} \\
&= \frac{v_{in}}{R_1+\frac{1}{s C_1}} - v_n \left(\frac{1}{R_1+s C_1} + \frac{A+1}{R_2} \right) \\
&= \frac{v_{in}}{R_1+\frac{1}{s C_1}} + \frac{v_{out}}{A} \left(\frac{1}{R_1+s C_1} + \frac{A+1}{R_2} \right) \\
&= \frac{v_{in}}{R_1+\frac{1}{s C_1}} - v_n \left(\frac{1}{R_1+\frac{1}{s C_1}} + \frac{A+1}{R_2} \right) \\
&= \frac{v_{in}}{R_1+\frac{1}{s C_1}} + \frac{v_{out}}{A} \left(\frac{1}{R_1+\frac{1}{s C_1}} + \frac{A+1}{R_2} \right) \\
\end{split}
\end{align}
@ -236,10 +265,9 @@ So,
\begin{align}
\begin{split}
\frac{v_{out}}{A} \left(\frac{1}{R_1+s C_1} + \frac{A+1}{R_2} \right) &= -\frac{v_{in}}{R_1+\frac{1}{s C_1}} \\
\frac{v_{out}}{A} \left(\frac{1}{R_1+\frac{1}{s C_1}} + \frac{A+1}{R_2} \right) &= -\frac{v_{in}}{R_1+\frac{1}{s C_1}} \\
H(s) = \frac{v_{out}}{v_{in}} &= -A \frac{R_2}{R_2+(A+1)(R_1+\frac{1}{s C_1})} \\
&= -A\frac{\SI{10}{\kohm}}{\SI{10}{\kohm} + (A + 1) (\SI{100}{\ohm} + \frac{1}{s \SI{39}{\nano\farad}})} \\
&= -A\frac{\SI{E4}{}}{\SI{E4}{} + (A + 1) (\SI{E2}{} + \frac{\SI{2.56E7}{}}{s})}
&= -A\frac{\SI{10}{\kohm}}{\SI{10}{\kohm} + (A + 1) (\SI{100}{\ohm} + \frac{1}{s \SI{3.9}{\nano\farad}})} \\
\end{split}
\end{align}
@ -284,11 +312,16 @@ approximately $\omega_c = 25$ rad/s. This yields:
In this case, the peak gain $A_0$ is still approximately $\SI{2E5}{}$
(agreeing with earlier content of the datasheet, noting
$A_{VD}= 200V/mV$ typical). The cut-off frequency, however, is much
higher, closer to $\omega_c = 210$ rad/s.
$A_{VD}= 200V/mV$ typical). The cut-off frequency, however, is higher,
closer to $\omega_c = 210$ rad/s.
\subsection{A whole new world: the OPA818}
The TI OPA818 is a much higher-cost, much higher-bandwidth, and much
less common op-amp than either the LM741 or the TL08X. It was selected
in case neither of the previous two op-amps were able to perform well
enough in this application.
\begin{figure}[h]
\caption{The open-loop gain of the OPA818}
\label{img:opa818_olgm}
@ -311,11 +344,161 @@ $-\frac{AR_2}{A(R_1 + \frac{1}{sC_1})} = -\frac{R_2}{R_1 +
function seen in Equation \ref{eqn:hpf_tf} for the inverting ideal
active high-pass filter.
\section{A complete comparison}
MATLAB is used to compare the final transfer functions of all four
cases; the result is shown in Figure \ref{img:bode_plots}. Of
particular note: MATLAB's symbolic handling does some heavy lifting to
generate Bode plots without having to further simplify the transfer
functions, beyond substituting each $A(s)$ into place.
\begin{figure}[h]
\caption{Bode plots of our four filters}
\label{img:bode_plots}
\centering
\includegraphics[width=0.5\textwidth]{bode_plots}
\end{figure}
Figure \ref{img:bode_plots} clearly shows that while the ideal op-amp
has no problem enabling a 'proper' $\omega_c = \SI{400}{\kHz}$, both
the LM741 and TL081 miss the mark substantially, and act in practice
more like band-pass filters with upper cutoff frequencies of around
$\SI{1.2E5}{}$ rad/s and $\SI{3.0E5}{}$ rad/s, respectively, both
\textit{below} the intended \textit{lower} cutoff frequency of our
high-pass filter.
Ultimately, even the much more capable OPA818 acts as a band-pass
filter rather than a high-pass filter. Nevertheless, it succeeds in
reaching an upper $\omega_{c_2} = \SI{4E7}{}$ rad/s
$\approx \SI{6.37}{\MHz}$, making it functional, if not necessarily
suitable, for the intended application.
\subsection{Gain response}
In terms of the frequency response of gain, it is clear that only
exceptional op-amps succeed in higher-frequency circumstances. Note as
well that the application described -- AM radio reception -- is,
compared to other types of radio amplification, relatively
low-frequency. Compare, for example, FM radio, which operates in the
$\SI{87.9}{\MHz}$ through $\SI{107.9}{\MHz}$ band. Of our models, the
only one that comes out unscathed is the ideal op-amp model.
\subsection{Phase response}
The phase response of the three non-ideal op-amps is consistent: below
the cutoff frequency, the phase is delayed by 270 degrees; in the
effective pass-band, the phase shift moves towards 180 degrees, and
after the pass-band, it reduces to 90 degrees.
Note that for any time-sensitive applications, where the bandwidth of
the signal being carried is a substantial portion of the frequency of
the signal, this phase shift would be problematic. An additional
phase-shift is entirely expected since we are using it as an inverting
op-amp, obligating an 180 degree phase-shift in the passband.
The ideal op-amp yields a true high-pass filter, which is reflected as
well in the fact that the phase shift never decreases past 180
degrees.
\section{Summary}
The exploration of op-amps applied to a higher-frequency application
makes for an excellent illustration for the extremely limited
applicability of some op-amps in this application. While an LM741 or
TL081 would make for a fine low-power audio amplifier, they are, due
to their low cutoff frequency, not suitable for $\ge \SI{100}{\kHz}$
applications; indeed, even more performant op-amps have limits in
amplifying higher-frequency signals.
\section{Code appendix}
\begin{lstlisting}[style=mystyle,breaklines=true,language=Octave]
% Define symbolic variables
syms s
r_1 = 1.e2
r_2 = 1.e5
c_1 = 3.9e-9
% Transfer function setup: LM741 %
A_0_LM741 = 2.e5;
tau_LM741 = 1./25;
A_sym_LM741 = A_0_LM741 / (tau_LM741 * s + 1);
% Symbolic setup and consumption: LM741 %
sym_LM741 = -A_sym_LM741 * (r_2) / (r_2 + (A_sym_LM741 + 1) * (r_1 + 1 / (s * c_1)));
tf_sym_LM741 = subs(sym_LM741, A_sym_LM741, A_0_LM741 / (tau_LM741 * s + 1));
% Perform sym extraction for LM741, and generate polynomial num/denom %
[nH_LM741um_sym_LM741, den_sym_LM741] = numden(tf_sym_LM741);
num_coeffs_LM741 = sym2poly(num_sym_LM741);
den_coeffs_LM741 = sym2poly(den_sym_LM741);
% Create the numeric transfer function object for LM741 %
LM741 = tf(num_coeffs_LM741, den_coeffs_LM741);
% Transfer function setup: TL081 %
A_0_TL081 = 2.e5;
tau_TL081 = 1./210;
A_sym_TL081 = A_0_TL081 / (tau_TL081 * s + 1);
% Symbolic setup and consumption: TL081 %
sym_TL081 = -A_sym_TL081 * (r_2) / (r_2 + (A_sym_TL081 + 1) * (r_1 + 1 / (s * c_1)));
tf_sym_TL081 = subs(sym_TL081, A_sym_TL081, A_0_TL081 / (tau_TL081 * s + 1));
% Perform sym extraction for TL081, and generate polynomial num/denom %
[num_sym_TL081, den_sym_TL081] = numden(tf_sym_TL081);
num_coeffs_TL081 = sym2poly(num_sym_TL081);
den_coeffs_TL081 = sym2poly(den_sym_TL081);
% Create the numeric transfer function object for TL081 %
TL081 = tf(num_coeffs_TL081, den_coeffs_TL081);
% Transfer function setup: OPA818 %
A_0_OPA818 = 4.e4;
tau_OPA818 = 1./7.E5;
A_sym_OPA818 = A_0_OPA818 / (tau_OPA818 * s + 1);
% Symbolic setup and consumption: OPA818 %
sym_OPA818 = -A_sym_OPA818 * (r_2) / (r_2 + (A_sym_OPA818 + 1) * (r_1 + 1 / (s * c_1)));
tf_sym_OPA818 = subs(sym_OPA818, A_sym_OPA818, A_0_OPA818 / (tau_OPA818 * s + 1));
% Perform sym extraction for OPA818, and generate polynomial num/denom %
[num_sym_OPA818, den_sym_OPA818] = numden(tf_sym_OPA818);
num_coeffs_OPA818 = sym2poly(num_sym_OPA818);
den_coeffs_OPA818 = sym2poly(den_sym_OPA818);
% Create the numeric transfer function object for OPA818 %
OPA818 = tf(num_coeffs_OPA818, den_coeffs_OPA818);
% Create a numeric TF for the ideal op-amp %
% Symbolic consumption: ideal %
tf_sym_ideal = - (r_2) / (r_1 + 1 / (s * c_1));
% Perform sym extraction for OPA818, and generate polynomial num/denom %
[num_sym_ideal, den_sym_ideal] = numden(tf_sym_ideal);
num_coeffs_ideal = sym2poly(num_sym_ideal);
den_coeffs_ideal = sym2poly(den_sym_ideal);
% Create the numeric transfer function object for the ideal op-amp %
ideal = tf(num_coeffs_ideal, den_coeffs_ideal);
% Generate the Bode plot
hold on;
bode(LM741);
bode(TL081);
bode(OPA818);
bode(ideal);
legend('LM741','TL081','OPA818','ideal');
grid on;
title('Frequency responses of op-amp driven high-pass filters');
hold off;
\end{lstlisting}
\begin{thebibliography}{00}
\bibitem{b1} J. Karki, ``Understanding Operational Amplifier Specifications.'' Accessed: May 05, 2025. [Online]. Available: https://www.ti.com/lit/an/sloa011b/sloa011b.pdf, p. 14.
\bibitem{b2} J. W. Nilsson and S. A. Riedel, Electric Crircuits, 12th ed., Hoboken: Pearson, 2022, p.168
\bibitem{b3} J. W. Nilsson and S. A. Riedel, Electric Crircuits, 12th ed., Hoboken: Pearson, 2022, p.576
\bibitem{b4} ``MSF radio time signal,'' NPLWebsite. https://www.npl.co.uk/msf-signal
\bibitem{b4} N. US Department of Commerce, ``NOAA Weather Radio Frequently Asked Questions,'' www.weather.gov. https://www.weather.gov/phi/nwrfaq
\end{thebibliography}
\vspace{12pt}
\end{document}