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Final.tex
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@ -54,29 +54,32 @@ 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{100}{kHz} is selected for the high-pass filter design. To compare,
three other op-amps are selected beyond the LM741:
$\SI{40}{kHz}$ is selected for the high-pass filter design. To compare,
three op-amps are selected beyond the LM741:
\begin{enumerate}
\item
The TI TL08x series op-amp,
The TI TL081,
\item
The TI OPA818 op-amp, and
The TI OPA818, and
\item
The ``ideal'' op-amp.
\end{enumerate}
TODO Lead into circuit analysis
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.
\section{TODO rename: circuit analysis section}
TODO Describe the circuit
\section{Design of a High-pass Active Filter}
A design for a first-order inverting active high-pass filter with
amplification is depicted below.
amplification is depicted below in Figure \ref{fig:hpf}.
\begin{figure}[h]
\caption{A first-order high-pass filter}
\label{fig:hpf}
\begin{circuitikz}[american voltages]
\draw
(0,2) node[left=0cm]{$v_{in}$} to [short, o-] [R, l_=$R_1$] (1.5,2) coordinate (IN)
@ -96,8 +99,8 @@ amplification is depicted below.
\end{figure}
Ordinarily, the analysis of this filter is easy. With an ideal op-amp
configured in negative feedback, $v_n = v_p$: with awareness of the
properties of an ideal op-amp, this is easy to see by inspection, as
configured in negative feedback, $v_n = v_p$: this is easy to see by
inspection with awareness of the properties of an ideal op-amp, as
when $v_p > v_n$, $v_{out}$ is driven as high as possible, and
vice-versa when $v_p < v_n$.
@ -111,10 +114,16 @@ divider, and so
\frac{v_{out}}{v_{in}} = - \frac{R_2}{R_1+Z_{C_1}}
\end{equation}
To achieve a cutoff frequency of $\omega_c = \SI{100}{kHz}$,
TODO select cap and resistors for cutoff frequency
The 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
suffice. Only $R_2$ remains to be selected to determine the limit on
gain; selecting $R_2 = \SI{100}{\kohm}$ yields
$K = 1000 = \SI{30}{dB}$.
\subsection{Toward a more perfect Model}
A more accurate representation of the op-amp foregoes the
assumption, oft made regarding op-amps wired in a feedback
configuration, that $v_n = v_p$. Avoiding this assumption requires a
@ -124,36 +133,35 @@ the relationship between the input and output - the open-loop gain
$A$. Figure \ref{img:opamp_internal} depicts such a model.
\begin{figure}[h]
\caption{A more accurate depiction of an op-amp}
\caption{A more accurate depiction of an op-amp \cite{b2}}
\label{img:opamp_internal}
\centering
\includegraphics[width=0.4\textwidth]{opamp_internal}
\end{figure}
TODO Add citation for this image
An ideal op-amp operates as though $R_i \to \infty$, $R_o \to 0$, and
$A \to \infty$: it is these three properties which in the ideal case
allow $v_n$ to be considered equal to $v_p$ in the closed-loop
feedback configuration.
feedback configuration \cite{b2}.
% TODO: Cite Nilsson file:///home/mkennedy/Sync/Books/James W. Nilsson/ELECTRIC CIRCUITS,12e (49)/ELECTRIC CIRCUITS,12e - James W. Nilsson.pdf#page=194
% > Note that Eq. 5.22 reduces to Eq. 5.4 as R 0,o → R ,i → ∞ and A ∞
This paper focuses only on perturbing $A$ in our model; the
This paper focuses only on adjusting $A$ in our model; the
assumptions that $R_o = 0$ and $R_i \to \infty$ remain in place.
\subsection{Notes on Terms related to Open-Loop Gain}
Manufacturers document the value of $A$ primarily as a function of
frequency in an attribute called the ``open-loop gain''.
frequency in an attribute called the ``open-loop gain''. This is the
gain of the op-amp when no feedback is applied to connect the output
and input of the op-amp. This measure is useful, as it describes the
absolute maximum gain performance of the op-amp: notice, for example,
that the feedback resistor $R_2$ in the selected high-pass filter
design only serves to limit the gain and has no bearing on the cutoff
frequency.
In some cases, manufacturers give more precise details about the
conditions under which the op-amp amplifies higher and lower
frequencies than would be specified under generic ``open-loop gain'':
In some cases, manufacturers give more precise details about op-amp
operation than would be specified under generic ``open-loop gain'':
for example, the LM741 datasheet documents the \textit{Open-Loop
Large-Signal Differential Voltage Amplification} as a function of
Large-Signal Differential Voltage Amplification} as a function of
frequency, as seen in Figure \ref{img:lm741_oclsg}. Despite being
closely related to ``open-loop gain'', it is distinct in that it is
measured with an output load (in this case, $R_L = \SI{2}{\kohm}$),
@ -178,31 +186,92 @@ complex. The following observations simplify the task:
source has voltage $A(-v_n)$
\item $R_i \to \infty$ is, in effect, an open circuit, so the $R_i$
branch need not be considered
\item $R_o = 0$ is a wire, and $R_o$ can be ignored.
\item $R_o = 0$ acts as a wire, meaning $R_o$ can be ignored.
\end{itemize}
The simplified circuit can be depicted as such:
The simplified circuit can be depicted as seen in Figure
\ref{fig:simp_model}. Solving this system is not daunting; there is
only a single branch.
\begin{figure}[h]
\caption{A simplification of the combined model}
\label{fig:simp_model}
\begin{circuitikz}[american voltages]
\draw (0,0) node[left=0cm]{$v_{in}$} to [short, o-] [R, l_=$R_1$, i=$i$] (1.5,0);
\draw (1.5,0) to [C, l_=$\frac{1}{j\omega C_1}$] (2.75,0);
\draw (1.5,0) to [C, l_=$\frac{1}{s C_1}$] (2.75,0);
\draw (2.75,0) node [above=0.1cm]{$v_n$} to [short, o-] [R, l_=$R_2$] (5,0);
\draw (5,0) node [above=0.1cm]{$v_{out}$} to [short, o-] [american controlled voltage source, label=$A(-v_n)$] (7,0);
\draw (7,0) node[ground]{};
\end{circuitikz}
\end{figure}
\subsection{Maintaining the Integrity of the Specifications}
By Kirchoff's Current Law,
The IEEEtran class file is used to format your paper and style the text. All margins,
column widths, line spaces, and text fonts are prescribed; please do not
alter them. You may note peculiarities. For example, the head margin
measures proportionately more than is customary. This measurement
and others are deliberate, using specifications that anticipate your paper
as one part of the entire proceedings, and not as an independent document.
Please do not revise any of the current designations.
\begin{equation}\label{eqn:kcl}
\frac{v_{in}-v_n}{R_1+\frac{1}{s C_1}} - \frac{v_n - v_{out}}{R_2} = 0
\end{equation}
By observation,
\begin{equation}\label{eqn:known}
v_{out} = A(-v_n), \quad \frac{-v_{out}}{A} = v_n
\end{equation}
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) \\
\end{split}
\end{align}
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}} \\
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})}
\end{split}
\end{align}
$H(s)$ is our \textit{transfer function}, representing how the
signal changes from the input to the output (specifically, the ratio
of the output to the input, as a function of frequency).
There remains an unspecified term $A$; this term depends on which
op-amp is being used. It remains to be shown how the overall transfer
function responds as the properties of each op-amp is applied in turn.
\subsection{The original case: the LM741}
As previously seen in Figure \ref{img:lm741_oclsg}, the open-loop gain
$A$ of the LM741 decreases logarithmically as the frequency increases
logarithmically. The rate of reduction matches that which is seen in
normal first-order filters, approximately 20dB per order of magnitude
of frequency increase (also known as ``per decade''). This standard
decrease means that a transfer function can be used to represent $A$ as
one might represent a first-order low-pass filter. For given values of
$\tau = \frac{1}{\omega_c}$ and $A_0$:
\begin{equation}
A(s) = \frac{A_0}{\tau s + 1}
\end{equation}
$A_0$ represents the peak gain; here, $\SI{106}{dB} \approx
\SI{2E5}{}$. Since $\omega_c = 25$ rad/s.
\subsection{An improvement: The TL081}
\begin{figure}[h]
\caption{The large-signal open-loop gain of the TL081}
\label{img:tl08xx_oclsg}
\centering
\includegraphics[width=0.4\textwidth]{tl08xx_oclsg}
\end{figure}
\section{Prepare Your Paper Before Styling}
Before you begin to format your paper, first write and save the content as a
@ -396,9 +465,9 @@ citation first, followed by the original foreign-language citation \cite{b6}.
\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. Clerk Maxwell, A Treatise on Electricity and Magnetism, 3rd ed., vol. 2. Oxford: Clarendon, 1892, pp.68--73.
\bibitem{b3} I. S. Jacobs and C. P. Bean, ``Fine particles, thin films and exchange anisotropy,'' in Magnetism, vol. III, G. T. Rado and H. Suhl, Eds. New York: Academic, 1963, pp. 271--350.
\bibitem{b4} K. Elissa, ``Title of paper if known,'' unpublished.
\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{b5} R. Nicole, ``Title of paper with only first word capitalized,'' J. Name Stand. Abbrev., in press.
\bibitem{b6} Y. Yorozu, M. Hirano, K. Oka, and Y. Tagawa, ``Electron spectroscopy studies on magneto-optical media and plastic substrate interface,'' IEEE Transl. J. Magn. Japan, vol. 2, pp. 740--741, August 1987 [Digests 9th Annual Conf. Magnetics Japan, p. 301, 1982].
\bibitem{b7} M. Young, The Technical Writer's Handbook. Mill Valley, CA: University Science, 1989.