From 7e1afa2ad0bbb5c7745774b512d8d557b642746b Mon Sep 17 00:00:00 2001 From: Martin Kennedy Date: Mon, 5 May 2025 07:09:31 -0400 Subject: [PATCH] MSF --- Final.tex | 153 +++++++++++++++++++++++++++++++++++++++--------------- 1 file changed, 111 insertions(+), 42 deletions(-) diff --git a/Final.tex b/Final.tex index 1575006..352b686 100644 --- a/Final.tex +++ b/Final.tex @@ -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.