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\begin{document}

\title{An Amplified Review of Operational Amplifiers in High-Frequency Applications}

\author{\IEEEauthorblockN{Martin Kennedy}
\IEEEauthorblockA{\textit{College of Engineering and Mathematical Sciences} \\
\textit{University of Vermont}\\
So. Burlington, Vermont, USA \\
martin.kennedy@uvm.edu}
}

\maketitle

\begin{abstract}
  Operational amplifiers see wide use in modern electronics thanks to
  their convenient properties. However, commonplace op-amps like the
  LM741 have limitations which make them unsuitable in some
  applications. This document discusses an active first-order
  high-pass filter design, and uses a more realistic model of the
  op-amp to analytically investigate the importance of op-amp
  selection in this type of application.
\end{abstract}

\begin{IEEEkeywords}
op-amp, filter, signaling
\end{IEEEkeywords}

\section{Introduction}
The operational amplifier is one of the most important building blocks
in electronic circuits. Discrete op-amp integrated circuits (ICs) are
produced by nearly every general semiconductor manufacturer, but not
all op-amps perform equally.

In this paper, a simple high-pass filter based on an op-amp is
introduced, and used to review a particular property of the op-amp:
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:

\begin{enumerate}
\item
  The TI TL08x series op-amp,
\item
  The TI OPA818 op-amp, and
\item
  The ``ideal'' op-amp.
\end{enumerate}

TODO Lead into circuit analysis

\section{TODO rename: circuit analysis section}

TODO Describe the circuit

A design for a first-order inverting active high-pass filter with
amplification is depicted below.

\begin{figure}[h]
  \caption{A first-order high-pass filter}
  \begin{circuitikz}[american voltages]
   \draw
    (0,2) node[left=0cm]{$v_{in}$} to [short, o-] [R, l_=$R_1$] (1.5,2) coordinate (IN)
    (3,3) to [R, l_=$R_2$] (5.375,3) coordinate (FB)
    (3,2)
    node[op amp, noinv input down, anchor=-](OA) {}
    (OA.-) to [short, o-] ++(0, 1)

    (OA.-) to [C=$C_1$] (IN)
    (OA.+) to  [short, o-] ++(0,-0.5) node[ground]{}
    (OA.out) to ++(0, 0) coordinate (OUT) to (FB)
    (OUT) to [short, *-o] ++(0.5,0) node [right=0.2cm]{$v_{out}$};
    \node (v_p) [above=0cm] at (OA.+) {$v_p$};
    \node (v_n) [below=0cm] at (OA.-) {$v_n$};
    ;
\end{circuitikz}
\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
when $v_p > v_n$, $v_{out}$ is driven as high as possible, and
vice-versa when $v_p < v_n$.

At this point, it would be easy to derive the relationship between
$v_{out}$ and $v_{in}$: $v_n = v_p = \SI{0}{V}$; since all current
which flows from $v_{in}$ into $v_n$ then flows from $v_n$ into
$v_{out}$, the network of $R_1$ and $C_1$ into $R_2$ forms a voltage
divider, and so

\begin{equation}
  \frac{v_{out}}{v_{in}} = - \frac{R_2}{R_1+Z_{C_1}}
\end{equation}

To generate a cutoff frequency of $\omega_c = \SI{100}{kHz}$,

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
more precise description of the properties of the op-amp itself,
describing its input resistance $R_i$, output resistance $R_o$, and
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}
  \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.

% 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
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''.

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'':
for example, the LM741 datasheet documents the \textit{Open-Loop
  Large-Signal Differential Voltage Amplification} as a function of
frequency, as seen in Figure \ref{img:lm741_oclsg}. This differs from
``open-loop gain'' in that the operating output is known to be a
meaningful fraction of the supply voltage; here, it is
$V_o = \SI{10}{V}$ for $V_{CC} = \pm \SI{15}{V}$.

\begin{figure}[h]
  \caption{The large-signal open-loop gain of the LM741}
  \label{img:lm741_oclsg}
  \centering
  \includegraphics[width=0.4\textwidth]{lm741_oclsg}
\end{figure}

\section{Ease of Use}

\subsection{Maintaining the Integrity of the Specifications}

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.

\section{Prepare Your Paper Before Styling}
Before you begin to format your paper, first write and save the content as a
separate text file. Complete all content and organizational editing before
formatting. Please note sections \ref{AA}--\ref{SCM} below for more information on
proofreading, spelling and grammar.

Keep your text and graphic files separate until after the text has been
formatted and styled. Do not number text heads---{\LaTeX} will do that
for you.

\subsection{Abbreviations and Acronyms}\label{AA}
Define abbreviations and acronyms the first time they are used in the text,
even after they have been defined in the abstract. Abbreviations such as
IEEE, SI, MKS, CGS, ac, dc, and rms do not have to be defined. Do not use
abbreviations in the title or heads unless they are unavoidable.

\subsection{Units}
\begin{itemize}
\item Use either SI (MKS) or CGS as primary units. (SI units are encouraged.) English units may be used as secondary units (in parentheses). An exception would be the use of English units as identifiers in trade, such as ``3.5-inch disk drive''.
\item Avoid combining SI and CGS units, such as current in amperes and magnetic field in oersteds. This often leads to confusion because equations do not balance dimensionally. If you must use mixed units, clearly state the units for each quantity that you use in an equation.
\item Do not mix complete spellings and abbreviations of units: ``Wb/m\textsuperscript{2}'' or ``webers per square meter'', not ``webers/m\textsuperscript{2}''. Spell out units when they appear in text: ``. . . a few henries'', not ``. . . a few H''.
\item Use a zero before decimal points: ``0.25'', not ``.25''. Use ``cm\textsuperscript{3}'', not ``cc''.)
\end{itemize}

\subsection{Equations}
Number equations consecutively. To make your
equations more compact, you may use the solidus (~/~), the exp function, or
appropriate exponents. Italicize Roman symbols for quantities and variables,
but not Greek symbols. Use a long dash rather than a hyphen for a minus
sign. Punctuate equations with commas or periods when they are part of a
sentence, as in:
\begin{equation}
a+b=\gamma\label{eq}
\end{equation}

Be sure that the
symbols in your equation have been defined before or immediately following
the equation. Use ``\eqref{eq}'', not ``Eq.~\eqref{eq}'' or ``equation \eqref{eq}'', except at
the beginning of a sentence: ``Equation \eqref{eq} is . . .''

\subsection{\LaTeX-Specific Advice}

Please use ``soft'' (e.g., \verb|\eqref{Eq}|) cross references instead
of ``hard'' references (e.g., \verb|(1)|). That will make it possible
to combine sections, add equations, or change the order of figures or
citations without having to go through the file line by line.

Please don't use the \verb|{eqnarray}| equation environment. Use
\verb|{align}| or \verb|{IEEEeqnarray}| instead. The \verb|{eqnarray}|
environment leaves unsightly spaces around relation symbols.

Please note that the \verb|{subequations}| environment in {\LaTeX}
will increment the main equation counter even when there are no
equation numbers displayed. If you forget that, you might write an
article in which the equation numbers skip from (17) to (20), causing
the copy editors to wonder if you've discovered a new method of
counting.

{\BibTeX} does not work by magic. It doesn't get the bibliographic
data from thin air but from .bib files. If you use {\BibTeX} to produce a
bibliography you must send the .bib files.

{\LaTeX} can't read your mind. If you assign the same label to a
subsubsection and a table, you might find that Table I has been cross
referenced as Table IV-B3.

{\LaTeX} does not have precognitive abilities. If you put a
\verb|\label| command before the command that updates the counter it's
supposed to be using, the label will pick up the last counter to be
cross referenced instead. In particular, a \verb|\label| command
should not go before the caption of a figure or a table.

Do not use \verb|\nonumber| inside the \verb|{array}| environment. It
will not stop equation numbers inside \verb|{array}| (there won't be
any anyway) and it might stop a wanted equation number in the
surrounding equation.

\subsection{Some Common Mistakes}\label{SCM}
\begin{itemize}
\item The word ``data'' is plural, not singular.
\item The subscript for the permeability of vacuum $\mu_{0}$, and other common scientific constants, is zero with subscript formatting, not a lowercase letter ``o''.
\item In American English, commas, semicolons, periods, question and exclamation marks are located within quotation marks only when a complete thought or name is cited, such as a title or full quotation. When quotation marks are used, instead of a bold or italic typeface, to highlight a word or phrase, punctuation should appear outside of the quotation marks. A parenthetical phrase or statement at the end of a sentence is punctuated outside of the closing parenthesis (like this). (A parenthetical sentence is punctuated within the parentheses.)
\item A graph within a graph is an ``inset'', not an ``insert''. The word alternatively is preferred to the word ``alternately'' (unless you really mean something that alternates).
\item Do not use the word ``essentially'' to mean ``approximately'' or ``effectively''.
\item In your paper title, if the words ``that uses'' can accurately replace the word ``using'', capitalize the ``u''; if not, keep using lower-cased.
\item Be aware of the different meanings of the homophones ``affect'' and ``effect'', ``complement'' and ``compliment'', ``discreet'' and ``discrete'', ``principal'' and ``principle''.
\item Do not confuse ``imply'' and ``infer''.
\item The prefix ``non'' is not a word; it should be joined to the word it modifies, usually without a hyphen.
\item There is no period after the ``et'' in the Latin abbreviation ``et al.''.
\item The abbreviation ``i.e.'' means ``that is'', and the abbreviation ``e.g.'' means ``for example''.
\end{itemize}
An excellent style manual for science writers is \cite{b7}.

\subsection{Authors and Affiliations}
\textbf{The class file is designed for, but not limited to, six authors.} A
minimum of one author is required for all conference articles. Author names
should be listed starting from left to right and then moving down to the
next line. This is the author sequence that will be used in future citations
and by indexing services. Names should not be listed in columns nor group by
affiliation. Please keep your affiliations as succinct as possible (for
example, do not differentiate among departments of the same organization).

\subsection{Identify the Headings}
Headings, or heads, are organizational devices that guide the reader through
your paper. There are two types: component heads and text heads.

Component heads identify the different components of your paper and are not
topically subordinate to each other. Examples include Acknowledgments and
References and, for these, the correct style to use is ``Heading 5''. Use
``figure caption'' for your Figure captions, and ``table head'' for your
table title. Run-in heads, such as ``Abstract'', will require you to apply a
style (in this case, italic) in addition to the style provided by the drop
down menu to differentiate the head from the text.

Text heads organize the topics on a relational, hierarchical basis. For
example, the paper title is the primary text head because all subsequent
material relates and elaborates on this one topic. If there are two or more
sub-topics, the next level head (uppercase Roman numerals) should be used
and, conversely, if there are not at least two sub-topics, then no subheads
should be introduced.

\subsection{Figures and Tables}
\paragraph{Positioning Figures and Tables} Place figures and tables at the top and
bottom of columns. Avoid placing them in the middle of columns. Large
figures and tables may span across both columns. Figure captions should be
below the figures; table heads should appear above the tables. Insert
figures and tables after they are cited in the text. Use the abbreviation
``Fig.~\ref{fig}'', even at the beginning of a sentence.

\begin{table}[htbp]
\caption{Table Type Styles}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{Table}&\multicolumn{3}{|c|}{\textbf{Table Column Head}} \\
\cline{2-4}
\textbf{Head} & \textbf{\textit{Table column subhead}}& \textbf{\textit{Subhead}}& \textbf{\textit{Subhead}} \\
\hline
copy& More table copy$^{\mathrm{a}}$& &  \\
\hline
\multicolumn{4}{l}{$^{\mathrm{a}}$Sample of a Table footnote.}
\end{tabular}
\label{tab1}
\end{center}
\end{table}

% \begin{figure}[htbp]
% \centerline{\includegraphics{fig1.png}}
% \caption{Example of a figure caption.}
% \label{fig}
% \end{figure}

Figure Labels: Use 8 point Times New Roman for Figure labels. Use words
rather than symbols or abbreviations when writing Figure axis labels to
avoid confusing the reader. As an example, write the quantity
``Magnetization'', or ``Magnetization, M'', not just ``M''. If including
units in the label, present them within parentheses. Do not label axes only
with units. In the example, write ``Magnetization (A/m)'' or ``Magnetization
\{A[m(1)]\}'', not just ``A/m''. Do not label axes with a ratio of
quantities and units. For example, write ``Temperature (K)'', not
``Temperature/K''.

\section*{Acknowledgment}

The preferred spelling of the word ``acknowledgment'' in America is without
an ``e'' after the ``g''. Avoid the stilted expression ``one of us (R. B.
G.) thanks $\ldots$''. Instead, try ``R. B. G. thanks$\ldots$''. Put sponsor
acknowledgments in the unnumbered footnote on the first page.

\section*{References}

Please number citations consecutively within brackets \cite{b1}. The
sentence punctuation follows the bracket \cite{b2}. Refer simply to the reference
number, as in \cite{b3}---do not use ``Ref. \cite{b3}'' or ``reference \cite{b3}'' except at
the beginning of a sentence: ``Reference \cite{b3} was the first $\ldots$''

Number footnotes separately in superscripts. Place the actual footnote at
the bottom of the column in which it was cited. Do not put footnotes in the
abstract or reference list. Use letters for table footnotes.

Unless there are six authors or more give all authors' names; do not use
``et al.''. Papers that have not been published, even if they have been
submitted for publication, should be cited as ``unpublished'' \cite{b4}. Papers
that have been accepted for publication should be cited as ``in press'' \cite{b5}.
Capitalize only the first word in a paper title, except for proper nouns and
element symbols.

For papers published in translation journals, please give the English
citation first, followed by the original foreign-language citation \cite{b6}.

\begin{thebibliography}{00}
\bibitem{b1} G. Eason, B. Noble, and I. N. Sneddon, ``On certain integrals of Lipschitz-Hankel type involving products of Bessel functions,'' Phil. Trans. Roy. Soc. London, vol. A247, pp. 529--551, April 1955.
\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{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.
\end{thebibliography}
\vspace{12pt}
\color{red}
IEEE conference templates contain guidance text for composing and formatting conference papers. Please ensure that all template text is removed from your conference paper prior to submission to the conference. Failure to remove the template text from your paper may result in your paper not being published.

\end{document}
