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Martin Kennedy 2025-05-05 07:54:38 -04:00
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29
Final.tex
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@ -114,23 +114,23 @@ divider, and so
\frac{v_{out}}{v_{in}} = - \frac{R_2}{R_1+Z_{C_1}}
\end{equation}
The high-pass filter design is a common one, with a well-known cutoff
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
suffice. Only $R_2$ remains to be selected to determine the limit on
gain; selecting $R_2 = \SI{100}{\kohm}$ yields
$K = 1000 = \SI{30}{dB}$.
$K = 1000 = \SI{60}{dB}$.
\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
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.
A more accurate representation of the op-amp foregoes the assumption,
often 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
$A$ between the input and output, also known as the open-loop
gain. Figure \ref{img:opamp_internal} depicts such a model.
\begin{figure}[h]
\caption{A more accurate depiction of an op-amp \cite{b2}}
@ -156,19 +156,24 @@ 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.
frequency; its value can be increased until the gain is restricted by
the op-amp itself instead of the resistor (otherwise, an op-amp
would have infinite gain if it lacked a feedback resistor).
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, $R_L = \SI{2}{\kohm}$),
and under conditions such that the load is significant, i.e. that the
operating output is known to be a meaningful fraction of the supply
voltage \cite{b1}. Here, it is $V_o = \SI{10}{V}$ for
$V_{CC} = \pm \SI{15}{V}$.
$V_{CC} = \pm \SI{15}{V}$. Still, for the purposes of this paper,
``open-loop gain'' is treated as ``open-loop large-signal differential
voltage amplification'', with caution made to stay away from
specifically small-signal measurements.
\begin{figure}[h]
\caption{The large-signal open-loop gain of the LM741}