Section 3.6 RC High-pass filter
Let’s look at some circuit behavior for sinusoidal signals. Figure 3.6.1 shows the circuit that we will examine.
There is nothing in the formulation of Kirchhoff’s laws that would limit them to only DC circuits. In fact, they apply equally well to all circuits (DC, AC, non-linear). Assume that all quantities with a tilde over them are complex quantities expressed as exponentials. So,
\begin{equation*}
\tilde{V}_\text{in}(t) = \tilde{V}_{\text{in}_0} e^{i\omega t} = V_0 e^{i\delta} e^{i\omega t}
\end{equation*}
where the real signal \(V_\text{in}(t)=\Re\left[\tilde{V}_\text{in}(t)\right]\text{.}\) Using Kirchhoff’s Voltage Law, we find that
\begin{equation*}
\tilde{V}_\text{in} - \tilde{I} Z_C - \tilde{I} Z_R = \tilde{V}_\text{in} - \tilde{I} \left(Z_C - Z_R\right) = 0
\end{equation*}
so that
\begin{equation*}
\tilde{I} = \frac{\tilde{V}_\text{in}}{Z_R + Z_C} = \frac{\tilde{V}_\text{in}}{R-\frac{i}{\omega C}}\text{.}
\end{equation*}
We can then find \(\tilde{V}_\text{out}\) using Ohm’s Law
\begin{equation*}
\tilde{V}_\text{out}=\tilde{I} R = \frac{R}{R-\frac{i}{\omega C}}\tilde{V}_\text{in}\text{.}
\end{equation*}
Expanding our voltage signals,
\begin{equation*}
\tilde{V}_{\text{out}_0} e^{i\omega t} = \frac{R}{R-\frac{i}{\omega C}}\tilde{V}_{\text{in}_0} e^{i\omega t}
\end{equation*}
so that
\begin{equation*}
\tilde{V}_{\text{out}_0} = \frac{R}{R-\frac{i}{\omega C}}\tilde{V}_{\text{in}_0}\text{.}
\end{equation*}
Then,
\begin{equation*}
\frac{\tilde{V}_{\text{out}_0}}{\tilde{V}_{\text{in}_0}}
= \frac{R}{R-\frac{i}{\omega C}}
= \frac{R}{R-\frac{i}{\omega C}} \frac{\left(R+\frac{i}{\omega C}\right)/R^2}{\left(R+\frac{i}{\omega C}\right)/R^2}
= \frac{1+\frac{i}{\omega R C}}{1+\left(\frac{1}{\omega R C}\right)^2}\text{.}
\end{equation*}
Expanding the complex amplitude,
\begin{equation*}
\frac{\tilde{V}_{\text{out}_0}}{\tilde{V}_{\text{in}_0}}
= \frac{V_{\text{out}_0}e^{i\delta'}}{V_{\text{in}_0}e^{i\delta}}
= \frac{V_{\text{out}_0}}{V_{\text{in}_0}}e^{i\phi}
\end{equation*}
where \(\phi=\delta' - \delta\text{.}\) We can calculate the voltage gain \(G_V=V_{\text{out}_0}/V_{\text{in}_0}\text{:}\)
\begin{align*}
G_V \amp
= \frac{V_{\text{out}_0}}{V_{\text{in}_0}}\\
\amp
= \left| \frac{V_{\text{out}_0}}{V_{\text{in}_0}} e^{i\phi}\right|
= \left| \frac{1+\frac{i}{\omega R C}}{1+\left(\frac{1}{\omega R C}\right)^2}\right|
\end{align*}
so
\begin{equation}
G_V= \frac{1}{\sqrt{1+\left(\frac{1}{\omega R C}\right)^2}}
= \frac{1}{\sqrt{1+\left(f_0/f\right)^2}}\tag{3.6.1}
\end{equation}
where \(f_0=1/2\pi RC\) and \(f=\omega/2\pi\text{.}\) We can also determine the phase difference \(\phi\) between the output and input voltages. Since
\begin{equation*}
\frac{V_{\text{out}_0}}{V_{\text{in}_0}} e^{i\phi}
= \frac{1+\frac{i}{\omega R C}}{1+\left(\frac{1}{\omega R C}\right)^2}\text{,}
\end{equation*}
we can rearrange to find
\begin{equation*}
e^{i\phi}
= \frac{V_{\text{in}_0}}{V_{\text{out}_0}}
\frac{1+\frac{i}{\omega R C}}{1+\left(\frac{1}{\omega R C}\right)^2}\text{.}
\end{equation*}
Now,
\begin{equation*}
\tan(\phi)
= \frac{\sin(\phi)}{\cos(\phi)}
= \frac{\Im\left(e^{i\phi}\right)}{\Re\left(e^{i\phi}\right)}
= \frac{1}{\omega R C}
\end{equation*}
so that
\begin{equation}
\phi
= \tan^{-1}\left(\frac{1}{\omega R C}\right)
= \tan^{-1}\left( f_0/f\right)\text{.}\tag{3.6.2}
\end{equation}
- \(0^\circ \lt \phi \lt 90^\circ\text{,}\) meaning \(\tilde{V}_\text{out}\) leads \(\tilde{V}_\text{in}\text{.}\)
- \(V_{\text{out}_0}\rightarrow V_{\text{in}_0}\) as \(f\rightarrow \infty\text{.}\)
- \(V_{\text{out}_0} \rightarrow 0\) as \(f\rightarrow 0\text{.}\)
Based on these behaviors, the circuit in Figure 3.6.1 is called a RC high-pass filter. Thus, if a voltage input signal contains many sinusoidal signal components with a variety of frequencies, then this filter circuit can eliminate the low-frequency components while leaving the high-frequency components unaffected.
