RC highpass filter

An RC highpass filter is the RC lowpass with its two components swapped: a capacitor $C$ in series with the signal path and a resistor $R$ from the output to ground, output taken across $R$. It blocks low-frequency (and DC) signals and passes high-frequency ones — the canonical Highpass filter.

The highpass RC filter: series capacitor, shunt $R$, $V_{out}$ across $R$; rises at $+20$ dB/decade below $f_c=1/(2\pi RC)$.

Intuition

Reuse the Capacitive reactance picture, but now the capacitor is in series between input and output rather than shunting the output to ground.

• At low frequency ($\omega \to 0$): $Z_C=1/(j\omega C)\to \infty$. The series capacitor looks like an open circuit and disconnects the input from the output. $V_{out}\approx 0$. In particular it blocks DC completely ($\omega =0$ gives infinite impedance).
• At high frequency ($\omega \to \infty$): $Z_C\to 0$. The series capacitor looks like a wire, so $V_{out}\approx V_{in}$. The signal passes through unimpeded.

The behaviour is the exact mirror of the lowpass: low frequencies blocked, high frequencies passed.

Transfer function — derive it

Again a voltage divider, but now the output is across $R$ and the series element is $Z_C$:

$H(j\omega )=\frac{V_{out}}{V_{in}}=\frac{R}{R+Z_C}=\frac{R}{R+\frac{1}{j\omega C}}$

Multiply numerator and denominator by $j\omega C$:

$H(j\omega )=\frac{j\omega RC}{j\omega RC+1}=\frac{j\omega RC}{1+j\omega RC}$

Check the limits: at $\omega =0$, $H=0$ (blocked); as $\omega \to \infty$, the $j\omega RC$ terms dominate and $H\to 1$ (passed). The magnitude is

$|H(j\omega )|=\frac{\omega RC}{\sqrt{1+(\omega RC{)}^2}}$

Same cutoff, opposite slope

The corner is at the same frequency as the lowpass — it depends only on $R$ and $C$, not on which one is in series:

${\omega }_c=\frac{1}{RC},f_c=\frac{1}{2\pi RC}$

At ${\omega }_c$ the magnitude is $1/\sqrt{2}\approx 0.707$ (the −3 dB point). The difference from the lowpass is the direction of the slope: below $f_c$ the response rises at $+20\text{dB/decade}$ (every decade lower in frequency cuts the output by a factor of 10) until it flattens to unity above $f_c$. Compare with the Bode plot of the lowpass, which rolls off above $f_c$.

Phase

The numerator has a factor of $j$, which contributes $+90°$, and the denominator subtracts a frequency-dependent angle. The net phase is

$\phi =\arctan \left(\frac{1}{2\pi fRC}\right)$

At very low frequency the argument blows up and $\phi \to 90°$ (the output leads the input by nearly a quarter cycle); far above $f_c$ the argument $\to 0$ and $\phi \to 0°$ (output in phase with input). At exactly $f_c$, $\phi =45°$.

This is a coupling capacitor

The “Note” on the slide makes the key practical point: every amplifier later in the course puts a series capacitor between stages to pass the AC signal while blocking the DC bias of one stage from disturbing the next. That series capacitor, together with the input resistance it sees, is an RC highpass — see Coupling capacitor. Its cutoff $f_c=1/(2\pi RC)$ sets the low end of the amplifier’s useful frequency range: signals below it are attenuated, so coupling capacitors are sized to put $f_c$ safely below the lowest frequency of interest.