How is the cutoff frequency of a first-order RC high-pass filter defined?

The cutoff frequency is f_c = 1/(2 pi R C). Larger R or C lowers f_c, so even lower frequencies are passed. At f = f_c the gain is about -3.01 dB and the phase is +45 degrees (lead).

How does a high-pass differ from a low-pass RC?

The circuit topology is swapped: in a high-pass C sits in series and R in parallel to ground. At DC the capacitor blocks the signal, while at high frequency C looks like a short and the input appears at the output. The phase response is also inverted - a high-pass introduces a phase lead from +90 degrees down to 0 degrees.

Where is a high-pass filter actually used?

Removing DC offset and 1/f noise, ECG baseline removal, audio AC coupling between stages, blocking the gravity (DC) component on accelerometer signals, and PWM edge detection. Any time you want to keep the signal variation but discard slow drift, a high-pass is the simplest answer.

Is the band below f_c completely blocked?

No. A first-order high-pass only attenuates at +20 dB/decade, so at f = 0.1 f_c the gain is -20 dB and at 0.01 f_c only -40 dB. To truly suppress DC drift you also need to consider capacitor leakage current and any DC gain in subsequent stages.

High-Pass Filter Simulator — First-Order RC Filter Frequency Response

Parameters

Resistance R

Capacitance C

Observation frequency f

Quantization bits N

bit

Results

—

Cutoff frequency f_c

—

Gain at observation f

—

Phase

—

Time constant τ = RC

SNR (theory): —

RC high-pass circuit (I/O waveforms)

Bode plot (magnitude)

Theory & Key Formulas

Transfer function:

$$H(j\omega) = \dfrac{j\omega\tau}{1 + j\omega\tau},\quad \tau = R\,C$$

Cutoff frequency, gain and phase:

$$f_c = \dfrac{1}{2\pi R C},\quad |H(f)|_{\rm dB} = 20\log_{10}\!\dfrac{f/f_c}{\sqrt{1+(f/f_c)^2}},\quad \phi = 90° - \arctan(f/f_c)$$

At $f = f_c$, $|H| = 1/\sqrt{2}\approx -3.01\,\text{dB}$ and the phase is $+45°$. For $f \ll f_c$ the magnitude rolls off at +20 dB/decade and for $f \gg f_c$ it asymptotes to 0 dB.

What is a first-order RC high-pass filter?

🙋

Low-pass made sense to me — kill the high frequencies. But why would I ever want a high-pass?

🎓

Classic example: blocking DC. A microphone capsule carries a DC bias from its power rail. If you feed that straight into an amplifier it saturates. Put a series capacitor in front (= RC high-pass) and the DC is blocked while the audio AC passes through. We call this "AC coupling".

🙋

How does the circuit differ from a low-pass?

🎓

You just swap the parts. Low-pass has R in series and C to ground; high-pass has C in series and R to ground. At DC the capacitor impedance 1/(jωC) is infinite so the output is zero. At high frequency C is essentially a wire and the input voltage shows up across R. That asymmetry kills low frequencies.

🙋

The phase is +45° at f_c — that surprised me. Low-pass was -45°, right?

🎓

Right, a high-pass introduces phase lead. At f_c it is +45° and at very low frequencies it approaches +90°. The circuit also acts as a differentiator for τ short compared to the signal period, which is how PWM edge detectors and ECG QRS picks work.

Physical model and key equations

Apply Kirchhoff's voltage law to the series RC, with the output taken across R. From $i_C = C\,d(V_{in}-V_{out})/dt = V_{out}/R$ we obtain a first-order linear ODE.

$$RC\,\dfrac{dV_{out}}{dt} + V_{out} = RC\,\dfrac{dV_{in}}{dt}$$

Substituting $s = j\omega$ yields the transfer function $H(j\omega) = j\omega\tau/(1+j\omega\tau)$, from which the gain and phase follow.

$$|H(f)| = \dfrac{f/f_c}{\sqrt{1+(f/f_c)^2}},\quad \phi = 90° - \arctan(f/f_c),\quad f_c = \dfrac{1}{2\pi RC}$$

In decibel form, the response is 0 dB for $f \gg f_c$, exactly -3.01 dB at $f = f_c$, and asymptotes to +20 dB/decade for $f \ll f_c$. The phase moves monotonically from +90° down to 0°.

Real-world applications

Audio AC coupling: Place a series capacitor between amplifier stages to remove DC offset. Setting f_c below 20 Hz keeps the audio band intact while protecting downstream gain stages from rail-to-rail clipping.

ECG baseline removal: Strip out 0.5 Hz drifts from breathing and posture changes while keeping the P, QRS and T waves. A cutoff around 0.05 to 0.5 Hz is typical for diagnostic ECG.

Accelerometer gravity removal: Subtract the DC gravity vector so that only the motion-induced acceleration remains. Step counters and gesture recognizers rely on this preprocessing.

Differentiator (PWM edge detection): With τ much shorter than the input period, the high-pass output becomes a series of sharp spikes at every rising and falling edge — perfect for timing or trigger generation.

Common misconceptions and caveats

Myth 1: below f_c the signal is fully blocked. A first-order high-pass only attenuates at 20 dB/decade. At f = 0.1 f_c you get -20 dB, at 0.01 f_c only -40 dB. Truly removing DC requires accounting for capacitor leakage and any downstream DC gain.

Myth 2: a high-pass leaves the phase untouched. It introduces a +45° lead at f_c. In control loops this is useful as a lead compensator, but for waveform-shaping applications the phase distortion can be a problem.

Myth 3: just pick the biggest capacitor. Electrolytics introduce leakage current, temperature drift and polarity constraints. For low-frequency high-pass stages, film capacitors, tantalum or large MLCC arrays are usually safer choices.

FAQ

For a step input the high-pass output jumps to the input height and decays to 36.8% (= 1/e) in one τ. This is the basic property exploited by edge-detecting differentiator circuits.

Cascade a high-pass and a low-pass with f_HP < f_LP. The combined response passes a band between the two cutoffs, with the centre frequency f_0 = √(f_HP·f_LP) carrying the maximum gain.

CR is smaller, cheaper and much more common. LR (R in series with L to ground) shows up in RF and on power lines, but inductor size, core loss and self-resonance usually rule it out for audio or low-frequency work.

High-Pass Filter Simulator — First-Order RC Filter Frequency Response

What is a first-order RC high-pass filter?

Physical model and key equations

Real-world applications

Common misconceptions and caveats

FAQ

Related Tools