How do I choose R and C values?

From the desired cutoff frequency: RC = 1/ω_c = 1/(2πf_c). For f_c = 100 Hz: RC = 0.00159s. Choose C = 100nF, then R = 15.9kΩ (use standard 16kΩ). The LAPLACE Calculator can verify your filter's frequency response.

What is the difference between first and second-order high-pass?

First-order: +20 dB/decade rolloff, 90° max phase shift, no resonance. Second-order: +40 dB/decade rolloff, 180° max phase shift, possible resonance peak if underdamped. Second-order gives sharper frequency separation.

How is a high-pass filter different from AC coupling?

AC coupling IS a high-pass filter. When you AC-couple a signal (series capacitor blocking DC), you're implementing a first-order high-pass filter. The cutoff frequency depends on the capacitor value and the input impedance of the next stage.

High Pass Filter Transfer Function

Quick Answer

A first-order high-pass filter has transfer function H(s) = s/(s + ω_c) = RCs/(RCs + 1), where ω_c = 1/(RC) is the cutoff frequency. It passes frequencies above ω_c while attenuating those below. The magnitude is −3 dB at ω_c, rises at +20 dB/decade below ω_c, and flattens to 0 dB above. Phase shifts from +90° at DC to 0° at high frequencies. For R = 10kΩ and C = 100nF, ω_c = 1000 rad/s (≈159 Hz). Second-order high-pass: H(s) = s²/(s² + 2ζω_ns + ω_n²).

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First-Order High-Pass Filter: Circuit and Transfer Function

The simplest high-pass filter places a capacitor in series with the signal path and a resistor to ground, with output taken across the resistor. Applying the voltage divider with impedances: H(s) = Z_R/(Z_C + Z_R) = R/(1/(sC) + R) = sRC/(sRC + 1). Normalizing: H(s) = s/(s + 1/(RC)) = s/(s + ω_c). At DC (s = 0), H = 0 — the capacitor blocks DC completely. At high frequencies (s → ∞), H → 1 — the capacitor acts as a short circuit, passing the signal through. The transition between these extremes occurs around ω_c = 1/(RC).

Key Formulas

Frequency Response and Bode Plot

Substituting s = jω gives H(jω) = jω/(jω + ω_c). The magnitude is |H| = ω/√(ω² + ω_c²). At ω = ω_c: |H| = 1/√2 = −3 dB (half-power point). Well below ω_c, magnitude increases at +20 dB/decade (the capacitor's impedance decreases). Well above ω_c, magnitude is flat at 0 dB. The phase is ∠H = 90° − arctan(ω/ω_c). At DC: +90° (capacitor dominates). At ω_c: +45°. At high frequencies: 0°. The Bode plot is the mirror image of the low-pass filter — everything that the low-pass attenuates, the high-pass passes.

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Second-Order High-Pass Filter

A second-order high-pass has H(s) = s²/(s² + 2ζω_ns + ω_n²), with −40 dB/decade rolloff below ω_n instead of −20 dB/decade for first-order. The damping ratio ζ affects the transition: ζ = 0.707 (Butterworth) gives the flattest passband. ζ < 0.707 creates a resonance peak near the cutoff — the magnitude exceeds 0 dB before settling. Second-order high-pass filters can be built with two capacitors and two resistors (Sallen-Key topology) or using an op-amp active filter. The steeper rolloff provides better rejection of unwanted low frequencies compared to first-order.

Applications: Audio, Instrumentation, and Communications

In audio, high-pass filters remove low-frequency rumble, handling noise, and wind noise from microphone signals — a 'high-pass filter at 80 Hz' is standard on mixing consoles. In instrumentation, high-pass (AC coupling) removes DC offset from sensor signals, allowing measurement of only the varying component. In communications, high-pass filtering separates the information-carrying frequencies from DC bias. In feedback control systems, high-pass compensators (lead compensators) add phase lead near the crossover frequency, improving stability margins.

Combining High-Pass and Low-Pass: Band-Pass and Notch

Cascading a high-pass filter (cutoff ω_L) with a low-pass filter (cutoff ω_H, where ω_H > ω_L) creates a band-pass filter that passes only frequencies between ω_L and ω_H. The transfer function is the product H_BP(s) = H_HP(s) · H_LP(s). Subtracting a band-pass from unity gives a band-stop (notch) filter that rejects a specific frequency range. These combinations are fundamental in audio crossover networks (separating bass from treble), radio receivers (selecting one station), and interference rejection (removing 50/60 Hz power line noise).

Frequently Asked Questions

First-order: H(s) = s/(s + ω_c) where ω_c = 1/(RC). It blocks DC, passes high frequencies, with −3 dB at the cutoff frequency ω_c. The rolloff below cutoff is +20 dB/decade (or equivalently, attenuation increases at −20 dB/decade going lower).

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