Engineering · Electrical Engineering · Signal Processing
High-Pass Filter Calculator
Calculates the cutoff frequency, impedance, and phase angle of a first-order RC or RL high-pass filter.
Calculator
Formula
For an RC high-pass filter: f_c is the cutoff frequency in Hz, R is resistance in ohms, and C is capacitance in farads. For an RL high-pass filter: R is resistance in ohms and L is inductance in henries. At the cutoff frequency the output signal power is reduced to half (−3 dB). The phase angle between input and output is given by \phi = \arctan\left(\frac{f_c}{f}\right) for the RC type.
Source: Sedra, A. S. & Smith, K. C. — Microelectronic Circuits, 7th Edition; Horowitz & Hill — The Art of Electronics, 3rd Edition.
How it works
A high-pass filter works by exploiting the frequency-dependent impedance of reactive components — capacitors and inductors. In an RC high-pass filter, the capacitor is placed in series with the signal path, and the resistor is connected to ground. At low frequencies, the capacitor presents a very high impedance and blocks the signal; at high frequencies, the capacitor impedance drops and signals pass freely. In an RL high-pass filter, the inductor is connected to ground and the resistor is in series — at low frequencies, the inductor's low impedance shunts the signal to ground, while at high frequencies it presents high impedance and allows the signal to pass.
The cutoff frequency is the critical design parameter. For an RC filter it is defined as f_c = 1 / (2πRC), and for an RL filter as f_c = R / (2πL). At the cutoff frequency, the output voltage is 1/√2 (approximately 70.7%) of the input voltage, which corresponds to a power reduction of one half, or −3 dB. The time constant τ = RC (or L/R for RL) describes how quickly the filter responds to transient signals. The phase angle describes the leading phase shift introduced by the high-pass filter — at the cutoff frequency the phase lead is 45°, approaching 90° for very low frequencies and 0° as frequency increases far above the cutoff.
High-pass filters are used across many domains: audio engineers use them to remove subsonic rumble and DC offsets from microphone inputs; RF engineers use HPFs in receiver front-ends to reject out-of-band interference; instrumentation engineers apply them to eliminate slow thermal drift in sensor signals; and power electronics engineers use RL high-pass topologies to filter switching noise. The −3 dB cutoff frequency, along with the filter's roll-off rate (−20 dB/decade for a first-order filter), defines its selectivity and suitability for a given application.
Worked example
Consider a first-order RC high-pass filter with a resistor of R = 1,000 Ω (1 kΩ) and a capacitor of C = 1 µF (1 × 10⁻⁶ F), driven by a signal at f = 5,000 Hz.
Step 1 — Cutoff Frequency:
f_c = 1 / (2π × 1000 × 1×10⁻⁶) = 1 / (6.2832 × 10⁻³) ≈ 159.15 Hz
Step 2 — Time Constant:
τ = RC = 1000 × 1×10⁻⁶ = 0.001 s (1 ms)
Step 3 — Impedance at 5,000 Hz:
Capacitive reactance X_C = 1 / (2π × 5000 × 1×10⁻⁶) ≈ 31.83 Ω
|Z| = √(1000² + 31.83²) ≈ 1,000.51 Ω
Step 4 — Phase Angle:
φ = arctan(X_C / R) = arctan(31.83 / 1000) ≈ 1.82°
This small phase lead confirms the signal frequency is well above the cutoff, so the filter has minimal effect on the phase.
Step 5 — Voltage Gain:
The normalised gain = f / √(f² + f_c²) = 5000 / √(5000² + 159.15²) ≈ 0.9995
Gain in dB = 20 × log₁₀(0.9995) ≈ −0.004 dB
This near-unity gain confirms that a 5,000 Hz signal passes through with negligible attenuation since it is far above the 159.15 Hz cutoff.
If instead the signal were at 159.15 Hz (exactly at the cutoff), the gain would be exactly −3 dB and the phase angle would be exactly 45° — the defining characteristics of the cutoff point.
Limitations & notes
This calculator models ideal, first-order single-pole high-pass filters only. Real components have tolerances (typically ±5% to ±20% for capacitors, ±1% to ±5% for resistors) that shift the actual cutoff frequency from the calculated value. Capacitors exhibit equivalent series resistance (ESR) and parasitic inductance, while resistors have parasitic capacitance, all of which alter filter behaviour at very high frequencies. The model does not account for component loading — connecting a load resistance in parallel with the output will modify both the cutoff frequency and the gain. For higher roll-off rates, multi-pole (second-order or higher) filters such as Butterworth, Chebyshev, or Bessel designs are required, which have more complex transfer functions than those computed here. At frequencies many decades above the cutoff, practical operational amplifier (op-amp) based active HPF designs are limited by the amplifier's gain-bandwidth product and slew rate. RL filters assume an ideal inductor; real inductors have winding resistance that adds to the series resistance and shifts the effective cutoff frequency.
Frequently asked questions
What is the cutoff frequency of a high-pass filter?
The cutoff frequency (also called the −3 dB frequency or corner frequency) is the frequency at which the output signal power drops to half of the input power, corresponding to a voltage reduction to 1/√2 ≈ 70.7% of the input. For an RC high-pass filter it equals 1/(2πRC), and for an RL high-pass filter it equals R/(2πL). Signals above this frequency pass with less than 3 dB of attenuation.
What is the difference between an RC and an RL high-pass filter?
Both achieve high-pass filtering but use different reactive components. An RC high-pass filter places the capacitor in series and the resistor to ground — capacitors block DC and low frequencies naturally. An RL filter places the inductor to ground and the resistor in series — inductors present high impedance at high frequencies, so low-frequency signals are shunted to ground. RC filters are far more common in electronics due to the compact size and low cost of capacitors compared to inductors.
Why is the phase angle 45° at the cutoff frequency?
At the cutoff frequency, the reactive impedance of the capacitor (or inductor) equals the resistance, making the resistive and reactive parts equal. The phase angle is arctan(X/R), and when X = R, arctan(1) = 45°. Below the cutoff, the phase approaches 90° (strongly leading); above it, the phase approaches 0° (no shift). This 45° phase shift at the cutoff is a universal property of all first-order high-pass filters.
How do I convert the cutoff frequency from Hz to rad/s?
Multiply the cutoff frequency in Hz by 2π (approximately 6.2832). So ω_c = 2πf_c. For example, a cutoff frequency of 159.15 Hz corresponds to ω_c = 2π × 159.15 ≈ 1,000 rad/s. The angular frequency form is commonly used in transfer function and Laplace domain analysis.
What roll-off rate does a first-order high-pass filter have?
A first-order (single-pole) high-pass filter has a roll-off rate of −20 dB per decade (or equivalently −6 dB per octave) below the cutoff frequency. This means that for every tenfold decrease in frequency below the cutoff, the output signal drops by 20 dB. For steeper roll-off, a second-order filter provides −40 dB/decade, and higher-order designs can achieve even sharper transitions.
Last updated: 2025-01-15 · Formula verified against primary sources.