Introduction to Electronics - Filters: RC high-pass filter behaviour

An RC high-pass filter is designed to allow high-frequency signals to pass through while attenuating low-frequency signals. In this configuration, a resistor and a capacitor are arranged so that at low frequencies the capacitor behaves like an open circuit, blocking most of the signal. As the frequency increases, the capacitor’s impedance decreases, allowing the high-frequency components to pass with minimal attenuation. This filter is especially useful in applications where it is necessary to eliminate low-frequency noise or DC offset from a signal.

Circuit Diagram

Below is a simple schematic representation of an RC high-pass filter:

In this circuit, the capacitor (C) is connected in series with the input voltage, and the resistor (R) is connected from the node between the capacitor and resistor to ground. This configuration forms a voltage divider in which the capacitor’s impedance decreases with increasing frequency, thereby allowing high frequencies to appear at the output.

Mathematical underpinning

As with the low-pass filter in the previous article, I am providing you with the underlying mathematics that describe the behaviour of the RC high-pass filter here. If you are interested in these details, then continue to read this segment. Otherwise, you will find the most important take-away formula (the cut-off frequency) at the end of the segment. The behaviour of the RC high-pass filter is described by its transfer function. The transfer function for an RC high-pass filter is given by:

[math]H(s)=\frac{sRC}{1+sRC}[/math]

To analyse the filter’s response to sinusoidal inputs, we substitute [math]s[/math] with [math]j\omega[/math], where [math]\Omega[/math] is the angular frequency in radians per second. Remember, from the previous article, [math]s[/math] is the complex frequency used in the Laplace transform. This substitution gives:

[math]H(j\omega)=\frac{j\omega RC}{1+j\omega RC}[/math]

The magnitude of the transfer function, which indicates the filter’s frequency response, is calculated as:

[math]|H(j\omega)|=\frac{\omega RC}{\sqrt{1+(\omega RC)^2}}[/math]

At low frequencies (when [math]\omega RC \ll 1[/math]), the magnitude approaches 0, meaning the output is greatly attenuated. Conversely, at high frequencies (when [math]\omega RC \gg 1[/math]), the magnitude approaches 1, allowing the high-frequency signals to pass through. The cutoff frequency, which marks the transition between these two regions, is defined as:

[math]f_c=\frac{1}{2\pi RC}[/math]

This mathematical framework is derived from the voltage division principle, taking into account the frequency-dependent impedance of the capacitor, expressed as [math]\frac{1}{j\omega C}[/math].

Frequency Response Plot

The following Python code snippet demonstrates how to plot the frequency response of an RC high-pass filter. The plot shows the magnitude of the output signal in decibels (dB) as a function of frequency.

import numpy as np
import matplotlib.pyplot as plt

# Define resistor and capacitor values
R = 1e3 # Resistance in ohms
C = 1e-6 # Capacitance in farads

# Calculate the cutoff frequency
fc = 1 / (2 * np.pi * R * C)

# Frequency range for the plot (10 Hz to 1 MHz)
frequencies = np.logspace(1, 6, num=500)
omega = 2 * np.pi * frequencies

# Calculate the magnitude of the transfer function for the high-pass filter
H_mag = (omega * R * C) / np.sqrt(1 + (omega * R * C)**2)
H_db = 20 * np.log10(H_mag)

# Create the plot
plt.figure(figsize=(8, 4))
plt.semilogx(frequencies, H_db)
plt.xlabel("Frequency (Hz)")
plt.ylabel("Magnitude (dB)")
plt.title("Frequency Response of RC High-Pass Filter")
plt.grid(True, which="both", linestyle="--")

# Mark the cutoff frequency
plt.axvline(x=fc,
   color="red",
   linestyle="--",
   label=f"Cutoff Frequency: {fc:.1f} Hz")
plt.legend()

plt.show()

Executing this code produces a plot that illustrates how the filter attenuates low frequencies while allowing high frequencies to pass, with the transition occurring around the cutoff frequency [math]f_c=\frac{1}{2\pi RC}[/math]. The red vertical line marks the cutoff frequency, which is also calculated by the script.

Key Takeaways

• The RC high-pass filter uses a capacitor in series with the input and a resistor to ground to block low-frequency signals and pass high-frequency signals.
• Its transfer function is given by [math]H(j\omega)=\frac{j\omega RC}{1+j\omega RC}[/math], which shows that the filter’s behaviour is frequency dependent.
• The magnitude of the transfer function, [math]|H(j\omega)|=\frac{\omega RC}{\sqrt{1+(\omega RC)^2}}[/math], confirms that the output is nearly zero at low frequencies and approaches unity at high frequencies.
• The cutoff frequency, [math]f_c=\frac{1}{2\pi RC}[/math], serves as the boundary between the stopband and the passband.

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