Introduction to Electronics - Filters: RL low-pass filter behaviour

An RL low-pass filter is designed to allow low-frequency signals to pass through while attenuating high-frequency signals. In this configuration, a resistor and an inductor form a voltage divider. At low frequencies, the inductor exhibits low impedance, so most of the input voltage appears across the resistor (the output). As the frequency increases, the inductor's impedance rises, diverting more voltage away from the resistor and thereby reducing the output. This gradual transition between the passband and the stopband is characterised by the filter's cutoff frequency.

Circuit Diagram

Below is a simple schematic of an RL low-pass filter:

In this circuit, the resistor (R) is connected in series with the input voltage, while the inductor (L) is connected from the node after the resistor to ground. The output is taken across the resistor.

Mathematical Underpinning

The behaviour of the RL low-pass filter can be described by its transfer function. The transfer function is given by:

[math]H(s)=\frac{R}{R+sL}[/math]

By substituting the complex frequency [math]s[/math] with [math]j\omega[/math], where [math]\Omega[/math] is the angular frequency in radians per second, the transfer function becomes:

[math]H(j\omega)=\frac{R}{R+j\omega L}[/math]

The magnitude of this transfer function, which represents the filter’s frequency response, is:

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

At low frequencies (where [math]\omega L \ll R[/math]), the magnitude is nearly 1, meaning the output closely follows the input. As [math]\omega[/math] increases, the term [math](\omega L)^2[/math] becomes dominant, and the magnitude decreases. The cutoff frequency, defined as the frequency at which the output falls to [math]\frac{1}{\sqrt{2}}[/math] of the input (approximately -3 dB), is calculated as:

[math]f_c=\frac{R}{2\pi L}[/math]

This model is derived using the voltage division principle, where the inductor's impedance, [math]j\omega L[/math], increases with frequency, thereby reducing the voltage across the resistor.

Frequency Response Plot

The following Python script demonstrates how to plot the frequency response of the RL low-pass filter and marks the cutoff frequency on the plot:

import numpy as np
import matplotlib.pyplot as plt

# Define resistor and inductor values
R = 1e3 # Resistance in ohms
L = 1e-3 # Inductance in henrys

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

# Define 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 low-pass filter
H_mag = R / np.sqrt(R**2 + (omega * L)**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 RL Low-Pass Filter")
plt.grid(True, which="both", linestyle="--")

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

plt.show()

When executed, this script produces a plot that displays how the RL low-pass filter maintains a high output at low frequencies and attenuates the signal as the frequency increases, with the cutoff frequency clearly marked.

Key Takeaways

• In an RL low-pass filter, the resistor and inductor form a voltage divider. Low frequencies see a low impedance in the inductor, resulting in most of the input voltage being dropped across the resistor.
• The filter's transfer function is [math]H(j\omega)=\frac{R}{R+j\omega L}[/math], which determines how the output varies with frequency.
• The magnitude [math]|H(j\omega)|=\frac{R}{\sqrt{R^2+(\omega L)^2}}[/math] shows that at low frequencies the output is nearly equal to the input, while at high frequencies, the output is reduced.
• The cutoff frequency [math]f_c=\frac{R}{2\pi L}[/math] defines the transition between the passband and the stopband, serving as a key parameter in the filter's performance.

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