Introduction to Electronics - Filters: Cutoff frequency

The cutoff frequency is a key parameter in filter design. It marks the boundary between the passband, where signals are transmitted with little attenuation, and the stopband, where signals are significantly reduced. In both RC and RL circuits, the cutoff frequency is defined as the frequency at which the output signal drops to [math]\frac{1}{\sqrt{2}}[/math] (approximately -3 dB) of the maximum value. Understanding and calculating the cutoff frequency allows engineers to design filters that effectively separate desired signals from unwanted noise or interference.

Mathematical Underpinnings

For RC and RL circuits, the cutoff frequency can be derived from the filter's transfer function by setting the magnitude of the output-to-input ratio equal to [math]\frac{1}{\sqrt{2}}[/math].

RC Circuits:

In an RC filter, the cutoff frequency is given by:

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

Here, [math]R[/math] is the resistance and [math]C[/math] is the capacitance. This formula is derived by analysing the transfer function [math]H(j\omega)=\frac{1}{1+j\omega RC}[/math] and finding the frequency at which the magnitude, [math]\left|H(j\omega)\right|=\frac{1}{\sqrt{1+(\omega RC)^2}}[/math], equals [math]\frac{1}{\sqrt{2}}[/math].

RL Circuits:

In an RL filter, the cutoff frequency is defined as:

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

In this case, [math]R[/math] is the resistance and [math]L[/math] is the inductance. The transfer function for an RL low-pass filter is [math]H(j\omega)=\frac{R}{R+j\omega L}[/math], and the cutoff frequency is obtained in a similar manner by setting the magnitude equal to [math]\frac{1}{\sqrt{2}}[/math].

Comparison Table

The following table summarises the cutoff frequency formulas for both RC and RL filters:

Demo Python Script

The script below demonstrates how to calculate and plot the cutoff frequencies for both an RC and an RL circuit. The script also shows a comparison by marking the cutoff frequency on frequency response plots.

import numpy as np
import matplotlib.pyplot as plt

# Define component values for RC and RL circuits
R = 1e3 # Resistance in ohms
C = 1e-6 # Capacitance in farads
L = 1e-3 # Inductance in henrys

# Calculate cutoff frequencies
fc_RC = 1 / (2 * np.pi * R * C)
fc_RL = R / (2 * np.pi * L)

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

# RC filter transfer function magnitude and plot in dB
H_RC = 1 / np.sqrt(1 + (omega * R * C)**2)
H_RC_db = 20 * np.log10(H_RC)

# RL filter transfer function magnitude and plot in dB
H_RL = R / np.sqrt(R**2 + (omega * L)**2)
H_RL_db = 20 * np.log10(H_RL)

# Create subplots for comparison
fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(8, 8))

# RC filter plot
ax1.semilogx(frequencies,
   H_RC_db,
   label="RC Filter")
ax1.axvline(x=fc_RC,
   color="red",
   linestyle="--",
   label=f"RC Cutoff: {fc_RC:.1f} Hz")
ax1.set_xlabel("Frequency (Hz)")
ax1.set_ylabel("Magnitude (dB)")
ax1.set_title("RC Filter Frequency Response")
ax1.grid(True, which="both", linestyle="--")
ax1.legend()

# RL filter plot
ax2.semilogx(frequencies,
   H_RL_db,
   label="RL Filter")
ax2.axvline(x=fc_RL,
   color="red",
   linestyle="--",
   label=f"RL Cutoff: {fc_RL:.1f} Hz")
ax2.set_xlabel("Frequency (Hz)")
ax2.set_ylabel("Magnitude (dB)")
ax2.set_title("RL Filter Frequency Response")
ax2.grid(True, which="both", linestyle="--")
ax2.legend()

plt.tight_layout()
plt.show()

When executed, this script produces two plots: one for the RC filter and one for the RL filter, with each plot displaying the cutoff frequency as a red dashed vertical line.

Key Takeaways

• The cutoff frequency is the point where the filter’s output falls to [math]\frac{1}{\sqrt{2}}[/math] of the maximum, defining the boundary between the passband and the stopband.
• For RC circuits, [math]f_c=\frac{1}{2\pi RC}[/math].
• For RL circuits, [math]f_c=\frac{R}{2\pi L}[/math].

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