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

An RC low-pass filter is designed to allow low-frequency signals to pass through with little attenuation while gradually reducing the amplitude of higher frequency signals. In this configuration, a resistor and a capacitor work together so that at low frequencies the capacitor presents a high impedance, resulting in the output voltage closely following the input voltage. As the frequency increases, the impedance of the capacitor decreases, which causes more of the high-frequency signal to be shunted to ground. This behaviour creates a smooth transition between the passband and the stopband, with the cutoff frequency marking the point where the output drops to about 70.7% (or -3 dB) of the input signal.

What is “passband”?

The range of frequencies that a filter allows to pass through with minimal attenuation. These frequencies remain largely unchanged by the filter.

What is “stopband”?

The range of frequencies that a filter significantly attenuates or blocks, reducing their amplitude substantially.

Circuit Diagram

The following schematic represents a basic RC low-pass filter:

In this circuit, the resistor (R) is connected in series with the input voltage, while the capacitor (C) is connected between the node after the resistor and ground. This arrangement forms a voltage divider where the capacitor's impedance decreases with increasing frequency, leading to signal attenuation at higher frequencies. In this example, the input voltage source is an alternating sine wave.

Mathematical underpinning

If you are curious about the underlying mathematics that describe the behaviour of the RC low-pass filter, 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 low-pass filter is characterised by its transfer function. The transfer function is given by:

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

To analyse the filter's response to sinusoidal inputs, we substitute the complex frequency [math]s[/math] with [math]j\omega[/math], where [math]\Omega[/math] is the angular frequency in radians per second.

What is a “complex frequency”?

The complex frequency [math]s[/math] is a variable used in the Laplace transform, defined as [math]s = \sigma + j\omega[/math]. It combines an exponential decay or growth component ([math]\sigma[/math]) with an oscillatory component ([math]\omega[/math]), making it essential for analysing the behaviour of linear time-invariant systems. Thankfully, as a new student of first-order filters, you don’t need to learn the concept of complex frequency [math]s[/math] to be able to understand how these filters work, and to be to use them.

What is the Laplace Transform? The Laplace transform is a mathematical operation that converts a time-domain function, [math]f(t)[/math], into a complex frequency-domain function, [math]F(s)[/math], where [math]s=\sigma+j\omega[/math]. This transformation simplifies the analysis of linear time-invariant circuits by turning differential equations into algebraic ones. For circuits such as first-order filters, the Laplace transform allows us to derive transfer functions, analyse stability, and predict transient and steady-state behaviour efficiently.

This substitution results in:

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

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

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

At low frequencies, where [math]\omega RC \ll 1[/math], the magnitude approaches 1, meaning the output is nearly equal to the input. As the frequency increases, the term [math](\omega RC)^2[/math] becomes significant, and the magnitude decreases accordingly. The cutoff frequency, defined as the frequency at which the output drops to [math]\frac{1}{\sqrt{2}}[/math] of the input (approximately -3 dB), is given by:

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

This mathematical model is derived using the voltage division principle, where the capacitor's impedance is expressed as [math]\frac{1}{j\omega C}[/math], making the filter's response inherently dependent on frequency.

Frequency Response Plot

Below is a Python code snippet that generates a plot of the filter's frequency response. The plot illustrates the signal attenuation (in decibels) as a function of frequency, highlighting the gradual roll-off characteristic of the RC low-pass filter.

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)

# 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 = 1 / 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 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 code produces a plot showing that the filter maintains a flat response at low frequencies and begins to attenuate the signal progressively above the cutoff frequency.

The red vertical line marks the cutoff frequency, which is also calculated by the script. Using this code, you can experiment with various values for R and C to see how they effect the frequency response of the filter.

Key takeaways

In summary, the RC low-pass filter is a foundational circuit that illustrates several core concepts in signal processing:

• The filter consists of a resistor in series and a capacitor to ground, forming a voltage divider whose behaviour changes with frequency.
• At low frequencies, the capacitor acts as an open circuit, allowing most of the input signal to appear at the output. At high frequencies, its decreasing impedance diverts the signal to ground.
• The transfer function, [math]H(j\omega)=\frac{1}{1+j\omega RC}[/math], and its magnitude, [math]|H(j\omega)|=\frac{1}{\sqrt{1+(\omega RC)^2}}[/math], mathematically describe this behaviour.
• The cutoff frequency, [math]f_c=\frac{1}{2\pi RC}[/math], defines the boundary between the passband (where signals pass with little attenuation) and the stopband (where signals are significantly reduced).

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