Introduction to Electronics - Filters: What is a first-order filter?

A first-order filter is one of the simplest types of electronic filters. It uses a single reactive component—a capacitor in RC filters or an inductor in RL filters—paired with a resistor to shape the frequency content of an input signal. The term "first-order" comes from the fact that the mathematical description of these filters involves a first-order differential equation. This simplicity leads to a gradual transition between the passband and the stopband, typically with a slope of 20 dB per decade.

Will you need to learn differential equations to understand first-order filters?

No! While first-order filters are derived from linear differential equations, a student can understand their behaviour and practical use without needing to solve such equations. You will need to know basic algebra, understand the concept of frequency, phasors (more about this later in this guide) and basic circuit laws, like Ohm’s and Kirchhoff’s Laws.

Key characteristics

The main attributes of a first-order filter include:

• Single Energy Storage Element: Only one capacitor or one inductor is used.
• Time Constant: The response is defined by the time constant. In RC filters, the time constant is given by [math]\tau = RC[/math], while in RL filters, it is [math]\tau = \frac{L}{R}[/math].
• Cutoff Frequency: The cutoff frequency is the frequency at which the output signal falls to 70.7% (or -3 dB) of the input. For RC filters, it is calculated as [math]f_c = \frac{1}{2\pi RC}[/math]; for RL filters, it is [math]f_c = \frac{R}{2\pi L}[/math].
• Gentle Roll-Off: With a slope of ±20 dB per decade, the filter gradually attenuates frequencies beyond the cutoff, unlike the steeper slopes seen in higher-order designs.

First-Order vs. Higher-Order Filters

First-order filters use a single reactive element, resulting in a smooth transition between passed and rejected frequencies. In contrast, second-order or higher-order filters incorporate two or more reactive components, which leads to:

• Steeper Roll-Off: Second-order filters typically offer a roll-off rate of 40 dB per decade, providing a sharper distinction between the passband and stopband.
• Complex Response Characteristics: Higher-order filters can exhibit resonant peaks and more pronounced phase shifts, which can be advantageous in applications requiring precise filtering but add complexity to the design.

What is “Roll-Off”?

Roll-off is the rate at which a filter attenuates the signal amplitude beyond its cutoff frequency, typically measured in decibels per decade or per octave.

Summary Table of First-Order Filters

The following table summarises the main attributes of common first-order filters:

Note: In these expressions, [math]s[/math] represents the complex frequency (with [math]s = j\omega[/math], where [math]\omega[/math] is the angular frequency in radians per second).

Adjusting the cutoff frequency

Imagine turning a knob that gradually adjusts how much of a signal is allowed to pass through a circuit. In a first-order filter, this adjustment is controlled by the resistor and the reactive component working together. At frequencies well within the passband, most of the signal passes through unchanged. As the frequency approaches the cutoff value, the filter starts to attenuate the signal gradually. Beyond this point, the filter increasingly reduces the signal's amplitude.

For example, an RC low-pass filter acts like a barrier to high frequencies, whereas an RC high-pass filter behaves like a gate that favours higher frequencies and reduces lower ones. RL filters operate on similar principles, but the reactive element is an inductor, which naturally opposes changes in current.

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