RLC Low Pass Filter Design: Theory, Math, and Real-World Component Picks

People commonly confuse the RLC low pass filter with two other topologies: the RC low pass (which lacks the inductor and only achieves a first-order slope) and the RLC bandpass filter (which uses the exact same three components but takes the output voltage across the resistor instead of the capacitor). Getting the output node wrong completely changes the frequency response from a low-pass to a band-pass.

The Core Mechanics: How an RLC Low Pass Filter Works

In the standard series RLC low pass configuration, the input signal passes through a resistor (R) and an inductor (L) in series, while the capacitor (C) is placed in parallel (shunt) to ground. The output voltage is measured across the capacitor.

To visualize the physics, use this single water-flow analogy: The inductor acts like a heavy water wheel that resists sudden changes in flow (blocking high-frequency AC spikes). The capacitor acts like a flexible rubber bladder connected to the pipe that absorbs high-pressure transients (shunting high frequencies to ground). The resistor acts as a deliberate pipe constriction that dampens the system, preventing the water wheel and rubber bladder from bouncing back and forth and creating pressure rings (electrical resonance).

The governing equation for the cutoff frequency ($f_c$) is:

$$f_c = \frac{1}{2\pi\sqrt{LC}}$$

However, the true power of the RLC topology lies in the Quality Factor (Q), which dictates the damping. If Q is too high, you get a massive voltage spike (peaking) right at the cutoff frequency. If Q is too low, the filter acts sluggishly and rolls off too early. For a flat Butterworth response (maximally flat passband with no peaking), you target a Q of exactly 0.707.

Worked Numeric Example: Smoothing an ESP32 PWM Signal

Let’s build a real filter. You are using an ESP32 DevKit v1 to generate a 5kHz PWM signal, and you need to smooth it into a clean analog DC voltage to drive a 10kΩ input on an external amplifier. You want a cutoff frequency ($f_c$) of 1kHz to aggressively kill the 5kHz PWM carrier and its harmonics.

Step 1: Select the Inductor (L)

Inductors are bulky and expensive, so we pick a standard, readily available value first. Let’s choose a 10mH shielded radial inductor (e.g., Bourns 78F103J-RC, rated for 130mA DC bias). Shielded is critical here to prevent the inductor's magnetic field from coupling into nearby high-speed ESP32 GPIO traces.

Step 2: Calculate the Capacitor (C)

Rearranging the cutoff formula to solve for C:

$$C = \frac{1}{(2\pi f_c)^2 \times L}$$

$$C = \frac{1}{(2\pi \times 1000)^2 \times 0.01} = 2.53\mu F$$

The nearest standard E12 capacitor value is 2.2µF. Using a 2.2µF X7R ceramic capacitor (e.g., KEMET C322C225K5R5TA) shifts our actual cutoff frequency slightly to 1,074 Hz, which is perfectly acceptable for this application.

Step 3: Calculate the Damping Resistor (R)

To achieve our target Butterworth Q of 0.707, we use the damping formula for a series RLC low pass:

$$R = \frac{1}{Q} \sqrt{\frac{L}{C}}$$

$$R = \frac{1}{0.707} \sqrt{\frac{0.01}{2.2 \times 10^{-6}}} = 1.414 \times 67.42\Omega = 95.3\Omega$$

The nearest standard E24 resistor value is 91Ω. By picking 91Ω instead of 100Ω, we slightly over-damp the circuit (Q drops to ~0.67). In practical PWM smoothing, slight over-damping is preferred to guarantee zero ringing on fast step-changes in the DAC output.

Where You Meet This in Practice

You won't just find RLC low pass filters in textbook exercises; they are mandatory in several real-world power and signal domains:

• Class-D Audio Amplifiers: The output of a Class-D amp is a high-power, high-frequency PWM square wave. An RLC low pass filter (often called an LC filter with the speaker's voice coil acting as the R) reconstructs the analog audio waveform while blocking the 300kHz+ switching carrier.
• Subwoofer Crossovers: Passive 2nd-order crossovers use an RLC topology to block mid-range and treble frequencies from reaching a woofer, protecting the driver from mechanical damage and preventing acoustic phase cancellation.
• DC Motor EMI Suppression: Brushed DC motors generate massive high-frequency commutator noise. An RLC filter on the power rails prevents this noise from back-feeding into sensitive microcontroller ADC lines.
• Sensor Anti-Aliasing: Before an analog signal hits an ADC, a 2nd-order RLC filter ensures that high-frequency RF interference is attenuated by at least -40dB before the Nyquist limit, preventing aliased ghost-signals in your digital data.

Common Confusions: RLC Low Pass vs. The Rest

When debugging a board that isn't passing signals correctly, verify you haven't fallen into one of these common traps:

• Confusing it with an Active Sallen-Key Filter: An active 2nd-order low pass uses an op-amp, two resistors, and two capacitors. It achieves the same -40dB/decade roll-off without the need for a physically large, EMI-generating inductor. If your circuit has an op-amp and no inductor, it's an active filter, not a passive RLC.
• Taking the Output Across the Wrong Component: If you build a series R-L-C circuit and measure the voltage across the resistor, you have built a bandpass filter. The low-pass output must strictly be taken across the capacitor.
• Ignoring Load Impedance: The math above assumes an unloaded filter (infinite load impedance). If you connect a 50Ω load directly across the capacitor, that load acts in parallel with your damping resistor, destroying your Q-factor and shifting the cutoff frequency. Always buffer an RLC filter with an op-amp voltage follower if driving a low-impedance load.

Decision Tree: Which Filter Topology Should You Build?

Don't default to an RLC filter just because it sounds more advanced. Use this decision matrix to pick the right topology for your specific bench constraint.

Frequently Asked Questions