What Is an LR Circuit? Topology and Node Definitions
An LR circuit (or RL circuit) is a fundamental network consisting of a resistor (R) and an inductor (L). While they can be arranged in parallel, the series RL topology is the workhorse for filtering, current limiting, and transient suppression. Unlike RC circuits where the time constant is $\tau = RC$, the time constant for an LR circuit is defined as $\tau = L/R$ (measured in seconds). This inverse relationship means increasing resistance actually speeds up the circuit's transient response, a common trap for beginners transitioning from RC design.
Let's define the standard RL Low-Pass Filter topology, which passes DC and low-frequency signals while attenuating high-frequency noise.
- Node IN: The input signal source.
- Series Element (L): The inductor connects Node IN to Node OUT. It presents low impedance to DC and high impedance to high-frequency AC ($X_L = 2\pi fL$).
- Node OUT: The junction between the inductor and the shunt resistor. This is where you measure your filtered output.
- Shunt Element (R): The resistor connects Node OUT to Node GND, providing a DC path to ground and setting the output impedance.
The RL vs RC Decision Path
Why choose an RL topology over the much cheaper and more common RC (Resistor-Capacitor) alternative? The decision hinges on source impedance, current requirements, and the nature of the load. Capacitors fight voltage changes; inductors fight current changes.
| Design Constraint | RC Topology (Series R, Shunt C) | RL Topology (Series L, Shunt R) | Winner & Default Pick |
|---|---|---|---|
| High Current / Power Lines | Poor. Capacitors draw massive inrush currents and can fail short, causing fires. | Excellent. Inductors naturally limit $di/dt$ and handle high DC current with minimal $I^2R$ loss. | RL (Use a power choke) |
| Low-Frequency Filtering (<10Hz) | Requires massive, expensive electrolytic capacitors. | Requires massive, heavy, and expensive iron-core inductors. | RC (or active op-amp filter) |
| Audio Crossovers | Capacitors block DC but can introduce phase shift and dielectric absorption distortion. | Inductors pass DC, handle high wattage from amplifiers, and naturally roll off tweeters/woofers. | RL / LC (Standard for passive crossovers) |
| Snubber / Flyback Protection | RC snubbers dissipate energy as heat in the resistor. | Inductors store energy; without a diode path, they will arc and destroy switching transistors. | RC (for AC snubbing) / Diode (for DC flyback) |
| Cost & PCB Footprint | Cheap, tiny MLCCs available in 0402 packages. | Inductors are physically large, prone to EMI coupling, and cost 5x-10x more. | RC (Always default for low-power signals) |
The Verdict: If you are filtering a low-power analog sensor signal (e.g., <20mA), use an RC filter. If you are filtering a PWM power line, smoothing a high-current DC-DC converter output, or building an audio crossover, choose the RL topology.
Parameter Shifts and Failure Extremes
Understanding how an LR circuit behaves when components drift or fail is critical for troubleshooting. Here is the behavior matrix for the Series-L, Shunt-R low-pass topology.
| Component Change | Effect on Cutoff Freq ($f_c$) | Effect on Time Constant ($\tau$) | Real-World Consequence |
|---|---|---|---|
| Increase R | Increases (Passes more high-freq noise) | Decreases (Faster step response) | Output voltage rises, but high-frequency attenuation degrades. |
| Increase L | Decreases (Better noise rejection) | Increases (Slower step response, more phase lag) | Better filtering, but the circuit takes longer to settle to a new DC level. |
| Short L (Failure) | $f_c \to \infty$ | $\tau \to 0$ | Filter is bypassed. Full input noise reaches Node OUT. System operates but with high EMI. |
| Open L (Failure) | $f_c \to 0$ | $\tau \to \infty$ | Signal path broken. Node OUT reads 0V (or floats if parasitic capacitance exists). |
| Short R (Failure) | $f_c \to 0$ | $\tau \to 0$ | Node OUT shorted to GND. Signal destroyed. Source may overcurrent and trigger protection. |
| Open R (Failure) | $f_c \to \infty$ | $\tau \to \infty$ | No DC path to ground. Node OUT floats. The circuit accidentally becomes a high-pass filter via parasitic capacitance. |
Design Walkthrough: 500Hz RL Low-Pass PWM Filter
Let's design a concrete RL filter to smooth a 12V, 1kHz PWM signal from a microcontroller into a clean DC analog voltage to drive a 0-10V industrial actuator. We want a cutoff frequency ($f_c$) of 500Hz to heavily attenuate the 1kHz fundamental and its harmonics.
Step 1: Choose the Shunt Resistor (R)
We need a resistor that won't draw excessive current from our 12V source but is low enough to provide a stiff DC path. Let's pick $R = 100\Omega$. At 12V DC, this draws 120mA, which is acceptable for a power bus but too high for a raw GPIO pin. (Assume this is driven by a MOSFET or op-amp buffer).
Step 2: Calculate the Inductor (L)
Using the cutoff formula $L = R / (2\pi f_c)$:
$L = 100 / (2 \cdot \pi \cdot 500) = 100 / 3141.59 = 0.0318\text{ H}$, or 31.8mH.
Step 3: Select Real Components
- Inductor: Bourns 78FR33K (33mH, 10% tolerance, 150mA max DC current). The slightly higher 33mH value pushes our actual cutoff down to ~482Hz, which improves 1kHz rejection.
- Resistor: Vishay MRS25000C1000FCT00 (100$\Omega$, 0.6W, 1% metal film). The 0.6W rating provides a safety margin over the $P = V^2/R = 1.44W$ peak, though at 50% PWM duty cycle, average power is ~0.72W. Correction for thermal safety: Let's upgrade to a 2W carbon film resistor (e.g., TE Connectivity LR2F100R) to handle the continuous 12V DC dissipation without burning up.
Step 4: Verify Attenuation
At our 1kHz PWM frequency, the inductive reactance is $X_L = 2\pi(1000)(0.033) = 207\Omega$. The voltage divider ratio is $100 / \sqrt{100^2 + 207^2} = 100 / 229 = 0.43$. Our 12V PWM ripple is attenuated by more than half before it even hits the actuator. For deeper smoothing, cascade a second identical RL stage or add a small shunt capacitor (e.g., 10$\mu$F) at Node OUT to create an RLC (second-order) filter.
Step-by-Step Breadboard Verification
Do not trust the datasheet blindly; parasitic capacitance in physical inductors creates a self-resonant frequency (SRF) that can ruin your filter's high-frequency performance. Here is how to validate your LR build on the bench.
- Prep the Board: Insert the 33mH Bourns inductor so its leads span the center trench. Connect one leg to your signal rail (Node IN) and the other to an empty bus (Node OUT).
- Place the Resistor: Insert the 100$\Omega$ 2W resistor with one leg in the Node OUT bus and the other in the ground rail. (Note: 2W resistors are thick; you may need to bend the leads or use a proto-board with wider holes).
- Connect the Source: Hook up a function generator to Node IN and GND. Set it to output a 1kHz square wave, 0V to 5V (to keep power dissipation low during testing).
- Probe Node OUT: Connect your oscilloscope probe to Node OUT. Ensure the probe is set to 1X or 10X correctly in the scope menu. Connect the probe ground spring (not the long alligator clip, to avoid ground loops) to the GND rail.
- Measure the Step Response: Trigger on the rising edge. You should see an exponential curve rising to 5V, not a sharp square edge. Measure the time it takes to reach 63.2% of 5V (3.16V). This is your empirical time constant $\tau$. It should read approximately $\tau = 33mH / 100\Omega = 330\mu s$.
- Sweep for SRF: Switch the function generator to a sine wave. Sweep the frequency from 1kHz up to 1MHz while monitoring the peak-to-peak voltage at Node OUT. The voltage should drop steadily. If you see the voltage suddenly spike at a high frequency (e.g., 500kHz), you have hit the inductor's parasitic self-resonance. For signals near that frequency, this specific inductor will fail as a low-pass filter.
For deeper theoretical grounding on inductive reactance and phase angles, refer to the All About Circuits chapter on Series Resistor-Inductor Circuits, or consult Electronics Tutorials on AC Inductance for phasor diagram breakdowns. When designing for power applications, always verify the inductor's saturation current ($I_{sat}$) rating; if your DC current exceeds $I_{sat}$, the core magnetizes fully, the inductance collapses to near-zero, and your LR circuit becomes a simple, unfiltered wire.
