The fundamental LR circuit formula defines the time constant as τ = L/R (in seconds) and the low-pass cutoff frequency as fc = R / (2πL) (in Hertz). When you need to smooth a high-current PWM signal into a steady DC voltage, an inductor-resistor (LR) topology drops virtually zero DC voltage and wastes minimal power as heat, unlike its RC counterpart. If you are driving a 5A+ load like a heater or motor, the LR configuration is your default choice. Below is the complete design framework for sizing, testing, and protecting an LR filter in real-world applications.
Topology, Node Labels, and the Behavior Matrix
In a standard LR low-pass filter used for PWM smoothing, the inductor is placed in series with the signal path, while the load itself acts as the shunt resistor to ground. This is highly efficient because the DC resistance (DCR) of a power inductor is typically under 50 milliohms, meaning almost all the source voltage reaches the load.
Topology Node Map:
- Node A (Input): Raw PWM signal from the driving MOSFET or microcontroller gate driver.
- Node B (Junction/Output): The connection point between the Inductor (L1) and the Load (R_load). This is where you probe for the smoothed DC waveform.
- Node C (Ground): Common ground for the load, the PWM driver, and your oscilloscope.
Understanding how component drift or design changes affect the circuit is critical. Here is the behavior matrix for the LR low-pass topology:
| Parameter Changed | Effect on Time Constant (τ = L/R) | Effect on Cutoff Freq (fc = R/2πL) | Real-World Consequence |
|---|---|---|---|
| Inductance (L) Increases | Increases (slower response) | Decreases (better filtering) | Smoother DC output, but slower reaction to PWM duty cycle changes; higher physical size and cost. |
| Inductance (L) Decreases | Decreases (faster response) | Increases (worse filtering) | More PWM ripple reaches the load; can cause audible whine in motors or flicker in LED arrays. |
| Load Resistance (R) Increases | Decreases (faster response) | Increases (worse filtering) | Less current draw, but the filter becomes less effective at smoothing the voltage. |
| Load Resistance (R) Decreases | Increases (slower response) | Decreases (better filtering) | Higher current draw; filter performs better, but inductor must be rated for the higher RMS current to avoid saturation. |
Failure Modes: What Breaks at the Extremes?
Unlike simple resistor networks, LR circuits store energy in a magnetic field. When things fail, the inductor will fight to keep current flowing, which creates destructive voltage spikes. Here is the failure-mode contrast for the extremes:
- Shorted Inductor (L1 fails short): The filtering action is completely lost. Node A and Node B become electrically identical. The load receives raw PWM. While the load might survive, the sudden switching edges will generate massive EMI, and the load may overheat if it isn't rated for the peak PWM voltage.
- Open Inductor (L1 fails open): The circuit safely fails off. No current reaches Node B. The load loses power. This is a benign failure mode.
- Shorted Load (R_load shorts): The inductor initially limits the di/dt (rate of current rise), but within a few time constants, current will spike to the maximum limit of the power supply. This will typically blow the upstream fuse or destroy the driving MOSFET at Node A.
- Open Load (R_load opens): Critical Hazard. If the load disconnects while the PWM is in the 'ON' state, current is flowing through L1. When the PWM switches 'OFF', the inductor's collapsing magnetic field has no path to discharge. It will generate a massive high-voltage spike (hundreds of volts) at Node B, instantly destroying the driving MOSFET and any connected logic. This is why a flyback diode is mandatory.
Design Walkthrough: Sizing a 20kHz PWM Smoothing Filter
Let's design a filter for a 12V, 5A high-power LED array driven by a 20kHz PWM signal from an ESP32 and a logic-level MOSFET. We want to smooth the PWM into a steady DC current to eliminate strobing effects and reduce EMI.
Step 1: Define the Load Resistance (R)
Using Ohm's Law, the effective resistance of the LED array at operating temperature is R = V / I = 12V / 5A = 2.4Ω.
Step 2: Select the Target Cutoff Frequency (fc)
To achieve a clean DC output, the cutoff frequency should be at least one decade (10x) below the PWM frequency.
fc = 20,000 Hz / 10 = 2,000 Hz.
Step 3: Calculate the Required Inductance (L)
Rearranging the LR circuit formula for L:
L = R / (2π × fc)
L = 2.4 / (2 × 3.14159 × 2000)
L = 2.4 / 12566.37 = 0.000191 H (or 191 µH).
Step 4: Pick the Concrete Component
We need an inductor with at least 191 µH that can handle 5A RMS current without saturating.
The Pick: Coilcraft MSS1210-184 (180 µH, shielded, 6.2A saturation current, 15mΩ DCR). The 180 µH value is close enough to 191 µH (shifting fc slightly to 2.12 kHz, which is perfectly acceptable). The 15mΩ DCR will only drop 75mV at 5A, wasting a negligible 0.37W as heat.
Step 5: Add the Flyback Diode
Place an SS34 Schottky diode (3A continuous, 40V reverse voltage—wait, we need >5A. Let's upgrade to an SS54 or MBR1045 10A Schottky). Place the MBR1045 with the cathode at Node A (PWM In) and the anode at Node B (Junction). This clamps the inductive spike to roughly -0.5V when the MOSFET turns off.
Breadboard Testing: Step-by-Step Verification
A standard solderless breadboard is rated for roughly 1A to 3A max per contact strip. Pushing 5A through it will melt the plastic and oxidize the contacts, causing a voltage drop that ruins your measurements. Here is how to safely verify the LR formula on the bench before soldering.
- Scale the Test Current: Swap the 5A LED array for a 24Ω, 10W power resistor. This draws exactly 0.5A at 12V. The LR time constant remains mathematically valid, but the cutoff frequency shifts. Recalculate fc for the test: fc = 24 / (2π × 0.00018) = 21.2 kHz. To test the filter, drop your ESP32 PWM frequency down to 2 kHz so the 21.2 kHz cutoff can actually filter it.
- Wire the Driver: Connect your MOSFET drain to Node A. Connect the Coilcraft inductor between Node A and Node B. Connect the 24Ω dummy resistor between Node B and Ground (Node C). Install the MBR1045 flyback diode across the inductor.
- Probe the Nodes: Connect your oscilloscope Channel 1 to Node A (raw PWM) and Channel 2 to Node B (filtered output). Ensure the scope ground clip is attached to Node C. Keep the ground lead as short as possible to avoid picking up switching noise.
- Verify the Waveform: Set the PWM to 50% duty cycle. Channel 1 should show a clean 0-12V square wave. Channel 2 should show a smoothed DC level hovering around 6V with minor ripple.
- Check for Ringing: Zoom in on the falling edge of the PWM signal on Channel 2. If you see high-frequency oscillation (ringing), your layout parasitics are interacting with the inductor. Add a small 100nF ceramic capacitor directly across the load resistor to form an RLC snubber and kill the ringing.
The Decision Tree: LR vs. RC vs. LC
When designing a filter, you must choose the right topology based on your current requirements and acceptable power loss. Use this decision matrix to finalize your design.
| Condition / Constraint | RC Low-Pass Filter | LC Low-Pass Filter | LR Low-Pass Filter |
|---|---|---|---|
| Load Current is < 100mA | Winner. Cheap, tiny, easy to source. | Overkill, expensive. | Overkill, inductor DCR might drop too much voltage. |
| Load Current is 1A to 20A | Fails. Series resistor will overheat and waste massive power. | Good, but requires a bulky shunt capacitor. | Winner. Inductor passes DC with minimal loss. |
| Strict Space / Height Constraints | Winner (0805 SMD parts). | Fails (tall inductors + caps). | Fails (power inductors are physically large). |
| Needs Zero DC Voltage Drop | Fails. | Winner. Both L and C have near-zero DCR/ESR. | Acceptable (drops millivolts based on DCR). |
If your load draws more than 1A and you are smoothing a PWM signal, do not use an RC filter; the thermal dissipation will force you to use massive wirewound resistors. Choose the LR topology. Default Pick: Use the Coilcraft MSS1210 series shielded power inductors paired with an MBR1045 Schottky diode. Calculate your exact inductance using L = R / (2π × fc), verify the saturation current rating exceeds your peak load, and always test at scaled-down currents on the bench before committing to a PCB layout. For deeper theoretical background on transient responses, refer to the Electronics Tutorials LR Circuits guide or the All About Circuits DC textbook chapter on inductor transients.
