Mastering the LR Circuit Formula: Design, Topology, and Testing

The fundamental LR circuit formula for the time constant is τ = L / R, where L is inductance in Henries and R is resistance in Ohms. For a DC step response (charging), the current over time is calculated as I(t) = (V/R) * (1 - e^(-t/τ)). This first-order differential relationship governs everything from relay coil energization to PWM smoothing filters. While the math is straightforward, translating it to a physical bench design requires understanding parasitic resistance, core saturation, and transient failure modes.

The Series LR Topology and Core Formulas

When we discuss the LR circuit formula, we are almost exclusively referring to the series LR topology. In this configuration, the voltage source, inductor, and resistor share the same current path.

Why series over parallel? A parallel LR circuit simply acts as a frequency-dependent current divider. It does not create a useful first-order voltage transient across a shared load node in the same predictable way. The series LR topology is the standard for modeling motor windings, designing low-pass filters, and analyzing relay coil kickback because the inductor directly opposes changes in the loop current, creating a measurable exponential voltage drop across the resistor.

The governing equations for a series LR circuit are:

• Time Constant: τ = L / R (seconds)
• Charging Current: I(t) = (V/R) * (1 - e^(-Rt/L))
• Discharging Current: I(t) = I₀ * e^(-Rt/L)
• Cutoff Frequency (Low-Pass): f_c = R / (2πL)

For a deeper mathematical derivation of these calculus-based relationships, refer to the inductor calculus chapter on All About Circuits.

Design Walkthrough: 1kHz PWM Smoothing Filter

Let’s move off the whiteboard and onto the bench. Suppose you need to convert a 5V, 1kHz PWM signal from an ESP32 GPIO pin into a smooth DC voltage to drive an analog sensor input. We will design a 1st-order series LR low-pass filter.

Design Targets:

• PWM Frequency: 1,000 Hz
• Target Cutoff Frequency (f_c): ~150 Hz (roughly one decade below the PWM frequency to ensure severe attenuation of the AC ripple).
• Load Impedance: High (ESP32 ADC input is >1MΩ, so it won’t load down our filter).

Step 1: Pick the Resistor (R)
We need a resistor that is low enough to drive the ADC but high enough to prevent excessive current draw from the ESP32 pin (max ~40mA, but we want to stay under 10mA). Let’s choose a standard E12 value: R = 47Ω. At 5V, steady-state current is 5V / 47Ω = 106mA. Wait, that’s too high for a direct GPIO pin. Let’s adjust. We will drive this from a 5V rail through a buffer, or simply increase R. Let’s use R = 470Ω. Current is now ~10.6mA, perfectly safe for most 5V logic buffers.

Step 2: Calculate the Inductor (L)
Rearranging the cutoff frequency formula: L = R / (2π * f_c)
L = 470 / (2 * 3.14159 * 150) = 470 / 942.47 = 0.498 Henries (498mH).

Step 3: Select Real Components
A 500mH inductor is physically massive and has high DC resistance (DCR). Let’s compromise. If we use a standard 47mH power inductor (e.g., Würth Elektronik 74477420), we must recalculate R to maintain our 150Hz cutoff.
R = 2π * f_c * L = 2 * 3.14159 * 150 * 0.047 = 44.3Ω.
We will use a standard 47Ω metal film resistor (e.g., Vishay Dale RN55D47R0F). This pushes our actual cutoff frequency to f_c = 47 / (2π * 0.047) = 159 Hz.

Step 4: Verify the Time Constant
τ = L / R = 0.047H / 47Ω = 1 millisecond.
At 1kHz, the PWM period is 1ms. The time constant perfectly matches the period, meaning the output will exhibit a predictable, smoothed triangle-like ripple rather than a raw square wave. For more on practical inductor selection and DCR impacts, check out this guide on LR circuits from Electronics Tutorials.

Parameter Behavior and Extreme Failure Modes

Understanding how the LR circuit formula reacts to component drift or catastrophic failure is critical for troubleshooting. Below is a behavior matrix for the series topology.

Parameter Change	Effect on Time Constant (τ)	Effect on Cutoff Freq (fc)	Effect on Steady-State Current
Increase L	Increases (slower response)	Decreases	No change (dictated by R)
Increase R	Decreases (faster response)	Increases	Decreases
Increase V	No change	No change	Increases proportionally

What Breaks at the Extremes? (Failure Mode Contrast)

When components fail, the LR circuit formula breaks down, and physics takes over. Here is how the series LR topology handles extreme faults compared to parallel alternatives:

• Shorting the Inductor (L = 0): The time constant drops to zero. The circuit becomes purely resistive. Current instantly jumps to V/R. In a PWM filter, you lose all smoothing; the raw square wave passes directly to the load.
• Opening the Inductor (L = ∞): Current stops immediately. Danger: If the circuit is opened while carrying current, the formula V = L(di/dt) generates a massive voltage spike (inductive kickback). This will arc across switch contacts or destroy driving transistors unless a flyback diode is present.
• Shorting the Resistor (R = 0): Mathematically, τ approaches infinity. The inductor ramps current linearly (I = V*t / L) until the power supply hits its current limit, the inductor core saturates (dropping L drastically), or the wire melts.
• Opening the Resistor (R = ∞): Current ceases. Unlike a parallel LR circuit where opening one branch just redirects current to the other, opening the series resistor breaks the entire loop, safely halting operation but yielding zero output voltage.

Step-by-Step Breadboard Verification

Do not trust the datasheet blindly; verify the LR circuit formula on the bench. Here is how to measure the transient response of our 47mH / 47Ω design.

1. Build the Topology: Insert the inductor and resistor in series on the breadboard. Connect the function generator ground to the resistor’s free leg (Node C), and the signal output to the inductor’s free leg (Node A).
2. Configure the Source: Set the function generator to output a 0V to 5V square wave at 150 Hz. (We use a low frequency here to allow the current to fully charge and discharge, making the time constant easy to measure).
3. Probe the Circuit: Connect Oscilloscope Channel 1 to Node A (Vin) and Channel 2 to Node B (the junction between L and R). Set the timebase to 2ms/div.
4. Trigger and Capture: Trigger on the rising edge of Channel 1. You will see a flat 5V square wave on CH1, and an exponential curve rising from 0V to 5V on CH2.
5. Measure τ Empirically: Use the scope’s cursor tools. Measure the time it takes for the CH2 waveform to rise from 0V to 63.2% of its final value (0.632 * 5V = 3.16V). This measured time is your empirical τ.
6. Compare to Theory: Your scope should read approximately 1.0ms. If it reads significantly lower (e.g., 0.7ms), your inductor is likely saturating at higher currents, or the function generator’s 50Ω output impedance is adding to your 47Ω resistor (making total R = 97Ω, which halves your time constant!). Always set your function generator to "High-Z" load mode in the menu to prevent this exact bench error.

Frequently Asked Questions

How do you calculate the time constant using the LR circuit formula?

The time constant (τ) is calculated by dividing the inductance (L) in Henries by the resistance (R) in Ohms. For example, a 10mH inductor in series with a 100Ω resistor yields τ = 0.010 / 100 = 0.0001 seconds, or 100 microseconds. This value represents the time required for the current to reach 63.2% of its final steady-state value.

What is the LR circuit formula for discharging current?

When the voltage source is removed and the inductor discharges through the resistor (often via a flyback diode path), the current decays exponentially. The formula is I(t) = I₀ * e^(-Rt/L), where I₀ is the initial current flowing at the exact moment the source is disconnected. After 5 time constants (5τ), the current is considered effectively zero (less than 1% remaining).

Why use an LR circuit instead of an RC circuit for filtering?

RC circuits are cheaper and smaller for low-frequency, low-current signal filtering. However, LR circuits are preferred in high-current power applications (like buck converters or motor drives) because inductors pass DC with minimal loss (only the wire’s DCR causes heat), whereas an RC filter’s resistor constantly dissipates power as heat (I²R losses). If you are filtering a 10A power rail, an LR filter is vastly more efficient than an RC filter.

What happens to the LR circuit formula if resistance approaches zero?

If R approaches zero (a pure inductor across an ideal voltage source), the time constant τ = L/R approaches infinity. The exponential curve flattens into a linear ramp. The current will increase linearly at a rate of di/dt = V/L. In the real world, R can never be truly zero due to the inductor’s internal DCR and the power supply’s internal resistance, but in superconducting magnets, this linear ramp is exactly how they are charged over hours or days.