The current in a series LR circuit rises exponentially when a DC voltage is applied, governed by the time constant τ = L/R. At exactly 1τ, the LR circuit current reaches 63.2% of its maximum steady-state value (V/R). Unlike resistive circuits that react instantaneously, the inductor opposes changes in current by generating a back-EMF, creating a predictable delay widely used in solenoid drivers, relay snubbers, and power supply filtering.
The Series LR Topology: Node Labels and Current Path
To analyze the transient response, we use the standard series LR topology. Let's define the nodes for a standard DC step-response circuit:
- Node A (VCC): The positive DC supply terminal (e.g., 5.0V).
- Node B (Junction): The electrical connection point between the resistor and the inductor.
- Node C (GND): The ground return path completing the circuit.
When a switch closes between the supply and Node A, current flows from Node A, through the resistor (R) to Node B, and then through the inductor (L) to Node C. The governing equation for the rising LR circuit current is:
i(t) = (V / Rtotal) × (1 - e-t/τ)
Where τ = L / Rtotal, and Rtotal includes both the external resistor and the inductor's internal DC resistance (DCR).
Design Walkthrough: Sizing Real Components for a 1ms Delay
Let's design a passive LR delay circuit that reaches 63.2% of its final current in approximately 1 millisecond (τ ≈ 1ms), powered by a 5V logic-level supply.
- Select the Inductor: We choose the Bourns 78FR47K-RC, a 47mH radial inductor. According to its datasheet, it has a maximum DC current rating of 160mA and a typical DCR of 5.5Ω.
- Calculate Required Total Resistance: To achieve τ = 1ms (0.001s), we need Rtotal = L / τ = 0.047H / 0.001s = 47Ω.
- Size the External Resistor: Since Rtotal = Rext + DCR, we calculate Rext = 47Ω - 5.5Ω = 41.5Ω. The closest standard E12 resistor value is 39Ω.
- Verify the Actual Time Constant: With a 39Ω external resistor, Rtotal = 44.5Ω. Our actual τ = 47mH / 44.5Ω = 1.056ms. This is well within standard component tolerances.
- Check Steady-State Current and Power: Max current (Imax) = 5V / 44.5Ω = 112mA. This is safely below the inductor's 160mA saturation limit. The power dissipated by the 39Ω resistor is P = I²R = (0.112)² × 39 = 0.49W. We must select a 1W carbon film resistor (e.g., Yageo CFR-25JB-52-39R) to maintain a safe 50% derating margin.
Behavior Matrix: How Component Shifts Alter LR Circuit Current
Understanding how component degradation or failure affects the LR circuit current is critical for troubleshooting. The table below maps parameter shifts to their transient and steady-state outcomes.
| Parameter Changed | Effect on Time Constant (τ) | Effect on Max Steady-State Current | Effect on Initial di/dt (Slew Rate) |
|---|---|---|---|
| Increase R | Decreases (faster settling) | Decreases | Decreases |
| Increase L | Increases (slower settling) | No change | Decreases |
| Increase VCC | No change | Increases proportionally | Increases proportionally |
Failure-Mode Contrast: Series vs. Parallel Extremes
When analyzing what breaks at the extremes, the topology dictates the hazard:
- Series LR (Open Inductor): Current drops to zero instantly. However, breaking the circuit while current is flowing causes a massive voltage spike (V = L × di/dt) that will arc across switch contacts or destroy driving transistors. A flyback diode is mandatory.
- Series LR (Shorted Resistor): τ becomes extremely long (limited only by wire parasitics), and steady-state current spikes to V/DCR, likely burning out the inductor winding or tripping the power supply's overcurrent protection.
- Parallel LR (Shorted Inductor): In a parallel configuration, a shorted inductor simply bypasses the resistor, drawing maximum supply current and acting as a dead short across the source. Unlike series, parallel LR topologies do not inherently limit fault current, making them unsuitable for primary delay networks.
Step-by-Step Breadboard Testing and Verification
Oscilloscopes measure voltage, not current. To verify the LR circuit current transient on a bench, we use a low-value shunt resistor to convert the current waveform into a measurable voltage drop.
- Insert a Shunt Resistor: Place a 1.0Ω, 1% tolerance metal film resistor in series between Node C (the inductor's ground leg) and the actual breadboard ground rail. Because 1.0Ω is negligible compared to our 44.5Ω Rtotal, it will not alter the time constant.
- Configure the Signal Source: Do not use a manual switch; contact bounce will ruin the transient trace. Use a function generator set to a 50Hz square wave, 0V to 5V amplitude. A 50Hz wave gives a 10ms high-pulse, allowing roughly 10τ for the current to fully settle before the pulse drops low.
- Probe the Shunt: Connect Channel 1 of your oscilloscope across the 1.0Ω shunt resistor. Set the vertical scale to 200mV/div and the horizontal timebase to 2ms/div.
- Measure the 1τ Mark: Trigger on the rising edge. The peak voltage across the shunt should be roughly 112mV (representing 112mA). Move your cursor to the 63.2% mark (approx 71mV). The time delta from the trigger point to this cursor should read approximately 1.05ms, confirming your design.
LR vs. RC: Why Choose an Inductive Delay?
If you just need a time delay, an RC (resistor-capacitor) network is cheaper, smaller, and easier to source. So why use an LR topology? The decision hinges on energy storage and fault tolerance.
| Criteria | Series LR Circuit | Series RC Circuit |
|---|---|---|
| Primary Use Case | Current smoothing, solenoid/motor winding modeling, buck converter energy storage. | Signal filtering, timing oscillators, voltage snubbers, decoupling. |
| DC Steady-State Behavior | Acts as a short circuit (passes DC current freely, limited only by DCR). | Acts as an open circuit (blocks DC current entirely once charged). |
| Short-Circuit Fault Tolerance | High. The inductor inherently limits the rate of current rise (di/dt) during a dead short. | Low. A shorted capacitor results in instantaneous, massive inrush current spikes. |
| Component Cost & Size | High cost, large physical footprint, susceptible to EMI coupling. | Low cost, tiny footprint (especially MLCC), immune to magnetic interference. |
Choose the LR topology when the load itself is inherently inductive (like a relay coil) and you need to model or control the current ramp-up to prevent mechanical slamming or to ensure a smooth magnetic field buildup. For pure signal timing, stick to RC.
Frequently Asked Questions About LR Circuit Current
How do you calculate the exact LR circuit current at any given time?
Use the formula i(t) = (V / Rtotal) × (1 - e-t/τ) for a rising edge, where τ = L / Rtotal. If the source is removed and the circuit is shorted to decay, the formula changes to i(t) = Iinitial × e-t/τ. Always ensure your time (t) and time constant (τ) are in the same units (e.g., both in seconds or both in milliseconds) before calculating the exponent.
Why does LR circuit current decay faster with a higher resistance?
It seems counterintuitive since higher resistance restricts current flow, but in an LR decay scenario, the inductor acts as the temporary power source. A higher resistance in the decay path dissipates the inductor's stored magnetic energy (E = ½LI²) as heat much faster. Therefore, increasing the resistance decreases the time constant (τ = L/R), forcing the current to collapse to zero more rapidly. This is why placing a high-value resistor in parallel with an inductor (a snubber network) is an effective way to kill flyback voltage quickly.
What happens to the LR circuit current if the DC source is suddenly disconnected?
If the DC source is opened without an alternative path, the LR circuit current attempts to drop to zero instantly. Because di/dt approaches infinity, the inductor generates a massive voltage spike (V = L × di/dt) to force the current across the expanding air gap of the switch. This spike can reach hundreds or thousands of volts, causing arcing and component destruction. To prevent this, a freewheeling diode must be placed in reverse-bias across the inductor, providing a safe, low-resistance loop for the current to circulate and decay naturally.
For deeper mathematical derivations on transient responses, refer to the Electronics Tutorials inductor guides, and for practical application notes on managing inductive kickback in power designs, consult Texas Instruments' application reports on inductor fundamentals.
