When a switch opens in an RL (resistor-inductor) circuit, the collapsing magnetic field forces current to continue flowing, generating a voltage spike that can destroy semiconductors. The inductor discharge equation dictates exactly how fast that current decays and how high the transient voltage will climb. For a standard RL discharge path, the current at any time t is calculated as i(t) = I0 · e-(R/L)t. Understanding this exponential decay is mandatory for sizing flyback diodes, designing snubber networks, and preventing MOSFET avalanche failures on the bench.
The Inductor Discharge Equation and Symbol Definitions
The fundamental equations governing the discharge of an inductor through a resistive path are derived from Kirchhoff's Voltage Law (KVL). When the source is removed and the inductor discharges through a total resistance R, the current and voltage follow these exponential curves:
Current Decay: i(t) = I0 · e-(R/L)t
Inductor Voltage: vL(t) = -I0 · R · e-(R/L)t
Every variable in these equations must be strictly defined to avoid calculation errors. Below is the complete symbol reference table.
| Symbol | Parameter | Standard Unit | Practical Description |
|---|---|---|---|
| i(t) | Instantaneous Current | Amperes (A) | The current flowing through the inductor at time t after the switch opens. |
| I0 | Initial Current | Amperes (A) | The steady-state current flowing immediately before the switch opens (Vsource / Rcoil). |
| vL(t) | Inductor Voltage | Volts (V) | The back-EMF voltage across the inductor terminals at time t. The negative sign indicates polarity reversal. |
| R | Total Discharge Resistance | Ohms (Ω) | The sum of the inductor's DC resistance (DCR) plus any external snubber, load, or flyback path resistance. |
| L | Inductance | Henries (H) | The inductance value of the coil. Must be converted from mH or μH to base Henries for the exponent. |
| t | Time | Seconds (s) | Elapsed time since the switch opened. Must be in base seconds, not milliseconds or microseconds. |
| e | Euler's Number | Dimensionless | Mathematical constant ≈ 2.71828, the base of the natural logarithm. |
Real-World Inductor Discharge Parameters
Theoretical textbook problems often use round numbers like 1 H and 1 Ω. In practice, you are dealing with millihenries, parasitic resistances, and fast transient times. The table below provides real-world parameters for four common inductive loads you will encounter in the field, establishing a baseline for what realistic time constants (τ = L/R) look like.
| Component Type | Specific Model / Spec | Inductance (L) | DC Resistance (R) | Steady-State I0 | Time Constant (τ) |
|---|---|---|---|---|---|
| 12V Automotive Relay | Omron G8P-1A4P | 45 mH | 120 Ω | 100 mA | 0.375 ms |
| 24V Solenoid Valve | Parker Viking Extreme | 200 mH | 40 Ω | 600 mA | 5.00 ms |
| Buck Converter Inductor | Wurth 744774210 (10μH) | 10 μH | 0.045 Ω | 3.0 A (peak) | 0.22 ms (effective)* |
| Ignition Coil (Primary) | Standard Motor Products UF215 | 5.5 mH | 0.6 Ω | 6.0 A (dwell) | 9.16 ms |
*Note: Switch-mode power supply inductors discharge into the output capacitor and load, making the effective discharge path highly dynamic compared to a simple resistive RL decay. See RL Circuit Theory for complex impedance variations.
Rearranged Forms for Circuit Design
On the bench, you rarely need to find i(t). Usually, you know the desired decay time or the maximum voltage your MOSFET can survive, and you need to solve for the required snubber resistance or the exact time delay. Here are the algebraically rearranged forms of the inductor discharge equation for practical design work.
- Solve for Time (t): Use this to find how long it takes for the current to drop to a specific threshold (e.g., relay drop-out current).
t = -(L / R) · ln(i(t) / I0) - Solve for Resistance (R): Use this to size a dynamic braking resistor or snubber to achieve a specific decay time.
R = -(L / t) · ln(i(t) / I0) - Solve for Inductance (L): Use this to identify an unknown inductor's value by measuring its decay time through a known resistor.
L = -(R · t) / ln(i(t) / I0) - Solve for Initial Current (I0): Use this to back-calculate the steady-state current if you capture a mid-decay current measurement on a scope.
I0 = i(t) / e-(R/L)t
Worked Examples with Unit Tracking
The most common point of failure in these calculations is unit mismatch. The exponent -(R/L)t must be entirely dimensionless. Below are two step-by-step bench scenarios with explicit unit tracking.
Problem 1: Relay Drop-Out Time Calculation
Scenario: You are driving an Omron G8P 12V relay (L = 45 mH, R = 120 Ω) with a microcontroller. When the BJT switch turns off, the coil discharges through its own internal resistance (assuming no flyback diode is installed for the sake of calculating the raw natural decay). How long does it take for the current to drop to 10% of its initial value (the approximate drop-out threshold)?
Step 1: Convert to base SI units.
- L = 45 mH = 0.045 H
- R = 120 Ω
- Target i(t) = 0.10 · I0
Step 2: Select the rearranged formula for time.
t = -(L / R) · ln(i(t) / I0)
Step 3: Substitute and track units.
- t = -(0.045 H / 120 Ω) · ln(0.10 · I0 / I0)
- t = -(0.000375 s) · ln(0.10) (Note: H/Ω simplifies to seconds)
- t = -(0.000375 s) · (-2.3025)
- t = 0.000863 s
Answer: The current drops to 10% in 0.863 milliseconds. This is why relays without flyback diodes release quickly, but generate massive voltage spikes.
Problem 2: Sizing a Snubber Resistor for a Solenoid Valve
Scenario: A 24V solenoid valve (L = 200 mH, R = 40 Ω, I0 = 0.6 A) is switched by a MOSFET rated for 100V maximum drain-source voltage (VDSS). You want to use a Zener diode and resistor network to clamp the flyback spike. If you use a standard flyback diode in series with a resistor to speed up decay, what is the maximum resistance you can add in the discharge path before the initial voltage spike exceeds 80V (leaving a 20V safety margin)?
Step 1: Identify the initial voltage spike formula.
At t = 0, the exponential term e0 = 1. The magnitude of the voltage spike is simply:
|Vspike| = I0 · Rtotal
Step 2: Define limits and convert units.
- I0 = 0.6 A
- Max |Vspike| = 80 V
- Rtotal = Rcoil + Rsnubber
Step 3: Solve for Rsnubber.
- 80 V = 0.6 A · (40 Ω + Rsnubber)
- 133.33 Ω = 40 Ω + Rsnubber
- Rsnubber = 93.33 Ω
Answer: You must use a snubber resistor of 93 Ω or less. If you use a 150 Ω resistor, the initial spike will be 0.6 A · 190 Ω = 114 V, which will avalanche and likely destroy your 100V MOSFET.
Assumptions, Unit Traps, and Realistic Magnitudes
The inductor discharge equation is a highly reliable model, but it relies on specific physical assumptions. Ignoring these assumptions or falling for common unit traps will lead to catastrophic hardware failures.
When the Formula Applies (and Its Assumptions)
- Linear Inductance: The equation assumes L is constant. In reality, ferrite and iron-powder cores saturate. If the inductor was heavily saturated during the ON state, its effective inductance drops as current decreases, altering the decay curve. For precision timing, use air-core or gapped-ferrite inductors.
- Constant Resistance: It assumes R does not change. In high-current, high-frequency snubbers, skin effect and temperature coefficients can alter the resistance during the transient event.
- Instantaneous Switching: The math assumes the switch opens in zero time. Real MOSFETs and BJTs have turn-off times (nanoseconds to microseconds). During this transition, the circuit is neither fully charging nor fully discharging, which slightly blunts the theoretical peak voltage spike.
Unit Mistakes That Break the Math
The most frequent error made by hobbyists and junior engineers is failing to convert prefixes to base SI units before plugging them into the exponent. The term (R/L)t must evaluate to a pure number.
- The Millihenry Trap: Plugging L = 45 instead of L = 0.045 makes the time constant 1000x smaller, predicting a decay time in microseconds instead of milliseconds.
- The Microsecond Trap: Oscilloscope cursors often read in μs. If your scope says the decay took 860 μs, you must enter t = 0.00086 into the algebraic rearrangements. Entering 860 will yield a calculated inductance that is off by a factor of one million.
Realistic Answer Magnitudes
Developing an intuition for realistic magnitudes helps you catch calculation errors immediately. According to standard inductor theory and application guides, expect the following timeframes:
- Electromechanical Relays & Contactors: Decay times are in the milliseconds (ms). If your math yields 2 seconds or 2 nanoseconds for a standard 12V relay, your units are wrong.
- Solenoids & Actuators: Decay times range from 5 ms to 50 ms, heavily dependent on the mechanical mass and magnetic gap.
- Switch-Mode Power Supplies (Buck/Boost): Discharge events occur in the microseconds (μs) or even nanoseconds. The high frequency (e.g., 500 kHz) demands incredibly fast transient responses.
- Ignition Coils: The primary coil discharge (dwell) is in the milliseconds, but the secondary high-voltage spark discharge occurs in microseconds due to the massive turns ratio and plasma arc resistance.
