The inductor discharging equation defines how current decays in an RL (resistor-inductor) circuit when the driving voltage source is removed or switched off. The core formula for current over time is i(t) = I0 · e-(R/L)t. In practical 2026 circuit design, this equation is not just academic; it dictates flyback diode selection for relays, snubber sizing for solenoids, and freewheeling periods in switch-mode power supplies (SMPS). Misjudging this decay time leads to arced switch contacts, fried MOSFETs, or mechanical chatter in valves.

The Core Formula and Symbol Definitions

When a charged inductor is disconnected from its source and allowed to discharge through a resistive path, the magnetic field collapses, inducing a voltage that drives current through the resistor. This current decays exponentially. The governing equation is:

i(t) = I0 · e-(R/L)t

Alternatively, using the time constant τ = L/R, the equation is written as i(t) = I0 · e-t/τ. Below is the strict definition of every variable required to use this formula correctly.

Symbol Parameter SI Unit Practical Notes
i(t) Instantaneous current at time t Amperes (A) The decaying current value you are solving for.
I0 Initial current at t = 0 Amperes (A) The steady-state current flowing immediately before the switch opens.
R Total discharge path resistance Ohms (Ω) Includes coil DCR, flyback diode dynamic resistance, and any external bleed resistors.
L Inductance Henries (H) Assumes linear inductance; real cores may drop in value if saturated prior to discharge.
t Time elapsed since discharge began Seconds (s) Must be tracked from the exact microsecond the source is removed.
e Euler's number Dimensionless Mathematical constant ≈ 2.71828.
τ Time constant (L/R) Seconds (s) Time required for current to decay to ~36.8% of I0.

Real-World Component Discharge Profiles

Abstract formulas become useful only when mapped to physical components on your bench. The magnitude of the discharge time varies wildly depending on the application. In a buck converter, the inductor discharges in microseconds; in an industrial solenoid, it takes tens of milliseconds. For practical engineering, we consider an inductor 'fully discharged' at 5τ, where the current has fallen to less than 1% of I0.

The table below profiles four common 2026-era components, tracking their real-world discharge characteristics through their standard freewheeling or flyback paths.

Component Type Example Part Inductance (L) Discharge R Initial I0 τ (L/R) Time to <1% (5τ)
12V Automotive Relay Bosch 0332014150 150 mH 75 Ω 160 mA 2.0 ms 10.0 ms
5V PCB Signal Relay Omron G5V-2-DC5 70 mH 125 Ω 40 mA 0.56 ms 2.8 ms
SMPS Buck Inductor Wurth 74477420 10 μH 0.5 Ω* 2.0 A 20 μs 100 μs
24V Solenoid Valve ASCO 8210G015 800 mH 40 Ω 600 mA 20.0 ms 100.0 ms

*Discharge R includes MOSFET RDS(on), inductor DCR, and Schottky diode forward dynamic resistance.

According to foundational circuit theory documented by Electronics Tutorials, recognizing these time scales is critical. If you are designing a 500 kHz SMPS (2 μs period), a 100 μs discharge time means the inductor current never reaches zero, resulting in Continuous Conduction Mode (CCM). Conversely, a 100 ms solenoid decay time will cause severe mechanical chatter if the valve is commanded to cycle faster than 10 Hz.

Rearranged Forms for Circuit Design

On the bench, you rarely solve for i(t) directly. Usually, you have a target decay time and need to size a bleed resistor, or you have a fixed resistor and need to know when the current drops below a microcontroller's logic threshold. Here are the algebraically rearranged forms of the inductor discharging equation:

  • Solve for Time (t): t = -(L / R) · ln(i(t) / I0)
  • Solve for Resistance (R): R = -(L / t) · ln(i(t) / I0)
  • Solve for Inductance (L): L = -(R · t) / ln(i(t) / I0)
  • Solve for Initial Current (I0): I0 = i(t) / e-(R/L)t

These rearranged forms are heavily utilized when designing snubber networks. For instance, if a relay contact is arcing, you use the 'Solve for R' form to calculate the exact bleed resistor needed to force the coil energy to dissipate before the contacts physically separate.

Worked Examples with Unit Tracking

The most common point of failure in RL calculations is unit mismatch. The SI base units must be strictly maintained: Henries, Ohms, Amperes, and Seconds. Below are two bench-realistic problems demonstrating strict unit tracking.

Problem 1: Relay Flyback Decay Current

Scenario: You are driving a 12V automotive relay with a microcontroller. The coil has an inductance of 150 mH and a DC resistance of 75 Ω. The steady-state current is 160 mA. You open the driver transistor. What is the coil current exactly 4 ms later?

Step 1: Convert to SI Base Units

  • L = 150 mH = 0.15 H
  • I0 = 160 mA = 0.16 A
  • t = 4 ms = 0.004 s
  • R = 75 Ω (Already base unit)

Step 2: Calculate the Time Constant (τ)

  • τ = L / R = 0.15 H / 75 Ω = 0.002 s (or 2 ms)

Step 3: Calculate the Exponent

  • -t / τ = -0.004 s / 0.002 s = -2

Step 4: Solve for i(t)

  • i(0.004) = 0.16 A · e-2
  • i(0.004) = 0.16 A · 0.13534
  • i(0.004) = 0.02165 A or 21.65 mA

Bench Insight: At 4 ms (2τ), the current has dropped to ~13.5% of its initial value. If your flyback diode is rated for 50 mA continuous, it will easily survive this transient.

Problem 2: Sizing a Bleed Resistor for a Solenoid

Scenario: A 24V pneumatic solenoid valve has an inductance of 400 mH. To prevent the valve from chattering during rapid cycling, the pneumatic engineer requires the magnetic field to collapse to 5% of its initial current within exactly 20 ms. The coil's internal DCR is 12 Ω. What value external bleed resistor must you place in parallel with the coil?

Step 1: Convert to SI Base Units and Define Targets

  • L = 400 mH = 0.4 H
  • t = 20 ms = 0.02 s
  • Target ratio i(t) / I0 = 0.05 (5%)

Step 2: Use the Rearranged Formula for Total R

  • Rtotal = -(L / t) · ln(i(t) / I0)
  • Rtotal = -(0.4 / 0.02) · ln(0.05)
  • Rtotal = -20 · (-2.9957)
  • Rtotal = 59.91 Ω

Step 3: Calculate the External Bleed Resistor

The total discharge resistance is the parallel combination of the coil DCR and the external bleed resistor. However, in a standard flyback configuration where the bleed resistor is placed in series with a flyback diode across the coil, the discharge path resistance is simply the sum of the coil DCR, the diode resistance, and the external resistor. Assuming a standard series flyback loop:

  • Rtotal = Rcoil + Rdiode + Rbleed
  • 59.91 Ω = 12 Ω + 1 Ω (approx diode) + Rbleed
  • Rbleed = 59.91 - 13 = 46.91 Ω

Bench Insight: Select a standard 47 Ω resistor. Because it will dissipate the coil's stored energy (E = 0.5 · L · I2), ensure it is rated for the peak pulse power, typically requiring a 1W or 2W wirewound or metal oxide film resistor rather than a standard 1/4W carbon film.

Assumptions, Edge Cases, and Unit Traps

The inductor discharging equation is an idealized mathematical model. As documented in first-order circuit analysis by MIT OpenCourseWare, real-world physics introduces parasitics that break the pure exponential curve if ignored.

When the Formula Applies (and When It Doesn't)

This formula assumes a linear inductor. If your inductor core was heavily saturated just before the switch opened, its effective inductance L is much lower than the datasheet value. As the current decays and the core comes out of saturation, L increases dynamically, meaning the actual decay will start fast and then slow down, deviating from the pure exponential curve.

It also assumes a purely resistive discharge path. In high-speed SMPS or long cable runs, parasitic capacitance exists in parallel with the inductor. When the switch opens, the L and C form a resonant tank, causing the voltage to ring (RLC circuit behavior) rather than smoothly decaying. If you see high-frequency oscillation on your oscilloscope during the discharge phase, the pure RL equation is no longer sufficient; you must use the damped RLC differential equations.

Unit Mistakes That Break the Math

The most frequent errors on the bench occur when mixing metric prefixes. Memorize these conversions:

  • Inductance: 1 mH = 10-3 H. 1 μH = 10-6 H. Plugging '10' into the formula for a 10 μH inductor without converting to '0.00001' will result in a calculated time constant that is one million times too large.
  • Current: While the ratio i(t) / I0 is dimensionless (allowing you to use mA for both if you are just finding a percentage), calculating absolute voltage drops across the resistor (V = i(t) · R) requires i(t) to be in Amperes.
  • Time: Oscilloscope timebases are usually in μs or ms. Always convert to seconds before multiplying by R/L.

By strictly adhering to SI base units during the intermediate calculation steps and understanding the physical limitations of the magnetic core, you can reliably use the inductor discharging equation to design robust, arc-free, and efficient switching circuits.