Inductors and RL Circuits: Time Constant, Energy and Back EMF
An inductor is a coil of wire, often wound on a ferrite or iron core, that stores energy in a magnetic field. Where a capacitor resists sudden changes in voltage, an inductor resists sudden changes in current. This behaviour is the foundation of switching power supplies, filters, relays, motors and transformers, and it is also the reason a relay coil can destroy the transistor that drives it. This article covers the inductor equations, the RL time constant and the practical ratings you see on inductor datasheets.
Basic Equations
Voltage across inductor: v = L x di/dt Energy stored: E = 0.5 x L x I^2 (joules) Reactance (AC): XL = 2 x pi x f x L Series: L_total = L1 + L2 + ... (no mutual coupling) Parallel: 1/L_total = 1/L1 + 1/L2 + ...
Inductance L is measured in henries (H); practical values range from nanohenries in RF circuits to hundreds of millihenries in mains chokes. The equation v = L di/dt says the voltage across an inductor depends on how fast the current changes, not on the current itself. A steady DC current produces no voltage across an ideal inductor; only the coil's wire resistance drops any voltage.
Worked example: reactance
A 100 mH choke at 50 Hz has XL = 2 x 3.1416 x 50 x 0.1 = 31.4 Ω. At 5 kHz the same choke has XL = 3142 Ω. Inductive reactance rises with frequency, the opposite of a capacitor, which is why an inductor in series blocks high-frequency noise while passing DC.
The RL Time Constant
When a DC voltage Vs is applied to an inductor in series with a resistance R (which includes the coil's own resistance), current does not jump to its final value. It rises exponentially towards Vs / R with a time constant tau = L / R.
Final current: I_final = Vs / R Current rise: i(t) = I_final x (1 - e^(-t / tau)) Current decay: i(t) = I0 x e^(-t / tau) Time constant: tau = L / R (seconds, with L in H and R in Ω)
The percentages are the same as for an RC circuit: 63.2% after one tau, 86.5% after two, 95% after three and 99.3% after five.
Worked example: relay-type coil
A coil of 100 mH with a total series resistance of 10 Ω is connected to 12 V. Find the final current, the time constant, the current after 10 ms and the stored energy.
I_final = 12 / 10 = 1.2 A tau = 0.1 / 10 = 0.01 s = 10 ms i(10 ms) = 1.2 x (1 - e^-1) = 1.2 x 0.632 = 0.76 A i(50 ms) = 1.2 x 0.993 = 1.19 A (effectively full) Energy at full current = 0.5 x 0.1 x 1.2^2 = 0.072 J
Relays have exactly this delay: the armature only pulls in once the current has risen far enough, a few milliseconds after the drive voltage is applied.
Back EMF and the Flyback Diode
Now open the switch. The inductor tries to keep its 1.2 A flowing, and v = L di/dt tells us what happens if the current is forced to stop quickly. Suppose a transistor turns off in 1 µs.
v = L x di/dt = 0.1 x (1.2 / 0.000001) = 120 000 V (theoretical)
In practice the voltage never reaches that figure, because something breaks down first: the transistor's collector junction, the switch contacts (which arc) or the insulation. Either way, the stored 0.072 J is dumped into whatever fails. This is why every relay coil, solenoid and small DC motor driven by a transistor needs a flyback diode across it.
The diode is connected in reverse across the coil, cathode to the positive supply side. In normal operation it is reverse-biased and does nothing. At turn-off the coil voltage reverses, the diode conducts, and the current circulates through the coil and diode, decaying with the coil's own time constant. The switch sees only the supply voltage plus one diode drop. A 1N4007 or 1N4148 is common for small relays; for fast PWM motor drives a Schottky diode is preferred.
Common Mistakes
- Omitting the flyback diode on relays, solenoids and motors.
- Placing the flyback diode the wrong way round, which shorts the supply when the transistor turns on.
- Ignoring the coil's DCR when calculating the final current and time constant.
- Choosing a power inductor by inductance alone without checking saturation current.
- Measuring inductance with a basic multimeter; you need an LCR meter or an RL step-response measurement on an oscilloscope.
Summary
An inductor obeys v = L di/dt, stores 0.5LI^2 joules and has a reactance of 2 pi f L. In an RL circuit the current builds up and decays with time constant L / R. Interrupting inductor current quickly generates a large voltage spike, so always clamp switched coils with a flyback diode, and select power inductors by DCR, saturation current and heating current as well as inductance.