RL Circuits and the Time Constant ⏳

An RL circuit combines a resistor (R) and an inductor (L).
Unlike RC circuits (which control voltage change), RL circuits control how fast current changes.

👉 The key idea: the time constant (τ) defines how quickly current rises or falls when voltage is applied or removed.

---

RL Time Constant (τ)

The time constant for an RL circuit is:

$\tau = \frac{L}{R}$

Where:

• L = Inductance (Henries)
• R = Resistance (Ohms)
• τ = Time (seconds)

note

Large inductance or small resistance → slow current change
Small inductance or large resistance → fast current change

---

Example Calculation

Given:

• L = 10 mH (0.01 H)
• R = 100 Ω

$\tau = \frac{0.01}{100} = 0.0001 \text{ s} = 0.1 \text{ ms}$

What this means:

• After 1τ (0.1 ms) → current reaches 63% of its final value
• After 5τ (0.5 ms) → circuit is considered settled

important

Just like RC circuits, engineers use 5τ as “steady state”.

---

Current Growth (Switch Closed)

When a voltage is suddenly applied:

$I(t) = \frac{V}{R}\left(1 - e^{-t/\tau}\right)$

Behavior Over Time

• At t = 0 → inductor behaves like an open circuit
• At t = τ → current reaches 63%
• At t = 5τ → current is essentially at its final value
• As t → ∞ → inductor behaves like a short circuit

tip

Inductors don’t block current forever — they only delay it.

---

Current Decay (Switch Opened)

When the supply is removed, current decays exponentially:

$I(t) = I_{\text{initial}} \times e^{-t/\tau}$

• Current tries to keep flowing
• Energy stored in the magnetic field must go somewhere

warning

This is where voltage spikes are born if no protection exists.

---

Dangerous Transient Response ⚠️

When a switch opens:

• Current does not stop instantly
• Inductor generates a voltage spike: $V = L \frac{dI}{dt}$
• High $\frac{dI}{dt}$ → very high voltage
• Can destroy transistors, relays, and ICs

danger

Never switch an inductive load without protection — it will fail eventually.

---

Comparison: RC vs RL Time Constants

Feature	RC Circuit	RL Circuit
Controls	Voltage change	Current change
Time constant	$τ = R\times C$	$τ = \frac{L}{R}$
63% point	Capacitor voltage	Inductor current
Energy stored in	Electric field	Magnetic field
Common risk	Slow edges, noise	Voltage spikes

---

Practical Applications 🛠️

RL behavior is critical in real systems:

• Motor control → smooth acceleration and torque ramps
• PWM switching → reduces harsh current transitions
• Current limiting → prevents instant shorts
• Switching power supplies → controls efficiency and stress

---

Protection Strategies 🛡️

Technique	Purpose
Flyback diode	Safely clamps voltage spike
Parallel resistor	Dissipates stored energy
RC / LC snubber	Reduces switching noise
TVS diode	Protects against extreme transients

tip

For DC coils, flyback diode first — always.
For high-speed or AC switching, add snubbers or TVS.

---

Key Takeaway

• RL circuits control current timing
• τ defines how fast the system reacts
• Inductors are harmless only if respected
• Protection is not optional — it’s mandatory

Master RL circuits, and motors, relays, and power converters start behaving. 🚀