7.2 RL circuits: current growth and decay

Inductor Basics

Inductor Characteristics

An inductor stores energy in a magnetic field. When current flows through its coils, a magnetic field builds up around it. The key property that makes inductors interesting in transient circuits is their resistance to change: an inductor generates a back-EMF (electromotive force) that opposes any change in current flowing through it.

This opposition is described by the inductor voltage equation:

$v_L = L \frac{di}{dt}$

where $L$ is the inductance (in henrys) and $\frac{di}{dt}$ is the rate of change of current. If current is steady (not changing), $\frac{di}{dt} = 0$ and the voltage across the inductor is zero. That's why, at steady state, an inductor behaves like a short circuit (just a wire).

Why This Matters for RL Circuits

Because of this back-EMF behavior, current in an RL circuit can't jump instantly from one value to another. Instead, it follows a smooth exponential curve as it grows or decays. This is the core idea behind everything in this section.

RL Circuit Response

The Time Constant

The time constant of an RL circuit is:

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

This single value controls how fast the current changes. A larger inductance $L$ means more energy stored in the magnetic field, so the current changes more slowly. A larger resistance $R$ dissipates energy faster, so the current reaches its final value more quickly.

For example, a circuit with $L = 10 \text{ mH}$ and $R = 5 \text{ Ω}$ has $\tau = \frac{0.01}{5} = 2 \text{ ms}$.

Current Growth (Source Connected)

When you connect a DC voltage source $V$ to a series RL circuit, the current grows from zero toward its maximum value according to:

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

Here's what happens step by step:

1. At $t = 0$, the inductor fully opposes the change, so $i(0) = 0$.
2. Current rises quickly at first, then gradually slows as it approaches the final value.
3. At $t = \tau$, the current reaches about 63.2% of its final value $\frac{V}{R}$.
4. At $t = 3\tau$, the current is at about 95%.
5. By $t = 5\tau$, the current is within about 1% of its final value, and the circuit is considered to be at steady state.

Current Decay (Source Removed)

When the voltage source is removed (and the circuit has a path for current to flow, such as through the resistor), the current decays from its initial value:

$i(t) = I_0 \, e^{-t/\tau}$

where $I_0$ is the current at the moment the source is removed. The decay follows the same time constant $\tau$. After one time constant, the current has dropped to about 36.8% of $I_0$. After $5\tau$, it's effectively zero.

Notice the symmetry: growth uses $(1 - e^{-t/\tau})$ and decay uses $e^{-t/\tau}$. Both are governed by the same $\tau$.

Steady-State Behavior

Reaching Steady State

Steady state is the condition where current is no longer changing. Since $\frac{di}{dt} = 0$ at steady state, the voltage across the inductor drops to zero and it acts like a short circuit (a plain wire). All the source voltage then appears across the resistor, giving a steady-state current of:

$I_{ss} = \frac{V}{R}$

The practical rule: a circuit reaches steady state after approximately $5\tau$. For the example above ($\tau = 2 \text{ ms}$), that's about 10 ms.

Why Steady-State Matters

Knowing the steady-state current tells you the final operating point of the circuit. In a DC power supply, it determines the maximum current delivered to a load. In motor control, it sets the operating torque and speed. And in switching circuits, the time to reach steady state ($5\tau$) directly affects how fast you can switch the circuit on and off. Faster switching requires a smaller time constant, which means lower $L$ or higher $R$.