KAHIBARO
Table of Contents
Introducing RL Circuits
An RL circuit is a circuit that contains a resistor, with resistance $R$, and an inductor, with inductance $L$. These circuits are important because the inductor does not allow the current to change instantly. Instead, the current changes gradually over time.
This time-dependent behavior is the main feature of RL circuits. The resistor opposes current by dissipating energy as heat, while the inductor opposes changes in current by creating an induced emf.
Series RL Circuit
The simplest RL circuit has a battery, a resistor, and an inductor connected in series. When the switch is closed, current begins to flow, but it does not jump immediately to its final value. The inductor produces a back emf that resists the increase in current.
Using Kirchhoff's loop rule, the voltage equation is
$$
\mathcal{E} - iR - L\frac{di}{dt} = 0
$$
where $\mathcal{E}$ is the battery emf and $i$ is the current.
Current Growth After Switching On
Suppose the current is initially zero, and the switch is closed at $t = 0$. Solving the differential equation gives
$$
i(t) = \frac{\mathcal{E}}{R}\left(1 - e^{-t/\tau}\right)
$$
where
$$
\tau = \frac{L}{R}
$$
is called the time constant of the RL circuit.
At long times, the exponential term becomes very small, so the current approaches the final value
$$
i_{\infty} = \frac{\mathcal{E}}{R}
$$
This is the same current that would flow if the inductor acted like an ordinary wire in steady state.
For a series RL circuit connected to a dc source,
$$
i(t) = \frac{\mathcal{E}}{R}\left(1 - e^{-t/\tau}\right), \qquad \tau = \frac{L}{R}
$$
The current cannot change instantaneously in an ideal inductor.
Meaning of the Time Constant
The time constant $\tau = L/R$ sets the speed of the current change. A larger inductance makes the current change more slowly. A larger resistance makes the current settle more quickly.
At $t = \tau$,
$$
i(\tau) = \frac{\mathcal{E}}{R}\left(1 - e^{-1}\right) \approx 0.632\,\frac{\mathcal{E}}{R}
$$
So after one time constant, the current has reached about $63.2\%$ of its final value.
After several time constants, the current is very close to its final value.
| Time | Current as fraction of final value |
|---|---|
| $t = \tau$ | $0.632$ |
| $t = 2\tau$ | $0.865$ |
| $t = 3\tau$ | $0.950$ |
| $t = 5\tau$ | $0.993$ |
Current Decay After Switching Off
If the battery is removed and the resistor and inductor remain connected in a loop, the current does not vanish instantly. The energy stored in the inductor keeps the current flowing for a while.
In this case, the loop equation becomes
$$
L\frac{di}{dt} + iR = 0
$$
If the initial current is $I_0$ at $t = 0$, then the solution is
$$
i(t) = I_0 e^{-t/\tau}
$$
with the same time constant
$$
\tau = \frac{L}{R}
$$
The current decreases exponentially toward zero.
For current decay in a series RL loop,
$$
i(t) = I_0 e^{-t/\tau}, \qquad \tau = \frac{L}{R}
$$
The current falls to about $36.8\%$ of its initial value after one time constant.
Voltage Across the Resistor and Inductor
In a series RL circuit, the resistor voltage is
$$
V_R = iR
$$
and the inductor voltage is
$$
V_L = L\frac{di}{dt}
$$
When the current is building up, the resistor voltage starts at zero because the current starts at zero. The inductor voltage starts at its maximum value, equal to the battery emf, because the inductor initially strongly opposes the current increase.
As time passes, $V_R$ increases and $V_L$ decreases.
For the charging process,
$$
V_R(t) = \mathcal{E}\left(1 - e^{-t/\tau}\right)
$$
$$
V_L(t) = \mathcal{E}e^{-t/\tau}
$$
These always satisfy
$$
V_R + V_L = \mathcal{E}
$$
Physical Interpretation
The resistor and the inductor play different roles. The resistor converts electrical energy into thermal energy. The inductor stores energy in its magnetic field and resists changes in current.
At the instant the circuit is switched on, the inductor behaves like an open circuit because it prevents a sudden rise of current. After a long time in a dc circuit, the current becomes constant, so $\frac{di}{dt} = 0$, and the inductor behaves like a short circuit, assuming the inductor is ideal.
In a dc steady state,
$$
\frac{di}{dt} = 0 \quad \Rightarrow \quad V_L = L\frac{di}{dt} = 0
$$
An ideal inductor has zero voltage across it when the current is constant.
Energy in an RL Circuit
As current grows in the circuit, energy is stored in the magnetic field of the inductor. The stored energy is
$$
U = \frac{1}{2}LI^2
$$
where $I$ is the current at that moment.
When the circuit is switched off, this stored energy is released and then dissipated in the resistor.
Energy stored in an inductor:
$$
U = \frac{1}{2}LI^2
$$
Typical Graphs
In an RL circuit, the main graphs are exponential curves. The current rises exponentially toward a final value when connected to a battery, and decays exponentially toward zero when the source is removed.
Summary
An RL circuit contains a resistor and an inductor, usually in series. Its defining feature is that current changes gradually, not instantly. When connected to a dc source, the current grows as
$$
i(t) = \frac{\mathcal{E}}{R}\left(1 - e^{-t/\tau}\right)
$$
When the source is removed, the current decays as
$$
i(t) = I_0 e^{-t/\tau}
$$
The time constant is
$$
\tau = \frac{L}{R}
$$
and the energy stored in the inductor is
$$
U = \frac{1}{2}LI^2
$$
These ideas make RL circuits one of the simplest and most important examples of time-dependent electric behavior.