Inductance and RL circuits Knowledge-Based Study

Study notes

1. Self-Induction and Inductance

   When an electric current flows through a conducting coil or circuit, it establishes a magnetic field proportional to the current. If the current changes with time, the magnetic flux threading the circuit also changes. According to Faraday's law of electromagnetic induction and Lenz's law, this changing flux induces an electromotive force (EMF) in the circuit itself that opposes the change in current. This phenomenon is known as self-induction.

   The magnetic flux $\Phi_B$ through a coil with $N$ turns is directly proportional to the current $I$:

   $$N\Phi_B = LI$$

   Here, the constant of proportionality $L$ is defined as the self-inductance (or simply inductance) of the coil. The SI unit of inductance is the henry (H), where 1 H = 1 Wb/A = 1 V\cdot s/A.

   By applying Faraday's law, the self-induced EMF $\varepsilon_L$ (often called back EMF) is given by:

   $$\varepsilon_L = -L\frac{dI}{dt}$$

   The negative sign reflects Lenz's law: if the current is increasing, $\varepsilon_L$ opposes the increase; if the current is decreasing, $\varepsilon_L$ acts in the direction of the current to resist the decrease.

   Magnetic Energy of an Inductor

   Establishing a current in an inductor requires work against the opposing self-induced EMF. This work is not dissipated as thermal energy; instead, it is stored in the magnetic field created within and around the inductor.

   The rate at which work is done by the source against the back EMF is the instantaneous power:

   $$P = \frac{dW}{dt} = -\varepsilon_L I = L I \frac{dI}{dt}$$

   Integrating this expression from an initial current of zero to a final steady current $I$ yields the total stored magnetic potential energy $U_B$:

   $$U_B = \int_0^I L i \, di = \frac{1}{2}LI^2$$

   Key points regarding energy in inductors:

   • An ideal inductor stores energy purely in its magnetic field, analogous to how an ideal capacitor stores electric potential energy $U_E = \frac{1}{2}CV^2$ in its electric field.
   • When current is reduced to zero, this stored magnetic energy is returned to the circuit.

   Current Growth in a Series RL Circuit

   A series $RL$ circuit consists of a resistor of resistance $R$, an inductor of inductance $L$, and an ideal DC voltage source $\varepsilon$ connected in series via a switch.

   When the switch is closed at $t = 0$, Kirchhoff's loop rule gives:

   $$\varepsilon - IR - L\frac{dI}{dt} = 0$$

   Solving this first-order linear differential equation with the initial condition $I(0) = 0$ yields the current as a function of time:

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

   where $\tau$ is the inductive time constant of the circuit:

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

   Circuit Behavior at Limiting Times

   • At $t = 0$: The current is zero ($I = 0$). The inductor acts like an open circuit, and the entire source voltage drops across the inductor: $\varepsilon_L = -\varepsilon$.
   • At $t = \tau$: The current reaches approximately 63.2% of its maximum value: $I(\tau) = \frac{\varepsilon}{R}(1 - e^{-1}) \approx 0.632 \frac{\varepsilon}{R}$.
   • As $t \to \infty$: The current approaches its steady-state maximum value $I_{\max} = \frac{\varepsilon}{R}$. The rate of change $dI/dt \to 0$, meaning the inductor behaves as an ideal conductor (short circuit) with zero voltage drop across it.

   Current Decay in a Series RL Circuit

   Consider a circuit in which a steady current $I_0 = \frac{\varepsilon}{R}$ has been established through an inductor $L$ and a resistor $R$. If the source is instantly bypassed or removed at $t = 0$ while maintaining a closed loop containing only $R$ and $L$, the current does not drop to zero immediately.

   Applying Kirchhoff's loop rule to the closed loop gives:

   $$-IR - L\frac{dI}{dt} = 0$$

   Separating variables and integrating with the initial condition $I(0) = I_0$ yields:

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

   where $\tau = \frac{L}{R}$ is the same inductive time constant.

   Key Characteristics of Current Decay

   • At $t = \tau$, the current has decreased to about 36.8% of its initial value ($I(\tau) = I_0 e^{-1} \approx 0.368 I_0$).
   • As $t \to \infty$, the current approaches zero asymptotically.
   • The total energy dissipated as heat in the resistor during decay is equal to the magnetic energy initially stored in the inductor: $E_{\text{dissipated}} = \frac{1}{2}LI_0^2$.