Inductance and LC/RL Circuits: Concepts and Applications

Inductance and Circuits

Introduction to Inductance

Inductance is a fundamental property of electrical circuits, particularly those containing coils or solenoids. It describes the ability of a circuit element to induce an electromotive force (emf) as a result of a changing current. Inductance is measured in henries (H), where 1 H = 1 Wb/A (weber per ampere).

• Inductor: A passive circuit element that stores energy in its magnetic field when current flows through it.

• Symbol: L

• Unit: Henry (H)

Mutual Inductance

Definition and Principle

Mutual inductance occurs when a change in current in one coil induces an emf in a nearby coil. This is the principle behind transformers and many wireless charging systems.

• Mutual Inductance (M): The proportionality constant relating the emf induced in one coil to the rate of change of current in another coil.

• Formula: $\mathcal{E}_2 = -M \frac{dI_1}{dt}$

• Unit: Henry (H)

• Application: Used in transformers to transfer energy between circuits.

Self-Inductance

Definition and Formula

Self-inductance is the property of a single coil (or circuit) to induce an emf in itself as the current changes. This is a key concept in understanding how inductors resist changes in current.

• Self-Inductance (L): The proportionality constant relating the emf induced in a coil to the rate of change of current in itself.

• Formula: $\mathcal{E}_L = -L \frac{dI}{dt}$

• Unit: Henry (H)

Energy Stored in an Inductor

Magnetic Energy Storage

When current flows through an inductor, energy is stored in its magnetic field. The amount of energy stored depends on the inductance and the current.

• Formula: $U = \frac{1}{2} L I^2$

• Interpretation: The energy is proportional to the square of the current and the inductance.

• Example: If L = 2 H and I = 3 A, then $U = \frac{1}{2} \times 2 \times 9 = 9$ J.

RL Circuits

RL Circuit Behavior and Time Constant

An RL circuit consists of a resistor (R) and an inductor (L) in series with a voltage source. When the switch is closed, the current does not immediately reach its maximum value due to the inductor's opposition to changes in current. The time constant (τ) characterizes how quickly the current approaches its steady-state value.

• Time Constant: $\tau = \frac{L}{R}$

• Current Growth: $I(t) = \frac{\mathcal{E}}{R} \left(1 - e^{-t/\tau}\right)$

• Steady-State Current: $I_{\infty} = \frac{\mathcal{E}}{R}$

• At t = 0: $I(0) = 0$ (current starts at zero)

• At t → ∞: $I(\infty) = \frac{\mathcal{E}}{R}$ (current reaches maximum value)

Example Problem

• Given: $\mathcal{E} = 120\ \mathrm{V},\ R = 200\ \Omega,\ L = 10\ \mathrm{H}$

• (a) Initial rate of current change: $\left. \frac{dI}{dt} \right|_{t=0} = \frac{\mathcal{E}}{L} = \frac{120}{10} = 12\ \mathrm{A/s}$

• (b) Long-term current: $I_{\infty} = \frac{\mathcal{E}}{R} = \frac{120}{200} = 0.6\ \mathrm{A}$

LC Circuits

Oscillations in LC Circuits

An LC circuit consists of an inductor (L) and a capacitor (C) connected together. When the capacitor is charged and the switch is closed, energy oscillates between the electric field of the capacitor and the magnetic field of the inductor. This forms the basis for many oscillatory circuits, such as radio tuners.

• Oscillation Frequency: $f = \frac{1}{2\pi\sqrt{LC}}$

• Energy Exchange: Energy alternates between the capacitor (electric field) and the inductor (magnetic field).

• Applications: Used in radio receivers and transmitters for tuning to specific frequencies.

Energy in LC Circuits

• Maximum Energy in Capacitor: $U_E = \frac{1}{2} C V^2$

• Maximum Energy in Inductor: $U_B = \frac{1}{2} L I^2$

• At any instant, the total energy is conserved and oscillates between $U_E$ and $U_B$.

Summary Table: RL vs. LC Circuits

Property	RL Circuit	LC Circuit
Components	Resistor, Inductor	Inductor, Capacitor
Behavior	Current grows/exponentially decays	Oscillatory (sinusoidal) current and voltage
Energy Storage	Magnetic field in inductor	Alternates between electric field (capacitor) and magnetic field (inductor)
Time Constant / Frequency	$\tau = \frac{L}{R}$	$f = \frac{1}{2\pi\sqrt{LC}}$