AC Circuits: Inductors, Capacitors, and RLC Circuits

Inductors and Inductance

Introduction to Inductors

Inductors are passive circuit elements that store energy in their magnetic field when current flows through them. The property of an inductor to oppose changes in current is called inductance, denoted by L, and measured in henries (H).

• Induced emf (𝜀): When the current through an inductor changes, an emf is induced that opposes the change, according to Faraday's Law.

• Energy Storage: The energy stored in an inductor is in the form of a magnetic field.

Analogy with Capacitors

Inductors and capacitors are analogous in their energy storage properties, but while capacitors store energy in electric fields, inductors store energy in magnetic fields.

• Capacitor: Stores energy in an electric field between plates.

• Inductor: Stores energy in a magnetic field inside the coil.

Inductance of a Solenoid

The inductance of a solenoid depends on its physical properties: number of turns, length, and cross-sectional area. For an ideal solenoid:

• Formula: $L = \mu_0 n^2 \ell A$

• Where $n$ is the number of turns per unit length, $\ell$ is the length, $A$ is the cross-sectional area, and $\mu_0$ is the permeability of free space.

RL Circuits: Charging and Discharging

Current Growth and Decay in RL Circuits

When a switch is closed in an RL circuit, the current does not immediately reach its maximum value due to the inductor's opposition to change. The current increases exponentially with a time constant $\tau = \frac{L}{R}$.

• Charging (Current Increasing): $I(t) = I_{\text{max}} (1 - e^{-t/\tau})$

• Discharging (Current Decreasing): $I(t) = I_{\text{max}} e^{-t/\tau}$

Energy in Inductors

Energy Stored in a Magnetic Field

The energy stored in an inductor is given by:

• $U_B = \frac{1}{2} L I^2$

• This energy is stored in the magnetic field created by the current in the inductor.

LC Circuits: Oscillations

Energy Exchange in LC Circuits

In an LC circuit, energy oscillates between the electric field of the capacitor and the magnetic field of the inductor. The total energy remains constant (in the absence of resistance).

• Total Energy: $E_{\text{tot}} = \frac{1}{2} Q^2/C + \frac{1}{2} L I^2$

• At different times, all energy may be in the capacitor, in the inductor, or shared between both.

AC Circuits: Fundamentals

Alternating Current (AC) and Voltage

AC circuits are powered by sources that produce sinusoidally varying voltages and currents. The instantaneous voltage is given by:

• $\Delta V = \Delta V_{\text{max}} \sin(2\pi f t)$

• Root Mean Square (rms) Values: $I_{\text{rms}} = \frac{I_{\text{max}}}{\sqrt{2}}$, $\Delta V_{\text{rms}} = \frac{\Delta V_{\text{max}}}{\sqrt{2}}$

Power in AC Circuits

The average power dissipated in a resistor in an AC circuit is:

• $P_{\text{av}} = I_{\text{rms}}^2 R$

Capacitors and Inductors in AC Circuits

Capacitive Reactance

The opposition of a capacitor to AC is called capacitive reactance:

• $X_C = \frac{1}{2\pi f C}$

• Voltage lags current by 90° in a capacitor.

Inductive Reactance

The opposition of an inductor to AC is called inductive reactance:

• $X_L = 2\pi f L$

• Voltage leads current by 90° in an inductor.

Series RLC Circuits

Impedance and Phase Relationships

When a resistor, inductor, and capacitor are connected in series to an AC source, the total opposition to current is called impedance (Z):

• $Z = \sqrt{R^2 + (X_L - X_C)^2}$

• The phase angle $\phi$ between the current and voltage is given by $\tan \phi = \frac{X_L - X_C}{R}$

Phasor Diagrams

Phasor diagrams are used to represent the phase relationships between voltages across the resistor, inductor, and capacitor.

Resonance in RLC Circuits

Resonant Frequency

Resonance occurs in a series RLC circuit when the inductive and capacitive reactances are equal ($X_L = X_C$), resulting in maximum current. The resonant frequency is:

• $f_0 = \frac{1}{2\pi \sqrt{LC}}$

Transformers and Power Transmission

Transformer Principles

Transformers use electromagnetic induction to change the voltage and current levels in AC circuits. The voltage ratio is determined by the turns ratio:

• $\frac{\Delta V_2}{\Delta V_1} = \frac{N_2}{N_1}$

• Power is ideally conserved: $I_1 \Delta V_1 = I_2 \Delta V_2$

Power Loss in Transmission Lines

High-voltage, low-current transmission minimizes power loss due to resistance in transmission lines ($P_{\text{loss}} = I^2 R$).

Summary Table: Capacitor vs. Inductor

Capacitor	Inductor
$E = \frac{V_C}{d}$ (parallel-plate capacitor)	$B = \mu_0 n I$ (solenoid)
Electric field $E$ between plates	Magnetic field $B$ in coils
Voltage $V_C$ across plates	Current $I$ in wire
Charge $Q$ stored by capacitor	Magnetic flux $\Phi_B$ through inductor
Capacitance $C$	Inductance $L$
$Q = CV_C$	$\Phi_B = LI$
$C = \frac{\varepsilon_0 A}{d}$ (parallel-plate capacitor)	$L = \mu_0 n^2 \ell A$ (solenoid)
$U_E = \frac{1}{2} C V_C^2$	$U_B = \frac{1}{2} L I^2$