Alternating Current

Resonance is when inductive and capacitive reactances are equal, resulting in maximum current.

Derivation of resonant frequency:

• At resonance: $X_L = X_C$
• Substitute: $\omega L = \frac{1}{\omega C}$
• Solve for ω: $\omega^2 = \frac{1}{LC}$
• Therefore: $\omega_0 = \frac{1}{\sqrt{LC}}$ or $f_0 = \frac{1}{2\pi\sqrt{LC}}$

At resonance:

• Impedance is minimum: $Z = R$ (purely resistive)
• Current is maximum: $i_0 = \frac{E_0}{R}$
• Current and voltage are in phase: $\phi = 0$
• Power factor = 1 (maximum power transfer)
• Circuit has maximum power dissipation
• Voltages across L and C can be much larger than source voltage
• The smaller the R, the sharper the resonance peak ("high Q factor")

Applications: Radio tuning, frequency selection, oscillator circuits

An AC generator (dynamo) consists of three main components:

1. Magnet:

   • Creates a strong uniform magnetic field between N and S poles
   • Can be permanent or electromagnetic

2. Armature:

   • Coil wound on a soft-iron core
   • Connected to slip rings (C₁ and C₂)
   • Rotates within the magnetic field at angular velocity ω
   • The core increases magnetic field strength through magnetization

3. Brushes and Slip Rings:

   • Brushes (B₁, B₂): Made of graphite, maintain electrical contact
   • Slip rings (C₁, C₂): Connected to coil ends, rotate with armature
   • Conduct current to external circuit terminals (P, Q)

The rotation of the coil in the magnetic field induces a sinusoidal EMF according to Faraday's law, producing alternating current.

AC generator energy conversion process:

1. Mechanical to electrical conversion:

   • Coil with area A and N turns rotates at angular velocity ω in uniform magnetic field B
   • Rotation changes magnetic flux through coil (Φ = BA cos ωt)
   • Changing flux induces EMF by Faraday's law

2. EMF equation:

   • E = -dΦ/dt = NBA ω sin ωt = E₀ sin ωt
   • Where E₀ = NBA ω is the peak EMF

3. Working principle:

   • At t=0: Coil perpendicular to field, flux is maximum (Φ = BA)
   • As coil rotates: Flux decreases, inducing current
   • At 90° rotation: Flux becomes zero, EMF is maximum
   • At 180° rotation: Flux is negative maximum, EMF returns to zero
   • Continuous rotation produces sinusoidal EMF with period T = 2π/ω

The turbine (powered by steam, water, or wind) provides the mechanical energy to rotate the armature.

Purely resistive AC circuit vs. reactive circuits:

Phase relationships:

• Resistive: Current and voltage are in phase (φ = 0)
• Capacitive: Current leads voltage by 90° (φ = +π/2)
• Inductive: Current lags voltage by 90° (φ = -π/2)

Power transfer:

• Resistive: Power factor = cos(φ) = 1 (maximum power transfer)
• Reactive: Power factor = 0 (no average power consumed)
• Combined circuits: Power factor between 0 and 1

Impedance:

• Resistive: Z = R (constant across frequencies)
• Capacitive: Z = 1/ωC (decreases with frequency)
• Inductive: Z = ωL (increases with frequency)

Energy storage:

• Resistive: Energy dissipated as heat
• Reactive: Energy stored in electric field (capacitor) or magnetic field (inductor) and returned to source

Example: A 60W incandescent bulb (resistive) converts all power to heat/light, while a perfect inductor or capacitor would ideally consume zero power.

Vector Representation Method for AC Circuit Analysis:

1. Represent components vectorially:

   • Resistance (R): Vector along positive x-axis
   • Capacitive reactance (Xₗ = 1/ωC): Vector along positive y-axis
   • Inductive reactance (Xₗ = ωL): Vector along negative y-axis

2. Determine net reactance (X):

   • X = Xₗ - Xₗ = ωL - 1/ωC

3. Calculate impedance magnitude:

   • Z = √(R² + X²)

4. Find phase angle:

   • φ = tan⁻¹(X/R)
   • If X > 0 (inductive dominance): Current lags voltage
   • If X < 0 (capacitive dominance): Current leads voltage
   • If X = 0 (resonance): Current in phase with voltage

5. Determine current:

   • Peak current: i₀ = E₀/Z
   • Current equation: i = i₀sin(ωt + φ)

This method provides a powerful visual approach to understanding the relationship between voltage and current in any AC circuit.

• Consists of a split cylinder with two halves (C₁ and C₂) that rotate with the armature
• Carbon brushes (B₁ and B₂) press against rotating halves to make external connections
• As armature rotates, connections to external circuit reverse at precise moments
• Key mechanism: Gaps pass under brushes exactly when emf becomes zero
• Result: Although induced emf alternates between positive and negative, current in external circuit maintains one direction
• This differs from AC dynamo which uses continuous slip rings without commutation

• Starting with Faraday's law for each coil:

  • Primary: E₁ = N₁(dΦ/dt)
  • Secondary: E₂ = -N₂(dΦ/dt)

• Dividing equations:

  • E₂/E₁ = -N₂/N₁

• Physical meaning:

  • Voltage ratio equals turns ratio
  • Negative sign indicates 180° phase difference
  • Common magnetic flux links both coils
  • Energy transfer occurs through magnetic field in core
  • No direct electrical connection between circuits

• When secondary is open: Primary current is only magnetizing current (i_s), 90° out of phase with E₁, delivering zero power
• When secondary is loaded: Additional current i₁ appears in primary, in phase with E₁
• Total primary current becomes vector sum: i_s + i₁
• Physical principle: Magnetic fields from secondary current induce EMFs that must be counteracted
• The primary draws additional current i₁ that generates EMF to precisely cancel EMF induced by secondary current
• This maintains flux balance in the core
• Power relationship: E₁i₁ = E₂i₂ (neglecting losses)
• Current ratio: i₂ = -(N₁/N₂)i₁

Calculation:

• E₂/E₁ = N₂/N₁
• E₂ = (N₂/N₁) × E₁ = (18/660) × 220V = 6V

If connected to 6Ω load:

• Secondary current: i₂ = E₂/R = 6V/6Ω = 1A
• Power delivered to load: P = E₂i₂ = 6V × 1A = 6W
• Primary current: i₁ = (N₂/N₁) × i₂ = (18/660) × 1A = 0.0273A
• Power drawn from source: P = E₁i₁ = 220V × 0.0273A = 6W (neglecting losses)

This illustrates power conservation in ideal transformers where input power equals output power.

• AC dynamo: Supplies alternating current that changes direction periodically • DC dynamo: Supplies current in one direction only • Key design difference: DC dynamo uses split-ring commutator instead of simple slip rings • This commutator reverses connections at precisely the moment when EMF polarity would change • Result: Current in external circuit maintains consistent direction despite fluctuating magnitude