Alternating Current

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.

The vector representation method for CR circuit analysis:

1. Draw vectors on a coordinate system where:

   • Resistance R is represented along the x-axis (horizontal)
   • Capacitive reactance Xc = 1/ωC is represented along the y-axis (vertical, positive direction)

2. The impedance Z is the resultant vector of these two components:

   • Magnitude: Z = √(R² + Xc²)
   • Direction: Forms angle φ with the x-axis where tan φ = Xc/R = 1/ωCR

3. For a source voltage E₀sin ωt, the current is:

   • Peak current: i₀ = E₀/Z
   • Phase relationship: i = i₀sin(ωt + φ) where current leads voltage by angle φ

This method visually demonstrates why the current leads the voltage in a capacitive circuit and provides an intuitive way to calculate both the current magnitude and phase angle without complex algebra.

The impedance (Z) in an LR circuit is the total opposition to current flow when an AC voltage is applied, combining both resistance and inductive reactance.

Calculation:

• Z = √(R² + X_L²) where:
  • R = resistance in ohms (Ω)
  • X_L = inductive reactance = ωL (in ohms)
  • ω = angular frequency (2πf)
  • L = inductance in henries (H)

The impedance is measured in ohms (Ω) and determines the amplitude of current according to Ohm's law for AC: I₀ = E₀/Z

As frequency increases in a choke coil circuit:

1. Circuit impedance increases according to: $Z = \sqrt{R^2 + \omega^2L^2} = \sqrt{R^2 + (2\pi f)^2L^2}$

2. Current decreases following: $I_{rms} = \frac{V_{rms}}{\sqrt{R^2 + \omega^2L^2}}$

3. Phase difference increases: $\phi = \tan^{-1}\left(\frac{\omega L}{R}\right)$

Key relationships:

• Impedance is directly proportional to frequency
• Current is inversely proportional to frequency
• As f → ∞, Z approaches ∞ and current approaches 0
• As f → 0, Z approaches R (DC resistance) and inductive effect disappears

This frequency-dependent behavior makes choke coils particularly effective at AC power frequencies (50-60Hz) while allowing them to be physically smaller than equivalent resistive components.

• Cause: Single coil produces varying magnitude of emf that follows a sinusoidal pattern
• Although commutator keeps current unidirectional, magnitude still oscillates
• Solution: Add perpendicular coils to the system with their own commutators
• Principle: Second coil produces maximum emf when first coil's emf is zero (phase shift)
• Combined effect: Summing the outputs creates more consistent current with less variation
• Adding more coils at different angles further reduces oscillations
• This creates what's called a "smoother" DC output with less ripple

• Current from battery/DC source flows through armature coil via brushes and slip rings
• Magnetic field from field magnets interacts with current-carrying coil
• Fleming's left-hand rule: Force experienced by current-carrying conductor in magnetic field
• This interaction creates torque on the coil, causing rotation
• Torque magnitude varies with coil orientation: zero when perpendicular to field, maximum when parallel
• As coil rotates, back emf (e) is induced opposite to applied emf (E)
• Net current at any instant: i = (E-e)/R
• Mechanical load attached to shaft rotates due to this torque

Advantages:

• Measures both AC and DC currents without modification
• Reading shows true RMS value directly, regardless of waveform
• Not affected by external magnetic fields
• No frequency-dependent errors (works at high frequencies)
• Simple construction with few components

Disadvantages:

• Slow response time due to thermal inertia
• Lower accuracy compared to moving-coil instruments
• Susceptible to ambient temperature variations
• Non-linear scale (compressed at lower readings)
• Power consumption is higher than other meter types
• Limited sensitivity for very small currents

• Primary technique: Adding multiple armature coils positioned at different angles • A second coil is placed perpendicular to the first coil • The perpendicular coil produces maximum EMF when the first coil's EMF is zero • Each coil has its own commutator arrangement • The combined output current contains fewer oscillations • The resulting "net current" has smaller amplitude variations • Adding more coils at different angles further smooths the output • Modern designs may use additional electronic filtering components

• A transformer works on the principle of mutual induction between two electrically isolated coils sharing a common magnetic circuit
• When an alternating current flows through the primary coil, it creates a changing magnetic flux in the iron core
• This changing flux induces an EMF in the secondary coil according to Faraday's law (E = -N·dΦ/dt)
• The voltage ratio between primary and secondary is directly proportional to the turns ratio: E₂/E₁ = N₂/N₁
• The current ratio is inversely proportional to the turns ratio: i₂/i₁ = N₁/N₂
• Example: A 1000-turn primary coil connected to 240V will induce 24V in a 100-turn secondary coil