Alternating Current

Power factor: $\cos\phi$ where $\phi$ is phase angle between voltage and current

• Equals $\frac{R}{Z} = \frac{R}{\sqrt{R^2 + X^2}}$ in series circuits

Average power: $P_{avg} = V_{rms}I_{rms}\cos\phi = I_{rms}^2 R$

Why reactive elements don't consume real power:

• For purely reactive components: $\phi = \pm 90°$, so $\cos\phi = 0$
• Energy is stored in capacitors (electric field) or inductors (magnetic field) during part of cycle
• Stored energy is returned to the circuit in another part of cycle
• Over complete cycle, net energy transfer is zero
• Only resistive components convert electrical energy to heat (irreversible)

Example: In a circuit with Z = 50Ω and R = 40Ω, power factor is 0.8, meaning 80% of apparent power is real power

LC oscillation principle:

• Energy oscillates between electric field (capacitor) and magnetic field (inductor)
• No energy is dissipated in an ideal LC circuit
• Results in self-sustained oscillations at frequency f = 1/(2π√LC)

Process when charged capacitor is connected to inductor:

1. Initially: All energy stored in capacitor's electric field
2. Capacitor begins discharging through inductor
3. Current increases, storing energy in inductor's magnetic field
4. Capacitor voltage decreases to zero (all energy now in inductor)
5. Inductor maintains current flow, charging capacitor in opposite polarity
6. Current decreases as capacitor charges
7. When current reaches zero, all energy is back in capacitor
8. Process repeats in opposite direction
9. Circuit oscillates at natural frequency ω = 1/√LC

Mathematically: This system follows differential equation LC(d²q/dt²) + q = 0, with solution q = q₀cos(ωt)

A capacitor blocks DC but passes AC because:

• DC case: With constant voltage, current only flows momentarily while the capacitor charges. Once charged to the source voltage, current stops completely (i = C·dV/dt = 0 when dV/dt = 0).

• AC case: With continuously changing voltage, the capacitor continuously charges and discharges, allowing current to flow throughout the circuit. The current is given by i = C·dV/dt, which is non-zero for sinusoidal voltage.

• The 90° phase lead of current before voltage demonstrates this principle: current must flow first to build up charge (and thus voltage) on the capacitor plates.

• At higher frequencies, capacitive reactance (X_C = 1/ωC) decreases, allowing more current to flow—the capacitor becomes less resistive to current flow as frequency increases.

Example: A coupling capacitor in audio equipment blocks DC bias voltages while allowing AC audio signals to pass between stages.

The impedance (Z) in an LCR series circuit is:

$Z = \sqrt{R^2 + \left(\frac{1}{\omega C} - \omega L\right)^2}$

Where:

• R is the resistance in ohms
• ωL is the inductive reactance
• 1/ωC is the capacitive reactance
• ω is the angular frequency (2πf)

The impedance:

• Represents the total opposition to current flow
• Is measured in ohms
• Forms a right triangle with resistance (R) and net reactance (X = 1/ωC - ωL)
• Determines the peak current: $i_0 = \frac{E_0}{Z}$

Example: In a circuit with R = 100Ω, L = 50mH, C = 10μF at 60Hz, calculate Z using the formula to find the total opposition to current flow.

A choke coil is an inductor with high inductance but low resistance that:

• Reduces current in AC circuits without significant power loss
• Creates impedance through inductive reactance (X_L = ωL)
• Limits current without dissipating energy as heat
• Functions effectively at AC frequencies while being ineffective for DC

The primary advantage over using resistors is energy efficiency - the inductor stores energy in its magnetic field rather than converting it to heat, significantly reducing power waste in the circuit.

Example: In a 220V, 50Hz fluorescent tube circuit, a choke coil provides the necessary current limitation while maintaining energy efficiency.

A choke coil reduces the RMS voltage across a resistive load by:

• Creating a voltage divider effect through impedance
• The RMS voltage across the resistive load is calculated as:

$V_{R,rms} = \frac{R}{\sqrt{R^2 + \omega^2L^2}} \cdot V_{rms}$

Where:

• V_{R,rms} is the RMS voltage across the resistor
• R is the resistance of the load
• L is the inductance of the choke coil
• ω is the angular frequency (2πf)
• V_{rms} is the applied RMS voltage

The reduction factor (R/√(R² + ω²L²)) can be adjusted by changing the inductance value to achieve the desired voltage across the load.

• 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

A hot-wire ammeter operates based on thermal expansion:

• Current passing through a thin wire causes heating proportional to i²rms
• The heated wire expands, reducing its tension
• A spring attached to the wire maintains opposing tension
• As tension decreases, the spring rotates a cylinder with an attached pointer
• The pointer moves along a graduated scale calibrated to show RMS current directly
• A small parallel shunt resistance keeps the instrument's total resistance low, preventing significant circuit disturbance

The key principle: Current measurement through the thermal effect (i²R heating) rather than magnetic effects used in other meters, making it suitable for both AC and DC measurements.

• The armature coil ends connect to separate halves (C₁, C₂) of a split cylinder • Carbon brushes (B₁, B₂) press against the rotating split cylinder • As the armature rotates, the contacts to external circuit reverse • The commutator gaps pass under brushes exactly when EMF becomes zero • When EMF polarity reverses, the connection also reverses, maintaining current direction • External circuit terminals P and Q thus receive unidirectional current • This produces pulsating DC rather than true constant current

• Unloaded transformer (open secondary):

• Only has primary current iₛ (source current)
• iₛ is 90° out of phase with primary EMF ℰ₁
• Power delivered = 0 (purely inductive circuit)

• Loaded transformer (secondary connected to load):

• Primary current has two components: iₛ and i₁
• i₁ is in phase with ℰ₁ and contributes to power transfer
• Total primary current = iₛ + i₁
• Power delivered = ℰ₁i₁ = ℰ₂i₂

• When load increases:

• i₂ increases
• i₁ increases correspondingly
• The total power transferred increases
• A bulb in series with primary glows brighter when secondary is loaded