Alternating current (AC) is generated when a coil rotates in a magnetic field, or when a magnetic field rotates around a stationary coil. This process is based on Faraday's Law of Electromagnetic Induction.
When a conductor moves through a magnetic field, an EMF (voltage) is induced in the conductor. As the coil rotates, the direction of the induced EMF reverses every half rotation, creating an alternating voltage.
Coil rotating in magnetic field produces sinusoidal AC voltage
Or with phase angle:
Where:
One complete set of positive and negative values of an alternating quantity. The waveform repeats after each cycle.
The time taken to complete one cycle. Measured in seconds (s).
Number of cycles completed per second. Measured in Hertz (Hz). In India, standard frequency is 50 Hz.
Rate of change of angle in radians per second.
The maximum value reached by the alternating quantity during one cycle.
The difference between maximum positive and maximum negative values.
The average value of an AC quantity over a complete cycle is zero (positive and negative halves cancel out). However, the average value over a half-cycle is meaningful.
The average value represents the DC equivalent that would transfer the same amount of charge over a half-cycle.
The RMS value is the effective value of an AC quantity. It is defined as the DC equivalent that would produce the same heating effect (power dissipation) in a resistance.
This is the most important value for AC circuits! When we say "220V AC supply", we mean 220V RMS.
| Quantity | Formula | Value for Sine Wave |
|---|---|---|
| Peak Value | Vm | Vm |
| RMS Value | Vm/√2 | 0.707 Vm |
| Average Value | 2Vm/π | 0.637 Vm |
| Form Factor | Vrms/Vavg | 1.11 |
| Peak Factor (Crest Factor) | Vm/Vrms | 1.414 (√2) |
An AC voltage has a peak value of 340V. Find: (a) RMS value, (b) Average value, (c) Form factor
A phasor is a rotating vector that represents a sinusoidal quantity. It rotates at angular velocity ω and its projection on the vertical axis gives the instantaneous value.
Phasors allow us to represent magnitude and phase of AC quantities graphically, making AC circuit analysis much simpler.
Phase difference is the angular displacement between two phasors (or two sinusoidal waveforms) of the same frequency.
Two voltages are given as: V₁ = 100 sin(ωt) and V₂ = 50 sin(ωt + 90°). Find the resultant voltage.
In a pure resistive circuit, voltage and current are in phase (phase difference = 0°).
V and I in phase
In a pure inductive circuit, current lags voltage by 90°. The inductor opposes change in current.
Power = 0 (Average power consumed is zero - energy is stored and returned)
Remember: "ELI the ICE man" - In inductor (L), voltage (E) leads current (I)
In a pure capacitive circuit, current leads voltage by 90°. The capacitor opposes change in voltage.
Power = 0 (Average power consumed is zero)
Remember: "ELI the ICE man" - In capacitor (C), current (I) leads voltage (E)
In an R-L series circuit, current lags voltage by angle φ (between 0° and 90°).
Z² = R² + XL²
In an R-C series circuit, current leads voltage by angle φ (between 0° and 90°).
The R-L-C series circuit combines all three elements. The net reactance determines whether the circuit is inductive or capacitive.
A series RLC circuit has R = 10Ω, L = 0.1H, C = 100μF connected to 230V, 50Hz supply. Find impedance, current, and power factor.
The circuit is nearly at resonance (XL ≈ XC), so impedance is almost purely resistive!
In parallel AC circuits, voltage is common across all branches. We use admittance (Y) for easier calculations.
Where:
Also called Active Power or True Power. This is the actual power consumed by the resistance and converted to heat/work.
Unit: Watt (W)
Power that oscillates between source and reactive elements (L, C). It does no useful work.
Unit: VAR (Volt-Ampere Reactive)
The total power supplied by the source. It's the product of RMS voltage and current.
Unit: VA (Volt-Ampere)
A load draws 10A from a 230V supply at 0.8 power factor lagging. Find real power, reactive power, and apparent power.
Verify: S = √(P² + Q²) = √(1840² + 1380²) = √(3385600 + 1904400) = √5290000 = 2300 VA ✓
Admittance is the reciprocal of impedance. It represents how easily a circuit allows current to flow.
Components:
Series resonance occurs when XL = XC in a series RLC circuit. At this condition:
At resonance: ωrL = 1/(ωrC)
Parallel resonance occurs in a parallel LC circuit when the susceptances are equal (BL = BC). At this condition:
| Property | Series Resonance | Parallel Resonance |
|---|---|---|
| Impedance at resonance | Minimum (= R) | Maximum |
| Current at resonance | Maximum | Minimum |
| Power factor | Unity (1) | Unity (1) |
| Resonant frequency | fr = 1/(2π√LC) | fr = 1/(2π√LC) |
| Application | Tuning circuits, filters | Oscillators, tank circuits |
Three-phase voltage is generated by three identical coils placed 120° apart, rotating in a uniform magnetic field. This produces three sinusoidal voltages of equal magnitude but displaced by 120° in phase.
Three phases displaced by 120° each
In star connection, one end of each phase winding is connected to a common point called the neutral point (N). The other ends are the line terminals (R, Y, B).
Where:
In delta connection, the three phase windings are connected end-to-end to form a closed triangle. The line terminals are taken from the junction points.
These formulas apply to both Star and Delta connections!
| Parameter | Star (Y) Connection | Delta (Δ) Connection |
|---|---|---|
| Line Voltage (VL) | √3 × Vph | Vph |
| Line Current (IL) | Iph | √3 × Iph |
| Neutral | Available | Not available |
| Phase Voltage | VL/√3 (lower) | VL (higher) |
| Used for | Transmission, distribution | Motors, transformers |
A three-phase, star-connected load draws a line current of 20A from a 415V, 50Hz supply at 0.8 power factor lagging. Calculate: (a) Phase voltage, (b) Total power consumed.
| Quantity | Formula | Unit |
|---|---|---|
| RMS Value | Vrms = Vm/√2 = 0.707 Vm | V or A |
| Average Value | Vavg = 0.637 Vm | V or A |
| Inductive Reactance | XL = 2πfL = ωL | Ω |
| Capacitive Reactance | XC = 1/(2πfC) = 1/(ωC) | Ω |
| Impedance (Series RLC) | Z = √(R² + (XL − XC)²) | Ω |
| Resonant Frequency | fr = 1/(2π√LC) | Hz |
| Real Power | P = VI cos φ | W |
| Reactive Power | Q = VI sin φ | VAR |
| Apparent Power | S = VI | VA |
| 3-Phase Power | P = √3 VL IL cos φ | W |