Chapter-2_AC Circuits

Generation of AC

  • AC Generation:

    • Alternating voltage can be generated by rotating a coil in a magnetic field.

    • Alternatively, it can be generated by rotating a magnetic field inside a stationary coil.

    • Key Components:

      • Coil: A rectangular coil with N turns rotating in a uniform magnetic field.

      • Angular velocity: Represented by ( w ) radians/second.

Magnetic Flux and Coil Rotation

  • When the coil's plane is aligned with the magnetic field, maximum flux ( \Phi_m ) is linked.

  • The angle of rotation can be described with the formula:( \Phi = \Phi_m \cos(\theta) ) where ( \theta ) is the angle of rotation over time ( t ).

  • Induced EMF: According to Faraday's Law, the induced electromotive force (emf) in the coil is proportional to the rate of change of magnetic flux:

    • Formula: ( \text{emf} = -\frac{d\Phi}{dt} )

Instantaneous EMF

  • The instantaneous value of induced emf is given as:

  • ( e = N \Phi_m w \sin(wt) )

    • When the coil is turned 90°, ( \sin(\theta) = 1 ) and the emf reaches its maximum ( E_m ).

Sine Wave Characteristics

  • The maximum and instantaneous voltage equations can be represented as:

    • ( e = E_m \sin(\omega t) )

    • The current formulas are similar, showing how they function as sine functions over time.

AC Waveforms

  1. Sine Waveform:

    • Represents the fundamental waveform of AC.

  2. Complex Waveforms: Includes triangular and square waves. These differ in their shape and characteristics.

  3. Cycle and Frequency:

    • One complete cycle of AC waveform spans 360°.

    • Frequency (f) is the number of cycles per second, measured in Hertz (Hz).

Time Period and Frequency Relationship

  • The time period (T) represents how long it takes to complete one cycle and is inversely proportional to frequency:

    • ( f = \frac{1}{T} )

    • For a 50 Hz alternating current, T = 1/50 seconds.

RMS and Average Values of AC

  • RMS Value: Represents the effective current that provides the same power as DC, defined by( I_{rms} = 0.707 I_{max} )

  • Average Value: Unlike RMS, it averages over a cycle. For sinusoidal AC, the average value is zero over one complete cycle; thus, values for half-cycles are often calculated.

Power in AC Circuits

  • Power Measurement: Active (real) power is calculated based on current and voltage in phase, while reactive power is measured separately.

Phase Relationships in AC Circuits

  • Distinction of lagging and leading quantities based on which reaches zero first. Phase differences can also be quantified using sinusoidal relationships.

Resonance in RLC Circuits

  • At resonance in series circuits, the impedance is minimized to resistance (R), resulting in maximal current flow and power dissipation.

  • Quality Factor (Q): Indicates the selectivity of the circuit and is influenced by the ratio of reactive components to resistance.

Summary of AC Properties

  • Voltage and Current Relationship:

    • Current generally lags in purely inductive circuits, while it leads in purely capacitive setups.

  • Power Factor: Cosine of the phase angle (( \cos(\phi) )) provides insight into efficiency and real power consumption.

Three-Phase Supply

  • Designed for efficiency, three-phase systems involve interconnections (Star or Delta) to provide balanced loads and simultaneous power delivery with minimized copper use.

  • Phase Sequence: The order of phase voltages reaching peak values defines the rotation direction and effective operation within a three-phase generator.