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AExE0103 Alternating-current fundamentals

AC generation and waveform

A coil rotating in a magnetic field experiences changing flux. For an ideal sinusoidal generator,

\[ e(t)=E_p\sin(\omega t+\phi), \]

where \(E_p\) is peak emf, \(\omega=2\pi f\), and \(\phi\) is the phase at \(t=0\). For an ideal \(N\)-turn coil of area \(A\) rotating in uniform flux density \(B\), \(E_p=NBA\omega\).

AC periodically reverses direction. DC is unidirectional; it need not be perfectly constant to remain DC.

Peak, RMS, and average

Peak, RMS, and average values of a sinusoid

For a sine wave with peak \(V_p\):

Quantity Definition Sine-wave value
Peak Maximum magnitude \(V_p\)
Peak-to-peak Positive peak to negative peak \(2V_p\)
RMS \(\sqrt{\frac1T\int_0^T v^2(t)dt}\) \(V_p/\sqrt2=0.707V_p\)
Full-cycle average \(\frac1T\int_0^T v(t)dt\) \(0\)
Rectified/half-cycle average Average magnitude \(2V_p/\pi=0.637V_p\)

RMS is the DC-equivalent heating value for a resistor. Unless a question says otherwise, quoted AC supply voltages are RMS values.

Two easily confused ratios:

\[ \text{peak factor} = \frac{V_p}{V_{rms}}, \qquad \text{form factor} = \frac{V_{rms}}{V_{avg,rect}}. \]

For a sine wave:

\[ \text{peak factor}=\sqrt2\approx1.414, \qquad \text{form factor}=\frac{\pi}{2\sqrt2}\approx1.111. \]

Frequency and phase

\[ T=\frac1f, \qquad \omega=2\pi f, \qquad \theta=\omega t+\phi. \]
  • Frequency \(f\): cycles per second, in hertz (Hz).
  • Period \(T\): time per cycle, in seconds.
  • Phase difference: angular displacement between equal-frequency sinusoids.
  • A positive phase in \(x(t)=X_p\sin(\omega t+\phi)\) means that waveform leads the zero-phase reference under the usual convention.

Nepal's nominal power-system frequency is 50 Hz, so one cycle lasts

\[ T=1/50=20\ \text{ms}. \]

Balanced three-phase system

Three equal-frequency, equal-amplitude sinusoids separated by \(120^\circ\) form a balanced three-phase set:

\[ v_a=V_p\sin\omega t, \]
\[ v_b=V_p\sin(\omega t-120^\circ), \qquad v_c=V_p\sin(\omega t-240^\circ). \]

Their instantaneous sum is zero in a balanced set.

Balanced connection Voltage relation Current relation
Star/wye (Y) \(V_L=\sqrt3V_{ph}\) \(I_L=I_{ph}\)
Delta (\(\Delta\)) \(V_L=V_{ph}\) \(I_L=\sqrt3I_{ph}\)

Do not apply \(V_L=\sqrt3V_{ph}\) to a delta connection.

Balanced three-phase active power:

\[ P=\sqrt3V_LI_L\cos\phi=3V_{ph}I_{ph}\cos\phi. \]

Advantages include nearly constant total instantaneous power for a balanced load, efficient conductor use, and a naturally rotating field for motors. Swapping any two line conductors reverses phase sequence and motor rotation direction.

AC examples

  1. A \(230\ \text{V RMS}\) sine has \(V_p=\sqrt2(230)\approx325\ \text{V}\).
  2. A balanced wye system with \(V_{ph}=230\ \text{V}\) has \(V_L=\sqrt3(230)\approx398\ \text{V}\).
  3. A balanced delta load with \(I_{ph}=10\ \text{A}\) has \(I_L=\sqrt3(10)\approx17.3\ \text{A}\).

AC revision box

  • Full sine average is zero; \(0.637V_p\) is half-cycle/rectified average.
  • RMS sine value is \(0.707V_p\).
  • Peak factor uses peak/RMS; form factor uses RMS/average.
  • Nepal: \(50\ \text{Hz}\), so \(T=20\ \text{ms}\).
  • Wye: line voltage gets \(\sqrt3\); delta: line current gets \(\sqrt3\).