In one line
These are the basic terms that describe a repeating AC waveform: how big it gets (amplitude), how long one repetition takes (time period), how many repetitions per second (frequency), and its value at any moment (instantaneous value).
Explained simply
Think of a child on a swing. One full forward-and-back swing is a cycle. The time for one swing is the period. How many swings happen per second is the frequency. The highest point the swing reaches from the centre is the amplitude. The swing's position at any chosen moment is the instantaneous value.
Technically, for \( v = V_m\sin(\omega t + \phi) \): \( V_m \) is the amplitude (peak value), \( \omega \) is the angular frequency, \( \phi \) is the phase angle, and \( v \) is the instantaneous value at time \( t \).
How it works
- Waveform: graph of the quantity against time or angle.
- Cycle: one complete set of positive and negative values (360° or \( 2\pi \) rad).
- Time period (T): time taken for one cycle, in seconds.
- Frequency (f): number of cycles per second, in hertz (Hz). \( f = 1/T \).
- Amplitude / peak / maximum value (\( V_m \)): largest value reached in either half-cycle.
- Instantaneous value (v): value at a particular instant, shown in lower-case letters.
- Angular frequency (\( \omega \)): \( 2\pi f \) rad/s.
- Peak-to-peak value: \( 2V_m \) for a symmetrical wave.
Key formula
\( f = \dfrac{1}{T}, \qquad \omega = 2\pi f, \qquad v = V_m \sin \omega t \)
\( f \) in hertz (Hz), \( T \) in seconds (s), \( \omega \) in rad/s, \( V_m \) and \( v \) in volts (V). For an alternator, \( f = \dfrac{PN}{120} \), where \( P \) = number of poles and \( N \) = speed in rpm.
Quick example
Given \( v = 325\sin 314t \) V: amplitude \( V_m = 325 \) V; \( \omega = 314 \) rad/s, so \( f = 314/(2\pi) = 50 \) Hz; \( T = 1/50 = 20 \) ms. At \( t = 5 \) ms, \( v = 325\sin(314\times0.005) = 325\sin(1.57\text{ rad}) = 325 \) V, the positive peak. This is the Indian domestic supply (230 V RMS, 50 Hz).
Common mistakes
- Putting \( \omega t \) in degrees into a calculator set to radians (or vice versa).
- Confusing amplitude with peak-to-peak value.
- Writing frequency in rad/s; frequency is in Hz, angular frequency is in rad/s.
How MU asks it
Usually a 3–5 mark short: "Define cycle, frequency, time period, amplitude, instantaneous value" or "For \( v = 141.4\sin(314t + 30^\circ) \) find the peak value, frequency, time period and RMS value." For full marks, give each definition with its symbol and unit, sketch a labelled sine wave, and show substitutions clearly.