Induced Emf Due to Rotation of Conductor

Alternating Current

On rotating a rectangular current loop in a magnetic field at constant angular frequency ω a sinusoidal voltage (emf) is induced in the loop. This instantaneous voltage is

V = V_M sin ωt

V₀ is the maximum output voltage of the ac generator, ω is the angular frequency, ω = 2pf (f = 50 Hz). Time period is

A sinusoidal current can be described as,

i = i_M sin ωt

The average value of current for one cycle is,

Remember, as the averages for whole cycle is zero, the dc instrument will indicate zero deflection. So, average is taken over half the cycle.

Similarly,

A dc meter can be used in an ac circuit if it is connected in the full wave rectifier circuit showing same value.

An important way to represent current is through its rms (root mean square) value.

Thus,

Similarly,

Power of 220V is supplied in our houses,

which is the rms value. Corresponding maximum value V_m = 311 volt

Form Factor

The ratio, is known as or factor,

Note:

In a positive cycle:

• The average value of sin ωt, cos ωt, sin2ωt, cos2ωt, etc. is zero .
• The average value of sin² ωt and cos² ωt is 1/2

Example:

If the current in an ac circuit is represented by the equation,

i = 5 sin (300t - ∏/4)

Here, t is in second and i in ampere. Calculate:

1. peak and rms value of current.
2. frequency of ac
3. average current

Solution:

1. An in case of ac,
   i = i₀ sin (ωt ±Φ)
   ∴ The peak value i₀ = 5A
   and
   
2. Angular frequency ω = 300 rad/s