Equation of sinusoidally varying emf and current (sinusoidal functions)

EQUATION OF SINUSOIDALLY VARYING EMF AND CURRENT (SINUSOIDAL FUNCTIONS)

Consider a rectangular coil with N turns. It is rotated in an uniform magnetic field in anticlockwise direction.

Let B = flux density of the field in weber/m² (or) tesla

ω = angular velocity of the coil in radians/sec.

The time is measured from the positive x-axis. After 't' seconds of rotation, the coil is rotated through an angle θ = ωt.

In this position, resolve the flux into two components. One component is perpendicular to the plane of the coil. The other one is parallel to the plane of the coil. The component of flux at right angles to the plane of the coil is

ϕ = ϕm cos θ

= ϕm cos ωt

Only this flux links with the coil. The instantaneous emf induced

When sin ot is maximum (=1), then e is maximum and is denoted by E_m

i.e., E_m = ω Nϕ_m … (2)

Therefore, the above equation for e becomes

e = Em sin ωt  = Em sin θ ... (3)

The above equation is called the standard sinusoidal equation of the voltage. The value of e depends upon sin θ.

The above equation can be graphically represented as below. It is called voltage wave form.

Note:

1. The magnitude of maximum emf depends upon (a) angular velocity (b) number of turns and (c) maximum flux. It does not depend upon the angle.

2. The value of instantaneous emf e depends upon 0. In other words it depends upon the time for a given speed (since θ = wt).

3. The standard equation of sinusoidally varying voltage is, instantaneous voltage = maximum voltage x sin θ.

4. The standard form for a sinusoidally varying current is

i = Im × sin θ