# Alternating Current Generator: its principle, construction, and working also derived expression for induced emf

## Alternating Current Generator

It is a device that converts mechanical energy into electrical energy and generates alternating current is called an alternating current generator.

### Principle

It principle is based on Faraday’s law, when a coil rotates by any means in the magnetic field, the magnetic flux through the coil changes and induced emf in coil.

### Construction

A rectangular loop abacda of aera A placed in a uniform magnetic field B. The lop is rotated abut the Z axis through its center at constant angular speed W. One of the loop is attached to a metal ring and the other to ring A. These rings are concentric with the axis of the loop and rotate with it and are called sleep rings. Rings A, ́A slide against station ary carbon bushes B,B to which the external circuit is connected. This is called the armature.

Alternating Current Generator

### Working

the figure graph shows the current at different positions during the one rotation of the loop of the coil.
When the angle between V and B θ=0°

Perpendicular to B, the current is zero. As θ increases, the current increases, and at θ =90° =π/2 rad, the loop is parallel to B, current is maximum directed along abcda. On further increase in θ current decreases and at θ =180 =π rad the current becomes zero as the loop is again perpendicular to B. For 180°< θ<270° current increases but reverses its direction and directly tends along dcbad. At θ=270°=3π/2 rad, the current is maximum in the reverse direction as the loop is parallel to B. At θ=360°=2π rad, one rotation is completed, the loop is perpendicular to B, and the current decreases to zero. In this way, the cycle repeats itself. The current alternates in direction once in one cycle. Such a current is called alternating current which reverses in the direction f time per second.

### Expression for induced emf

consider the position of the loop rotating anti-clockwise direction as shown in the figure. The vertical side ab of the loop is moving with velocity v in the magnetic field b, if the angle between, the motional emf induced in the side ab has the magnitude,

εab =VBLSinθ

The direction of induced current in the wire ab is the same as that of force F on ve change from top to bottom. The same amount of emf is induced in the side cd but the direction of current is from the bottom to the top

εcd=VBLSinθ

The net contribution to emf by sides be and da is zero because the force acting on the charge inside bc and da is not along the wire.

εbc= εda=0

Both the emf in the sides ab and cd drive current in the same direction around the loop. Then the total emf in the loop will be

ε= εab+ εcd

ε=VBL Sinθ+VBL Sinθ

ε=2VBLSinθ →eq1

If the loop is replaced by a coil of N turn, the total emf in the coil will be,

ε=2NVBLSinθ→eq2

we know that v=rw putting values in eq2

ε=2N (rw)BL Sin θ

When r is the distance of the vertical wires from the center of the coil. From about equ.

ε=Nw (2rL)BSinθ

where 2rL=A=Area of the coil

ε=NwAB Sinθ→eq3

putting θ=wt and it gives induced emf varies sinusoidally with the time

ε=NwAB Sin wt→eq4

It has a maximum value when θ=wt=90° and Sin90°=1

εₒ=Nw AB (1)=Nw AB →eq5

eq 4 becomes

ε= εₒSin wt →eq 6

If R is the resistance of the coil, then by Ohm’s law, induce of current in the coil will be,

I=ε/R=εₒSin wt/R=εₒ/R Sin wt

I=Iₒ Sin ( wt)→eq 7

we know that w=2πf then ε=εₒSin (2πft)→eq8

and I=Iₒ Sin (2πft)→eq 9

Equations 8 and 9 show that ε and I are functions of θ=2πft.