CBSE Class 12-science Answered

The diagram of AC generator:

Basic parts of the AC generator:

i. Rectangular coil mounted on a rotor shaft. The coil also called armature is mechanically rotated in the uniform magnetic field by some external means.

ii. The axis of rotation of the coil is perpendicular to the direction of the magnetic field. The rotation of the coil causes the magnetic flux through it to change, so an emf is induced in the coil.

iii. The ends of the coil are connected to an external circuit by means of slip rings and brushes.

iv. N and S are two permanent magnets which provide a constant magnetic field region in which the coil rotates.

Principle of working:

A wire loop of area A is free to rotate about an axis which is perpendicular to a uniform magnetic field B. If the normal to the loop makes an angle with then, flux through the loop .

If this loop rotates with a constant angular velocity

The flux through it changes at the rate,

where C_o is a constant

emf is induced between ends A and B given by:

, here Peak value of emf generated.

Since the emf induced in the coil is varying as a function of sine, it is alternating in value and direction

Or

If I is the instantaneous current through LCR circuit, the instantaneous voltage across the resistor is, across the inductor is and across the capacitor is .

From conservation of energy, when all of them are connected in series, the total work required is just the sum of these.

So,

The current will have the same frequency as V, but in general will not be in phase with V.

The current can be written as

with the peak valueand the relative phase fixed by equation.

Hence the voltage equations across each of the components will be

where with .

In the phasor diagram, the external source of EMF is (peak value), which is the x component of the phasor, this vector will be along the X axis.

The current phasorwill be at an anglerelative to this.

So, since, it will be parallel to this current phasor I, at anglerelative to.

will make an angleandwill make anglerelative to.

This also implies thatandwill lie in opposite directions, as the following figure shows.

The vectoris perpendicular to the vector, and also, of course,

Taking the dot product

gives

From this, we get

The factor Z is the analog of resistance in a purely resistive circuit, and is called Impedance.

The phase angle is found as

Condition for resonance:

The amplitude or peak value of the current through the circuit will be

The expression clearly shows thatis maximum if the term inside the bracket in the denominator is zero, since then the denominator is minimum and the impedance is completely resistive.

This means that the current is maximum at frequency, such that

The maximum current is, as if the reactive elements are not there at all!

This phenomenon, in which the peak value of current shoots up at this particular frequency is called 'resonance'.

The condition for resonance is hence:

Power factor:

Average power dissipated per cycle, is given by

The factor is called the 'Power Factor', since the power crucially depends on it. This factor depends on the applied frequency, apart from circuit parameters like L,C and R.

At resonance,, so that. This implies that, or that the power factor attains its maximum value, equal to unity.

For the (ii) capacitive and (iii) inductive circuits, the current and voltage differ in phase by, so either way, it is minimum, ie., zero.