# Generation and Equations of Alternating Voltages and Currents

Generation of Alternating Voltages and Currents

Alternating voltages may be generated in the following two ways :

1. By rotating a coil in a stationary magnetic field, as shown in Fig. 1.

2. By rotating a magnetic field within a stationary coil, as shown in Fig. 2. The value of the voltage generated in each case depends upon the following factors :

(i) The number of turns in the coils ;

(ii) The strength of the field ;

(iii) The speed at which the coil or magnetic field rotates.

• Out of the above two methods the rotating-field method is mostly used in practice.

Equations of Alternating Voltages and Currents

Fig. 3 shows a rectangular coil of N turns rotating clockwise with n angular velocity w radians per second in a uniform magnetic field.

Since by Faraday’s law, the voltage is proportional to the rate at which the conductor its across the magnetic field or to the rate of change of flux linkages, the shape of the wave of voltage applied to the external circuit will be determined by the flux distribution in the air gap. For a uniform field between the poles it is evident that maximum flux will link with the coil when its plane is in vertical position i.e., perpendicular to the direction of flux between the poles.

Also it is obvious that when the plane of coil is horizontal no flux will link with the coil. If the position of the coil with reference to the vertical axis be denoted bye the flux linking with the coil at any instant, as the coil rotates may be determined from the relation,

ϕ = ϕ max cos θ

= ϕ max cos wt

where,    ϕ max = Maximum flux which can link with the coil, and

t = Time taken by the coil to move through an angle e from vertical position.

Using Faraday’s -law to eqn. (i), in order to determine the voltage equation,

e = N d ϕ / d t (where e is the instantaneous value of the induced e.m.f.)

= – N d / dt ( ϕ max cos wt ) wN  ϕ max sin wt

e = wN ϕ  max sin θ

As the value of e will be maximum when sin 9 = 1,

E max =wN ϕ max

The eqn. (ii) can be written in simpler form as

e = E max sin θ

Similarly the equation of induced alternating current (instantaneous value) is

i =I max sin θ                          (if the load is resistive)

Waveforms. A waveform (or wave-shape) is the shape of a curve obtained by plotting the instantaneous values of voltage or current as ordinate against time as abscissa.

Fig. 4 (a, b, c, d, e) shows irregular waveforms, but each cycle of current/voltage is an exactly replica of the previous one. Alternating e.m.fs and currents produced by machines usually both have positive and negative half waves, the same shape as shown. Fig. 4(/) represents a sine wave of A.C. This is the simplest possible waveform, and alternators are designed to give as nearly as possible a sine wave of e.m.f. • In general, an alternating current or voltage is one the circuit direction of which reverses at regularly recurring intervals.

• The waves deviating from the standard sine wave are termed as distorted waves.

• Complex waves are those which depart from the ideal sinusoidal form. All alternating complex waves, which are periodic and have equal positive and negative half cycles can be shown to be made up of a number of pure sine waves, having different frequencies but all these frequencies are integral multiples of that of the lowest alternating wave, called the fundamental (or first harmonic). These waves of higher frequencies are called harmonics.  