A current (or voltage) is called *alternating* if it reverses periodically in direction, and its magnitude undergoes a definite cycle of changes in definite intervals of time. Each cycle of alternating current (or voltage) consists of two half cycles, during one of which the current (or Voltage) acts in one direction; while during the other in opposite direction.

In more restricted sense, alternating current is a periodically varying current, the average value of which, over a period, is zero. The direct current always flows in one direction, and its magnitude remains unaltered. In order to produce an alternating current through an electric circuit, a source capable of reversing the emf periodically (ac generator) is required while for generating dc in an electric circuit, a source capable of developing a constant emf is required such as a battery or dc generator. The graphical representations of alternating current and direct current are given in Figs. 1(a) and (b) respectively.

At present a large percentage of the electrical energy (nearly all) being used for domestic and commercial purposes is generated as alternating current. In fact, almost the whole of the vast amount of electrical energy used throughout the world for every imaginable purpose is generated by alternating current generators. This is not due to any superiority of alternating current over direct current in the sphere of applicability to industrial and domestic use. In fact, there are certain types of works for which alternating current is unsuitable and, therefore, direct current is absolutely necessary such as for electroplating, charging of storage batteries, refining of copper, refining of aluminum, electrotyping, production of industrial gases by electrolysis, municipal traction etc. In some power applications, the ac motor is unsatisfactory such as for metal rolling mills, paper making machines, high speed gearless elevators, automatic machine tools and high-speed printing presses. Direct current required for these applications is nowadays derived from an ac supply by the use of suitable convertors or rectifiers. For lighting and heating dc and ac are equally useful. The reasons for generation of electrical energy in the form of alternating current are given below:

- AC generators have no commutator and can, therefore, be built in very large units to run at high speeds producing high voltages (as high as 11,000 volts), so that the construction and operating cost per kW is low, whereas dc generator capacities and voltages are limited to comparatively low values.
- Alternating current can be generated at comparatively high voltages and can be raised and lowered readily by a static machine called the transformer which makes the transmission and distribution of electrical energy economical. In direct current use of transformers is not possible.
- AC induction motor is cheaper in initial cost and in maintenance since it has got no commutator and is more efficient than dc motor for constant speed work, so it is desirable to generate power as alternating current.
- The high transmission efficiency in ac makes the generation of electrical energy economical by generating it in large quantities in a single station and distributing over a large territory.
- The switchgear (e.g. switches, circuit breakers etc.) for ac system is simpler than that required in a dc system.
- The maintenance cost of equipment is less.

## GENERATION OF ALTERNATING EMF

We know that an alternating emf can be generated either by rotating a coil within a stationary magnetic field, as illustrated in Fig. 2 (a) or by rotating a magnetic field within a stationary coil, as illustrated in Fig. 2 (b). The emf generated, in either case, will be of sinusoidal waveform. The magnitude of emf generated in the coil depends upon the number of turns on the coil, the strength of magnetic field and the speed at which the coil or magnetic field rotates. The former method is employed in case of small ac generators while the later one is employed for large sized ac generators.

Now consider a rectangular coil of N turns rotating in counter-clockwise direction with angular velocity of ω radians per second in a uniform magnetic field, as illustrated in Fig. 3.

Let the time be measured from the instant of coincidence of the plane of the coil with the X-axis. At this instant maximum flux, Φ_{max} links with the coil. Let the coil assume the position, as shown in Fig. 3, after moving in counter-clockwise direction for *t* seconds.

The angle θ through which the coil has rotated in *t* seconds = *ωt*

In this position,

the component of flux along perpendicular to the plane of coil = Φ_{max }cos* ωt*.

Hence flux linkages of the coil at this instant = Number of turns on coil × linking flux

i.e; instantaneous flux linkages = N Φ_{max }cos* ωt*.

Since emf induced in a coil is equal to the rate of change of flux linkages with minus sign,

EMF induced at any instant,

……(1)

When *ωt* = 0, sin* ωt* = 0, therefore, induced emf is maximum, which is denoted by E_{max} and is equal to Φ_{max }N* ω*

Substituting Φ_{max }N* ω* = E_{max} in Eq. (1) we have

Instantaneous emf, …….(2)

So, the emf induced varies as the sine function of the time angle *ωt*, and if emf induced is plotted against time, a curve of sine wave shape is obtained as illustrated in Fig. 4. Such an emf is called the *sinusoidal* emf. The sine curve is completed when the coil rotates through an angle of radians. The induced emf e will have maximum value, represented by E_{max}, when the coil has turned through radians (or 90^{0}) in counterclockwise direction from the reference axis (i.e. OX axis).

**Example 1: A coil having 200 turns and area of x-section 250 cm ^{2} is rotated about its axis at right angle to a uniform magnetic field of strength 0.5T at a speed of 1,200 rpm. Determine**

*a). maximum value of emf induced*

*b). Equation of instantaneous induced emf*

*c). Instantaneous values of induced emf when*

*(i). The plane of the coil is at right angle to the field.*

*(ii). The plane of the coil is parallel to the field and*

*(iii). The plane of the coil is at an angle of 60 ^{0} to the field.*

**Solution:** Angular velocity,

Maximum flux linking with the coil,

* a).* Maximum value of induced emf,

* b).* Equation for instantaneous induced emf,

* c).* Instantaneous value of induced emf when

* (i).* the plane of the coil is at right to the field

i.e. when

* (ii).* The plane of the coil is parallel to the field,

i.e. when

.

* (iii).* The plane of the coil makes an angle of 60

^{0}to the field

i.e. when

.

**Example 2: A square coil of 100 turns is rotated at a uniform speed of 1,000 revolutions per minutes, about an axis at right angle to a uniform magnetic field of 0.5 Wb/m ^{2}. Calculate the instantaneous value of the induced electromotive force, when the plane of the coil is**

*At right angle to the field**In the plane of the field.*

**Solution:** Angular velocity,

Maximum flux linking with the coil,

Number of turns on the coil, N = 100

Maximum value of emf induced,

Instantaneous value of induced emf when the plane of the coil is,

- At right angle to field i.e. when

- In the plane of the field

i.e. When

## SINUSOIDAL QUANTITIES (EMF, VOLTAGE OR CURRENT)

It is not an accident that the bulk of electric power generated in electric power stations throughout the world and distributed to the consumers appears in the form of sinusoidal variations of voltage and current. There are many technical and economical advantages associated with the use of sinusoidal voltages and currents. For example, it will be learned that the use of sinusoidal voltages applied to appropriately designed coils results in a revolving magnetic field which has the capacity to do work. As a matter of fact, it is this principle which underlies the operation of almost all the electric motors found in home appliances and about 90% of all electric motors found in commercial and industrial applications. Although other waveforms can be used in such devices, none leads to an operation which is as efficient and economical as that achieved through the use of sinusoidal quantities.

The other advantages of using sinusoidal voltages and Currents are:

- The waveform from generation to utilization remains the same if a sinusoidal waveform is generated.
- Electromagnetic torque developed in three phase machines (generators and motors) with balanced three phase currents is uniform (constant), and therefore, there are no oscillations in developed torque and absence of noise in operation.
- Non-sinusoidal voltages which contain harmonic frequencies, according to Fourier analysis, are harmful to the system on account of
- increased losses in generators, motors, transformers, and transmission and distribution systems,
- more interference (noise) to nearby communication circuits,

- resonance may result in over-voltages or over-currents at many pockets on the way from generating station to consumer’s premises which may damage the equipment and increase losses.
- increased current through power factor improvement capacitors.

In practical electrical engineering, it is assumed that the alternating voltages and currents are sinusoidal, though they may slightly deviate from it. The advantage of this assumption is that calculations become simple. *It may be noted that alternating voltage and current mean sinusoidal voltage and current unless stated otherwise.*

Alternating emf following sine law (i.e. sinusoidal emf) is illustrated in Fig. 4 and is expressed in the form

…….(3)

where e, is the instantaneous value of alternating emf (or voltage), E_{max} is the maximum value of the alternating emf (or voltage) and *ω* is angular velocity of the coil.

The rotating coil moves through an angle of radians in one cycle, so angular velocity where f is the number of cycles completed per second.

Substituting in Eq. (3) we have

……(4)

If the alternating emf (or voltage) given by Eq. (3) is applied across a load, alternating current flows through the circuit which would also vary sinusoidal i.e. following a sine law. The expression for alternating current is given as

…..(5)

provided the load is pure resistive (The load may be resistive, inductive or capacitive. When load is inductive or capacitive the current equation differs in the time angle).