In this article we will discuss about: 1. Meaning of Alternating Current 2. Generation of Alternating Emf 3. Sinusoidal Quantities (Emf, Voltage or Current) 4. Average and Effective (RMS) Values of Alternating Voltage and Current 5. Average and Effective (RMS) Values of Sinusoidal Current and Voltage 6. Form Factor and Peak Factor of Sinusoidal Wave and Other Details.
Contents:
- Meaning of Alternating Current
- Generation of Alternating Emf
- Sinusoidal Quantities (Emf, Voltage or Current)
- Average and Effective (RMS) Values of Alternating Voltage and Current
- Average and Effective (RMS) Values of Sinusoidal Current and Voltage
- Form Factor and Peak Factor of Sinusoidal Wave
- RMS Value, Average Value, Peak Factor and Form Factor of Half Wave Rectified Alternating Current
- RMS And Average Values of a Triangular Waveform Alternating Current
1. Meaning of Alternating Current:
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.
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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. 3.1(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 aluminium, electrotyping, production of industrial gases by electrolysis, municipal traction etc.
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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:
1. 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.
2. 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.
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3. 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.
4. 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.
5. The switchgear (e.g., switches, circuit breakers etc.) for ac system is simpler than that required in a dc system.
6. The maintenance cost of ac equipment is less.
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2. 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. 3.2 (a) or by rotating a magnetic field within a stationary coil, as illustrated in Fig. 3.2 (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 co radians per second in a uniform magnetic field, as illustrated in Fig. 3.3.
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, ɸmax links with the coil. Let the coil assume the position, as shown in Fig. 3.3, 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 = ɸmax cos ωt.
Hence flux linkages of the coil at this instant = Number of turns on coil × linking flux i.e., instantaneous flux linkages = N ɸmax cos ωt
Since emf induced in a coil is equal to the rate of change of flux linkages with minus sign:
When ωt = 0, sin ωt = 0, therefore, induced emf is zero, when ωt = π/2, sin π/2 =1, therefore, induced emf is maximum, which is denoted by Emax and is equal to ɸmax N ω sin ωt
Substituting ɸmax Nω = Emax in Eq. (3.1) we have:
Instantaneous emf, e = Emax sin ω t … (3.2)
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. 3.4. Such an emf is called the sinusoidal emf. The sine curve is completed when the coil rotates through an angle of 2π radians.
The induced emf e will have maximum value, represented by Emax, when the coil has turned through π/2 radians (or 90°) in counter-clockwise direction from the reference axis (i.e., OX axis).
3. 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 economic 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:
1. The waveform from generation to utilization remains the same if a sinusoidal waveform is generated.
2. 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.
3. Non-sinusoidal voltages which contain harmonic frequencies, according to Fourier analysis, are harmful to the system on account of-
(i) increased losses in generators, motors, transformers, and transmission and distribution systems,
(ii) More interference (noise) to nearby communication circuits,
(iii) 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, and
(iv) 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. 3.4 and is expressed in the form:
e = E max sin ωt … (3.3)
Where e is the instantaneous value of alternating emf (or voltage), Emax 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 2π radians in one cycle, so angular velocity ω = 2πf where f is the number of cycles completed per second.
Substituting ω = 2π f in Eq. (3.3) we have:
e = Emax Sin 2πft … (3.4)
If the alternating emf (or voltage) given by Eq. (3.3) 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:
i = Imax sin ωt = Imax sin 2πft … (3.5)
Provided the load is pure resistive.
Plotting of Sine Waveform:
Sine curve may be graphically drawn, as illustrated in Fig. 3.8. Draw a circle of radius equal to the maximum value of sinusoidal quantity. Divide the circumference of the circle drawn so into any number of equal parts, say 12, and draw a horizontal line AB (the base on which the sine wave is to be drawn) passing through the centre of the circle.
Divide the line AB into the same number of equal parts i.e., 12 and number the points correspondingly. Draw perpendicular ordinates from each point. Project the points on the circle horizontally to meet the perpendicular ordinates having corresponding numbers. Draw smooth curve through these points. Curve so drawn will be of sine waveform.
4. Average and Effective (RMS) Values of Alternating Voltage and Current:
In a dc system, the voltage and current are constant and, therefore, there is no problem in specifying their magnitude. But in case of ac system, an alternating voltage or current varies from instant to instant and so poses a problem how to specify the magnitude of an alternating voltage or current. An alternating voltage or current may possibly be expressed in terms of peak (maximum) value, average (mean) value or effective (rms) value.
In specifying an alternating voltage or current, its peak or maximum value is rarely used because it has that value only twice each cycle. Furthermore, the average or mean value cannot be used because it is positive as much as it is negative, so the average value is zero.
Although the average value over half cycle might be used, it would not be as logical a choice as what we shall find effective (virtual or rms) value which is related to the power developed in a resistance by the alternating voltage or current.
Average Value of Alternating Current:
The average (or mean) value of an alternating current is equal to the value of direct current which transfers across any circuit the same charge as is transferred by that alternating current during a given time.
Since in the case of a symmetrical alternating current (i.e., one whose two half cycles are exactly similar, whether sinusoidal or non-sinusoidal) the average or mean value over a complete cycle is zero hence for such alternating quantities average or mean value means the value determined by taking the average of instantaneous values during half cycle or one alternation only. However, for unsymmetrical alternating current, as half wave rectified current, the average value means the value determined by taking the mean of instantaneous values over the whole cycle.
The average value is determined by measuring the lengths of a number of equidistant ordinates and then taking their mean i.e. of i1, i2, i3 … in etc. which are mid-ordinates.
Using the integral calculus the average (or mean) value of a function f (t) over a specific interval of time between t1 and t2 is given by:
Any function, whose cycle is repeated continuously, irrespective of its wave shape, is termed as periodic function, such as sinusoidal function, and its average value is given by:
Where T is time period of periodic function.
In case of a symmetrical alternating current, whether sinusoidal or non-sinusoidal the average value is determined by taking average of one half cycle or one alternation only.
RMS Value or Effective Value of Alternating Current:
The rms or effective value of an alternating current or voltage is given by that steady current or voltage which when flows or applied to a given resistance for a given time produces the same amount of heat as when the alternating current or voltage is flowing or applied to the same-resistance for the same time.
Consider an alternating current of waveform shown in Fig. 3.10 flowing through a resistor of R ohms. Divide the base of one alternation into n equal parts and let the mid-ordinates be i1, i2, i3 …in. etc.
Now if Ieff is the effective current, then heat produced by this current in time T = I2eff RT joules. By definition these two expressions are equal-
Hence the effective or virtual value of alternating current or voltage is equal to the square root of the mean of the squares of successive ordinates and that is why it is known as root-mean-square (rms) value.
Using the integral calculus the root mean square (rms) or effective value of an alternating quantity over a time period is given by:
5. Average and Effective (RMS) Values of Sinusoidal Current and Voltage:
i. Average Value for Sinusoidal Current or Voltage:
The average value of a sine wave over a complete cycle is zero. Therefore, the half cycle average value is intended.
ii. Effective (RMS) Value for Sinusoidal Current or Voltage:
A sinusoidal alternating current is represented by:
6. Form Factor and Peak Factor of Sinusoidal Wave:
Form Factor:
In certain cases it is convenient to have calculations at first upon the mean value of the emf over half a period, therefore, it becomes essential to have some means of connecting this mean value with the effective or rms value. The knowledge of form factor, which is defined as the ratio of effective value to the average or mean value of periodic wave is, therefore, necessary.
Mathematically form factor is given by the relation:
Peak Factor:
Knowledge of peak factor of an alternating voltage is very essential in connection with determining the dielectric strength since the dielectric stress developed in any insulating material is proportional to the maximum value of the voltage applied to it.
Peak or crest or amplitude factor of a periodic wave is defined as the ratio of maximum or peak to the effective or rms value of the wave:
7. RMS Value, Average Value, Peak Factor and Form Factor of Half Wave Rectified Alternating Current:
Half wave rectified alternating current is one whose one-half cycle has been suppressed i.e. one which flows for half the time during one cycle. It is illustrated in Fig. 3.16 where suppressed half cycle is shown dotted.
For determining rms and average values of such an alternating current summation would be carried over the period for which current actually flows i.e., 0 to π but would be averaged for the whole cycle i.e., from 0 to 2π.
8. RMS And Average Values of a Triangular Waveform Alternating Current:
Let the maximum value of the current be Imax amperes.
Since I = Imax when θ = π.
Hence, expression for the instantaneous current can be written as:
