An alternating quantity (voltage or current) is one which changes continuously in magnitude and alternates in direction at regular intervals of time. It rises from zero to maximum positive value, falls to zero, increases to a maximum in the reverse direction and falls back to zero again, as illustrated in Fig. 1. The emf (or voltage) and current repeat the procedure.
The important ac terms are defined below:
The shape of the curve of the voltage or current when plotted against time as abscissa (base) is called the waveform. The waveform of an alternating voltage varying sinusoidal is shown in Fig. 1. The waveform of the induced emf in an alternator differs slightly from that of sine wave but for calculation purposes it is treated as such.
Though efforts are made to generate and maintain sinusoidal waveform of supply voltage, the voltage wave pattern does deviate from sinusoidal variation. This is partly due to the conditions that may exist in ac generators, transformers, and other equipment. However, main distortions are on account of non-linear loads such as electronic and power electronic equipment’s, arc furnaces, arc welding equipment’s and electromagnetic machines operating with saturated magnetic circuits. Problems are further aggravated due to involvement of reactive elements in the path of power flow from mains to load.
A distorted wave pattern is made up of a main or fundamental wave which is the frequency of the circuit and other waves of a higher frequency, called harmonics, which are superimposed on the fundamental Wave. The exact appearance of the distorted wave will depend upon the particular frequency and magnitude of the ac voltage wave(s) superimposed on the fundamental wave. For example, let a harmonic wave of three times the frequency of the fundamental wave be superimposed on the fundamental wave. The resultant distorted wave patterns differ depending on the phase relation between the harmonic wave and the fundamental wave. In Fig. 2 the harmonic wave is shown in reference to the zero-axis and is also shown superimposed on the fundamental wave. Note that the distorted wave pattern of the fundamental wave is different in the two diagrams because the phase relationship of the triple harmonic wave in the two illustrations is different.
Active or passive power line conditioners (or filters) are installed to filter out harmonics or clean the waveform to contain tolerable limit of distortion.
With certain electronic equipment like signal generators it is possible to obtain non-sinusoidal waves like square or rectangular wave, triangular or sawtooth wave.
The value of alternating quantity (emf, voltage or current) at any particular instant is called the instantaneous value and is designated by a small italic letter (e for emf, v for voltage and i for current). The instantaneous values of an alternating quantity can be determined either from the curve or from an equation of the alternating quantity. For example, the instantaneous values of emf represented by the curve shown in Fig. 1 at 0, , and are zero, +Emax, zero and -Emax respectively.
Alternation and Cycle.
When a periodic wave, such as sinusoidal wave, goes through one complete set of positive or negative values, it completes one alternation and when it goes through one complete set of positive and negative values it is said to have completed one cycle.
Alternation and cycle can also be defined in terms of angular measure. One alternation corresponds to 180° (or radians) while one cycle corresponds to 360° (or radians).
Time Period and Frequency.
The time taken in seconds by an alternating quantity to complete one cycle is known as time period or periodic time and is denoted by T.
The number of cycles completed per second by an alternating quantity is known as frequency and is denoted by f. In SI system, the frequency is expressed in hertz (pronounced as hurts). One hertz (or Hz) is equal to one cycle per second. The number of cycles completed per second = f.
Time period, T = Time taken in completing one cycle =
The commercial ac power is generated at frequency of 50 Hz or 60 Hz. The reasons of suitability of frequency of this range are:
- The output of the equipment increases with the increase in frequency. For a given output, smaller size machines are required as compared to those for lower frequency output. Because of high power-weight ratio, relative cost of the equipment is also reduced.
- Lower regulation, lower skin effect resulting in lower ohmic losses, lower magnetic and dielectric losses resulting in higher efficiency, lower corona loss and higher power transmission line capacity as compared to those at higher frequency.
Angular Velocity and Frequency.
A glance at Fig. 1 indicates that each cycle spans radians. Hence, if this quantity is divided by the time period, angular velocity of the sine function is obtained. It is denoted by ω and is expressed in radians per second.
radians per second …….(2)
Electrical Time Degrees and Mechanical Degrees.
It is seen that the coil must revolve past a pair of poles in order to carry the generated emf through one complete cycle. In circuit work, one complete cycle of voltage or current is designated as 360 electrical degrees or electrical radians. To correlate with this designation the arc through which a coil of dynamo must rotate in order to generate one cycle of emf is called 360 electrical degrees. In a 2-pole machine one complete revolution of coil produces one cycle of emf. But in a multipolar machine, such as four, six or eight pole machine, the emf completes one cycle or 360 electrical degrees or electrical radians as Soon as the coil passes a pair of poles and a mechanical degree will be equal to as many electrical degrees as there are pairs of poles in the dynamo structure. In a multipolar machine, the number of cycles completed per second by generated emf.
f = Pair of poles number of revolutions made per second.
where N is the speed of rotation of the coil in rpm.
The maximum value, positive or negative, which an alternating quantity attains during one cycle is called the amplitude of the alternating quantity. The amplitude of an alternating emf (or Voltage) and current is designated by Emax (or Vmax) and Imax respectively.
Example 1. What is the time period of the wave produced by a 6-pole alternator which is driven at 1,000 rpm?
Solution: Number of poles on alternator, P = 6
Speed of alternator, N = 1,000 rpm
Number of cycles completed per second by generated emf.
Time period, Second.
DETERMINATION OF MAXIMUM VALUE AND FREQUENCY FROM EMF OR CURRENT EQUATIONS
From above expression we observe that
- the maximum value ofan alternating emfis given by the coefficient of the sine of the time angle.
- the frequency is given by coefficient of time t divided by 2π.
Similarly, we can also find the maximum value and frequency of the current from the equation of instantaneous values of current.
Example 2: An ac voltage of 50Hz frequency has a peak value of 220 V. Write down the expression for the instantaneous value of this voltage.
Supply frequency, f = 50Hz
Peak value of ac voltage, vmax = 220V
Expression for instantaneous value of ac voltage (assumed sinusoidal) with as zero, is given as
Example 3: An alternating current of frequency 50 Hz has a maximum value of 100 A. Calculate
- its value second after the instant the current is zero and its value decreasing there afterward.
- How many second after the instant the current is zero (increasing there afterward) will the current attain the value of 86.6 A?
Solution: The current waveform is shown in figure 2.
The equation of the alternating current (assumed sinusoidal) with respect to the origin O is
- Since the current is measured from the instant the current is zero and is decreasing there afterward (i.e. from point A in figure. 3), the equation for the alternating current with respect to point A becomes
Substituting second in above equation we get the instantaneous value of current second after the instant the current is zero and decreasing there afterward
2. Let the current attain the value of 86.6 A, t seconds after the zero value of the current. Now substituting i = 86.6 A in Eq. (i) we get
PLOTTING OF SINE WAVEFORM
Sine curve may be graphically drawn, as illustrated in Fig. 13.8. 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.