Representation of Alternating Quantities
Representation of Alternating Quantities
It has already been pointed out that an attempt is made to obtain alternating voltages and currents having sine waveform (Representation of Alternating Quantities). In any case, a.c. computations are based on the assumption of sinusoidal voltages and currents. It is, however, cumbersome to continuously handle the instantaneous values in the form of equations of waves like e = Em sin ωt etc.
A conventional method is to employ vector method of representing these sine waves. These vectors may then be manipulated instead of the sine functions to achieve the desired result. In fact, vectors are a shorthand for the representation of alternating quantities like voltages and currents and their use greatly simplifies the problems in a.c. work.
A vector is a physical quantity which has magnitude as well as direction. Such vector quantities are completely known when particulars of their magnitude, direction and the sense in which they act, are given. They are graphically represented by straight lines called vectors. The length of the line represents the magnitude of the alternating quantity, the inclination of the line with respect to some axis of reference gives the direction of that quantity and an arrow-head placed at one end indicates the direction in which that quantity acts.
The alternating voltages and currents are represented by such vectors rotating counter-clockwise with the same frequency as that of the alternating quantity. In Figure (a), OP is such a vector which represents the maximum value of the alternating current and its angle with X axis gives its phase. Let the alternating current be represented by the equation e=Emsin ωt. It will be seen that the projection of OP and Y-axis at any instant gives the instantaneous value of that alternating current.
∴ OM = OP sin ωt or
e = OP sin ωt = Emsin ωt
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It should be noted that a line like OP can be made to represent an alternating voltage or current if it satisfies the following conditions :
(i) Its length should be equal to the peak or maximum value of the sinusoidal alternating current to a suitable scale.
(ii) It should be in the horizontal position at the same instant as the alternating quantity is zero and increasing.
(iii) Its angular velocity should be such that it completes one revolution in the same time as taken by the alternating quantity to complete one cycle.
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