Use of Vectors

Author: E.E. Kimberly

The action of a sinusoidal voltage applied to a circuit can best be analyzed by means of vectors. All physical quantities are either scalars or vectors. A quantity that can be described by one attribute only, such as weight, power, energy, length, emf, or current, is a scalar quantity. A directed quantity which cannot be described by less than two attributes, such as force, velocity, or acceleration, is a vector quantity. A vector quantity has magnitude and direction.

Fig. 4-1. Generation of Sinusoidal emf

Vectors were first used to define space relationships; when so used they may be called space vectors. The same devices may also be used to define time relationships of two or more events; when so used they are called time vectors. It has been recently proposed to call a time vector by the term phasor in order to retain the term vector for its original meaning applying to space only. The term vector is so firmly entrenched in the language of engineering, however, that no confusion seems to arise from its use in a time sense. Therefore, the term vector will be used in this book and will be construed to mean a time vector unless otherwise indicated.

In Fig. 2-2 it was shown that the emf generated in a conductor moving with constant velocity in a circular path in a uniform magnetic field may be represented by a sinusoid. In Fig. 4-1 the conductor a is shown located at the end of a radius space vector Oa revolving counter-clockwise with constant velocity about the origin 0 of a rectangular-coordinate system with axes Ox and Oy. With the x-axis as reference, the generated emf in a is a sine function of the angle θ₁ by which the space vector Oa is displaced from that axis. Thus,

The sinusoidally varying emf E_a in the conductor a is similar to the sinusoidally varying projection On of the space vector Oa on the y-axis. Thus, the sinusoid marked ea is a plot of Oa sin θ₁. Hence, although emf is a scalar quantity, it may be treated as a vector quantity. Similarly, current may be treated as a vector and hereafter will be so used.