The Use of Vectors

Abstract

A brief reference was made (Section 4.5) to the fact that there is a special technique for manipulating quantities which vary sinusoidally with time. It makes use of the concept that such a quantity may be completely represented by a vector rotating at a constant speed. The diagram (Figure 5.1) shows a line of length V _m rotating at an angular speed ω radians per second. If time t is measured from the datum X′X the projection of the line on to the vertical axis Y′Y at any instant will be V _m sin ωt. If V _m represents the maximum amplitude of a voltage wave it follows that the value of voltage at any instant will be completely defined by the position of the line. The whole wave-shape may be developed from a sequence of lines spaced by equal time intervals (Figure 5.2). Developing this idea further, let us consider two sine waves displaced in time

$$\begin{array}{*{20}{l}} {}&{{v_1}}& = &{{v_{1m}}\sin \omega t}& = &{{V_{1m}}{\text{Sin}}\,\beta } \\ {{\text{and}}}&{{v_2}}& = &{{v_{2m}}\sin \omega (t - {t_0})}& = &{{V_{2m}}{\text{Sin}}\,{\text{(}}\beta - \alpha ).} \end{array}$$

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