By Vladimir Karapetoff

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4. , and which reaches its positive maxima at the same instants in which the first curve passes through zero. drawn. Show that with these data two distinct curves can be ^ Prob. 6. , and which lags by 30 degrees with respect to the current in problem 1. Prob. 6. The current mentioned in problem 1 is generated by a At what 12-pole alternator, that in problem 3 by a 14-pole machine. speeds must these machines be driven in order to give the required Ans. m. frequencies? Prob. 7. Express the currents given in problems 1 to 5 by equa= tions of the form of eq.

This proves the proposition, because we see from eq. (44) that i eq is a sine-wave having the same u = 2 irft for the variable angle, hence the same frequency as the component waves. The amplitude and the phase position of this resultant wave are determined by eqs. (46) and (47) , . When two their sum currents or voltages are represented by vectors, or difference is also a vector, because, as proved before, SINE-WAVES AND VECTORS CHAP. ] 39 The problem is also a sine-wave of the same frequency.

FIG. 12. Let From OA and the end B OB Addition of vectors. be the given vectors to be added together. of the vector OA. to the projection of OC upon BC OB draw and Connecting in magnitude and position. It lel C a line equal and paralgives the resultant vector OC, will be seen from the figure that the F-axis is equal to the sum of the projections of OB and BC upon the same axis. But BC is equal and parallel to OA, so that the projection of OC on the vertical axis is equal to the sum of the projections of the given vectors on the same This construction holds true for any instant drawing AC, the parallelogram OBCA is completed, so that the construction is identical with that for finding the resultant of two mechanical forces.