CHAPTER V 



THE USE OF COMPLEX QUANTITY 



41. Methods in alternating currents. The current-voltage relations in any net- 

 work of circuits, or in any generator, motor, or transformer, may be completely 

 represented by the geometric relations of the lines in a clock diagram, on the assump- 

 tion that all voltages and currents are harmonic. Therefore any problem in alternating 

 currents may be solved in either of two ways, namely, (a] Graphically, by drawing 

 the clock diagram carefully, so as to represent the given data to scale, and measuring 

 off the desired result, or (Z>) Trigonometrically, by establishing the trigonometric 

 equations which express the conditions represented in the clock diagram, and solving 

 these equations so as to express the desired results in terms of the given data. 



The difficulty with the trigonometric method is that the equations are often very 

 complicated and the necessary transformations are often very tedious. This difficulty 



X Axis of reference 



is greatly reduced by the use of complex quantity, inasmuch as in this case the initial 

 equations for the solution of a problem can be written out with the utmost ease, 

 especially if a rough sketch of the clock diagram is used as a guide, and all trigono- 

 metric transformations are greatly simplified. It is to be remembered, however, that 

 the use of complex quantity in alternating-current problems does not constitute a 

 distinct method ; it is merely a systematic scheme for carrying out method (<5) above. 



42. The use of complex quantity in alternating-current problems involves two 

 distinct ideas, namely, (a) The specification of a vector in the clock diagram by giv- 

 ing its components in the directions of chosen axes of reference, and (b) The idea of 

 impedance as an algebraic operator. 



(a) Specification of current and electromotive force -vectors by giving their com- 

 ponents. Consider the clock diagram Fig. 83, in which an axis of reference is 

 indicated. This axis of reference is supposed to rotate with E and / so that E 



88 



