298 



THE POLYPHASE CIRCUIT. 



the third is to draw curves representing the simul- 

 taneous values of the currents. This has been done in 

 Fig. 145. Taking at random any instant represented on 

 the curve it will be seen that the algebraic sum of the 

 ordiuates of two curves is equal to the height of the 

 ordinate of the third, and is opposite to it in sign. 



The mathematical proof of the proposition is simple 

 when the quantities are represented by vectors, as in 

 Fig. 146, where C 1} C 2 , C 3 show the three currents. 



The angles referred to are plainly marked. C is the 

 common maximum value of the currents. 



Ni x 



FIG. 146. 



The instantaneous values of the currents are obtained 

 by horizontal projection. It should be remembered that 

 the cosine of an angle is equal to the sine of its complement. 



(1) ON l =C cos < = C sin 6. 



(2) ON 2 =C cos (120 -<). 



=C (cos 120 cos $ + sin 120 sin <). 



=<7 cos 120 sin 0+ (7 sin 120 cos .. (A) 



(3) ON S =C cos (120+ <). 



=C (cos 120 cos < sin 120 sin <). 

 =C cos 120 sin 0- C sin 120 cos 6 . . (B) 

 Therefore, from equations (A) and (B) above we get : 

 N, + O N 3 =2 C cos 120 sin B. 



= - 2 C sin 6 = - C sin = - O N,. 



Evidently, also, this equality is independent of the 

 value of and is true throughout the cycle. 



