THE USE OF COMPLEX QUANTITY. 



93 



and these expressions satisfy the fundamental relations : current is equal to 

 electromotive force divided by impedance, and electromotive force is equal to 

 current multiplied by impedance. On the other hand, the product of current 

 by voltage gives S3^~ Mt ~ ^ which is a vector which rotates at the angular 

 velocity 2. Evidently the impedance vector which is stationary cannot be 

 properly shown in the clock diagram, inasmuch as the clock diagram is sup- 

 posed to rotate at angular velocity , and the vector Sc/^^ 2<at ~ e ^ cannot be prop- 

 erly shown in the clock diagram because it rotates at angular velocity 2w. 

 The use of complex quantity does not lead to any graphical representation of 

 power because the vector & cannot be interpreted as a representa- 



tion of power. Products of current by current and of current by voltage are 

 foreign to the clock diagram, and these products as physical facts do not enter 

 into the complex quantity method of representation. 



44- Coils in series. Two coils are connected in series between alternating cur- 

 rent mains as shown in Fig. 85. The total electromotive force E, the resistances 



tnnin 



o 

 o 



#,o 

 o 

 o 



; 



--J- 



main 



Fig. 85. 



Fig. 86. 



and ^ 2 of the respective coils are given, and it is 

 required to find E l and E v Let / be the current flowing through the two coils. 

 The general relation between , E v E v and / is shown in the clock diagram 

 Fig. 86. This clock diagram is given in order that the following equations (i), (ii) 

 and (iii) may be easily understood. According to equation (15) Art. 42 we have : 



and E 2 (r 2 -f/r 2 )7 (ii) 



Furthermore, the electromotive force E is the vector sum of l and E^ as shown 

 in Fig. 86 so that, remembering that E, E v and 2 are complex quantities, we have : 



= (iii) 



