ELECTRIC CIRCUITS 



137 



A current multiplied by an impedance gives an e.m.f. displaced 

 from it in phase by an angle whose tangent is the ratio of the re- 

 actance to the resistance. 



A current divided by an admittance gives an e.m.f. displaced 

 in phase by an angle whose tangent is the ratio of the susceptance 

 to the conductance. 



Similarly an e.m.f. divided by an impedance or multiplied by 

 an admittance gives a current. 



By definition a vector multiplied by j is turned through 90 

 degrees in the counter-clockwise direction; when multiplied by 

 j 2 it is turned through 180 degrees and its sign is reversed. 



Therefore 



and 



3 = v- 



(195) 



Taking this value for j alternating quantities expressed in rect- 

 angular coordinates referred to a given axis can be added, sub- 

 tracted, multiplied and divided and the results obtained are 

 expressed in rectangular coordinates referred to the same axis. 



FIG. 104. Power in rectangular coordinates. 



It is not necessary to choose either the current in a circuit or 

 the e.m.f. as axis and any other line may be taken as shown in 

 Fig. 104, but the e.m.f must now be expressed as the sum of two 

 components along and perpendicular to the new axis, 



E = 



and similarly the current is 



I = ii + jit- - 



