20 ALTERNATING CURRENTS 



of reference in our vector diagram ; let this p.d. vector be laid off 

 horizontally as in Fig. 14. 



The value of the joint impedance being the same for all values of 

 the p.d., we may, for the sake of convenience, assume the p.d. to 

 have a value of unity. Since 



current = = - = p.d. X . 



impedance impedance 



we see that, on the assumption of unit p.d., the currents in the 

 various branches are given by the reciprocals of the impedances. The 



reciprocal of the impedance of a 

 circuit is termed its admittance. 

 Let AI, A 2 , A 3 . . . be the admit- 

 tances corresponding to the various 



branches, so that AI = ^-, A 2 



\ 



= ^= A 3 = -^ . . . ; then, with unit 



A, ^3 



p.d. across the terminals, the cur- 

 rents in the various branches are 

 equal to AI, A 2 , A 8 . . . . 



An admittance, like an impe- 

 dance, is a directed or vector quan- 

 tity, so that it is not completely 

 FIG. 14. Vector Diagram of Currents determined unless in addition to 

 in Branched Circuit. itg magn itude we are also given 



its phase angle. 



The phase angles of the admittances AI, A 2 , A 3 . . . are equal 

 in magnitude to B\, 2 , O s . 



Let us now, in our vector diagram, lay off vectors OIi, OI 2 , OI 3 . . . 

 (Fig. 14) of lengths AI, A 2 , A 3 . . . making angles 0i, 2 , 3 . . . 

 with the p.d. vector. These vectors will be the current vectors for 

 the various branches, and since the total current between the two 

 points is at any instant equal to the sum of the instantaneous currents 

 in the various branches, it is clear that the vector of total current is 

 obtained by compounding the vectors according to the polygon law, 

 as in Fig. 15, the closing side 01 of the polygon giving the vector of 

 total current. Since, however, the p.d. was assumed to have a value 

 of unity, it follows that 01 represents the joint admittance of the 

 various branches, and its phase angle is the angle which it makes 

 with the p.d. vector. 



The joint impedance is at once obtained by taking the reciprocal 

 of the joint admittance. 



With a parallel arrangement of impedances, therefore, the admit- 

 tances of the various branches are compounded according to the same 



