KENNELLY. — OSCILLATING-CURRENT CIRCUITS. 399 



where A and B are integration constants, while -i- and w follow from 

 the construction of the triangle OPQ, Figure 7. 



Choosing the constants consistently with the discharge of the con- 

 denser initially charged to potential U^ volts, the potential at time t is 



u = Uq cosec (fi €-"' sin (wt + 4>) volts (27) 



= U^i-^t sin (wt + <!>), volts (28) 



from which q follows by the relation q = u/s = uc coulombs. U^ is the 

 initial value of the vector discharging p. d. as obtained from Figures 

 8 and 12. 



The instantaneous current is 



i = Qqw cosec^ <j> €-^^ sin (ot amperes 



= UoCdiQ cosec 4> e-^^ sin cot amperes 



= -^ e~-" sin wt = Iq€-^^ sin wt, amperes 



where /o = Uq/Iu) = Uo/z^ = U^cwf^. amperes 



The emf. of self-induction in the circuit at any instant is 



e=. U^ cosec ^ e-^^ sin {wt — (f)) volts 



= Uoe-^'t sin {wt — <f>). volts 



The instantaneous power of the condenser in the circuit is 



p = ui = f7(,/Qe-2-'< sin wt • sin (u)t + <^) watts 

 = —^ e-2^' [cos ct> - cos (2 uit + <^)] watts 

 = UIe-2'^t [cos cf} — cos (2 wt + (fy)^. watts 



The instantaneous power of the reactance in the circuit is 



p' = ei = f/o/oe~2-'< gjjj (^f; . sin ^^^f _ <^-) watts 



= —^ e-2^« [cos <^ - cos (2 uit — <t>);\ watts 

 = Ule'^'-^ [cos <^ — cos (2 wt — cjj)]. watts 



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