Method of Measuring Energy dissipated in Condensers. 33 



and its energy Fig. 10. 



W=iOB 1 2 sin 2 0. r rs 



The expression for the power is 



JliEj cos (j>, 



But , ^^ . , 



I 1 = CE 1 sm</>.p. 



Hence the energy dissipated per second is 



•JCEi 2 sin (j> cos (j> .p, 



and the energy dissipated per half-period (that 

 is during the time of a single charge and dis- 

 charge) is 



zy = iCE 1 2 sin <j) cos cj> . it. 



The relative loss is therefore 



^ = 7TCOt(/>, 



and the net efficiency is 



6=1 — TTCOt (f>. 



For 



= 89°, 6 = 94-52%, 



= 88°, e=89-03 / , 



<£ = 87°, 6 = 83-54 0/0, 

 = 72° 20' 30", e=0. 



The net efficiency, e, is therefore slightly less than the 

 gross efficiency, rj, for values of cf> nearly 90° ; but, as the 

 angle <\) diminishes, e falls rapidly below rj, and for 

 = 72° 20' 30" the energy dissipated is equal to the energy 

 stored,' and the net efficiency is therefore zero, while the 

 gross efficiency is about 38 per cent. For greater angles of 

 lag the loss is greater than the maximum energy stored, and 

 e becomes negative (see fig. 9). 



For ordinary cases the angle <j) is greater than 88°, and e 

 and 7/ are nearly equal. Since the wattmeter method gives 

 directly the value of (f> it is much easier to express the value 

 of the net efficiency e (namely, 1 — 7r cot <£) than the value 

 of 7]. For small values of cot <£ this is sufficiently exact to 

 write it e = 1 — it cos (j>. 



Suppose that instead of assuming the effective resistance r s 

 of the condenser to be in series with the condenser, as we 

 have done in figs. 2, 3, 7, and 10, we consider that it is in 



Phil Mag. S. 5. Vol. 47. No. 284. Jan. 1899. D 



