KENNELLT. — OSCILLATING-CURRENT CIRCUITS. 389 



power Pm, of which the component h k is the maximum cyclic reactive 

 power, or the maximum cyclic power in the inductance ; while the real 

 component g h is the maximum cyclic power expended in resistance 

 by the discharging p. d. Uq, Figure 8. But a like dissipation of power 

 occurs under the influence of the emf. of self-induction 0L\, so that 

 the total undamped maximum cyclic dissipative power in the circuit is 

 gg', Figure 7, of 6666.6 watts. 



Finally, if we divide the P diagram by 2w, or twice the resistant 

 angular velocity, we obtain the W diagram of Figure 7, or the triangle 

 I m n, which may be drawn to a suitable scale of joules. The stationary 

 vector / n is the undamped maximum cyclic energy W in the oscillatory 

 circuit as measured, at condenser terminals, or 2.151,65 joules in the 

 case considered. The —J component, or 1.66 joules, is the undamped 

 maximum cyclic energy in the reactance, and the real component 

 / ni — 1.3608 is the undamped maximum cyclic energy dissipated by 

 the discharging p. d. Uo, Figure 8, on the oscillatory current. But a 

 like amount of energy will be dissipated by the self-inductive emf. OE. 

 Consequently, the total undamped maximum cyclic dissipative energy 

 in the circuit will be / /', Figure 7, or 2.7216 joules. 



The condition of either the vector p. d. OUq (Figure 7), the vector 

 self-inductive emf. OEq, or the vector current 01^ after t seconds, is 

 obtained by applying the exponential (e- (•'-»'), as in (19). This 

 exponential may be expressed as 6~-^K eJ'^\ the first of which is a damp- 

 ing-factor, and the secoad a rotating factor. The diagrams of Figure 7 

 apply only to the effects of the rotating factor, as though no damping 

 existed. That is, they represent undamped oscillating quantities, or 

 quantities which would be projected on —XOX by the rotation of the 

 vectors OU^, OEf^, OR^ in pure circles, instead of in spirals. The 

 damping-factor e^-'Ho be applied, is represented in Figure 9, which is 

 drawn on semi-logarithm paper, i.e., on paper ruled to a logarithmic 

 scale of ordinates, but to a uniform scale of abscissas. The ordinates 

 give the damping- factor, and the abscissas the time from release, to a 

 scale of degrees, and also of seconds. The straight line OA represents 

 the damping-factor for voltage and currents. The straight line OB 

 represents the damping-factor for powers and energies. The time in 

 which the voltage and current fall to 1/eth of their initial value is r, 

 the oscillatory time-constant, or 0.001 second in the case considered. 

 The number of radians through which the vectors of Figure 8 must 

 rotate in order to shrink to 1/eth of their initial values, i. e., the time- 

 angle for a damping-factor of e~^ = 0.367,88, is /w/p = tan ^, or, in the 

 case considered, 1.22475 radians = 70.2°. 



