380 PROCEEDINGS OF THE AMERICAN ACADEMY. 



The opposite sinusoid iVc, also of double frequency, and 1 joule am- 

 plitude, is the graph of energy in the condenser. It is evident that at 

 every instant 



«^'c + '^h = ^^0 joules (9) 



where TFo is the initial energy Uo^c/2 of the system, or 2 joules in the 

 case considered. 



All of the vectors in Figure 3 are stationary, and in the series /, P, 

 and W, are all drawn to the phase of the discharging p. d. Uo as 

 standard. This means that the vector current i is 90° behind that of 

 u (Figure 5), the condenser power p is quadrature power, or purely 

 reactive power, and the delivered energy W^ is also quadrature or 

 reactive energy. 



With respect to current-phase as standard, we have the series of 

 vectors in Figure 4, commencing with Zq the impedance of the 

 circuit, which is drawn in the +7 direction to a scale of ohms and a 

 length of Im on this scale. In the case considered, ^0 = ^158. 114 

 ohms. Zf^ is thus a purely reactive impedance, or reactance. 



The vector Uo, Figure 4, represents the amplitude or initial value of 

 the p. d. at condenser terminals. It is drawn in the -\-j direction, or 

 is 90° ahead of the current, and to a scale of volts, to a length of 

 I^Zq = IZfjV^ on that scale, where /is the r.m. s. value of the vector 

 Iq. In the case considered, Jo = 6.3246 amplitude amperes and 

 /= 4.472 r. m. s. amperes. In terms of the electric quantit}', however, 

 it may also be expressed as QqS = Qs'\/2 volts. In the case considered, 

 this vector amplitude p. d. [/q is 1 000 volts, with an effective or r. m. s. 

 value of L^ — 707.1 volts. 



The vector F^ in Figure 4 represents the amplitude, or maximum 

 cyclic value, of the oscillatory power of the condenser. It is drawn in 

 the + j direction, being leading quadrature power with respect to cur- 

 rent phase, to watt scale, and to a length of °^ ° — lU units on that 



scale. In the case considered, Pm. =i 3162.3 watts. 



The vector W^ in Figure 4 represents the amplitude or maximum 

 cyclic value of the oscillatory energy expended by the condenser. It 

 differs only from the vector Wm of Figure 3, by being drawn in the + j 

 instead of in the — j direction. This is because the energy and power 

 are leading quadrature quantities with reference to the current, but 

 lagging ({uadrature quantities with respect to the discharging p. d. 



The stationary vector-diagrams of Figures 3 and 4 may be considered 

 as graphically corresponding to the following vector equations: 



