KEl^NELLY. — OSCILLATING-CURRENT CIRCUITS. 395 



give the actual current at any instant. This is eqiiivalent to rotating 

 df in an ecjuiangular spiral of angle in the manner of Figure 8. 



Similar treatment will convert U into a rotating vector-triangle, 

 except that Z^/*^ should start from the position OUq of Figure 8. 



The P diagram, in either Figure 7 or Figure 12, can be made into 

 a rotating vector- diagram, by rotating the triangle with angular velocity 

 2w about the vertex k or K respectively. Thus, taking the power tri- 

 angle ghk of Figure 7, we mount it on an axis at k, Figure 13, and 

 draw /•;• equal and parallel to kg. The three- vector system, M, kg, and 

 kr is then allowed to rotate with the angular velocity 2w, in the case 

 considered 2449.5 rad/sec, starting from the position shown, when the 

 p. d. vector OUq, Figure 8, starts with angular velocity w from the posi- 

 tion AT/, or parallel to OY. The orthogonal projections on — XOX 

 of the three vectors kh, kg, and kr, then define at any instant the un- 

 damped reactive, total, and dissipative power, under the action of U^ 

 on the oscillatory current. The total undamped power ^^ =^ 5270 watts 



thus lags 7 4- <A behind the p. d. vector in terms of power phase, or 



- -f - in terms of p. d. jjhase (70° .23'). The total undamped power 

 4 2 



is measured on the OX axis from the point and oscillates between 



the limits Op = 8603.3 and 0—p = —1936.6 watts, as shown in 



Figure 10. 



The undamped dissipative power /v oscillates between the limits 

 Opr = 6666.6 watts and zero. The total undamped dissipative power 

 is 2 Pr = 13,333.3 watts, owing to the separate actions of Od^ and 

 OBo, Figure 8. 



The undamped reactive power pi is reckoned from o as zero, and 

 oscillates between ±4082.5 watts. 



The damping-factor e"^^' must be applied to p and pr as measured 

 from 0, and to j^i as measured from o, along the OX axis. 



If we employ the phase of the oscillating current as standard, and as 

 taken in Figure 10, we mount the power triangle GHK of Figure 12 

 on an axis at K, and rotate it, as before, at angular velocity 2oj, start- 

 ing from the position G HK shown in Figure 13 when the current vector 

 OJq of Figure 8 is passing through the position Kffo with angular ve- 

 locity w. The total power p will then lag - — ^ behind the current 



vector, in terms of power-phase, or - — -, in terms of current phase 



4 ^ 



