AN INTERESTING CASE OF RESONANCE 
195 
gram also shows how the voltage is first lagging, then leading, 
then lagging, and finally leading at the higher frequencies. 
It is also clear that the' power-factor passes through unity at 
three different values of frequency. 
The current in the circuit, as the frequency is varied (con- 
stant voltage of thirty volts), can he computed from the total 
impedances and is represented as a curve in Plate X (d) and 
vectorially in Plate X (f). The current passes through two 
maxima and one minimum and as the frequency is increased 
changes from leading to lagging to leading to lagging. The 
vector diagram shows clearly how the power-factor becomes 
unity at three different values of frequency. The current hodo- 
graph winds up on itself and is similar in form to the limacon. 
Several very interesting results have been accomplished with 
this circuit. The curves and diagrams have all been made sym- 
metrical. The current has been made to pass through two 
maxima and the minimum has been so controlled as to be prac- 
tically equal to the end values. The power-factor has been 
made to remain at practically unity for a range of frequency 
from forty-five to seventy-eight cycles and to drop off rapidly 
outside of these frequencies. 
We will now examine the conditions existing in the circuit 
in detail so as to see the causes which contribute to and control 
these results. We have already seen that the choice of con- 
stants, so as to secure balanced reactances at sixty cycles, is 
responsible for symmetrical results with a range of frequency 
of thirty to 120 cycles. 
The control of the relative magnitude of the current at ex- 
treme frequencies and the minimum value is through the choice 
of the resistance and reactance of the series circuit (a given 
parallel circuit being assumed). If the resistance be made 
greater there is very little difference in the total impedance at 
thirty and 120 cycles. Change of resistance is, however, of 
great importance in the neighborhood of sixty cycles for there 
the series impedance is made up entirely of resistance. Thus a 
variation of the series resistance can be made to control the 
minimum value of the total current. 
On the other hand the magnitude of the impedance of the 
series circuit at the extreme frequencies is changed enormously 
by any change in the inductance or capacity. Thus these values 
