AN INTERESTING CASE OF RESONANCE IN AN 
ALTERNATING CURRENT CIRCUIT. 
H. L. DODGE. 
The phenomena of voltage and current resonance are familiar 
to all students of alternating currents. The former occurs in 
series circuits and complete resonance is secured when the con- 
densive reactance is equal to the inductive reactance. The lat- 
ter occurs in connection with parallel circuits, the necessary 
condition being that the sum of all the susceptances, both con- 
densive and inductive, equals zero. The expression for the 
impedance of a series circuit is Z = V r 2 +(2 ff f L - 'Lhlb) 2 
This becomes a minimum when the condensive reactance, ‘LTfc? 
.just balances the inductive reactance, 2 K f L. This occurs at a 
frequency f = 2 V "vTA current is equal to E/Z, this is 
also the condition for maximum current and as the current is 
in phase with the voltage, the power-factor is unity. 
If the frequency is less than that determined by the above 
expression then the reactance of the condenser becomes greater 
and that of the inductance less. The result is that the current 
becomes smaller and smaller with decrease in frequency and 
leads by an increasing angle. If, on the other hand, the fre- 
quency is increased, the inductive reactance is made more prom- 
inent and the condensive reactance is reduced. The current be- 
comes smaller and smaller and lags by an increasing angle. 
Thus we see that in a series circuit, as the frequency is in- 
creased the current begins at a small value, increases to a maxi- 
mum and then returns to a small value again. At the same 
time the power-factor increases to unity and then decreases. 
The current leads for the lower frequencies and lags for the 
higher. Therefore, with a given voltage, as the frequency is 
increased every value of current or power-factor occurs twice, 
since each value of current or power-factor that is obtained 
at a frequency less than that required for resonance occurs 
again at some higher frequency. 
