Parallel resonance with voltage and current waveforms

Resonance in a parallel RLC circuit occurs at the frequency where the inductor current and capacitor current cancel each other, so the supply voltage and supply current are in phase and the circuit acts as a pure resistor R.

Phenomenon

For R, L, C connected across a source v = Vm sin ωt:

I_R = V/R             (in phase with v)
I_L = V/(ωL)          (lags v by 90°)
I_C = V·ωC            (leads v by 90°)
Y = 1/R + j(ωC − 1/ωL)

I_L and I_C are 180° out of phase. When ωC = 1/(ωL), they are equal and cancel, so the source supplies only I_R. The admittance is a minimum (1/R), the impedance is a maximum (R), and the line current is minimum.

Resonance frequency

Im(Y) = 0  →  ω0C = 1/(ω0L)
ω0 = 1/√(LC) rad/s      f0 = 1/(2π√(LC))

Q-factor: Q0 = R/(ω0L) = ω0CR, bandwidth BW = ω0/Q0 = 1/(RC).

Waveforms at resonance

 v, i
  ^      v and i (total) coincide in phase
  |    __        __
  |   /  \      /  \
--+--/----\----/----\---> ωt
  |       \__/      \__/
  |
 i_L: lags v by 90° (peak at ωt = π)
 i_C: leads v by 90°, i_C = −i_L at every instant

Here Im_L = Vm/(ω0L) and Im_C = ω0C·Vm are equal at resonance, so the two reactive currents have equal peaks and mirror each other. The total current is therefore a sine wave in phase with v, with peak Vm/R only.

Answer: f0 = 1/(2π√(LC)); at f0, v and the line current i are in phase, and i_L and i_C are equal and opposite.