Parallel RLC resonance and resonance frequency

A parallel RLC circuit is in resonance when the supply voltage and the total line current are in phase. This happens when the inductive susceptance equals the capacitive susceptance, so the circuit behaves as a pure resistance.

        +-----+-----+-----+
   I -->|     |     |     |
        R     L     C     |  V
        |     |     |     |
        +-----+-----+-----+

Explanation of the phenomenon

Total admittance of the parallel circuit (R, L and C all across the same voltage V):

Y = 1/R + 1/(jωL) + jωC
  = 1/R + j(ωC − 1/ωL)

• At low frequency the inductor takes a large lagging current, so the circuit is inductive.
• At high frequency the capacitor takes a large leading current, so the circuit is capacitive.
• At one frequency the two reactive currents are equal and opposite (I_L = I_C). They cancel, and the source supplies only the resistor current I_R = V/R.

At resonance Y = 1/R is minimum, so the impedance is maximum (Z = R), the line current is minimum, and the power factor is unity. A large current circulates between L and C (I_L = I_C = Q × I), so this is also called a current-magnification circuit.

Derivation of resonance frequency

Resonance needs the imaginary part of Y to be zero:

ω0C − 1/(ω0L) = 0
ω0² = 1/(LC)
ω0 = 1/√(LC) rad/s        f0 = 1/(2π√(LC)) Hz

The resonant frequency is independent of R.

Other results

Q0 = R/(ω0L) = ω0CR = R√(C/L)
BW = ω0/Q0 = 1/(RC) rad/s

Note: for a practical coil (resistance r in series with L) in parallel with C, the resonant frequency becomes ω0 = √(1/LC − r²/L²).

Answer: f0 = 1/(2π√(LC)), the impedance is maximum (= R) and the line current is minimum at f0.