Parallel RLC Resonance and the Tank Circuit
The one point where the susceptances cancel
Sweep the frequency across a parallel R, L, C. The capacitor’s susceptance ωC grows in proportion to frequency while the inductor’s 1/ωL shrinks in inverse proportion. At the resonance ω_0 = 1/√(LC) where the two are exactly equal, they cancel, so the total admittance is minimal and the impedance maximal (= R). Find the resonant frequency where the impedance seen by the source is largest.
In parallel, think in susceptance
Just as reactances added in series, admittances — the reciprocals of impedance — add in parallel. Their imaginary part is the susceptance. The inductor’s susceptance is −1/ωL, inverse to frequency, and the capacitor’s is +ωC, in proportion. Opposite in sign, they subtract when added, so the net susceptance is ωC − 1/ωL. Where series resonance was solved by the cancellation of reactances, parallel resonance is the same story told in susceptances — its mirror.
At resonance the impedance is maximal
The frequency where the net susceptance is zero — where ωC = 1/ωL — is the resonant point. Solving gives ω_0 = 1/√(LC), the same formula as series resonance. At this frequency the admittance loses its imaginary part and only the conductance 1/R remains, at its smallest. So the impedance is largest, equal to R, and with the same voltage applied the current the source sends out, V/R, is at its minimum. Where series resonance dropped the impedance to a minimum and opened the circuit wide, parallel resonance raises the impedance to a maximum and shuts the circuit tight from the source’s side.
A large current circulating in the tank
The source sends almost no current, yet the circuit is not quiet inside. At resonance the inductor and capacitor exchange a large current of opposite phase under the same voltage, and that circulating current reaches Q times the source current. Energy shuttles between the L’s magnetic field and the C’s electric field with almost no leak. So a parallel LC is called a tank circuit, used as the heart that holds a particular frequency in oscillators and RF filters. Where series resonance raised a voltage Q times the source across the resistor (voltage magnification), parallel resonance circulates a current Q times the source within the tank (current magnification). The two are exact mirrors.
Back to the first screen
As you slid the frequency, the impedance curve drew a peak, and its top was at ω/ω_0 = 1, the resonance. There the capacitor’s ωC and the inductor’s 1/ωL became exactly equal, the susceptances cancelled, the admittance fell to its minimum, and the impedance rose to its maximum of R. The current the source saw was at its smallest, yet inside the tank a current Q times larger circulated between L and C. Where series resonance dropped the impedance to a minimum and made the current surge, parallel resonance, its mirror, raises the impedance to a maximum and lulls the source current to rest.