Free Parallel RLC Resonant Tank & Impedance Calculator
Calculate parallel LC resonant frequency, inductor ESR damping, tank dynamic impedance ($R_p$), unloaded/loaded Q factor, and 3dB bandwidth.
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Parallel RLC Resonant Tank Circuit Theory
A parallel RLC resonant tank circuit consists of an inductor, a capacitor, and inherent loss resistances connected in parallel. At resonance, the inductive susceptance $B_L = 1/(\omega L)$ and capacitive susceptance $B_C = \omega C$ cancel each other out, causing the net tank impedance to peak at a purely resistive value known as the dynamic resistance ($R_p$).
Mathematical Governing Equations
- Ideal Resonant Frequency: $f_0 = \frac{1}{2\pi \sqrt{LC}}$
- ESR-Compensated Resonant Frequency: When coil winding series resistance $R_s$ is significant, the anti-resonant frequency where terminal voltage and current are in phase is: $$f_{damp} = \frac{1}{2\pi \sqrt{LC}} \sqrt{1 - \frac{R_s^2 C}{L}}$$
- Dynamic Tank Impedance ($R_p$): The equivalent parallel resistance transformed from the coil's series ESR is: $$R_p = \frac{L}{C \cdot R_s} = Q_0^2 \cdot R_s$$
- Unloaded Quality Factor ($Q_0$): Defined as the ratio of energy stored to energy dissipated per radian: $$Q_0 = \frac{\omega_0 L}{R_s} = \frac{R_p}{\omega_0 L} = R_p \omega_0 C$$
- Loaded Q ($Q_L$) & Bandwidth: With an external parallel load $R_{ext}$, the effective parallel resistance is $R_{eff} = R_p \parallel R_{ext}$, yielding: $$Q_L = \frac{R_{eff}}{\omega_0 L}, \quad \Delta f_{3dB} = \frac{f_0}{Q_L}$$
| Circuit Property | Series RLC Tank | Parallel RLC Tank |
|---|---|---|
| Impedance at Resonance | Minimum ($Z = R$) | Maximum ($Z = R_p = L/(CR_s)$) |
| Line Current at Resonance | Maximum ($I = V/R$) | Minimum ($I = V/R_p$) |
| Circulating Reactive Current | Equal to line current | $I_{circ} = Q \cdot I_{line}$ |
| Typical Application | Notch filters, series traps | RF oscillators, bandpass filters, IF amps |
Frequently Asked Questions
Why is the impedance of a parallel tank at maximum rather than zero?
At resonance, the inductor and capacitor exchange stored reactive energy between the electric field of the capacitor and the magnetic field of the inductor 180 degrees out of phase. The branch currents cancel out at the external nodes, so only enough external current flows to supply the resistive losses (ESR), resulting in an extremely high parallel dynamic resistance Rp.
What happens if the inductor ESR Rs is too large?
If Rs exceeds sqrt(L/C), the term under the square root (1 - Rs^2*C/L) becomes negative, meaning the circuit is over-damped and will not exhibit a true resonance peak or oscillation.
How does an external load resistor affect the tank bandwidth?
Adding a parallel load resistor R_ext lowers the effective parallel resistance (R_p || R_ext), decreasing the loaded quality factor Q_L and broadening the -3dB bandwidth (delta f = f_0 / Q_L).