Calculate Parallel Resonant Circuit

Calculator and formulas for calculating an RCL parallel resonant circuit

Parallel Resonant Circuit Calculator

RCL Parallel Connection

This calculator computes the important values of a parallel resonant circuit consisting of resistor, inductor and capacitor at resonance frequency. Parallel resonant circuits are often used as band-stop filters (notch filters).

Results at Resonance Frequency
Resonance frequency f₀:
Total current I₀:
Currents IL, IC:
Reactance XL/XC:
Quality factor Q:
Damping d:
Bandwidth b:
Upper cutoff freq. fo:
Lower cutoff freq. fu:

RCL Parallel Resonant Circuit

Parallel Resonant Circuit

Parallel resonant circuits are often used as band-stop filters (notch filters) to reject specific frequencies. The impedance Z is maximum at the resonance frequency when XL = XC.

Impedance at Resonance
\[Z = R \text{ (maximum)}\]

The impedance Z is maximum at resonance and is determined only by the ohmic resistance R.

Current Enhancement
\[I_L = I_C = \frac{U}{X_L} = \frac{U}{X_C}\]

The current in the supply line is minimum at resonance. Larger currents can flow through the inductor and capacitor.


RLC Parallel Resonant Circuit - Theory and Formulas

Fundamentals of Parallel Resonant Circuit

The total resistance of the resonant circuit is called impedance Z. Ohm's law applies to the complete circuit. The impedance Z is maximum at the resonance frequency when XL = XC.

Resonance Frequency

Resonance Condition
\[2\pi f L = \frac{1}{2\pi f C}\]

This gives us the resonance frequency:

\[f_0 = \frac{1}{2\pi\sqrt{LC}}\]

At resonance, the phase shift = 0°.

Impedance and Current

Impedance at Resonance
\[Z = \sqrt{R^2 + (X_L - X_C)^2}\]

At resonance: XL = XC

\[Z_0 = R\]

The impedance Z is maximum at resonance.

Currents at Resonance
\[I_0 = \frac{U}{Z_0} = \frac{U}{R}\]
\[I_L = \frac{U}{X_L} = \frac{U}{X_C}\]

The current in the supply line is minimum at resonance. Larger currents can flow through the inductor and capacitor.

Quality Factor and Damping

Quality Factor Q
\[Q = \frac{I_L}{I} = \frac{R}{X_C} = \frac{R}{X_L}\]

The quality factor Q indicates the current enhancement.

Damping d
\[d = \frac{1}{Q}\]

The damping is the reciprocal of the quality factor.

Bandwidth and Cutoff Frequencies

Bandwidth
\[b = \frac{f_0}{Q} = f_0 \cdot d = \frac{f_0 \cdot X_L}{R}\]

The bandwidth determines the frequency range between the upper and lower cutoff frequency. The higher the quality factor Q, the more narrow-band the resonant circuit.

\[f_{go} = f_0 + \frac{b}{2}\]

Upper cutoff frequency

\[f_{gu} = f_0 - \frac{b}{2}\]

Lower cutoff frequency

Practical Applications

Band-stop filters:
• Notch filters
• Interference suppression
• Mains hum filters
• Trap circuits
Tuned circuits:
• Antenna couplers
• Oscillator tanks
• Frequency doublers
• Impedance converters
Energy storage:
• Resonant storage
• Resonant converters
• Wireless power
• Induction heating

Differences to Series Resonant Circuit

Parallel vs. Series
Parallel resonant circuit:
  • Z maximum at resonance
  • I minimum at resonance
  • Band-stop (notch filter)
  • Current enhancement in L and C
Series resonant circuit:
  • Z minimum at resonance
  • I maximum at resonance
  • Band-pass (pass filter)
  • Voltage enhancement at L and C

Design Guidelines

Important Design Aspects
  • Quality factor Q: Determines bandwidth and rejection
  • Losses: Reduce the maximum impedance
  • Current enhancement: IL and IC can significantly exceed I
  • Loading: External load parallel to the circuit reduces quality factor
  • Coupling: Loose coupling maintains high quality factor
  • Component tolerances: Affect the resonance frequency

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