# Calculate the parallel resonant circuit

Calculator and formulas for calculating a parallel resonant circuit from inductor, capacitor and resistor

## Calculate the RCL parallel resonant circuit

This function calculates the most important values of a parallel resonant circuit consisting of a resistor, inductor and capacitor.

RCL parallel circuit calculator

 Input Inductor L H mH µH nH Capacitor C F mF µF nF pF Resistor R mΩ Ω kΩ MΩ Voltage U mV V kV Decimal places 0 1 2 3 4 6 8 10 Results at resonance frequency Frequency f0 Total current I0 Currents I L, I C Impedance XL/XC Q factor Damping d Bandwidth b Upper cut-off fH Lower cut-off fL

## Formulas for the RLC parallel circuit

Parallel resonant circuits are often used as a bandstop filter (trap circuit) to filter out frequencies.

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

### Impedance at resonance

The impedance is calculated according to the formula:

$$\displaystyle Z=\sqrt{R^2 + (X_L-X_C)^2}$$

At resonance, XL = XC . The phase of the voltage is opposite; the two values cancel each other out and the following applies:

$$\displaystyle Z=R$$

### Resonance frequency

The resonance frequency is given when XL = XC .

$$\displaystyle 2πf·L=\frac{1}{2πf·C}$$

This results in the formula for the resonance frequency

$$\displaystyle f_0=\frac{1}{2π\sqrt{L·C}}$$

In the case of resonance, the phase shift is = 0 °.

### Resistance and current

The impedance Z is greatest at resonance. It is then only determined by the ohmic resistance R .

$$\displaystyle Z_0=R$$

Tthe current is smallest at resonance. Larger currents can flow through the coil and the capacitor.

$$\displaystyle I_0=\frac{U}{Z_0}=\frac{U}{R}$$
$$\displaystyle I_L=\frac{U}{X_L}=\frac{U}{X_C}$$

### Cut-off frequencies

Upper cut-off frequency:   $$\displaystyle f_H=f_0+\frac{b}{2}$$
Lower cut-off frequency:   $$\displaystyle f_L=f_0-\frac{b}{2}$$

### Quality Q and damping d

The quality Q indicates the excess current

$$\displaystyle Q=\frac{I_L}{I}=\frac{X_C}{R}=\frac{X_L}{R}$$
Damping:   $$\displaystyle d=\frac{1}{Q}$$

### Bandwidth

The bandwidth determines the frequency range between the upper and lower cut-off frequency. The higher the quality Q, the narrower the resonant circuit.

Damping:   $$\displaystyle b=\frac{f_0}{Q}=f_0 ·d =\frac{f_0 · R}{X_L}$$

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