Calculator and formulas for a series circuit consisting of a coil, capacitor and resistor
This calculator returns the most important values of a series resonant circuit consisting of a resistor, coil and capacitor.
The ohmic resistance R is an external damping resistor or coil loss resistance.

The series oscillating circuit is a sieve or filter circuit. Frequencies close to the resonance frequency are allowed to pass
The current is the same at every measuring point.
Current and voltage are in phase at the ohmic resistance.
At the inductive reactance of the coil, the voltage leads the current by + 90 °.
At the capacitive reactance of the capacitor, the voltage lags the current by 90 °.
Therefore U_{L } and U_{C } are phase shifted by 180 °, i.e. out of phase
The total resistance of the resonant circuit is called Impedance Z. Ohm's law applies to the entire circuit. The impedance Z is smallest at the resonance frequency when X_{L } = X_{C }.
\(\displaystyle 2πf·L=\frac{1}{2πf·C} \)
This results in the formula for the resonance frequency
\(\displaystyle f_0=\frac{1}{2π\sqrt{L·C}} \)
The phase shift is 0°.
\(\displaystyle Z=\sqrt{R^2 + (X_LX_C)^2} \)
At resonance, X_{L } = X_{C }. The phase of the voltage is opposite; the two values cancel each other out and the following applies:
\(\displaystyle Z=R \)
The current is greatest at resonance
\(\displaystyle I_0=\frac{U}{Z_0}=\frac{U}{R} \)
If there is a resonance, there is an increase in voltage. The voltage at L and C can be greater than the applied voltage
The quality Q indicates the voltage increase
\(\displaystyle Q=\frac{U_L}{U}=\frac{U_C}{U}=\frac{X_L}{R}=\frac{X_C}{R} \)
Damping: \(\displaystyle d=\frac{1}{Q} \)
The bandwidth determines the frequency range between the upper and lower cutoff frequency. The higher the quality Q, the narrower the resonant circuit.
\(\displaystyle b=\frac{f_0}{Q}=f_0 ·d =\frac{f_0 · R}{X_L} =\frac{f_0 · U}{U_L} \)
Upper cutoff frequency: \(\displaystyle f_{go}=f_0+\frac{b}{2} \)
Lower cutoff frequency: \(\displaystyle f_{gu}=f_0\frac{b}{2} \)
\(\displaystyle f=f_{go}\) oder \(\displaystyle f=f_{gu}\)
\(\displaystyle φ=45° \)
\(\displaystyle I_g=\frac{I_0}{\sqrt{2}} \)
\(\displaystyle U_R=\frac{U}{\sqrt{2}} \)
\(\displaystyle Z_g=\sqrt{2}·Z_0 \)
