Calculate the RCL series resonant circuit

Formulas for the RLC series resonant circuit

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 .

Resonance frequency

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

This results in the formula for the resonance frequency

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

The phase shift is 0°.

Impedance at resonance

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

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

\(\displaystyle Z=R \)

Current and voltage

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

Quality Q and damping d

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:   \(\displaystyle d=\frac{1}{Q} \)

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.

\(\displaystyle b=\frac{f_0}{Q}=f_0 ·d =\frac{f_0 · R}{X_L} =\frac{f_0 · U}{U_L} \)

Cut-off frequencies

Upper cut-off frequency:   \(\displaystyle f_{go}=f_0+\frac{b}{2} \)

Lower cut-off frequency:   \(\displaystyle f_{gu}=f_0-\frac{b}{2} \)

The following applies to the cut-off frequency:

\(\displaystyle f=f_{go}\) oder \(\displaystyle f=f_{gu}\)

\(\displaystyle φ=45° \)

\(\displaystyle I_g=\frac{I_0}{\sqrt{2}} \)

\(\displaystyle U_R=\frac{U}{\sqrt{2}} \)

\(\displaystyle Z_g=\sqrt{2}·Z_0 \)