Calculate the RCL series resonant circuit
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.
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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.
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Current and voltage are in phase at the ohmic resistance.
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At the inductive reactance of the coil, the voltage leads the current by + 90 °.
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At the capacitive reactance of the capacitor, the voltage lags the current by -90 °.
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Therefore UL and UC 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 XL = XC .
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, XL = XC . 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 \(\displaystyle f=f_{gu}\)
\(\displaystyle φ=45° \)
\(\displaystyle I_g=\frac{I_0}{\sqrt{2}} \)
\(\displaystyle U_R=\frac{U}{\sqrt{2}} \)
\(\displaystyle Z_g=\sqrt{2}·Z_0 \)
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