# Calculate the RCL series resonant circuit

Calculator and formulas for a series circuit consisting of a coil, capacitor and resistor

## RCL series circuit calculator

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.

Series resonant 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 Voltage U0 at L/C Reactance XL/XC Q factor Damping d Bandwidth b Upper cut-off fH Upper cut-off fL Current at f0 Impedance at f0

## 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 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 $$\displaystyle f=f_{gu}$$
$$\displaystyle φ=45°$$
$$\displaystyle I_g=\frac{I_0}{\sqrt{2}}$$
$$\displaystyle U_R=\frac{U}{\sqrt{2}}$$
$$\displaystyle Z_g=\sqrt{2}·Z_0$$