Calculate RCL Series Resonant Circuit

Calculator and formulas for calculating a series resonant circuit consisting of inductor, capacitor and resistor

Series Resonant Circuit Calculator

RCL Series Connection

This calculator computes the important values of a series resonant circuit consisting of resistor, inductor and capacitor at resonance frequency. The ohmic resistance R is an external damping resistor or inductor loss resistance.

Results at Resonance Frequency
Resonance frequency f?:
Current at resonance I?:
Voltage U? at L / C:
Reactance XL/XC:
Quality factor Q:
Damping d:
Bandwidth b:
Upper cutoff freq. fo:
Lower cutoff freq. fu:
Current at fg:
Impedance at fg:

RCL Series Resonant Circuit

Series Resonant Circuit

The series resonant circuit is a filter circuit. Frequencies near the resonance frequency are passed through. The current is the same at every measurement point.

Phase Relationships
  • Resistor R: Current and voltage in phase
  • Inductor L: Voltage leads current by +90°
  • Capacitor C: Voltage lags current by -90°
  • UL and UC: 180° phase shifted
Resonance Condition
\[X_L = X_C\] \[Z = R \text{ (minimum)}\]

At resonance, the impedance Z is minimum and equals R.

RLC Series Resonant Circuit - Theory and Formulas

Fundamentals of Series Resonant Circuit

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

Resonance Frequency

Resonance Condition
\[2\pi f L = \frac{1}{2\pi f C}\]

This gives us the resonance frequency:

\[f_0 = \frac{1}{2\pi\sqrt{LC}}\]

The phase shift is 0°.

Impedance and Current

Impedance
\[Z = \sqrt{R^2 + (X_L - X_C)^2}\]

At resonance: XL = XC

\[Z = R\]
Current at Resonance
\[I_0 = \frac{U}{Z_0} = \frac{U}{R}\]

The current is maximum at resonance. There is voltage enhancement: The voltages across L and C can be larger than the applied voltage.

Quality Factor and Damping

Quality Factor Q
\[Q = \frac{U_L}{U} = \frac{U_C}{U} = \frac{X_L}{R} = \frac{X_C}{R}\]

The quality factor Q indicates the voltage enhancement.

Damping d
\[d = \frac{1}{Q}\]

The damping is the reciprocal of the quality factor.

Bandwidth and Cutoff Frequencies

Bandwidth
\[b = \frac{f_0}{Q} = f_0 \cdot d = \frac{f_0 \cdot R}{X_L} = \frac{f_0 \cdot U}{U_L}\]

The bandwidth determines the frequency range between the upper and lower cutoff frequency. The higher the quality factor Q, the more narrow-band the resonant circuit.

\[f_{go} = f_0 + \frac{b}{2}\]

Upper cutoff frequency

\[f_{gu} = f_0 - \frac{b}{2}\]

Lower cutoff frequency

Behavior at Cutoff Frequency

At f = fgo or f = fgu
\[\phi = 45°\]

Phase angle

\[I_g = \frac{I_0}{\sqrt{2}}\]

Current

\[Z_g = \sqrt{2} \cdot Z_0\]

Impedance

\[U_R = \frac{U}{\sqrt{2}}\]

Voltage across resistor

Practical Applications

Band-pass filters:
• Narrow-band filters
• Frequency selection
• Antenna technology
• Signal conditioning
Resonant circuits:
• Tuned circuits
• Oscillators
• Impedance matching
• Filter design
Measurement technology:
• Q measurement
• Frequency analysis
• Damping measurement
• Spectral analysis

Design Guidelines

Important Design Aspects
  • Quality factor Q: Determines bandwidth and selectivity
  • Losses: Real components have additional resistances
  • Voltage enhancement: UL and UC can exceed U
  • Temperature stability: L and C should be temperature stable
  • Loading: External load changes the quality factor
  • Parasitic effects: Consider self-resonances


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