Resonance Frequency Calculator
Enter values for two of the following fields (using SI units: Farads for capacitance, Henries for inductance, and Hertz for frequency). Leave the field you want to calculate blank. | Formulas
Enter values for two of the following fields (using SI units: Farads for capacitance, Henries for inductance, and Hertz for frequency). Leave the field you want to calculate blank. | Formulas
This calculator is based on well-established principles of electromagnetism, circuit theory, and resonance phenomena. It provides scientifically accurate computations for LC circuits using the following fundamental theories.
The natural frequency of a series or parallel LC circuit is determined using: \[ f = \frac{1}{2\pi\sqrt{L C}} \] Derived from Kirchhoff’s Voltage Law (KVL) and the governing differential equation of an LC oscillator: \[ L \frac{d^2q}{dt^2} + \frac{q}{C} = 0 \] which has a harmonic solution where the oscillation frequency is: \[ \omega_0 = \frac{1}{\sqrt{LC}}, \quad f = \frac{\omega_0}{2\pi} \]
In AC circuit analysis, reactance is frequency-dependent: \[ X_C = \frac{1}{\omega C}, \quad X_L = \omega L \] At resonance, their magnitudes cancel in a series LC circuit, making it behave as a pure resistor.
Unlike a series LC circuit, a parallel LC circuit exhibits infinite impedance at resonance. The impedance at resonance is given by: \[ Z_{\text{parallel}} = \frac{L}{RC} \] where \( R \) represents the equivalent resistance of the circuit.
The wavelength corresponding to the calculated frequency is: \[ \lambda = \frac{c}{f} \] where \( c \) is the speed of light (\( 3.00 \times 10^8 \) m/s). This is important for radio frequencies and electromagnetic wave propagation.
The Quality Factor (Q) quantifies energy loss in a resonant circuit. A higher \( Q \) means lower energy dissipation and a sharper resonance peak.
For a series LC circuit:
\[ Q_{\text{series}} = \frac{1}{R} \sqrt{\frac{L}{C}} \]For a parallel LC circuit:
\[ Q_{\text{parallel}} = R \sqrt{\frac{C}{L}} \]The Bandwidth (BW) of the circuit, which determines how sharply it responds to its resonant frequency, is given by:
\[ BW = \frac{f_0}{Q} \]
The voltage and current in an LC circuit oscillate sinusoidally: \[ V_L = L \frac{di}{dt}, \quad V_C = \frac{1}{C} \int i \, dt \] In a damped resonant circuit (when resistance \( R \) is present), the current follows: \[ I = I_0 e^{-\frac{R}{2L}t} \cos (\omega_d t + \phi) \] where: \[ \omega_d = \sqrt{\frac{1}{LC} - \left(\frac{R}{2L}\right)^2} \] If \( R \) is too high, the system becomes **overdamped** and does not oscillate.
The energy stored in an LC circuit fluctuates between the capacitor and inductor: \[ U_C = \frac{1}{2} C V^2, \quad U_L = \frac{1}{2} L I^2 \] The energy lost per cycle is given by: \[ \Delta U = \frac{U}{Q} \] where \( Q \) represents the quality factor.
Real LC circuits have losses, making them behave as damped oscillators: \[ L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{q}{C} = 0 \] The presence of resistance leads to a decaying oscillation over time.
Under strong external forcing, an LC circuit can exhibit **nonlinear resonance**, following the Duffing equation: \[ L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \alpha q + \beta q^3 = 0 \] Here:
Some nonlinear LC circuits follow the Duffing oscillator equation: \[ L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \alpha q + \beta q^3 = 0 \] leading to bifurcations and chaotic behavior in certain conditions.