Resonance 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.
v08022025 | Home
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.
v08022025 | Home
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: \[ f = \frac{1}{2\pi\sqrt{L C}} \] This is 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.
The Quality Factor (Q) measures energy storage vs. dissipation: \[ Q = \frac{1}{R} \sqrt{\frac{L}{C}} \] The Bandwidth (BW) is related to \( Q \) and the resonant frequency \( f_0 \): \[ BW = \frac{f_0}{Q} \] A higher \( Q \) means a sharper resonance peak and lower energy dissipation.
The energy in the electric field of a capacitor: \[ U_C = \frac{1}{2} C V^2 \] The energy in the magnetic field of an inductor: \[ U_L = \frac{1}{2} L I^2 \] These continuously exchange in an LC circuit, forming an oscillatory system.
Ideal LC circuits are lossless, but real components have resistive losses, leading to: \[ Z = \sqrt{R^2 + (X_L - X_C)^2} \] The system follows damped harmonic motion due to resistive losses, modeled as: \[ L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{q}{C} = 0 \] resulting in a decaying oscillation in real circuits.
From atomic transitions to electromagnetic waves, resonance is a universal principle in nature. The same physics govern **radio signals, lasers, quantum harmonic oscillators, and even planetary orbits**.
The **mathematical framework** of this calculator is **analogous** to solutions in **Schrödinger’s equation** for quantum harmonic oscillators: \[ \hat{H} \psi = E \psi \] where resonance corresponds to **eigenstates of energy levels**.
- Nonlinear LC circuits lead to **bifurcations and chaos** in **dynamical systems**. - Resonance plays a role in **neural oscillations, DNA resonance, and even biological rhythms**.