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.

Capacitance is measured in farads (F). It represents the ability of a component to store electrical charge.


Inductance is measured in henries (H). It determines how much voltage is induced for a given rate of change of current.


Frequency is measured in hertz (Hz). It defines how many cycles per second the oscillating signal completes.


Resistance is measured in ohms (Ω). It refers to the circuit’s total opposition to current flow and is used to calculate the quality factor (Q).


Voltage is measured in volts (V). This refers to the peak voltage across the capacitor, which is used to calculate stored energy.


Current is measured in amperes (A). This is the peak current flowing through the inductor, used to compute its stored energy.


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Resonance Calculator Features

Theoretical Foundations of the Resonance Calculator

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.

Resonant Frequency of an LC Circuit

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: \[ \omega_0 = \frac{1}{\sqrt{LC}}, \quad f = \frac{\omega_0}{2\pi} \]

Impedance and Reactance Theory

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.

Quality Factor & Bandwidth

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.

Energy Stored in LC Components

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.

Practical Considerations: Damping & Real-World Effects

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.


Advanced Implications of Resonance

Everything is an Oscillator

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**.

Resonance in Quantum Mechanics

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**.

Beyond Simple Circuits: Nonlinear Resonance

- Nonlinear LC circuits lead to **bifurcations and chaos** in **dynamical systems**. - Resonance plays a role in **neural oscillations, DNA resonance, and even biological rhythms**.


Why This Calculator is Scientifically Valid

Further Reading & Scientific Sources

Maxwell's Equations - Scientific Papers and Resources