LC Circuit Oscillation Simulator

The core behavior is described by a second-order differential equation for the charge $q(t)$ on the capacitor, derived from Kirchhoff's voltage law.

Where $q$ is the charge, $\frac{dq}{dt}=i$ is the current, $L$ is inductance (H), $R$ is resistance (Ω), and $C$ is capacitance (F). This is the equation of a damped harmonic oscillator.

The solution for the underdamped case (most common oscillatory behavior) gives the decaying voltage waveform you see in the simulator.

Here, $V_0$ is the initial voltage, $\alpha = R/(2L)$ is the damping coefficient, and $\omega_d = \sqrt{\omega_0^2 - \alpha^2}$ is the damped natural frequency. The exponential term $e^{-\alpha t}$ is the envelope of decay controlled by R and L.

Radio Tuners and Filters: LC circuits are the heart of analog radio receivers. By adjusting the capacitor (tuning), the circuit resonates at the frequency of a specific radio station, amplifying its signal while rejecting others. The sharpness (Q) of the resonance determines the selectivity of the tuner.

Switch-Mode Power Supplies: LC "tank" circuits are used to store and transfer energy efficiently at high frequencies in power converters. The oscillation is carefully controlled to minimize energy loss (requiring high Q) when switching transistors on and off, leading to compact and efficient power adapters.

Wireless Power Transfer: Systems like Qi chargers for phones use a pair of magnetically coupled LC resonators. The transmitter and receiver coils are each part of an LC circuit tuned to the same resonant frequency, enabling efficient energy transfer over a small air gap.

Signal Conditioning and Clock Generation: Crystal oscillators, which provide precise timing signals for computers and microcontrollers, fundamentally rely on the resonant properties of a piezoelectric crystal, electrically modeled as a very high-Q LC circuit. This provides the stable clock signal that synchronizes all digital operations.

When you start using this simulator, there are a few points that are easy to misunderstand. First, while it's correct that "the resonant frequency is determined solely by L and C," you often forget that whether the oscillation sustains is dominated by R. For example, with L=1mH and C=1μF, the resonant frequency is about 5kHz, but simply changing R from 10Ω to 100Ω drastically changes the oscillation decay from a "beautiful sine wave" to a "curve that decays immediately." In practice, the DC resistance of the coil and wiring resistance are also included in "R," which can cause a discrepancy between ideal simulations and actual measurement results.

Next, it's not always true that "a higher Q factor is better." Certainly, for radio tuning, you want a high Q factor (sharp resonance characteristic). However, in an LC filter for a switching power supply, if the Q factor is too high, the circuit can become unstable and oscillate due to switching noise. If you use the simulator to increase the Q factor by keeping L and C constant and only decreasing R, you'll see the peaks on the graph become sharper and more numerous. That's a visualization of the "sharpness of resonance." It's important to design the optimal amount of damping (i.e., Q factor) according to the application.

Also, be mindful of assumptions about initial conditions. This simulator often starts from a state where the capacitor has an initial charge, right? But in a real circuit, the state the moment a switch is turned on or the influence of noise also become initial conditions. When simulation results don't match textbook examples, try questioning "what are the true initial conditions?"