This applet explores the differential equation modeling an LRC circuit with a driving voltage. The differential equation is \[Lx'' + Rx' + \frac{x}{C} = f'(t)\]

Here \(x(t)\) represents the current and \(f(t)\) is the impressed (driving) voltage in the circuit. \(L,\, R,\, C\) are the inductance, resistance and capacitance respectively. We assume compatible units

In this model we consider the function \(f(t)\) to be the input to the system. It is given as a sum of sinusoids as described below.

Graphs
Both the input and the response are plotted in the large graphing graphing window on the left. The linear gain of the system is plotted in the graphing window on the right.

Setting system parameters
The system parameters \(L,\, R,\, C\) are all settable using their respective sliders.

Input modes
There are two modes of input which are controlled by the N terms mode checkbox.

• When N terms mode is checked:
  • The input \(f(t)\) is a finite sum of \(N\) sine terms with base frequency \(\omega\). \[f(t) = c\,\sin(\omega t) + c\,\sin(2\omega t) + c\,\sin(3\omega t) + \ldots + c\,\sin(N\omega t) = c\,\sum_{k=1}^N\sin(k\omega t)\]
  • Note that all the terms have the same amplitude \(c\).
  • The input parameters are controlled as follows:
    \(N\) and \(c\) are set using respective sliders.
    \(\omega\) is set by dragging the small 'knob' on the \(\omega\) axis of the gain graphing window (right hand window). The value of \(\omega\) and the corresponding gain \(g\) are displayed below the graphing window.
• When N terms mode is unchecked:
  • The input \(f(t)\) is a sum of two sine terms with independent frequencies \(\omega_1\) and \(\omega_2\) \[ f(t) = c_1\sin(\omega_1 t) + c_2\sin(\omega_2 t). \]
  • Note that the terms have independent amplitudes \(c_1\) and \(c_2\).
  • The input parameters \(c_1,\, c_2,\, \omega_1,\, \omega_2\) are controlled as follows:
    \(c_1\) and \(c_2\) are set using respective sliders.
    \(\omega_1\) and \(\omega_2\) are set by dragging the small 'knobs' on the \(\omega\) axis of the gain graphing window (right hand window). The values of \(\omega_1, \, \omega_2\) and the corresponding gains \(g_1,\, g_2\) are displayed below the graphing window.

Time
The time \(t\) can be changed by sliding the small 'knob' on the \(t\) axis of the time graphing window (left hand window). The values of \(t\), the input \(f(t)\) and output \(x(t)\) are all displayed below the graphing window.

Graph value readouts
For both graphs: if your mouse is in the graphing window the coordinates of the mouse position are displayed above the graph.

© 2015-2024 J. Orloff, J.-M. Claus