SERIES RLC CIRCUIT

The parameters of a sinusoidally driven RLC circuit are controlled by sliders at lower right: the resistance \(R\), inductance \(L\), capacitance \(C\), maximum voltage \(V_0\) of the source, and circular frequency \(\omega\). The selected value is read out to the right of the slider. The source voltage is given by \(V = V_0\sin(\omega t)\).

Check boxes to the left of the sliders toggle the presence of a resistor, an inductance coil, and a capacitor. Further to the right are check boxes which toggle the representation of the voltage drops across the resistor, the coil, and the capacitor, the voltage increase across the voltage source, and the current through the circuit. Each box admits three settings: black, which leaves the quantity unrepresented; pale gray, which causes the quantity to be represented by the same color; and vivid yellow (for \(V_R\)), orange (for \(V_L\)), red (for \(V_C\)), cyan (for the source voltage), or green (for the current).

The representation occurs in three places. At bottom left, the circuit is represented. When the circuit element is removed using the checkbox, its representation in the circuit diagram is removed. When the value of the voltage drop or increase is requested, leads appear in the appropriate location and a two headed arrow, of the appropriate color, indicates the gap across which the voltage is measured.

At upper right there is a graphing window, showing the graphs of the various quantities as functions of time. Use the slider handle below the graphing window to select a value of t. Animate the system using the [>>] key. During the animation the key changes to [||], and selecting it will stop the animation. At the end of the animation the key changes to [<<], which resets \(t\) to zero. A scale of volts appears at left, and (if toggled) a scale of Amperes appears at right. A cursor rollover creates a set of crosshairs and a readout of the time and "\(x\)" (voltage, and amperage if appropriate) values.

The [Phasor diagram] key toggles display of a phasor diagram at upper left. Once selected, it gives a third representation of the voltage drops and the current. The vectors are absent, ghosted, or colored, according to the selections made at lower right. The vectors rotate as time increases. The imaginary parts of the phasor complex numbers are the corresponding values of the system functions. This relationship is indicated by ghosted horizontal lines which are toggled by the [Relate diagrams] key.

The equations controlling the voltage drops are:

\[L V_C'' + R V_C' + \frac{1}{C} V_C = \frac{1}{C} V\]

\[L V_R'' + R V_R' + \frac{1}{C} V_R = R V'\]

\[L V_L'' + R V_L' + \frac{1}{C} V_L = L V''\]

The three voltage drops are related to \(V\), the voltage increase across the source, by

\[V_L + V_R + V_C = V\]

The current \(I\) satisfies the equation:

\[L I'' + R I' + \frac{1}{C} I = V'\]

In all cases, only the steady state, periodic responses are shown.

© 2007 P. Dourmashkin, H. Miller and J.-M. Claus