Differential Equation of an Driven RLC Series Circuit

Part I - Building an LC Circuit with external voltage

The Applet is at: http://www.falstad.com/circuit/

Building the Circuit:

Under Circuits, open up a Blank Circuit
Build a series LC circuit using the following components:

Capacitor Inductor AC Source (2-terminal) inductors and capacitors are in Passive components; AC Source is in Inputs/Outputs

Right click on the background; then drag with your left mouse button to draw the component.

sample circuit

Put in the component values specified below for your group:

To Edit right click on the component and type in a value or use the slider. milli=m and micro=u.
Apply it to the component.

The following table gives the values to use for each group: Note units!

Group 1

Group 2

Group 3

Group 4

Group 5

Group 6

AC source at 5V;
use f_r = resonant frequency

Inductance L (mH)

50

25

7.9

3

9.4

5

Capacitance C (mF)

5

10

2

6.9

3

8.1

To get a graph of current I, right click on the capacitor and View in Scope.
Right click on the graph to show which quantities to display.

1a. Sketch a graph of current vs. time at the resonant frequency f_r by doing the following.

Right click on the capacitor and View in Scope.
Right click on the graph and show Current, Peak Value, and Frequency.
Hit the RESET button in the upper right to restart the graph.

(sketch Graph) Show you scales! The scale continuously adjusts so watch the vertical scale carefully.

1b. Repeat using a frequency 4 Hz below the resonant frequency.

(sketch Graph)

1c. Repeat using a frequency 10 Hz above the resonant frequency.

(sketch Graph)

Part II - Building the differential equation for an undamped LC Circuit

The differential equation for an RLC Series Circuit is: *see below for derivation

where the external AC voltage V will be given by: and I(t) is the current in the circuit at time t.
L = Inductance in henries (H); C = capacitance in farads (F); R = Resistance in ohms; Forcing V is in volts; I or I(t) is in amps

Notes on units:
mH is millihenries - multiply by 10^-3; uF = microF is microfarads - multiply by 10^-6;
is the "angular" frequency in radians/sec; f is measured in Hz - cycles/second

2a. Write the differential equation using the external voltage as: using the component values for your group from part I.
Leave omega for now.

2b. Find the roots s, of the homogenous equation and write I_h:

s =

I_{h =}

2c. What value of the angular frequency (omega) gives resonance?

2d. Write the differential equation using the value of omega you calculated.

s* =

I_p =

2e. Solve your differential equation I(t) = I_h + I_p assuming initial conditions I(0) = 0; I'(0) = 0

2f. Graph the solution I(t) of your differential equation.
Note that the frequency is high and the period is small (milliseconds); to get a good graph your t range will need to be small also.

(sketch Graph) Show you scales! The scale continuously adjusts so watch the vertical scale carefully.

2g. From omega, find the value of the frequency f in Hz. From frequency f find the period T in milliseconds.

Compare to the period of the applet graph.

Part III - Building a damped RLC Circuit

3a. Add a resistor, R = 1 Ohm, to build an LRC series circuit.
Sketch a graph of capacitor current vs. time using the same values L, C, and f, as your resonant circuit in part I.

(sketch Graph) Show you scales! The scale continuously adjusts so watch the vertical scale carefully.

Part IV - Building the differential equation for a damped RLC Circuit

Differential equation for and RLC series circuit with external voltage:

4a. Write the differential equation using your group's values with R = 1 Ohm using omega and V₀ from part I.

4b. Find the roots s, of the homogeneous equation and write s* (omega) of the forcing function. Then write I_hand I_p:

s = s =*

I_{h =}I_{p =}

4c. Solve the differential equation finding the solution I(t) assuming initial conditions I(0) = 0; I'(0) = 0

4d. Graph the solution I(t) of your differential equation.

(sketch Graph) Show you scales! The scale continuously adjusts so watch the vertical scale carefully.

* Derivation: From Kirchoff's Laws in terms of charge q for a series LRC circuit with an applied voltage: L q'' + R q' + q/C = V_ext
I = q' so I' = q'' hence LI' + R I + q/C = V_ext

Taking derivatives, substituting I = q' and dividing by L: