Chapter 6
RLC Circuits

m6.2 Natural Response of the Series RLC Circuit

The SPST switch in the circuit of Figure m6.2 opens at t = 0 after it had been closed for a long time.

1. Determine v_C(t) for t ≥ 0.
2. Plot v_C(t) over the time range 0 ≤ t ≤ 1 ms with a plotting tool such as MathScript or MATLAB.
3. Determine the following numerical values; use either the equation v_C(t) or take cursor measurements from the plot you created in the previous step:
   • Initial voltage v_C,
   • v_C(0),
   • Maximum value of v_C,
   • Damped oscillation frequency f_d = ω_d/2π in Hz, and
   • Damping coefficient α.

Use these component values: R₁ = 220 Ω, R₂ = 330 Ω, L = 33 mH, C = 0.01 μF, and V _src = 3.0 V.

NI Multisim Measurements

1. Enter the circuit of Figure m6.2 using the SPST switch for interactive simulation. Select Simulate → Interactive Simulation Settings and set “Maximum time step (TMAX)” to 1e-007 to obtain the needed resolution for this circuit.
2. Connect the two-channel oscilloscope to plot v_C(t) over the time range 0 ≤ t ≤ 1 ms.
3. Determine the following numerical values with cursor measurements:
   • Initial voltage v_C(0) just before the switch opens,
   • Minimum value of v_C,
   • Maximum value of v_C,
   • Damped oscillation frequency f_d in Hz, and
   • Damping coefficient α.

Additional Multisim tips:

• Measure the damped oscillation frequency by using the cursors to measure the time between an integer number of oscillation cycles; zero crossings are the easiest to identify. Determine the measured oscillation period T and then take the reciprocal of this value for the oscillation frequency.
• Measure the damping coefficient with the following procedure:
  1. Place a cursor at the first peak value after the transient begins; choose either a positive peak or a negative peak,
  2. Place a second cursor at a peak value several cycles after the first peak; choose the same type of peak (positive or negative) as you did in the previous step,
  3. Record the voltage of the first peak as V ₁,
  4. Record the voltage of the second peak as V ₂,
  5. Measure the time difference between the two cursors and record its value as T₁₂, and
  6. Calculate α = ln(V ₁/V ₂)/T₁₂.

NI myDAQ Measurements

1. Construct the circuit of Figure m6.2 using the following components and NI ELVISmx instruments (do not place resistors R_sw and R_w because they simply model the finite resistance of the analog switch and the inductor):
   • Normally-open Switch 1 contained in the Intersil DG413 quad analog switch described in Appendix D. Refer to the pinout diagram of Figure D.1 and connect power according to the photograph of Figure D.2.
   • 3.0 volt source created with the LM317 variable voltage circuit of Figure B.2 in Appendix B.
   • AO0 (Analog Output 0) to the switch control input of Switch 1.
   • AI0 (Analog Input 0) to display the switch control voltage for Switch 1; connect AI0+ to the switch control input and connect AI0- to ground.
   • AI1 (Analog Input 1) to display the capacitor voltage v_C(t).
   • Function Generator to create the switch control waveform: choose “Squarewave,” set the peak-to-peak amplitude to 5 V and the offset to 2.5 V.
   • Oscilloscope to view the switch control and capacitor voltage waveforms.
2. Display the switch control waveform and v_C(t) over the range 0 ≤ t ≤ 1 ms. Adjust the oscilloscope settings to display the voltage waveforms filling a reasonable amount of the available display. Use a combination of edge triggering and the “Horizontal Position” control. You may find it helpful to set the “Acquisition Mode” to “Run Once” and then click the “Run” button repeatedly until you capture a good trace. Choose a function generator frequency that allows the natural response to occupy most of the display.
3. Determine the following numerical values with cursor measurements:
   • Initial voltage v_C(0) just before the switch opens,
   • Minimum value of v_C,
   • Maximum value of v_C,
   • Damped oscillation frequency f_d in Hz, and
   • Damping coefficient α.

   NOTE: Do not expect close agreement with your earlier analytical and simulation results. Refer to the “Further Exploration” section below to learn why and the steps you can take to achieve closer agreement.

Further Exploration with NI myDAQ

The finite wire resistance of the physical inductor significantly contributes to the total resistance of the series-connected loop. In fact, you should have observed an unusually large mismatch between the physical circuit measurements and the analytical as well as simulated results. Try reducing the 330-ohm resistor value to account for the inductor resistance and obtain closer agreement between the physical circuit and the mathematical models.

Measure and record the resistance of your inductor with the DMM ohmmeter. How does this value compare on a percentage basis with the 330-ohm resistor?

Next, connect a 10K potentiometer in parallel with the 330-ohm resistor. Measure the resistance of this combination in series with the inductor; remember to disconnect the other circuit elements. Adjust the potentiometer until the total measured resistance is 330 ohms. Reconnect your original series RLC circuit including the potentiometer.

Repeat your earlier cursor measurements for initial voltage, minimum and maximum values, damped oscillation frequency, and damping coefficient. Discuss the degree of improved match between the physical circuit and its mathematical model.