Chapter 6
RLC Circuits

m6.3 s-Domain Circuit Analysis

Determine v(t) of the circuit shown in Figure m6.3 for t ≥ 0, given that the switch is opened at t = 0 after having been closed for a long time. Use the following component values: V _src = 8 V, R₁ = 470 Ω, R₂ = 100 Ω, R_w = 90 Ω, C = 1.0 μF, and L = 33 mH.

1. Plot v(t) from 0 to 5 ms using a tool such as MathScript or MATLAB. Include hardcopy of the script used to create the plot.
2. Determine the following values for v(t):
   • Initial value v(0),
   • Final value of v(t),
   • Minimum value of v(t), and
   • Time to reach the minimum value of v(t).

NI Multisim Measurements

1. Enter the circuit of Figure m6.3 using the same component values listed in the problem statement. Implement the switch with a
   VOLTAGE_CONTROLLED_SWITCH operated by a PULSE_VOLTAGE source configured to open the switch at time 1 ms; this delay makes the initial transition easier to see.
2. Plot v(t) from 0 to 5 ms with a Simulate → Analyses → Transient analysis.
3. Use the Grapher View cursors to measure the following values for v(t):
   1. Initial value v(0),
   2. Final value of v(t),
   3. Minimum value of v(t), and
   4. Time to reach the minimum value of v(t).

Helpful tip for this problem:

• Remember that Multisim voltages are all node voltages, i.e., a voltage with respect to ground. The voltage v(t) in this problem exists between two nodes, however. Name the nets on either side of R₂ (or display their default net numbers), click “Add expression” in the “Output” tab of the transient analysis setup panel, and enter an expression of the form “v(pos)-v(neg)” where pos and neg denote the two net names that connect to R₂ with positive and negative polarity; the expression forms the mathematical difference between the two node voltages.

NI myDAQ Measurements

1. Construct the circuit of Figure m6.3 using the following components and NI ELVISmx instruments:
   • One normally-closed Switch 2 contained in the Intersil DG413 quad analog switch described in Appendix D.
   • 8.0 volt source created with the LM317 variable voltage circuit of Figure B.2 in Appendix B.
   • 1.0 μF electrolytic capacitor. IMPORTANT: Observe proper polarity of the capacitor by connecting the negative terminal of the capacitor to ground.
   • AIO0 (Analog Output 0) to the “Logic Control” (switch control) input for Switch 2.
   • AI0 (Analog Input 0) to display the switch control voltage for Switch 2; connect AI0+ to the switch control input and connect AI0- to ground.
   • AI1 (Analog Input 1) to display v(t).
   • Function Generator to create the switch control waveform; set the frequency to 100 Hz and adjust the amplitude and offset to place the control waveform between 0 and 5 volts.
   • Oscilloscope to view the Switch 2 control waveform and the voltage v_t(t).
2. Display v(t) from 0 to 5 ms.
3. Use the oscilloscope cursor to measure the following values for v(t):
   1. Initial value v(0),
   2. Final value of v(t),
   3. Minimum value of v(t), and
   4. Time to reach the minimum value of v(t).

Further Exploration with NI myDAQ

Switching circuits such as the one in this problem generally demand high current from the power supply for brief periods of time. These high-current pulses can cause spikes on the supply line that could disrupt proper operation of other connected devices such as digital microcontrollers. Connecting a capacitor between the power supply rail and ground provides a local supply of temporary current for the switching circuit to stabilize the power supply rail for other devices.

1. Observe the V _SRC rail created by the LM317 on the oscilloscope; it should still be set to 8.0 V. Estimate the magnitude of the voltage spike and express its value as a percentage of 8.0 V.
2. Continue to observe the power supply rail as you connect a 10 μF capacitor between the V _SRC rail and ground; place the capacitor in close proximity to the switching circuit and remember to observe its polarity. Discuss the improvement in the stability of the power supply rail.