Chapter 5
RC and RL First-Order Circuits

m5.3 Response of the RC Circuit

Figure m5.3a shows a resistor-capacitor circuit with a pair of switches and Figure m5.3a shows the switch opening-closing behavior as a function of time. The initial capacitor voltage is -9 V. Component values are R₁ = 10 kΩ, R₂ = 3.3 kΩ, and R₃ = 2.2 kΩ, C = 1.0 μF, V ₁ = 9 V and V ₂ = -15 V.

1. Determine the equation that describes v(t) over the time range 0 to 50 ms.
2. Plot v(t) over the time range 0 to 50 ms.
3. Determine the values of v(t) at the times 5, 15, 25, 35, and 45 ms.

NI Multisim Measurements

1. Enter the circuit of Figure m5.3a using the following components:
   • VOLTAGE_CONTROLLED_SWITCH
   • ABM_VOLTAGE (Analog Behavioral Modeling) voltage source; use step functions (u(TIME)) to create the switch control waveforms of Figure m5.3b.
   • Capacitor with initial value of -9 volts.
2. Name the net that connects the two switches to the capacitor. Set up a Simulate → Analyses → Transient analysis with the end time set to 0.05 seconds and with “Initial Conditions” set to “User-defined.” Select the “Output” tab and add the capacitor voltage to the list of analysis variables. Run the simulator to plot v(t).
3. Use the oscilloscope cursor to measure the values of v(t) at the times 5, 15, 25, 35, and 45 ms.

NI myDAQ Measurements

1. Construct the circuit of Figure m5.3a using the following components and NI ELVISmx instruments:
   • Two normally-open Switches 1 and 4 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.
   • 9.0 volt source created with the LM317 variable voltage circuit of Figure B.2 in Appendix B.
   • 1.0 μF electrolytic capacitor. Connect the negative terminal of the capacitor to ground.
   • AO0 (Analog Output 0) to the switch control input of Switch 1.
   • AO1 (Analog Output 1) to the switch control input of Switch 4.
   • 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(t); connect AI1+ to the positive side of the electrolytic capacitor and connect AI1- to ground.
   • Arbitrary Waveform Generator to create the switch control waveforms of Figure 5.3b.
   • Oscilloscope to view the Switch 1 control waveform and the capacitor voltage v(t). Adjust the Oscilloscope settings to display the voltage v(t) so that the waveform fills 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.
2. Use the oscilloscope cursor to measure the values of v(t) at the times 5, 5, 25, 35, and 45 ms.

Further Exploration with NI myDAQ

The circuit of Figure m5.3a permits the capacitor to be charged to a desired voltage by closing one of the switches that connects the capacitor to a source. After charging, opening both switches should in principle allow the capacitor to maintain its “charge” (or stored energy) indefinitely. However, the physical capacitor contains a nonideal dielectric material between its plates that allows a slow trickle of current that eventually depletes the stored energy. The nonideal dielectric can be modeled as a resistor in parallel with the capacitor plates.

Devise a method to estimate the value of the equivalent resistance that connects the capacitor plates. Consider the half-life measurement technique of Figure E.1 in Appendix E to measure the time constant. Connect the switch control inputs to DIO0 and DIO1 and use the NI ELVISmx Digital Writer to manually operate the switches. Use NI ELVISmx Oscilloscope to display the capacitor voltage, taking special care to enable only one active oscilloscope channel. Enabling both oscilloscope channels greatly reduces the effective input resistance of the myDAQ analog inputs due to the rapid switching between these channels to a common analog-to-digital converter.