Chapter 5
RC and RL First-Order Circuits

m5.1 Capacitors

The voltage v(t) across a 10-μF capacitor is given by the waveform shown in Figure m5.1.

1. Determine the equation for the capacitor current i(t) and plot it over the time 0 to 50 ms.
2. Calculate the values of capacitor current at times 0, 25, and 30 ms.

NI Multisim Measurements

Oscilloscopes display a time-varying voltage as a function of time. The current through a component such as the capacitor in this problem can also be displayed on an oscilloscope with a small-valued “shunt resistor” placed in series with the component. The shunt resistor produces a proportional voltage according to Ohm’s Law v = Ri where the resistance R serves as the proportionality constant. A trade-off exists here: a small shunt resistance minimizes disruption to the surrounding circuit, but a large shunt resistance maximizes the available signal to the oscilloscope.

1. Enter a circuit that contains the following components:
   • 10-μF capacitor and 10-Ω shunt resistor connected in series
   • ABM (Analog Behavioral Modeling) voltage source (ABM_VOLTAGE) connected across the capacitor-resistor combination; set up the voltage value to match the waveform of Figure m5.1.
   • Two-channel oscilloscope showing the capacitor voltage on Channel A and the shunt resistor voltage on Channel B.

   Run interactive simulation, adjusting the oscilloscope settings to display the capacitor voltage and current with each waveform filling a reasonable amount of the available display.

2. Use the oscilloscope display cursors to measure the capacitor current at times 0, 25, and 30 ms. Divide the cursor measurement by the shunt resistor value.

Additional Multisim tips for this problem:

• Build the ABM voltage source “Voltage Value” string by combining the following functions:
  • u(TIME) – Step function u(t)
  • uramp(TIME) – Ramp function r(t)
  • exp(TIME) – Exponential function e^t

  For example, the string 800⋆(uramp(TIME) - uramp(TIME-0.01)) implements the first 20 milliseconds of the capacitor voltage waveform.

• Place a DC voltage source someplace on the schematic sheet to enable interactive simulation. Do not connect the DC source to the capacitor circuit itself.

NI myDAQ Measurements

The myDAQ analog outputs AO0 and AO1 cannot source more than 2 mA and still maintain the expected voltage output. Use an op amp voltage follower (see Ulaby Section 4-7) to create a “strengthened” copy of the myDAQ analog output.

1. Construct a circuit similar to the Multisim circuit you created earlier, i.e., place a 10-ohm shunt resistor in series with the capacitor, and connect the capacitor-resistor combination between the voltage follower output and ground.

   IMPORTANT: Electrolytic capacitors are polarized; ensure that the positive-labeled terminal connects to the op amp output. Alternatively, if the capacitor is marked with a negative-labeled terminal, connect this terminal to the shunt resistor.

   Establish the following myDAQ connections:

   • AO0 (Analog Output 0) to the voltage follower input,
   • AI0 (Analog Input 0) to display the capacitor voltage; connect AI0+ to the positive capacitor terminal and AI0- to the negative capacitor terminal,
   • AI1 (Analog Input 1) to display the shunt resistor voltage; connect AI1- to ground.

   Create the capacitor voltage waveform of Figure m5.1 with the NI ELVISmx Arbitrary Waveform Generator; use 50 kS/s as the sampling rate.

   Adjust the NI ELVISmx Oscilloscope settings to display the capacitor voltage and current with each waveform filling a reasonable amount of the available display. Use a combination of edge triggering on Channel 0 and the “Horizontal Position” control to place the upper left corner of the voltage waveform at time 10 ms.

2. Use the oscilloscope display cursors to measure the capacitor current at times 0, 25, and 30 ms. Divide the cursor measurement by the shunt resistor value. Improve your measurement accuracy by using the measured shunt resistance obtained by the myDAQ DMM ohmmeter.

Additional tips for this problem:

• Use the Texas Instruments TL072 op amp described in Appendix C for the voltage follower. Follow the pinout diagram of Figure C.1 for either of the two available op amps in the package. You may also use an equivalent dual-supply op amp.
• Power the op amp with myDAQ +15V to V _CC+ and -15V to V _CC-. Use AGND for the circuit ground.
• Use Cursor 1 to take measurements on the shunt resistor voltage. Leave Cursor 2 at time zero to make the “dT” indicator show time directly.
• The shunt resistor voltage signal is relatively low amplitude. Expect the waveform to jump vertically somewhat. Simply click the oscilloscope “Stop” button to freeze the display.
• Create three waveform segments with the ARB “Waveform Editor,” two of length 10 ms and the third of length 30 ms. Choose the “Expression” option for each segment and enter expressions that include the time variable t as needed. Set the “X Range From” value of the 30 ms segment to 0.03 seconds and set “Cycles” to 1. You can then enter the exponential function exp() as shown in Figure m5.1. For convenience, create the complete waveform with a unit amplitude and then adjust the “Gain” value of the Arbitrary Waveform Generator to 8.