Chapter 5
RC and RL First-Order Circuits

m5.2 Inductors

The voltage v(t) across a 33-mH inductor is given by the sinusoidal pulse waveform shown in Figure m5.2.

1. Determine the equation for the inductor current i(t) and plot it over the time period from 0 to 0.4 ms. Assume zero initial inductor current.
2. Determine the time at which the inductor current reaches its maximum value.
3. Calculate the total peak-to-peak range of inductor current, i.e., the maximum value minus the minimum value.

NI Multisim Measurements

Oscilloscopes display a time-varying voltage as a function of time. The current through a component such as the inductor 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:
   • 33-mH inductor and 10-Ω shunt resistor connected in series
   • Two AC voltage sources (AC_VOLTAGE) connected in series and also connected across the inductor-resistor combination. Flip the orientation of one of the sources so that the series combination forms the difference of the two voltages. Set one voltage for a delay of 0.1 ms and the other for a delay of 0.3 ms; in this way only a single cycle of the sinusoid appears across the pair of sources as in the waveform of Figure m5.2.
   • Two-channel oscilloscope showing the inductor voltage on Channel A and the shunt resistor voltage on Channel B.

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

2. Use the oscilloscope display cursors to measure the time at which the inductor current reaches its maximum value.
3. Use the oscilloscope cursors to determine the total range of inductor current, i.e., the maximum value minus the minimum value.

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 inductor, and connect the inductor-resistor combination between the voltage follower output and ground.

   Establish the following myDAQ connections:

   • AO0 (Analog Output 0) to the voltage follower input,
   • AI0 (Analog Input 0) to display the inductor voltage; connect AI0+ to inductor terminal connected to the op amp output and connect AI0- to the other inductor terminal,
   • AI1 (Analog Input 1) to display the shunt resistor voltage; connect AI1- to ground.

   Create the inductor voltage waveform of Figure m5.2 with the NI ELVISmx Arbitrary Waveform Generator; use 200 kS/s as the sampling rate.

   Adjust the NI ELVISmx Oscilloscope settings to display the inductor 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 center the inductor voltage pulse.

2. Use the oscilloscope display cursors to measure the time at which the inductor current reaches its maximum value; use the same time reference as Figure m5.2.
3. Use the cursors to determine the total range of inductor current, i.e., the maximum value minus the minimum value. Divide the cursor measurement by the shunt resistor value. Improve your measurement accuracy by using the measured shunt resistance obtained from 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.
• Create three waveform segments with the ARB “Waveform Editor,” two of length 0.1 ms on either end and the middle segment of length 0.3 ms. Choose the “Library” option for the middle segment and experiment with parameters until you obtain a single cycle with unit amplitude. For convenience, create the complete waveform with a unit amplitude and then set the “Gain” value of the Arbitrary Waveform Generator to 9.