Chapter 6
RLC Circuits

m6.1 Initial and Final Conditions

The SPST switch in the circuit of Figure m6.1 opens at t = 0 after it had been closed for a long time. Draw the circuit configurations that appropriately represent the state of the circuit at t = 0^-, t = 0, and t = ∞ and use them to determine:

1. v_C(0), i_C(0), and v_C(∞), and
2. i_L(0), v_L(0), and i_L(∞).

Component values are: R₁ = 680 Ω, R₂ = 100 Ω, R₃ = 100 Ω, switch resistance R_sw = 10 Ω, wire resistance R_w = 10 Ω L = 3.3 mH, C = 0.1 μF, V _s = 4.7 V.

NI Multisim Measurements

Enter the circuit of Figure m6.1 using the SPST switch for interactive simulation. Select Simulate → Interactive Simulation Settings and set “Maximum time step (TMAX)” to 1e-006 to obtain the needed resolution for this circuit.

1. Connect the two-channel oscilloscope to plot v_C(t) and the voltage across resistor R₃; divide by the value of R₃ to obtain the capacitor current i_C(t). Start interactive simulation and adjust the oscilloscope settings to clearly show the two waveforms when the switch opens; operate the switch with the space bar. Choose two different colors for the oscilloscope traces to make them easy to identify. Take cursor measurements to determine v_C(0) and i_C(0) just before the switch opens. Take another cursor measurement to determine v_C(∞). Remember that “t = ∞” means the circuit has settled to it new steady-state value.
2. Reconnect the two-channel oscilloscope to plot v_L(t) and the voltage across resistor R₂; divide by the value of R₂ to obtain the inductor current i_L(t). Repeat the techniques from the previous step to display the two waveforms. Take cursor measurements to determine i_L(0) and v_L(0) just before the switch opens. Take another cursor measurement to determine i_L(∞).

NI myDAQ Measurements

1. Construct the circuit of Figure m6.1 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-closed Switch 3 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.
   • 4.7 volt source created with 5V and DGND. The loading effect of this circuit reduces the unloaded 5-volt source to about 4.7 volts.
   • AO0 (Analog Output 0) to the switch control input of Switch 2.
   • AI0 (Analog Input 0) to display the voltage across the current-sensing resistor R₂ for inductor current or R₃ for capacitor current.
   • AI1 (Analog Input 1) to display the capacitor voltage v_C(t) or inductor voltage v_L(t).
   • Function Generator to create the switch control waveform: choose “Squarewave,” set the peak-to-peak amplitude to 5 V, the offset to 2.5 V, and the frequency to 1 kHz.
   • Oscilloscope to view the current and voltage waveforms. 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.
2. Establish oscilloscope connections to display v_C(t) and i_C(t). Take cursor measurements to determine v_C(0) and i_C(0) just before the switch opens, and also measure v_C(∞).
3. Modify the connections to display v_L(t) and i_L(t). Take cursor measurements to determine i_L(0) and v_L(0) just before the switch opens, and also measure i_L(∞).