Chapter 6
RLC Circuits

m6.4 Step Response

For the circuit of Fig. m6.4:

  1. Determine the transfer function H(s) = Vo(s)/ Vs(s). Write the transfer function in simplified standard form with symbolic values.
  2. Determine the output response vo(t) to the input vs(t) = 4u(t) by working in the Laplace domain. Assume the capacitor is initially discharged.
  3. Plot vs(t) and vo(t) on the same graph from 0 to 5 ms using a tool such as MathScript or MATLAB for R = 5.6 kΩ and C = 0.1 μF. Include a hardcopy of the script used to create the plot.
  4. Determine the following values for vo(t):

PIC


Figure m6.4: Circuit for Problem m6.4

NI Multisim Measurements

  1. Enter the circuit of Figure m6.4 using the same component values listed in the problem statement. Drive the circuit input with a PULSE_VOLTAGE source configured to produce vs(t) = 4u(t). Delay the pulse by 1 ms to make the initial step transition visible.
  2. Plot vs(t) and vo(t) on the same graph from 0 to 5 ms with a Simulate → Analyses → Transient analysis.
  3. Use the Grapher View cursors to measure the following values for vo(t):
    1. Initial value vo(0+),
    2. Time to reach 50% of the initial value, and
    3. Final value.
NI Multisim video tutorials:

NI myDAQ Measurements

  1. Build the circuit of Figure m6.4 using the same component values listed in the problem statement. Drive the circuit input with AO0 and use the ELVISmx Function Generator to produce a zero-to-four volt step transition with a period of 10 ms. Monitor the input voltage vs(t) with AI0 and the output voltage vo(t) with AI1.
  2. Display vs(t) and vo(t) on the ELVISmx Oscilloscope.
  3. Use the oscilloscope cursors to measure the following:
    1. Initial value vo(0+),
    2. Time to reach 50% of the initial value, and
    3. Final value.
NI myDAQ video tutorials:

Further Exploration with NI myDAQ

The circuit in this problem represents one implementation of an all-pass filter. Set up the ELVISmx Bode Analyzer to measure the frequency response of the circuit; the necessary myDAQ connections should already be in place. Set up the Bode analyzer controls as follows:

After running the frequency sweep set the “Gain” axis range to a minimum of zero and a maximum of 2; double-click the numerical values at the top and bottom of the axis display to set these values.

Study the response and then discuss the following questions:

  1. Why is the circuit called an “all-pass” filter?
  2. What is the general behavior of the phase response? More specifically, what are the maximum and minimum values of phase shift?
  3. Use the cursor to measure the frequency at the midpoint between the maximum and minimum phase shift values. Compare this frequency to the critical frequencies in the transfer function H(s) you derived in the analytical section. Hint: Remember to account for angular frequency versus oscillation frequency.