ENGR225 Circuit Analysis Name_______________________________

**Lab
#7 - Second Order RLC Circuits** Date________________________________

**Objectives:**

To see the actual voltage and current responses in second-order RLC circuits.

To see how our calculations agree with the observed responses.

**Procedures:**

1. Measurement Sequence Part 1

· Use the LCR meter to measure your values for the inductor and capacitor required for the circuit in Figure 1. Note: Use the Ls-Rs and Cs-Rs equivalent circuits.

·
Build the RLC circuit shown in Figure 1. Set up the Wavetek
for a 0-to-8V square wave into the circuit.
Note: **Adjust the Wavetek amplitude before
you connect it to the circuit. Or use the technique that I will explain to you
so you can monitor what in effect is the Thevenin
voltage before the internal 50Ω resistor of the Wavetek.**

· Adjust the frequency of the square wave so that the transient waveforms are allowed to go to their final values after each square wave transition (~ 100Hz).

·
Measure the voltage across the capacitor at **both the rising and the falling** of the
input voltage waveform.

·
After interchanging the capacitor and inductor, measure
the voltage across the inductor at **both
the rising and the falling** of the input voltage waveform.

·
After interchanging the inductor and the 51Ω resistor
R, measure the current waveform at **both
the rising and the falling** of the input voltage waveform by measuring the
voltage across the resistor and then dividing the amplitude scale by 51 to
convert it to current.

·
**Try to keep the Wavetek
frequency and the oscilloscope time scale the same for all of your waveforms so
that they will be easy to compare. **Be
sure to label these six figures as you go along or you may not be able to tell
which are which later.

· Comment on the character of the voltages and current. Do they indicate an over- or under-damped response? Do all of the waveforms exhibit the same basic characteristics? In the first-order RL and RC circuits we looked at last week, we saw that all of the waveforms for a given circuit had a simple exponential form with the same time constant.

2. Calculations and Measurement Sequence Part 2

· Compute the resistance that would need to be added to the circuit to produce a critically-damped response (find the total R needed to make R/2L = 1/ÖLC).

· Add a resistor to the circuit so that the total resistance will be such as to give a critically-damped response.

· Measure the voltage across the capacitor for both transitions of the input. Record these responses. Are they what you expect for a critically damped response?

3. Measurement Sequence Part 3

· Add an additional resistor of 680Ω to the circuit. This should make the response over-damped.

· Again measure the voltage across capacitor for both transitions of the input. Record these responses. Are they what you expect for an over damped response?

4. V(t) Calculations to Compare with Measurements

· Determine the mathematical expressions for the capacitor voltage Vc(t) for the series circuit for only the positive-going transition of the input square wave in each of the 3 cases measured above - under damped, critically damped and over damped. You should neglect the effect of the small capacitor resistance (RCs in the schematic) when determining the voltage across the capacitor.

· Compare the waveforms represented by these expressions with the measured capacitor voltage waveforms. I suggest plotting your mathematical expressions and comparing the time to steady-state (final value) or the peak amplitude and times of final-value crossings to show the agreement.

Figure 1 - RLC Circuit for Lab 7

A graph showing all three Vc(t) responses for comparison