March 30 Due April 6 and April 20
Longer lab report with peer editing this time
EGR 220: Engineering Circuit Theory
Lab 7: Second Order Circuits and Resonance
Second-order circuits, consisting of resistors and two, distinct storage elements
(capacitors and/or inductors), play a fundamental role in numerous engineering applications, including communication systems and control systems. The mathematics used to describe second-order circuits is identical to the mathematics that describe dynamic mechanical systems ( e.g., mechanical vibrations), acoustical systems, and many other dynamic systems in nature.
In this set of laboratory experiments, you will analyze the voltage response of a series
RLC circuit to both square wave and sinusoidal input signals . You will see that the response of this circuit depends on the frequency of the input signal, as well as the value of the resistance. In particular, at the frequency defined as the resonant frequency f r
, the magnetic field of the inductor and the electric field of the capacitor exchange relatively large amounts of energy during each cycle of the input signal, resulting in the possibility of surprisingly large output signals.
Note that for this lab you will write a longer lab report than the one-page lab memos you have completed for the previous labs. This will allow you to report everything you do and answer the questions at the end of this handout. Be as concise as possible, while still answering all the questions, and also providing your observations and statement about your learning in the lab (perhaps 2 or 3 pages of text, plus figures and tables).
• For the week of April 6, part of the lab time will be used for peer editing – you must bring THREE copies of your complete lab report (one report per team) to lab
– one for me, and two to be used for peer editing o The other part of lab April 6 will be for Knowledge Building work
• The week of April 13 is Midterm Exam 2, and this lab material will be on the exam, so you want to have all the intellectual work for this lab done before the exam. However…
•
You can wait to turn in your final, completed lab report (including responses to the peer editing) until April 20. That week’s lab time will be entirely for knowledge building work, with the lab report due at the start of lab, at 1:10.
•
Read through the entire lab a few times so that you are certain you know what you will be doing. This lab will take planning and coordination to complete in time.
• Complete all parts labeled “pre-lab.” o The purpose is to make sure you know what you will be doing when you show up to do the lab, and that will take more time this week than for other labs.
•
Bring a jump/flash drive to save the input/output signal displays from the oscilloscope.

• You might also want to bring graph paper – regular and semilog – to help in recording your observations, and making notes to remind yourself what you are saving from the scope.
For the next week, April 6, bring three (3) copies of your lab report to lab period.
One copy is to hand in, and the other two will be shared with classmates. You will each edit two other labs, and have your lab edited by two other people.
The intent is for you to all learn from each other, in terms of what is most effective in the lab reports for explaining what you did, how you did it, and what you learned/conclusions drawn. If you include observations from your peers in your final report, you must cite these source appropriately. (… personal communication with…)
The final report is due April 20. Be sure to have all the intellectual work completed by April 13 as that is the week of midterm exam 2, and this material will be on the midterm exam.
To Begin: Construct and Play With Your Circuit
Construct the circuit shown in Figure 1.
•
Note that you must clip the function generator and oscilloscope probes directly to the circuit element leads, and NOT to extra wires, in order to minimize the ‘parasitic’ inductance in your circuit.
• Also, you need to maintain a common ground for all connections throughout the lab.
This means the power supply and oscilloscope probes need to be connected to the same node at all times, which will require you to move the elements of your series circuit around so that the element being measured is in the far right position, connected directly to the ground node.
R = 2k Ω
L = 47mH
C = 100pF
Figure 1: The element values should be R = 2k Ω , L = 47mH, and C = 100pF. You will use the function generator to produce v
S
both as a square wave and as a sine wave. page 2

• Pre-lab : Using the values stated above for the circuit elements, calculate the circuit’s theoretical resonant frequency f r
(using 2 π f r
= ω
0
) and the damping factor, α , and identify the type of response (in terms of damping).
Task 1a: Observe V c
– at different input frequencies
• Use the function generator for the input signal, and begin with a square wave input of 1Vpp and frequency = 1kHz. o Note that a square wave input is a step input (over and over again) o Therefore, the response of a circuit to the square wave input is the step response .
• Observe both the input and capacitor voltages, v s
and v c
, on the oscilloscope. o Make sure that channel 1 on the scope is connected to the input, v s
, for all measurements during this lab to ensure proper triggering by the scope. o You could initially set the oscilloscope display to be 500mV/div for both channels 1 and 2, and 40µs/div for the time (x-axis). o You will need to change these settings as you change the input frequency .
•
Increase the frequency of the squarewave input signal and observe the change in the magnitude and shape of v c
relative to v s
. o Play : Begin with f s
= 1kHz, and increase f s
by 1kHz until you reach 10kHz. o Increase f s
by 10kHz until you are close to the resonant frequency. Close to f o increase and decrease the frequency by 1kHz or even smaller increments in order to observe the details of the v frequency for v s
at which v c c
waveform at this point. Record the
has the maximum value. o Increase f s
above the resonant frequency until you are comfortable with what is happening. o Observe Details and Clarify Confusions : Go back and forth through all these frequencies a number of times until you can make interesting observations and can draw some conclusions. o Record Results : Save some screen shots of the signals at frequencies: below, close to, equal to, and above the resonant frequency.
• Pre-lab : Using the discussion above, select the specific frequencies you will use for this experiment, with a brief statement explaining why you selected these frequencies
( e.g., what behavior do you expect with each frequency).
Task 1b: Observe Vc – with different series resistors
• Pre-lab : Calculate the resistor values needed for the circuit to have both a critically damped and an overdamped response.
• Overdamped Response: In lab, exchange the 2k Ω resistor with a resistor of the value you calculated for an overdamped response. Using the square wave input frequencies selected above, observe the capacitor and source voltages, v c
and v s
, for each input frequency. If needed, use input frequencies different from those identified in the pre-lab, to allow you to make interesting observations and conclusions. page 3

• Critically Damped Response : Next, repeat the step above with a resistor value that will achieve critical damping, R crit
and compare the response to that obtained with with R overdamp
. o You do not need to find the precise resistor value for R crit
(since we do not know the precise values for L and C), but do find one such that you can observe and confirm the difference between critically- and over-damped responses as discussed in the text and in class .
•
Pre-lab : State briefly , the expected difference in circuit behavior when you have the resistor values for critical and over damped responses. See text pages 323-324.
Using the same circuit, with the 2k Ω resistor, switch to applying a sine wave at the input:
• Use the function generator to generate a sine wave for the input signal. o Note that with a pure sine wave input, with the frequency being changed over a large range of frequency values, the circuit response is referred to as the frequency response . This is discussed in chapter 14 of the text.
• The general plan is that first you will observe the input signal v s
and the voltage across the capacitor ( v c
) simultaneously on the oscilloscope. Note that you can write the sinusoidal waveforms as (1)cos(2 π f t)V. (Note the use of ‘cos’ versus ‘sin’) o Starting at 1kHz, increase the frequency of the input signal by slowly spinning the dial on the function generator. Observe the change in the magnitude of v . c o Notice how and for which input frequencies the phase between the input and output signal shifts as the frequency changes. The phase between two signals is the angular difference between each signal crossing an axis, which is also the angular difference between the peak value of two signals, etc.
•
Make at least one phase measurement not using the measurement menu. Instead make this measurement and calculation manually, measuring the period and phase difference with the cursors à Show this to me before you leave lab o pre-lab : State how you will use the oscilloscope, identifying how you will use the cursors to do this measurement.
TASK 2 DETAILS: Series of Voltage-Phasor Measurements :
Make a series of voltage-phasor measurements, as discussed below, to fill in the table you will create for the pre-lab à to hand in with your lab report. These measurements will be for the underdamped circuit, as defined in figure 1, with the 2k Ω resistor.
After observing the general pattern of circuit behavior with different input frequencies, make this series of voltage phasor measurements . The ‘input frequency’ refers to the frequency of the sine wave input from the function generator. page 4

Using two oscilloscope probes, first measure (or observe on the oscilloscope) v s
and v c
, then observe v s
and v
LC
, then v s
and v
R
, and finally v s
and v
L
. Make all these measurements (observations) for a range of frequencies (the same range).
Measure the magnitude ( i.e., amplitude – from 0V to the peak amplitude) of the voltages
AND measure the phase difference between v s
and the other voltage you are viewing, for the range of frequencies. o When you measure a voltage phasor, you are measuring both the magnitude and the phase angle of all voltage signals à you may want to make all magnitude measurements first, and then all phase measurements, or you can make them all at the same time.
§ Note that the phase angle is defined relative to the phase angle of the input signal.
Which is to say, by convention, we define the phase angle of the voltage source to be zero degrees. o Record the exact resonant frequency, f , that you find experimentally . The resonant frequency is the frequency for which v c
achieves its maximum amplitude.
§ Pre-lab : How do you identify the resonant frequency in terms of the voltage magnitudes? How do you identify the resonant frequency in terms of using the phase difference? Is this resonant frequency different from that found with the step response, above? Explain briefly. o See the end of the lab for NOTES on making the voltage magnitude and phase measurements. You will need to play around with the oscilloscope to make sure you know what you are doing. o Record what you do, make observations and draw conclusions throughout the experiment. Record data, observations and graphs as needed so that you will remember what you observed and to be able to convince a reader of the accuracy and truth of your observations and conclusions. o You will need to refer to this lab material in a few weeks in class, for chapter 14, so be sure to take good notes for yourself. o
: On a separate, clean sheet of paper, create the table that you will fill in with all these voltage magnitude and phase measurements. Your table should have the column (voltage magnitude or phase) and row (input frequencies) headings clearly identified, with the body of the table ready to receive your data.
Note: This lab report will be longer than previous memos, in order to be able to answer the questions below as well as those in the discussion above. Do not copy and paste any text from this lab handout. For the discussion, do not simply answer the questions and report the data requested as a list, referring to the numbers in this handout. Instead, construct a complete, stand-alone report that anyone could read. Write your report so that any reader could understand and be convinced of the objective of the lab, understand the summary of what you did, what you learned, and the accuracy and truth of your conclusions (perhaps for Prof. Ellis, Prof McKahn, or one of your siblings?) page 5

In addition to your data, graphs, results, observations and conclusions from the lab outline above, also include :
1.
Plot the data ( professionally created with Matlab ) from the experimental values for the magnitudes v
C v
S
, v
R v
S
, v
L v
S and v
LC v
S as a function of frequency.
On a separate graph , plot the values for the corresponding phase angles
∠ v
C
, ∠ v
R
, ∠ v
L and ∠ v
LC
(relative to the phase angle of v s
) as a function of frequency. a.
For both the magnitude and the angle plots, use a logarithmic frequency axis (there is a link for semilog graphing paper on the course webpage). b.
Comment upon the graph and what it tells you about the behavior of the circuit.
2.
Explain (in English rather than with mathematical expressions) why v
LC v
S
is a minimum at the resonant frequency. Also explain how it is possible for v
C v
S
to be greater than one.
3.
Explore further the effect of the value of the resistor R on the circuit’s response. a.
What if R = 50,000 Ω instead of R = 2000 Ω ? b.
How does the larger resistance value affect the circuit’s response? c.
How would the maximum value of v
C v
S
be affected (be quantitative)? d.
How would the resonant frequency f o
be affected? e.
Generalizing, what if R → ∞ or what if R → 0? f.
Explain what you mean in complete sentences and with equations.
4.
Include a carefully constructed paragraph that describes resonance to someone who does not know anything about resonance, being certain to talk about energy.
In your description, include examples of resonance that are familiar to you in your everyday life. Also, give at least one example where resonance is desirable and give at least one example where it is undesirable. Include this paragraph in the body of your lab report , where you discuss resonance, not as a separate paragraph . Do Not use an example discussed in class, but rather come up with your own example. Therefore, do not use the example of a swing or a weight on a spring or a simple pendulum.
5.
Include your well-commented Matlab script (or other program code if you use something other than Matlab) as an appendix. page 6

Notes for placing probes and constructing your circuit
• Remember that whichever element(s) you are measuring the voltage across must have one lead as the ground node for the entire circuit.
•
Note that to measure v
L
, you must, in this case only , use an extra piece of wire in the node shared with the grounded inductor lead (unless you have an inductor with long leads). In this situation, there is no other way to connect the inductor to ground, while also having the scope probes connected to the shared ground.
• Note that to measure v r you can simply swap the position of the + v s
and ground probes, such that the “+” clip from the function generator is now clipped to the capacitor and all the ground connections are clipped to the resistor. You do not need to move or relocate any of the RLC elements.
A suitable set of frequencies might include:
• 5 frequencies below f r
( e.g., 1000 Hz, 7500 Hz, 15 kHz, 30 kHz, 0 kHz),
• 5 to 10 frequencies near f r
( e.g., ranging from 60,000 to 80,000 Hz and focusing at your experimentally determined f r
), and o You cannot assume that the resonant frequency estimated from theory is exactly correct. Determine the actual resonant frequency from your measurements.
• 5 frequencies above f r
( e.g., 80 kHz, 100 kHz, 150 kHz, 300 kHz, 600 kHz).
Hints for using the ‘Measurement’ menu on the oscilloscope
• Clear the screen and the existing measurement selections (if any) to be ready for making your measurements o Push the ‘Measure’ button, upper middle of the oscilloscope control panel o Options appear for the buttons along the bottom and right of the screen o Select ‘Remove Measurement’ from the bottom buttons o Select ‘Remove All Measurements’ from the bottom of the options along the right of the screen o Push the small, round ‘Menu off’ button near the bottom right corner of the
‘scope screen, until the screen is cleared of the extra information you do not want to be viewing.
•
Set up the amplitude and/or phase measurements o Push ‘Measure’ o Select “Add Measurement” along the bottom buttons o Select “Measurement Type” from the top option along the right of the ‘scope screen.
§ Use the top knob (which you have used for the cursors) to select the measurement you will make.
§ You will, for example, want to use ‘amplitude,’ ‘phase,’ and possibly others during this lab. o For the source, select the ‘scope channel you want to be measuring o Push the “OK Add Measurement” option at the bottom right page 7

page 8 o If you want to add a second, or third measurement, then go back to measurement type and source, to select what you want, and add the measurement. o For the phase measurement you need to select the ‘source’ and the ‘phase to’ for a single measurement (because the phase is always a relative phase – one phase angle relative to another one)
§ Select the channel with your output voltage (i.e., vc or vR) for the ‘source’ and the channel with the voltage source for the ‘phase to’ (which is to say
– the ‘phase relative to:’). o Push ‘menu off’ as many times as needed in order to clear the screen of unwanted information.