Experimental Methods. Transient Behavior Analysis

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Experiment 6: Transient Behavior Analysis
Team Report Due: Fri. 4/30
Abstract
1. Introduction
In this experiment, transient system behavior is investigated for both first- and second-order
systems. Two types of first-order systems are investigated: thermal and electrical. In the thermal
system, the transient response of a thermocouple is examined. For the electrical system, a
resistor-capacitor (RC) circuit (see Experiment #5) is examined. For the second-order system, a
single system is investigated: an inductor-resistor-capacitor (LRC) circuit.
In each of the three systems, a step input is used for excitation to investigate the transient
responses. For each system, the transient behavior is measured experimentally and used to
determine time domain parameters which quantify the transient behavior; these include the time
constant, rise time, maximum overshoot percentage, peak time, and settling time. For the first
order systems, the system’s time constant and frequency bandwidth are also estimated.
2. Theory
3. Methods
4. Results
Figure #. Covered thermocouple vs time.
Figure #. Uncovered thermocouple vs time.
0
y = -1.4555x + 0.1083
R² = 1
-1
-2
ln(G)
-3
-4
-5
-6
-7
-8
0
1
2
3
4
5
Time (s)
Figure #. Natural log of Error fraction vs time for covered thermocouple.
0.5
y = -7.5101x + 0.0969
R² = 0.9944
0
ln(G)
-0.5
-1
-1.5
-2
-2.5
0
0.05
0.1
0.15
0.2
0.25
0.3
Time (s)
Figure #. Natural log of Error fraction vs time for uncovered thermocouple.
Table #. Time constants and frequency bandwidths for both thermocouples.
Thermocouple Time Constant (s)
Covered
0.69
Uncovered
0.13
Frequency Bandwidth (Hz)
0 = f = 0.231
0 = f = 1.224
Table #. Circuit properties and their uncertainties.
Circuit Type
RC
RC
LRC
LRC
LRC
Variable
Resistance (O)
Capacitance (F)
Resistance (O)
Capacitance (F)
Inductance (H)
Value
440
5.94E-06
440
7.00E-08
4.70E-01
Uncertainty
±13.2
±1.78E-08
±13.2
±2.10E-10
±2.35E-02
Table #. Theoretical time constant and its total uncertainty.
t (s)
(?t/?R*uR)2 (s)
(?t/?C*uC)2 (s)
ut (s)
0.0026
6.15E-09
6.15E-11
±7.88E-05
1.20E+00
1.00E+00
Signal (V)
8.00E-01
6.00E-01
4.00E-01
Input
2.00E-01
Output
0.00E+00
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (s)
Figure #. Input and output signal vs time for RC circuit of 8 cycles.
1.20E+00
1.00E+00
Signal (V)
8.00E-01
6.00E-01
4.00E-01
Input
2.00E-01
Output
0.00E+00
0
0.02
0.04
0.06
0.08
0.1
Time (s)
Figure #. Falling and rising edge for RC circuit of 1 cycle.
0.12
Table #. Time constant and response time for falling and rising edge.
Edge
Falling
Time constant (s)
0.0033
Response Time (s)
0.0076
Rising
0.0022
0.0051
Table #. Theoretical natural frequency and its total uncertainty.
fn (Hz)
877.449
(?fn/?C*uC)2 (Hz)
(?fn/?L*uL)2 (Hz)
1.732
481.199
ufn (Hz)
±21.976
1.20E+00
1.00E+00
8.00E-01
Signal (V)
6.00E-01
4.00E-01
2.00E-01
0.00E+00
-2.00E-01
Input
Output
-4.00E-01
-6.00E-01
0.00E+00
1.00E-03
2.00E-03
3.00E-03
4.00E-03
5.00E-03
Time (s)
Figure #. Input and output signal vs time for LRC circuit.
Table #. Experimental natural frequency for LRC circuit.
Td (s)
fd (Hz)
d
?
fn (Hz)
6.00E-03
8.92E-04
1121.08
1.492
0.231
1152.24
Table #. Theoretical parameters for LRC circuit.
Tr (s)
2.48E-04
Tp (s)
4.46E-04
OS%
47.43
Ts,10 (s)
1.38E-03
Table #. Percent error for LRC circuit natural frequency.
Theoretical Frequency (Hz)
877.45
Experimental Frequency (Hz)
1152.24
Percent Error (%)
31.3
Table #. Percent error for RC circuit time constant.
Theoretical time constant (s)
0.0026
Experimental time constant (s)
0.0028
Percent Error (%)
7.7
5. Discussion
Important. (regarding audience of the report)
The audience of this report is intended for a bright mechanical engineering student who is just
finished a course in Mechanical Vibrations and Control Systems. The audience is familiar with
the terms: natural frequency, damped natural frequency, damping ratio, logarithmic decrement,
and a mass-spring-damper system.
Part A: Thermal System Transient Response
1. Discuss how you determined the range of data that was used to measure the time constants for
the thermocouples. How important is it to select the appropriate range of data, i.e. how does it
affect your results? 2. Discuss the differences of the time constants and how this relates to their
bandwidth. Compare the covered to the uncovered thermocouple and if they are different,
explain why.
Part B: Electrical System Transient Response
1. (RC circuit) Compare (% error) the experimental time constants for falling and rising edges.
How did you predict they would compare? Compare (% error) and discuss the differences
between the theoretical and average experimental time constants (average the time constants for
the falling edge and rising edge). Are your experimental values within uncertainty bounds?
Page 16 of 17
2. (LRC circuit) Compare (% error) and discuss the differences between the theoretical and
experimental natural frequencies of the system. Are your experimental values within the
uncertainty bounds?
3. (LRC circuit) If the damping in the system is suddenly half of its original value, what is the
effect of on the system’s output signal with time?
6. Conclusion
7. References
Appendix A: Sample Calculations
The following sample calculations are for Part A:
?(?) – ?8 3.68 – 1.92 × 10-7 ?
?(?) =
=
=1
?0 – ?8
3.68 – 1.92 × 10-7 ?
?=-
1
-1
=
= 0.69 ?
????? -1.45
?=-
1
-1
=
= 0.13 ?
????? -7.51
?=
1
= 0.231 ??
2?(0.69)
?=
1
= 1.224 ??
2?(0.13)
The following sample calculations are for Part B:
? = ?? = 440?(5.94 × 10-6 ?) = 0.0026 ?
??
=?
??
??
=?
??
?
?
?
?
??
??
?? = ±v( * ?? ) + ( * ?? ) = ±v((5.94 × 10-6 ?)13.2?) + ((440?)1.782 × 10-8 ?)
??
??
?? = ±7.88 × 10-5 ?
?? =
1
1
1
1
v =
v
= 877.45 ??
-1
2? ?? 2? (4.7 × 10 ?)(7 × 10-8 ?)
90% = (1 – ?
-?90%
? )
× 100
-?90%
. 90 = (1 – ? 0.0033? )
-?90%
. 1 = ? 0.0033?
-2.3025 =
-?90%
0.0033?
?90% = 0.0076 ?
???
=-
??
1
1
4??v?? ? 2
???
=-
??
1
1
4?? v?? ?2
?
?
???
???
??? = ±v(
* ?? ) + (
* ?? )
??
??
???
?
1


(
(
* (2.10 × 10-10 )
1
4?(4.7 × 10-1 ?)v
(4.7 × 10-1 ?)(7 × 10-8 ?)
(7 × 10-8 ?)2
)
)
?
+
v (
1

1
(4.7 × 10-1 ?)2
4?(7 × 10-8 ?)v
-1
-8
(
)(7
(
4.7 × 10 ?
× 10 ?)
)
* (2.35 × 10-2 )
??? = ±21.98 ??
?? =
??+?- ?? 5.9 × 10-3 – 1.44 × 10-4
=
= 8.92 × 10-4 ?
?
5
?? =
1
1
=
= 1121.08 ??
?? 8.92 × 10-4 ?
1
??
1 4.77 × 10-1 ?
d = ln(
) = ln(
) = 1.492
? ??+?
4 1.22 × 10-3 ?
?=
d
v4? 2 + d2
=
1.492
v4? 2 + (1.492)2
= 0.231
)
?? =
??
v1 – ?2
?? =
=
1121.08 ??
v1 – (0.231)2
= 1152.24 ??
2?
2?
???
=
= 7043.93
-4
?? 8.92 × 10 ?
?
???
7043.93 ?
???
?? =
=
= 7239.74
?
v1 – ?2 v1 – (0.231)2
??
1.8
1.8
=
= 2.48 × 10-4 ?
?? 7239.74 ???
?
?
?
?? =
=
= 4.46 × 10-4 ?
?? 7043.93 ???
?
?? =
??% = ?
??,10% =
% ????? =

?p
v1-?2
× 100 = ?

0.231p
v1-(0.231)2
× 100 = 47.43 %
ln(0.10)
2.30
=
= 1.38 × 10-3 ?
???
???
0.231(7239.74 ? )
???????????? – ?h????
1152.24 ?? – 877.45 ??
× 100 =
× 100 = 31.3 %
?h???
877.45 ??
% ????? =
???????????? – ?h????
0.0028 ? – 0.0026 ?
× 100 =
× 100 = 7.7 %
?h???
0.0026 ?
Appendix B: Extra Figures/Tables
Appendix C: Contributions
Page 2 of 17
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3
1/4” Flat head screw driver
Data acquisition system
36” M-M Mini-B to A USB 1.0 cable
Laboratory computer with the following software:
o Windaq Data Acquisition Software
o Windaq Dashboard
o Microsoft Excel
K type thermocouple, uncovered tip
K type thermocouple, covered tip
Thermocouple connector
Thermocouple amplifier
Ice bath reservoir
Ice bath containment pan
Ice
Cooler
Experimental Procedure
During the experiment, be sure you are not leaning or bumping into the tables. Caution: be
careful taking data; part of your grade depends on how accurately your team can take
measurements. Not having good data can affect your report grade. Come prepared to lab!
Read the lab manual and watch the video on Blackboard prior to lab.
Prior to your laboratory session, all instrumentation is checked for function and accuracy. Record
the temperature and relative humidity at the start and the end of your experiment to ensure stable
room conditions (this is for your reference-do not include in the report). Give a copy of your
data to the TA at the end of the experiment.
Refer to Lab Manual #5 Filtering for screenshots on how to configure the DAQ (Data
Acquisition System) to record data from the function generator if you need help. Note that your
settings are different for this experiment than Experiment #5-be sure to read carefully and
follow each step in the manual to configure the DAQ properly.
3.1
Part A: Thermal System Transient Response
The transient thermal response is investigated in Part A using a thermocouple. A diagram of the
electrical setup for Part A is shown in Figure 2 and a more detailed illustration of the
thermocouple and the ice bath is shown in Figure 3.
Page 3 of 17
Figure 2. Diagram of the thermocouple system used in Part A.
Figure 3. Diagram of the ice bath and thermocouple setup for Part A.
Page 4 of 17
To set up the experiment and configure the data acquisition system for Part A: Thermal
Systems,
1. Use two 18” banana plug wires to connect the thermocouple amplifier’s white “VO-“ and
blue “VO+” output terminals to data acquisition system channel one “+” and “–”
terminals respectively.
2. Connect the thermocouple connector to thermocouple amplifier via the 3.5mm TRS
terminal labeled “TC”.
3. Connect the thermocouple amplifier’s red Vi+ and black Vi- terminals to the DC power
supply’s FIXED 5V terminals via the 36” long M-M dual banana plug coaxial cable.
a. The body of the dual banana plug has a nub indicating the ground plug.
b. Connect the one end of the dual banana plug coaxial cable to the 5V power
supply, inserting the ground plug to the BLACK (-) 5V jack.
c. Connect the other end of the dual banana plug coaxial cable to thermocouple
amplifier, inserting the ground plug into the BLACK (Vi-) jack.
4. Power on the power supply by pressing the power supply’s power switch.
5. Connect the covered K type thermocouple to the thermocouple connector.
a. Once connected, the red LED on the thermocouple amplifier should shut off. If
not, recheck the connections between the thermocouple, the thermocouple
connector, and the amplifier.
6. Launch Windaq Dashboard from the desktop of the lab computer.
7. In the Windaq Dashboard, select the DI-1110 data acquisition system and then click the
“Start Windaq” button.
8. Once launched, configure the Windaq software.
a. Enable Channel 1 by selecting “Edit” in the toolbar menu, then “Channels” in the
drop down menu.
b. In the “Channel Selection” pop window, enable Channel 1 by clicking in the box
in row “1”, column “1”. A check mark indicates that the channel is enabled.
Ensure all other boxes are empty. Close the window by clicking the “OK” button.
c. Set sample rate by selecting “Edit” in the toolbar menu, then “Sample Rate” in the
drop down menu.
d. In the sample rate pop up window, set the sample rate to 1,000 samples per
second. Then, close the window by clicking the “OK” button.
e. Format screen by selecting “View” in the toolbar menu, “Format Screen” in the
drop down menu, then click “1 Waveform” in the sub menu.
f. Enable channel limits by first clicking on channel number displayed on the left
side of the screen.
g. Set the limit for the selected channel by selecting “Scaling” in the toolbar menu,
then “Limits” in the drop down menu.
h. In the “Channel X Display Limits” pop-up window, set the top limit to 4 V, and
the bottom to -1 V. Close the window by clicking the “OK” button.
i. The data acquisition system is now configured for measurements.
Page 5 of 17
To begin taking measurements for Part A: Thermal Systems,
9. Record the initial voltage of the system; this is the initial condition.
10. Record a set of data as a .WDQ file.
a. Prepare to submerge the thermocouple in the ice bath.
b. Record data by selecting “File” in the toolbar menu, then “Record” in the drop
down menu.
c. In the “Open” pop-up menu, name the data file and choose a location to save to
(either desktop or a flash drive). Close the window by clicking the “Open” button.
d. In the “File Size” pop-up window, set file length to 15 seconds. Click the “OK”
button to close the window. Note that data acquisition will begin immediately.
e. Submerge the thermocouple in the ice bath.
f. While the data acquisition system is recording, the status indicator in bottom of
waveform browser will display “RECORD”.
g. Once the acquisition is complete, the status indicator in bottom of waveform
browser will display “FILE FULL”. The thermocouple may be removed from the
ice bath.
11. Once data acquisition is complete, close the data file by selecting “File” in the toolbar
menu, then clicking “Close” in the dropdown menu.
12. View the acquired data and export the to a .CSV file.
a. .CSV is a standard file format compatible with common spreadsheet software,
including Microsoft Excel
b. Double click on the .WDQ file which was created in step 9. This will launch the
WINDAQ data browser.
c. In data browser, save the data as a .CSV file by selecting “File” in the toolbar
menu, and clicking “Save As…” in the drop down menu.
d. In the “Save As” pop-up window, name the file and type the file extension
“.CSV” at the end of the file name. Choose a save location (desktop or flash
drive) and click the radio button labeled “5) Spreadsheet Print (CSV)”. Save the
file and close the window by clicking the “Save” button.
e. In “Spreadsheet Comments” pop-up window, ensure the box for “Relative Time”
is checked, all other fields and checkboxes are optional. Close the window and
generate the .CSV by clicking the “OK” button.
f. Once the save is complete, the data browser can be closed.
13. When opened in Microsoft Excel, Column A lists time and Column B lists data from
Channel 1. Optional comments from Step 11f will also be visible.
14. The thermocouples must be allowed time to return to room temperature between
trials. The voltage must return to the initial voltage of the system before you proceed to
take data again; otherwise, you have not waited long enough and your time constant be
different for each trial (when it should remain the same). To accurately measure the time
constant, ensure that the initial condition (voltage at room temperature) is the same for all
trials.
15. Repeat Steps 9 through 12, for a total of three trials. Ensure that you have a complete
time series in your files and that they all start with the same initial voltage.
Page 6 of 17
16. Change the thermocouple to the uncovered tip. Record the initial voltage of the
system-this is the initial condition for the uncovered tip. Repeat Steps 9 through 12
three times to obtain three data trials.
17. To use lab time efficiently, it is recommended that the thermocouple types be tested
alternatively (i.e. covered, uncovered, covered, etc.). Be careful to ensure they have
returned to room temperature-check the initial conditions (voltage) prior to recording data
to ensure it has returned to room temperature. You can use your hands to warm the tips
(but, do not overwarm them) or put them in room temperature water baths. Failure to
return to room temperature will be seen in your data analysis for each thermocouple in
that the time constants will vary considerably (and you will lose points in your reports).
3.2 Part B: Electrical Systems
Part B investigates electrical systems using an RC circuit and an LRC circuit, both circuits are
used as filters. A diagram of the filter system is shown in Figure 4. The High-Z voltage
follower is a simple operational amplifier which does not affect the characteristics of the filter
circuits (proof is outside the scope of ME 341). It is placed in the circuit to ensure that the
impedance of the circuits is high enough to prevent signal distortions of the function generator’s
signal.
Figure 4. Diagram of the filter system used in the experiment.
To set up the experiment for Part B,
18. Connect the High-Z Voltage Follower’s red Vi+ and black Vi- terminals to the DC power
supply’s FIXED 5V terminals via the 36” long M-M dual banana plug coaxial cable.
a. The body of the dual banana plug has a nub indicating the ground plug.
b. Connect the one end of the dual banana plug coaxial cable to the 5V power
supply, inserting the ground plug to the BLACK (-) 5V jack.
Page 7 of 17
c. Connect the other end of the dual banana plug coaxial cable to High-Z Voltage
Follower, inserting the ground plug into the BLACK (Vi-) jack.
19. Power on the power supply.
20. Power on the function generator.
21. Verify that function generator Channel 1 and Channel 2 outputs are off.
a. Toggle their state by pressing the “CH1” or “CH2” button near the output BNC
posts.
b. The button will be illuminated when the output is on.
22. Set up function generator to display Channel 1
c. Toggle the selected channel by pressing the “CH1/2” button.
d. The displayed channel and waveform are listed at the top of the screen.
e. Channel 1 is also signified by a yellow screen background.
23. Set Channel 1 to output a square wave by pressing the “Square” button. The button will
illuminate. See Figure 5.
24. Set Channel 1 to the following settings. See Figure 5.
a. Frequency:
10 Hz
b. Amplitude:
1.0 Vpp
c. DC Offset:
1.5 Vdc
d. Phase:

e. Duty:
50%
25. Connect function generator’s Channel 1 positive terminal (red) to the High-Z Voltage
Follower’s input terminal (white) via a M-M banana plug cable. Connect the High-Z
Voltage Follower’s output terminal (blue) to the breadboard’s power rail, via 18” long
banana plug wire. Connect Function generator’s Channel 1 negative terminal (black) to
the breadboard’s power rail via 18” long banana plug wire. See Figures 6 and 7.
26. Use 6” jumper wires to construct low pass RC filter on the breadboard; the input signal is
across the power rails connected to the function generator. See Figures 6 and 7.
a. Record the exact values of components on the breadboard.
b. Be sure you connect the correct capacitor or you will not get the desired signal.
27. Connect the filter’s input signal to data acquisition system Channel 1 via the breadboard
and two 12” 22 AWG wires. See Figures 6 and 7.
28. Connect the filter’s output signal to the data acquisition system Channel 2 via the
breadboard and two 12” 22 AWG wires. See Figures 6 and 7.
29. Launch Windaq Dashboard from the desktop of the lab computer.
30. In …
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