# Expert Answer :electric engineering question

Solved by verified expert:i need you to write a lab report not that formal it should be basic and solve the circuit by 2nd order of differential equation everything you need u will find it down below , such as the equation and everything elseplease reed it carefully before u bitit should not be that long 3 pages it is good
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BME 206
BME Sophomore Lab
Spring 2018
Lab #7: 2nd Order Electrical Systems (Time Domain) Laboratory
The purpose of this laboratory is to understand the analysis of a second order system in the time
domain using resistor-inductor-capacitor (RLC) circuit analysis. There are a number of different methods
by which these circuits can be analyzed: measuring voltage output from a physical system, deriving and
solving for the output voltage using differential equations, and using MATLAB to solve the differential
equations.
Background
In order to derive the differential equations describing the behavior of RLC circuits, the relationships
between the current and voltage for each element (resistor, inductor, and capacitor), known as element
laws, must be used. The element law for a resistor is known as Ohm’s Law:
𝑽𝑹 = 𝒊𝑹
(1)
where 𝑽𝑹 is the voltage drop across the resistor, i is the current through the resistor, and R is the
resistance (ohm = volt/amp). Note that energy is lost across a resistor in the form of heat. The element
law for a capacitor can be written either as
iC  C
dVC
,
dt
(2)
where 𝒊𝑪 is the current across the capacitor, 𝑪 is the capacitance in farads (amp-s/volt), and 𝑽𝑪 is the
voltage across the capacitor. Note that energy is stored within a capacitor. An alternative element law
for a capacitor is given by integrating (2)
VC 
1
iC dt .
C
(3)
diL
dt
(4)
For an inductor, the element law is
VL  L
or
iL 
1
VL dt ,
L
Page 1 of 3
(5)
where 𝒊𝑳 is the current through the inductor, 𝑳 is the inductance in henrys (volt-s/amp), and 𝑽𝑳 is the
voltage across the inductor.
Using Kirchoff’s voltage law, Kirchoff’s current law, and these element laws, the differential equation
relating the current and voltage across the various elements can be derived. Since an RLC circuit has
two storage elements, the differential equation will be second order.
The behavior of second order systems depends on the coefficients in the differential equation. The
relationship between these coefficients determines if the solution is underdamped, critically damped, or
overdamped. You will derive the differential equation for the RLC series circuit as part of your analysis.
The equation is
𝑳𝑪
𝒅𝟐 𝑽𝑪
𝒅𝒕𝟐
+ 𝑹𝑪
𝒅𝑽𝑪
𝒅𝒕
+ 𝑽𝑪 = 𝑽𝑺 ,
(6)
where 𝑹, 𝑳, and 𝑪 are the values of the resistance, inductance, and capacitance, respectively, 𝑽𝑪 is the
voltage across the capacitor, and 𝑽𝑺 is the input voltage. This equation can be written as
𝒅𝟐 𝑽𝑪
𝒅𝒕𝟐
𝑹 𝒅𝑽𝑪
+𝑳
𝒅𝒕
𝟏
𝟏
+ 𝑳𝑪 𝑽𝑪 = 𝑳𝑪 𝑽𝑺 .
(7)
A general form of a second order differential equation is
d2 x
dt2
dx
+2ζωn dt +ω2n x=ω2n f(t),
(8)
where 𝜁 is the damping ratio and n is the natural frequency. The homogenous solution of (8) is written
as
𝑥 (𝑡) = 𝐾1 𝑒
2 )𝑡
(−𝜁𝜔𝑛+√(𝜁𝜔𝑛)2 −𝜔𝑛
+ 𝐾2 𝑒
2 )𝑡
(−𝜁𝜔𝑛−√(𝜁𝜔𝑛)2−𝜔𝑛
.
(9)
The value of 𝜻 determines the behavior of the system. For an underdamped system (𝜻 < 𝟏), the terms under the radical in (9) are less than zero, the roots of the characteristic equation are complex, and the solution takes the form 𝟐 𝟐 𝒙(𝒕) = 𝒆−𝜻𝝎𝒏 𝒕 (𝑨 𝐬𝐢𝐧 𝝎𝒏 √𝟏 − 𝜻 𝒕 + 𝑩 𝐜𝐨𝐬 𝝎𝒏 √𝟏 − 𝜻 𝒕). (10) For a critically damped system (𝜻 = 𝟏), the roots of the characteristic equation are real and equal; therefore, the solution takes the form 𝒙(𝒕) = 𝑨𝒆−𝜻𝝎𝒏 𝒕 + 𝑩𝒕𝒆−𝜻𝝎𝒏 𝒕 . (11) Finally, for an overdamped system, (𝜻 > 𝟏), the roots of the characteristic equation are real and distinct;
therefore, the solution has the form
Page 2 of 3
√𝜻
𝒙(𝒕) = 𝑨𝒆(−𝜻𝝎𝒏 +𝝎𝒏
𝟐 −𝟏)𝒕
+ 𝑩𝒆(−𝜻𝝎𝒏 −𝝎𝒏
√𝜻𝟐 −𝟏)𝒕
.
(12)
Comparing (7) to (8), the damping ratio for the series RLC circuit is
𝑹
𝑪
𝜻 = 𝟐 √𝑳
(13)
and
𝟏
𝝎𝒏 = √ .
𝑳𝑪
(14)
As you will see later in the semester, the RLC circuit can be used to model the behavior of a
catheter/transducer system to measure blood pressure.
In order to solve higher order differential equations using the ODE45 solver in MATLAB, you must use a
system of first order differential equations. For example, if you have the second order equation
𝒅𝟐 𝒙
𝒅𝒙
𝑨 𝒅𝒕𝟐 + 𝑩 𝒅𝒕 + 𝑫𝒙 = 𝒇(𝒕),
(15)
you must express this as two first order equations. Set up a matrix
{
𝒙(𝟏)
}
𝒙(𝟐)
(16)
where
𝒅𝒙(𝟏)
𝒅𝒕
= 𝒙(𝟐).
(17)
Then, your second order differential equation can be written as a first order equation
𝒅𝒙(𝟐)
𝒅𝒕
𝟏
= (𝒇(𝒕) − 𝑩𝒙(𝟐) − 𝑫𝒙(𝟏)).
𝑨
Page 3 of 3
(18)
Biomedical Engineering Department
Standard Operating Procedure
No. BME 206-S17-6
Title:
Rev.
2nd Order Electrical Systems (Time Domain) Laboratory Experiments
Effective Date:
March 7, 2017
PURPOSE
The purpose of this experiment is to investigate the behavior of second order resistorinductor-capacitor (RLC) circuit models using experimental measurements, hand-calculated
solutions of differential equations, and MATLAB.
SCOPE
This standard operating procedure covers second order electrical systems (time domain)
laboratory experiments performed in BME 206 (BME Sophomore Lab).
SAFETY REQUIREMENTS
Follow all laboratory safety procedures required when using electrical and electronic
equipment during these experiments. Specifically, be sure to turn off output from function
generator before connecting and disconnecting leads.
EQUIPMENT AND MATERIALS
The following equipment and supplies are required for these experiments:

Resistor, inductor, and capacitor
Wire kit
DC power supply
Function/arbitrary waveform generator
Oscilloscope and probe
Cables for connecting waveform generator to breadboard
MATLAB software
PROCEDURES
A. Determining natural frequency of system
1. Set up the circuit shown in Fig. 1 using a breadboard, 500 𝑘 fifteen turn potentiometer,
3.3 𝑚𝐻 inductor, 10.0 𝑝𝐹 capacitor, and function generator. Note that the oscilloscope
probe has a capacitance of 15.0 𝑝𝐹 that is in parallel to the capacitor.
Page 1 of 3
a
Biomedical Engineering Department
Standard Operating Procedure
R
Vin
2nd Order Electrical Systems (Time Domain)
No. BME 206-S17-6 Rev. A
L
Vc = Vout
+
C
Fig. 1. Second order RLC circuit with square wave input
2. Set the initial resistor value of your potentiometer to zero, i.e., set 𝑅 = 0.
3. Adjust the function generator to produce a 1 Volt step input by using a 1 𝑘𝐻𝑧 1 𝑉𝑝-𝑝 square
wave. Remember to select the proper output impedance for the function generator. Press
the “Utility” button, then the “Output Setup” key. Select the “Load” button, then choose
“High Z.” This step is crucial to ensure that the proper output is generated by the function
generator.
4. Connect the oscilloscope probe to measure 𝑉𝑜𝑢𝑡. You should observe an underdamped
response with significant ringing. Save your data to a USB thumb drive for analysis.
B. Determining resistance value for critical damping
1. Using the circuit from Part A, use your potentiometer to increase the resistance until the
system becomes critically damped. Notice that as you increase the resistance the amount of
ringing and overshoot decreases until it just disappears. This is the resistance that
corresponds to a critically damped system. Record the resistance of your potentiometer
that produces a critically damped system.
C. Observing underdamping, overdamping, and critical damping in RLC circuit
1. Using the circuit from Part A, and the resistance value determined in Part B, record the
output voltage for the critically damped system. Save your data to a USB thumb drive.
2. Adjust the potentiometer so that the resistance is approximately half that of the critically
damped value. This results in an underdamped system. Record and save the output
voltage for this underdamped system.
3. Adjust the potentiometer so that the resistance is approximately twice that of the critically
damped value. This results in an overdamped system. Record and save the output voltage
for this overdamped system.
Page 2 of 3
Biomedical Engineering Department
Standard Operating Procedure
2nd Order Electrical Systems (Time Domain)
No. BME 206-S17-6 Rev. A
ANALYSIS
A. Determining natural frequency of system
Using the data collected in part A, find the period 𝑇 of one oscillation. The natural frequency of
𝜔𝑛 = 2𝜋𝑓𝑛 =
2𝜋
𝑇
(1)
Compare this value to the theoretical value.
B. Determining resistance value for critical damping
Compare the resistance value measured in part B to the theoretical value obtained from your
differential equation.
C. Observing underdamping, overdamping, and critical damping in RLC circuit
Derive and solve the differential equations for underdamping, overdamping, and critical
damping, both by hand and using ode45 solver in MATLAB. Plot the results of your hand
calculations, results from MATLAB, and your experimental measurements. Note any
discrepancies between these solutions.
DOCUMENTATION
Write your laboratory report in the form of a technical report. Include your derivations and solutions of
the differential equations for Part C in an appendix.
Page 3 of 3
BME 306 Biomedical Engineering Lab II
Spring 2017
Instructions for Technical Report
1. Cover page
This section should be formatted according to the usual technical report format:
WESTERN NEW ENGLAND UNIVERSITY
SPRINGFIELD, MA
COLLEGE OF ENGINEERING
BME 306 BME Laboratory II
Spring 2017
Experiment Name
The report should begin on the page following the cover page.
2. Introduction
Brief discussion of the goals of the experiment, including why it is important or relevant to the
course you are studying. Introduce, if applicable, any similar work or studies that have been done
previously in this area. Discuss any relevant physiology that applies to this topic.
3. Materials & Methods
Explain the steps you took in completing the experiment, including your setup and analysis.
Identify any materials or equipment you used to solve the problem. Include technical figures, and
label all figures with appropriate dimensions and units.
4. Results & Discussion
State the factual findings of your experiment. Identify, if appropriate, the mean value of the data,
the standard deviation, the range, the maximum, the minimum, percent of increase, or decrease,
etc. Present results in text, tables, or graphs, depending on what format is the most appropriate.
Keep in mind that if you display data, it should be discussed. Explain why your results might
have turned out as they did. What differences or similarities exist between the findings and the
expected values? Explain why errors, unusual trends, or outlier points occurred among the results.
5. Conclusions
Summarize the highlights of the work and state how the findings may be helpful in future
engineering studies. Remember, these are brief concluding remarks. Data cannot be displayed
here for the first time, only repeated from the results section.
6. Acknowledgements
Use this section to thank those who have assisted with your work, including industrial sponsors,
equipment donator/supplier, professors (other than the course instructor) or others who gave you
guidance or assistance.
7. Appendix
Use appendices for information that is not central to the report, but important for a complete
understanding. Only include information that you are discussing in the main body of the text.
Each appendix should be labeled with a letter and should be cited within the body of the report.
Example appendix material:
 Long derivations
 Programming code that is relevant
 Alternative design schematics
 MSDS (material safety data sheets)
 IRB approval forms
Formatting Tables & Figures in Technical Reports

Captions for figures should be placed below the figure, whereas table heading are placed above
the table.
Be sure to mention the table or figure in the text before you display it. For example, if your
experiment was to conduct a survey, you would first describe the survey and then show the table
with the results:
“To evaluate the food options in the hospital cafeteria, a survey was given to male and
female subjects aged 18-35. The results from the survey are shown in Table 1.
Table 1. Results of food choice survey for males and females aged 18-35.
Respondent
Score (0-10)
1
1
2
1
3
4
4
3
5
1
6
2
The results from the survey strongly suggest that new food options should be explored. A
subsequent analysis of the cafeteria food offerings was conducted and the results
displayed in Figure 1 as a function of frequency.
7
Frequency of Offering (per week)

6
5
4
3
2
1
0
Pizza
Pasta
Sandwiches
Fruit
Soup
Food Offering
Figure 1. Analysis of food offerings as function of frequency (offerings per week).
The results confirm a low variety in food choice offerings. A follow up survey… ”
Common Technical Writing Mistakes
Abstract
Always include an abstract, even for a technical memo. The abstract should be less than 200
words. Include the study’s purpose, or objective, and summarize the methods, results, and
conclusions. By reading the abstract, the reader should be able to understand the study without
reading the body of the text.
Abstract mistakes
1.
2.
3.
4.
5.
6.
7.
8.
9.
Not including an abstract.
Using more than one paragraph.
Not stating the purpose or objective of the study.
Not summarizing the methods.
Not summarizing the results.
Not summarizing the conclusion(s).
Not defining all abbreviations used in the abstract the first time they are used.
Starting a sentence with an abbreviation or number.
Making reference to figures or tables in your abstract.
Introduction
The introduction should provide enough material to orient the reader to the subject of your
research. The last part of your introduction should outline the study.
Introduction mistakes
1.
2.
3.
4.
Not including an introduction.
Referencing web sites and web pages. Reference books, journal articles etc.
One sentence paragraphs.
Methods and Materials
The purpose of the methods and materials section is to give the reader enough information so
that they can repeat the experiment. The methods and materials should describe what was done
and what equipment and materials were used.
Methods and materials mistakes
1. Not including a methods and materials section.
2. Copying the lab procedure into the methods section.
3. Listing equipment. For example
 Oscilloscope
 Voltmeter
 Etc.
4. Incorrect equipment referencing. When describing equipment in the methods and
materials section use the generic name followed by the manufacturer and model number.
For example: An oscilloscope (Tektronix MPR304) measured the time dependent
voltages. A low pass filter was designed using filter circuit simulation software (Texas
Instruments FilterPro 3.1).
5. Not using a consistent tense.
Results
The results section should present the data to the reader using paragraphs, figures, and tables.
Results mistakes
1. Only including graphs, plots, and tables in the results section. You must have organized
2. Using titles on figure plots or graphs.
3. Not using sentence capitalization to write figure captions.
4. Not including descriptive sentence(s) following your figure caption.
5. Not defining all figure variables.
6. Starting a figure caption with an abbreviation.
7. Plot and figure labels that are difficult to read.
10. Missing plot axis labels or units. Units must be in brackets. For example Voltage (mV).
11. Too many plots making the flow of the results difficult to follow. Consider placing extra
data in appendices.
12. Not including key figures or data in the results section.
Conclusions
Conclusions mistakes
1. Using more than one paragraph for your conclusions.
2. Extrapolating your conclusions beyond the study. The conclusions should simply state
whether the theoretical systems analysis or hypothesis successfully predicted the measurements.
Do not extrapolate the conclusions beyond what the current study has actually demonstrated.
References
References mistakes
1. Incorrect formatting. Use IEEE format for books, journal articles etc.
2. References to web links. Web links change. Use books, journal articles etc that will not
change over time.
Miscellaneous mistakes
1. Using first and second person. That is, avoid I, we, our, you, they etc
2. Undefined abbreviations. If you use an abbreviation in the abstract it must be defined
there. If you use an abbreviation in the body of the paper define it the first time it is used
even if it is defined in the abstract.
3. Not including equations as part of a sentence.
4. Not numbering equations.
5. Not defining all equation variables.
6. Confusing tenses.
7. Very long sentences. If a sentence is over thirty words consider breaking it into two
smaller sentences.
8. Excess use of passive tense.
9. Mixing up introduction, methods and materials, results, and discussion sections.

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