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 Ohmβs Law:

π½πΉ = ππΉ

(1)

where π½πΉ 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 ο½ C

dVC

,

dt

(2)

where ππͺ is the current across the capacitor, πͺ is the capacitance in farads (amp-s/volt), and π½πͺ 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 ο½

1

iC dt .

Cο²

(3)

diL

dt

(4)

For an inductor, the element law is

VL ο½ L

or

iL ο½

1

VL dt ,

Lο²

Page 1 of 3

(5)

where ππ³ is the current through the inductor, π³ is the inductance in henrys (volt-s/amp), and π½π³ is the

voltage across the inductor.

Using Kirchoffβs voltage law, Kirchoffβs 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

π³πͺ

π
π π½πͺ

π
ππ

+ πΉπͺ

π
π½πͺ

π
π

+ π½πͺ = π½πΊ ,

(6)

where πΉ, π³, and πͺ are the values of the resistance, inductance, and capacitance, respectively, π½πͺ is the

voltage across the capacitor, and π½πΊ is the input voltage. This equation can be written as

π
π π½πͺ

π
ππ

πΉ π
π½πͺ

+π³

π
π

π

π

+ π³πͺ π½πͺ = π³πͺ π½πΊ .

(7)

A general form of a second order differential equation is

d2 x

dt2

dx

+2ΞΆΟn dt +Ο2n x=Ο2n f(t),

(8)

where π is the damping ratio and ο·n is the natural frequency. The homogenous solution of (8) is written

as

π₯ (π‘) = πΎ1 π

2 )π‘

(βπππ+β(πππ)2 βππ

+ πΎ2 π

2 )π‘

(βπππββ(πππ)2βππ

.

(9)

The value of π» determines the behavior of the system. For an underdamped system (π» < π), 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
π
π
π(π) = πβπ»ππ π (π¨ π¬π’π§ ππ βπ β π» π + π© ππ¨π¬ ππ βπ β π» π).
(10)
For a critically damped system (π» = π), the roots of the characteristic equation are real and equal;
therefore, the solution takes the form
π(π) = π¨πβπ»ππ π + π©ππβπ»ππ π .
(11)
Finally, for an overdamped system, (π» > π), the roots of the characteristic equation are real and distinct;

therefore, the solution has the form

Page 2 of 3

βπ»

π(π) = π¨π(βπ»ππ +ππ

π βπ)π

+ π©π(βπ»ππ βππ

βπ»π βπ)π

.

(12)

Comparing (7) to (8), the damping ratio for the series RLC circuit is

πΉ

πͺ

π» = π βπ³

(13)

and

π

ππ = β .

π³πͺ

(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

π
π π

π
π

π¨ π
ππ + π© π
π + π«π = π(π),

(15)

you must express this as two first order equations. Set up a matrix

{

π(π)

}

π(π)

(16)

where

π
π(π)

π
π

= π(π).

(17)

Then, your second order differential equation can be written as a first order equation

π
π(π)

π
π

π

= (π(π) β π©π(π) β π«π(π)).

π¨

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:

ο·

ο·

ο·

ο·

ο·

ο·

ο·

ο·

Breadboard

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 πο fifteen turn potentiometer,

3.3 ππ» inductor, 10.0 ππΉ capacitor, and function generator. Note that the oscilloscope

probe has a capacitance of 15.0 ππΉ 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 π
= 0.

3. Adjust the function generator to produce a 1 Volt step input by using a 1 ππ»π§ 1 ππ-π square

wave. Remember to select the proper output impedance for the function generator. Press

the βUtilityβ button, then the βOutput Setupβ key. Select the βLoadβ button, then choose

βHigh Z.β This step is crucial to ensure that the proper output is generated by the function

generator.

4. Connect the oscilloscope probe to measure πππ’π‘. 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 π of one oscillation. The natural frequency of

your system is given by

ππ = 2πππ =

2π

π

(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

Your name

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:

ο· Long derivations

ο· Programming code that is relevant

ο· Alternative design schematics

ο· MSDS (material safety data sheets)

ο· IRB approval forms

Formatting Tables & Figures in Technical Reports

ο·

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:

βTo 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)

ο·

6

5

4

3

2

1

0

Pizza

Pasta

Salad

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 surveyβ¦ β

Common Technical Writing Mistakes

Abstract

Always include an abstract, even for a technical memo. The abstract should be less than 200

words. Include the studyβs 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.

Using downloaded images off the web.

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

ο· Oscilloscope

ο· Voltmeter

ο· 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

paragraphs that lead the reader through the data shown in your results.

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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