electric engineering question


Question Description

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 else

please reed it carefully before u bit

it should not be that long 3 pages it is good

Unformatted Attachment Preview

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 = π’Šπ‘Ή (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 ο€½ C dVC , dt (2) where π’Šπ‘ͺ is the current across the capacitor, π‘ͺ is the capacitance in farads (amp-s/volt), and 𝑽π‘ͺ 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 ο€½ 1 iC dt . C (3) diL dt (4) For an inductor, the element law is VL ο€½ L or iL ο€½ 1 VL dt , L Page 1 of 3 (5) where π’Šπ‘³ 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 𝑳π‘ͺ π’…πŸ 𝑽π‘ͺ π’…π’•πŸ + 𝑹π‘ͺ 𝒅𝑽π‘ͺ 𝒅𝒕 + 𝑽π‘ͺ = 𝑽𝑺 , (6) where 𝑹, 𝑳, and π‘ͺ are the values of the resistance, inductance, and capacitance, respectively, 𝑽π‘ͺ is the voltage across the capacitor, and 𝑽𝑺 is the input voltage. This equation can be written as π’…πŸ 𝑽π‘ͺ π’…π’•πŸ 𝑹 𝒅𝑽π‘ͺ +𝑳 𝒅𝒕 𝟏 𝟏 + 𝑳π‘ͺ 𝑽π‘ͺ = 𝑳π‘ͺ 𝑽𝑺 . (7) A general form of a second order differential equation is d2 x dt2 dx +2ΞΆΟ‰n dt +Ο‰2n x=Ο‰2n f(t), (8) where 𝜁 is the damping ratio and n is the natural frequency. The homogenous solution of (8) is written as π‘₯ (𝑑) = 𝐾1 𝑒 2 )𝑑 (βˆ’πœπœ”π‘›+√(πœπœ”π‘›)2 βˆ’πœ”π‘› + 𝐾2 𝑒 2 )𝑑 (βˆ’πœπœ”π‘›βˆ’βˆš(πœπœ”π‘›)2βˆ’πœ”π‘› . (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) = π’†βˆ’πœ»πŽπ’ 𝒕 (𝑨 𝐬𝐒𝐧 πŽπ’ √𝟏 βˆ’ 𝜻 𝒕 + 𝑩 𝐜𝐨𝐬 πŽπ’ √𝟏 βˆ’ 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) = π‘¨π’†βˆ’πœ»πŽπ’ 𝒕 + π‘©π’•π’†βˆ’πœ»πŽπ’ 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(βˆ’πœ»πŽπ’ +πŽπ’ 𝟐 βˆ’πŸ)𝒕 + 𝑩𝒆(βˆ’πœ»πŽπ’ βˆ’πŽπ’ √𝜻𝟐 βˆ’πŸ)𝒕 . (12) Comparing (7) to (8), the damping ratio for the series RLC circuit is 𝑹 π‘ͺ 𝜻 = 𝟐 βˆšπ‘³ (13) and 𝟏 πŽπ’ = √ . 𝑳π‘ͺ (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Ÿ 𝒙 𝒅𝒙 𝑨 π’…π’•πŸ + 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) βˆ’ 𝑩𝒙(𝟐) βˆ’ 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: 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 π‘˜ο— fifteen turn potentiometer, 3.3 π‘šπ» 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 π‘˜π»π‘§ 1 𝑉𝑝-𝑝 square wave. Remember to select the proper output impedance for the function generator. Press the β€œUtility” button, then the β€œOutput Setup” key. Select the β€œLoad” button, then choose β€œHigh Z.” This step is crucial to ensure that the proper output is generated by the function generator. 4. Connect the oscilloscope probe to measure π‘‰π‘œπ‘’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 𝑇 of one oscillation. The natural frequency of your system is given by πœ”π‘› = 2πœ‹π‘“π‘› = 2πœ‹ 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 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: β€œTo 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 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. 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. ...
Purchase answer to see full attachment

Tutor Answer

School: University of Maryland


Lab report


Lab Report
Student’s Name

Lab report


To solve the circuit by second order differential equation, then second order circuit is required.
This cuicuits often include the capacitor, inductor, and the resistor all connected in parallel or in
series to be in second order circuits. The figure below show the second-order circuits which are
driven by the forcing function or input source.

By measuring the voltage of the capacitor (Vo), the values obtained are for the frequency of
oscillation, the damping ratio, the time constant of the decay envelope, and the resonant
frequency for different resistances. An op amp circuit will is developed in order to obtain an
output equal to 1000Ω RLC circuit, without utilizing inductors representing the least essential
elements within the RCL circuit. Without using Ξ± and Ϛ for the 100 Ω RLC circuit, the expected
results from the experiments should closely follow the theory, with the op amp circuit having
minimal errors than those of 1000 Ω RLC circuit.
In the circuit show below, to solve output voltage loop is used to get the formula 1.
Vo(t) = Vi(t) –RI(t) L(di(t)/dt) [formula 1]
The current can also be solved using formula 2 as shown below.
I(t) = C (dVo(t)/dt) [formula 2]

Lab report


Combining formula 2 into formula 1, the following equation is obtained.
LC (d2tVo(t)/dt2) + RC (dtVo(t)/dt) + Vo(t) = Vi(t) [formula 3]
This third formula can be used for homogenious and particular solutions. To use this equation,
we first set Vi(t) equated to zero and Vo equated to Voh, to get formula 4. Formula 5 is then
substituted into formula 4 to arrive at formula 6, which can be simplified to formula 7 shown
LC (d2tVoh(t)/dt2) + RC (dtVoh(t)/dt) + Voh(t) = 0 [formula 4]
Voh(t) = Aest[formula 5]
LCS2Aest+ RCSAest+ Aest= 0 [formula 6]
S2+ (R/L)S + (1/LC) = 0 [formula 7
Formula 7 is then rewritten in the damping ratio, ΞΆ,equal to (R/2L) * (LC)(1/2) and resonant
frequency, with is equal (1/LC)(1/2) to arrive at formula 8 below
S^2 + 2ΞΆ,Ο‰o s + Ο‰o2= 0 [formula 8]

The same equation c...

flag Report DMCA

Top quality work from this tutor! I’ll be back!

Heard about Studypool for a while and finally tried it. Glad I did caus this was really helpful.

Thank you! Reasonably priced given the quality


Brown University

1271 Tutors

California Institute of Technology

2131 Tutors

Carnegie Mellon University

982 Tutors

Columbia University

1256 Tutors

Dartmouth University

2113 Tutors

Emory University

2279 Tutors

Harvard University

599 Tutors

Massachusetts Institute of Technology

2319 Tutors

New York University

1645 Tutors

Notre Dam University

1911 Tutors

Oklahoma University

2122 Tutors

Pennsylvania State University

932 Tutors

Princeton University

1211 Tutors

Stanford University

983 Tutors

University of California

1282 Tutors

Oxford University

123 Tutors

Yale University

2325 Tutors