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# ELEC ENGR LAB I E C E 211

Clemson

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This 75 page Class Notes was uploaded by Eloy Ferry on Saturday September 26, 2015. The Class Notes belongs to E C E 211 at Clemson University taught by James Harriss in Fall. Since its upload, it has received 6 views. For similar materials see /class/214304/e-c-e-211-clemson-university in ELECTRICAL AND COMPUTER ENGINEERING at Clemson University.

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Date Created: 09/26/15

LABORATORY MANUAL ECE 211 Electrical Engineering Lab 1 In conjuction With ECE 202 Electric Circuits I Clemson University Department of Electrical and Computer Engineering Clemson SC 29634 Revised December 2005 QC Parks Contents I Course Information Introduction Use of Laboratory Instruments Laboratory Notebooks and Reports II Laboratory Meetings Laboratory 1 Orientation Laboratory 2 Workstation Characteristics naborasory 3 DC Measurements Laboratory 4 Introduction to PSpice Laboratory 5 Statistical Analysis Laboratory 6 Network Theorems I Laboratory 7 Network Theorems II Laboratory 8 The Oscilloscope Laboratory 9 RC and RL Circuits Laboratory 10 Series RLC Circuits Laboratory 11 Design Lab Laboratory 12 Lab Review and Presentations Laboratory 13 Final Exam 26 28 32 35 39 42 45 47 49 CONTENTS III Appendices Appendix A Safety Appendix B Fundamentals of Electrical Measurements Appendix C Fundamentals of Statistical Analysis Appendix D Resistor Identi cation 51 52 57 66 75 Part I Course Information Introduction This course is intended to enhance the learning experience of the student in topics encountered in ECE 202 In this lab students are expected to get hands on experience in using the basic measuring devices used in electrical engineering and in interpreting the results of measurement operations in terms of the concepts introduced in the rst electrical circuits course How the student performs in the lab depends on hisher preparation participation and teamwork Each team member must participate in all aspects of the lab to insure a thorough understanding of the equipment and concepts The student lab teaching assistant and faculty coordinator all have certain responsibilities toward successful completion of the lab s goals and objectives Student Responsibilities The student is expected to be prepared for each lab Lab preparation includes reading the lab experiment and related textbook material If you have questions or problems with the preparation contact your Laboratory Teaching Assistant LTA but in a timely manner Don t wait until an hour or two before and then expect to find the LTA immediately available Active participation by each student in lab activities is expected The student is expected to ask the teaching assistant any questions heshe may have DO NOT MAKE COSTLY MISTAKES BECAUSE YOU DID NOT ASK A SIMPLE QUESTION A large portion of the student s grade is determined in the comprehensive final exam so understanding the concepts and procedure of each lab is necessary for successful completion of the lab The student should remain alert and use common sense while performing a lab experiment He she is also responsible for keeping a professional and accurate record of the lab experiments in a laboratory notebook Students should report any errors in the lab manual to the teaching assistant Laboratory Teaching Assistant Responsibilities The Laboratory Teaching Assistant LTA shall be completely familiar with each lab prior to class The LTA shall provide the students with a syllabus and safety review during the rst class The syllabus shall include the LTA s of ce hours telephone number and the name of the faculty coordinator The LTA is reSponsible for insuring that all the necessary equipment and or preparations for the lab are available and in working condition LAB EXPERIMENTS SHOULD BE CHECKED IN ADVANCE TO MAKE SURE EVERYTHING IS IN WORKING ORDER The LTA should should fully answer any questions posed by the students and supervise 5 Introduction 6 the students performing the lab experiments The LTA is expected to grade the lab notebooks and reports in a fair and timely manner The reports should be returned to the students in the next lab period following submission The LTA should report any errors in the lab manual to the faculty coordinator Faculty Coordinator Responsibilities The faculty coordinator should insure that the laboratory is properly equipped ie that the teaching assistants receive any equipment necessary to perform the experiments The coordinator is responsible for supervising the teaching assistants and resolving any questions or problems that are identi ed by the teaching assistants or the students The coordinator may supervise the format of the nal exam for the lab He she is also reSponsible for making any necessary corrections to this manual The faculty coordinator is responsible for insuring that the software version of the manual is continually updated and available Lab Policy and Grading The student should understand the following policy ATTENDANCE Attendance is mandatory and any absence must be for a valid excuse and must be documented If the instructor is more than 15 minutes late students may leave the lab LAB RECORDS The student must 1 Keep all work in preparation of and obtained during lab in an approved NOTEBOOK and 2 Prepare a lab report on selected experiments GRADING POLICY The nal grade of this course is determined using the following criterion Laboratory notebook and inclass work 30 Lab reports 30 Final exam 40 In class work will be determined by the teaching assistant who at his her discretion may use team evaluations to aid in this decision The nal exam should contain a written part and a practical physical operations part PRE REQUISITES AND CO REQUISITES The lab course is to be taken during the same semester as ECE 202 but receives a separate grade 1f ECE 202 is dropped then ECE 211 must be dropped also Students are required to have completed both MTHSC 108 and PHYS 122 with a C or better grade in each Students are also assumed to Introduction 7 have completed a programming class and be familiar with the use of a computer based word processor THE INSTRUCTOR RESERVES THE RIGHT TO ALTER ANY PART OF THIS INFORMATION AT HIS HER DISCRETION IF CIRCUM STANCES SHOULD DICTATE Any changes should be announced in class and distributed in writing to the students prior to their effect Course Goals and Objectives The Electrical Circuits Laboratory I is designed to provide the student with the knowledge to use basic measuring instruments and techniques with pro ciency These techniques are designed to complement the concepts introduced in ECE 202 In addition the student should learn how to effectively record experimental results and present these results in a written report More explicitly the class objectives are 1 To gain pro ciency in the use of common measuring instruments 2 To enhance understanding of basic electric circuit analysis concepts including a Independent and dependent sources b Passive circuit components resistors capacitors inductors and switches c Ohm s law Kirchoff s voltage law and Kircho s current law d Power and energy relations 7 1 A A A 7 Annav s t bofe t1 e Th venin and Norton 3 f Superposition 3 To develop communication skills through a Maintenance of succinct but complete laboratory notebooks as perma nent written descriptions of procedures results and analyses b Verbal interchanges with the laboratory instructor and other students c Preparation of succinct but complete laboratory reports 4 To compare theoretical predictions with experimental results and to resolve any apparent differences Use of Laboratory Instruments One of the major goals of this lab is to familiarize the student with the proper equipment and techniques for making electrical measurements Some understanding of the lab instruments is necessary to avoid personal or equipment damage By understanding the device s purpose and following a few simple rules costly mistakes can be avoided Ammeters and Veltmeters The most common measurements are those of voltages and currents Through out this manual the ammeter and voltmeter are represented as shown in Figure 1 A V R IS VERY LARGE IDEALLY INFINITE V R IS VERY SMALL IDEALLY ZERO Figure 1 Ammeter and voltmeter Ammeters are used to measure the ow of electrical current in a circuit The oretically measuring devices should not affect the circuit being studied Thus for ammeters it is important that their internal resistance be very small ideally near zero so they will not constrict the ow of current However if the ammeter is con nected across a voltage difference it will conduct a large current and damage the ammeter Therefore ammeters must always be connected in series in a cir cuit never in parallel with a voltage source High currents may also damage the needle on an analog ammeter The high currents cause the needle to move too quickly hitting the pin at the end of the scale Always set the ammeter to the highest scale possible then adjust downward to the appropriate level Use of Laboratory Instruments 9 Voltmeters are used to measure the potential difference between two points Since the voltmeter should not affect the circuit the voltmeters have a very high ideally in nite impedance Thus the voltmeter should not draw any current and not affect the circuit In general all devices have physical limits These limits are Specified by the device manufacturer and are referred to as the device rating The ratings are usually expressed in terms of voltage limits current limits or power limits It is up to the engineer to make sure that in device operation these ratings limit values are not exceeded The following rules provide a guideline for instrument protection Instrument Protection Rules 1 Set instrument scales to the highest range before applying power 2 Be sure instrument grounds are connected properly Avoid accidental grounding of hot leads ie those that are above ground potential 3 Check polarity markings and connections of instruments carefully be fore connecting power 4 Never connect an ammeter across a voltage source Only con nect ammeters in series with loads 5 Do not exceed the voltage and current ratings of instruments or other circuit elements This particularly applies to wattmeters since the current or voltage rating may be exceeded with the needle still on the scale 6 Be sure the fuse and circuit breakers are of suitable value When connecting electrical elements to make up a network in the laboratory it is easy to lose track of various points in the network and accidently connect a wire to the wrong place A procedure to follow that helps to avoid this is to connect the main series part of the network rst then go back and add the elements in parallel As an element is added place a small check by it on your circuit diagram Then go back and verify all connections before turning on the power One day someone 3 life may depend upon your making sure that all has been done correctly Laboratory Notebooks and Reports The Laboratory Notebook The student records and interprets his her experiments via the laboratory notebook and the laboratory report The laboratory notebook is essential in recording the methodology and results of an experiment In engineering practice the laboratory notebook serves as an invaluable reference to the technique used in the lab and is essential when trying to duplicate a result or write a report Therefore it is important to learn to keep an accurate notebook The laboratory notebook should 1 Be kept in a sewn and bound or spiral bound notebook 2 Contain the experiment s title the date the equipment and instru ments used any pertinent circuit diagrams the procedure used the data often in tables when several measurements have been made and the analysis of the results 3 Contain plots of data and sketches when these are appropriate in the recording and analysis of observations 4 Bean accurate and permanent record of the data obtained during the experiment and the analysis of the results You will need this record when you are ready to prepare a lab report The Lab Report The laboratory report is the primary means of communicating your experience and conclusions to other professionals In this course you will use the lab report to inform your LTA what you did and what you have learned from the experience Engineering results are meaningless unless they can be communicated to others Your laboratory report should be clear and concise The lab report shall be typed on a word processor As a guide use the format on the next page Use tables diagrams sketches and plots as necessary to show What you did what was observed and what conclusions you draw from this Even though you will work with one or more lab partners your report will be the result of your individual effort in order to provide you with practice in technical communication You will be directed by your LTA to prepare a lab report on a few selected lab experiments during the semester Your assignment might be different from your lab partner s assignment 10 Laboratory Notebooks and Reports 11 Format of Lab Report LABORATORY XX TITLE Indicate the lab title and number NAME Give your name LAB PARTNERS Specify your lab partner s name DATE Indicate the date the lab was performed OBJECTIVE Clearly state the objective of performing the lab EQUIPMENT USED Indicate which equipment was used in performing the experiment The manufacturer and model number should be speci ed PROCEDURE Provide a concise summary of the procedure used in the lab Include any modi cations to the experiment DATA Provide a record of the data obtained during the experiment Data should be retrieved from the lab notebook and presented in a clear manner using tables OBSERVATIONS AND DISCUSSIONS The student should state What con clusions can be drawn from the experiment Plots charts other graphical medium and equations should be employed to illustrate the student s viewpoint Sources of error and percent error should be noted here QUESTIONS Questions pertaining to the lab may be answered here These questions may be answered after the lab is over CONCLUSIONS The student should present conclusions which may be logically deduced from his her data and observations 39 SIGNATURE Sign your report at the end Include the statement This report is accurate to the best of my knowledge and is a true representation of my laboratory results Part II Laboratory quotMeetings 12 Laboratory 1 Orientation Introduction In the rst lab period the students should become familiar with the location of equipment and components in the lab the course requirements and the teaching instructor Students should also make sure that they have all of the co requisites and pre requisites for the course at this time Objective To familiarize the students with the lab facilities equipment standard oper ating procedures lab safety and the course requirements Preparation Read the introduction and Appendix A Safety in this manual Equipment Needed ECE 211 lab manual Procedure 1 During the rst laboratory period the instructor will provide the students with a general idea of what is expected from them in this course Each student will receive a copy of the syllabus stating the instructor s of ce hours and telephone number In addition the instructor will review the safety concepts of the course 2 The instructor will indicate which word processor should be used for the lab reports The students should familiarize themselves with the preferred word processor software 3 During this period the instructor will briefly review the equipment which will be used throughout the semester The location of instruments equipment and components eg resistors capacitors connecting wiring will be indicated The guidelines for instrument use will be reviewed 13 LABORATORY l ORIENTATION 14 Probing Further 1 During the next period the instructor may ask questions or give a quiz to determine if you have read the introductory material As a professional engineer it Will be your responsibility to prepare yourself to do your job correctly Learn as much as you can up fron You Will nd that as a practicing professional if you wait until the last minute you might have to pay a very painful price emotionally nancially and professionally Report No report is due next time Laboratory 2 Workstation Characteristics Introduction Every engineer relies on equipment to drive and measure an electrical system under study These devices are rarely ideal and have their own internal characteristics which must be considered The internal characteristics of various devices often have a signi cant effect on circuit operation This may be accounted for in circuit design and analysis to more adequately predict actual operation in the lab Students should understand the internal characteristics of the equipment they are using particularly the NI ELVIS workstations used in this course Objective By the end of this lab the student should know how to determine the internal resistance of meters and sources The student should understand how the internal resistance of these instruments affects the measurements Preparation Read the section Use of Lab Instruments and Appendix B Fundamentals of Electrical Measurement in this manual In your laboratory notebooii sketch the circuit diagram for each part of the procedure along with tables formatted to enter your data Equipment Needed NI ELVIS workstation Resistance substitution box 15 LABORATORY 2 WORKSTATION CHARACTERISTICS 16 Procedure 0 Once an instrument scale setting has been selected and the internal re sistance determined with an initial measurement do not change the scale setting because the internal resistance will change 1 Set the null voltage of the NI ELVIS digital multimeter DMM by connect ing the VOLTAGE HI and VOLTAGE LO pins together Open the digital multimeter window and use the null control to obtain a reasonable zero voltage reading 2 Adjust the output of the DC power supply to 10V and verify with the digital multimeter Set up the circuit as shown in Figure 21 using the Nl ELVIS DMM Adjust the resistor R to 09 and record the meter reading Adjust the resistance so that the meter reading drOps by a signi cant fraction up to about one half of the Original value Using voltage division determine the equivalent internal resistance of the voltmeter from these readings 10V Figure 21 Circuit diagram for measurement of the internal resistance of the 39 voltmeter LABORATORY 2 WORKSTATION CHARACTERISTICS 17 3 Adjust the output of the DC power supply to 10V and verify with the digital multimeter Set up the circuit as shown in Figure 22 using the NI ELVIS DMM with the same scale setting used in Part 1 Adjust the variable resistance R to 47M 100kQ 220M 10MQ and 10MB Calculate the theoretical voltages using the internal resistance obtained in Step 2 and compare the readings 100kQ 10V R Figure 22 Circuit diagram for measurements to determine the effect of voltmeter internal resistance on measurement accuracy 4 Adjust the output of the DC power supply to 10V and verify with the digital multimeter Set up the circuit as shown in Figure 23 using the NI ELVIS DMM before energizing null the ammeter display value to obtain a reasonable zero reading Adjust the variable resistance R to in nite Ohms open the circuit and record the meter reading Then adjust R so that the meter reading drops to about one half of the original value Using current division determine the internal resistance of the ammeter from these readings 1k 10V R Figure 23 Circuit diagram for measurement of the internal resistance of the ammeter LABORATORY 2 WORKSTATION CHARACTERISTICS 18 5 Adjust the output of the DC supply to 1V and verify with the digital multimeter Set up the circuit as shown in Figure 24 using the NI ELVIS DMM Adjust the series resistance R to the following values 1000 400 209 109 and 59 and record the value of current indicated for each resistance value Then assuming that the internal resistance of the meter is zero calculate the theoretical values and compare those to your measured values From these data estimate the value of the internal resistance of the ammeter and compare this value to that obtained in Part 4 R Figure 24 Circuit diagram for measurement of theeffect of the internal resistance of the ammeter on measurements 6 Adjust the output of the DC supply to 10V and verify with the digital multimeter Set up the circuit as shown in Figure 25 using the NIELVIS DMM Adjust the resistor R to 10kt and record the meter reading Adjust the resistance so that the meter reading39drops by a signi cant fraction up to about one half of the original value Using voltage division determine the equivalent internal resistance of the voltage source from these readings 10V R Figure 25 Circuit diagram for measurement of the internal resistance of the workstation power supply LABORATORY 2 WORKSTATION CHARACTERISTICS 19 7 Repeat Part 6 using the Function Generator set to output a sine wave of magnitude 10V Use the NI ELVIS voltmeter to take your measurements Be sure to set the voltmeter to measure AC volts Keep in mind that the display shows the value of RMS voltage where for a sinusoidal waveform Vea VRMS 7 Probing Further 1 Based on your estimates of the internal resistance of the voltmeters and ammeters used in these experiments what would be the values of internal resistance of these meters if in each case the range used had been a factor of 10 larger than the range you actually used Refer to Appendix B for the equivalent circuit models of the meters to guide your thinking about the answer to this question 2 Given your values for the internal resistance of the DC source and function generator what do you think is the maximum amount of current that can be supplied by each source Report Your Laboratory Teaching Assistant will inform you when a report is due and on which experiment you will report Make sure that you have recorded all necessary information and data in your laboratory notebook to enable you to prepare a report on this experiment if so directed at some time in the future Lab oratory 3 DC Measurements Introduction Voltage and current values may be used to determine the power consumed or provided by an electrical circuit Electric power consumption is a very important factor in all electrical applications ranging from portable computers to megawatt industrial complexes Thus an understanding of power and how it is measured is vital to all engineers Objective By the end of this lab the student should know how to make DC measurements of voltages and currents to determine power dissipation delivery for circuit elements branches and various combinations of elements and branches Preparation Read the introductory material in the ECE 202 textbook which describes the passive sign convention for circuit elements Also review the lab manual section Use of Laboratory Instruments Prior to coming to lab class calculate the values of voltage current and power absorbed delivered for each circuit element in Figure 31 ie do Part 0 of the Procedure Also sketch in your lab notebook the circuit diagrams to be used in each part of the procedure and have a table prepared for each part in order to record data Equipment Needed NI ELVIS workstation Resistors as required 20 LABORATORY 3 DC MEASUREMENTS 21 Procedure 0 For the circuit given in Figure 31 calculate the voltages across and currents through each circuit element Using these values determine the power absorbed or delivered by each circuit element Include your calculations in your laboratory notebook Record all of your theoretical results in a table for later comparison with your experimental values 39 1kQ 10V 5109 5109 Figure 31 DC resistive network 1 Set up the circuit in Figure 31 Adjust the output of the DC power supply to 10V and verify with the digital multimeter Using the digital multimeter function in the NI ELVIS workstation set to measure DC Volts measure the voltage across each individual circuit element Before making each measurement use the connection scheme shown in Figure 32 to verify that your voltmeter connection method is correct Record the measurements in your laboratory notebook For each measured voltage determine the percent error from the theoretical value determined in Part 0 1kQ Figure 32 Connection scheme for circuit voltage measurements LABORATORY 3 DC MEASUREMENTS 22 2 For the circuit in Figure 31 measure the current through each circuit element using the digital multimeter function in the NI ELVIS workstation set to measure DC Amps Before each measurement the circuit will have to be turned off and rewired to insert the ammeter in series with the component under test Before making each measurement use the connection scheme shown in Figure 33 to verify that your ammeter connection method is correct Record the measurements in your laboratory notebook For each measured current determine the percent error from the theoretical value determined in Part 0 1k 2 5109 5109 Figure 33 Connection scheme for circuit current measurements 3 Using your measurements from Parts 1 and 2 calculate the power absorbed 39 39 39 4 D 1 Hanna rnIIIn 2v v 1nLAun4ww ALALAnL 0139 delivered by each ClI39C CLliS element LLGCOLU ULLCDC Vaiucb 111 yuui iduur atur y iichuuun Compare them with the values obtained through circuit analysis in Part 0 Calculate the percent error from the theoretical values Probing Further 1 What is the relationship between the power values obtained from your measured values of voltage and current and those calculated theoretically in Part 0 What do you think are sources of error Explain 2 How does the sum of power absorbed by the resistances in the circuit compare to the amount delivered by the source Report Your Laboratory Teaching Assistant will inform you when a report is due and on which experiment you will report Make sure that you have recorded all necessary information and data in your laboratory notebook to enable you to prepare a report on this experiment if so directed at some time in the future Laboratory 4 Introduction to PSpice Introduction The widespread availability of computers has enabled the development of soft ware which quite accurately models the behavior of electrical circuits The most widely used program is SPICE develOped by the University of California Berkeley in the mid 1970 s and updated several times since then Amongst the several com mercially available versions of SPICE is OrCAD PSpice which is loaded onto the ECE 211 laboratory computers As an introduction you will learn to use PSpice circuit simulation software with relatively39simple circuits Later in the introductory electronics courses ECE 320 and ECE 321 you will learn to use PSpice to simulate the operation of more advanced components such as diodes transistors and even a few integrated circuits Objective This lab should give the student a basic understanding of how to use PSpice to simulate circuit operating conditions After this lab the student should be able to use PSpice to solve or check basic circuit problems Preparation Prior to coming to lab class calculate the voltages and currents for each re sistor shown in the circuit of Figure 41 ie do Part 0 of the Procedure Equipment Needed A personal computer with the PSpice demonstration or student version loaded and ready to use or an engineering workstation with PSpice loaded and ready to use 23 LABORATORY 4 INTRODUCTION TO PSPICE 24 Procedure 0 Before coming to lab determine the voltages across and currents through each resistor in the circuit of Figure 41 1k 2kQ 3kQ 100V 4k 5m 6k 2 Figure 41 Resistive network for hand analysis and PSpice simulation 1 In the lab your LTA will go through the guidelines for using computer equipment in a College of Engineering amp Science laboratory Then the LTA will provide an introduction to the PSpice software environment During this instruction session you will learn how to 39 a Open the software and create a new project b Place circuit components in your project VV39orkSpace 0 Connect circuit components together d Set up circuit measurements e Change simulation execution settings f Run simulations g Display or export the simulation output 2 Use PSpice to solve for all of the resistor voltages and currents for the circuit of Figure 41 Your problem solutions using PSpice are to be turned in Compare your simulation measurements with the results of your calculations in Part 0 3 Use PSpice to simulate the circuit used in Laboratory 3 Figure 31 Deter mine the voltages and currents for each circuit component Your problem solutions using PSpice are to be turned in 4 Use PSpice to solve any additional circuitproblems assigned by your LTA Your problem solutions using PSpice are to be turned in LABORATORY 4 INTRODUCTION TO PSPICE 25 Probing Further 1 How do the PSpice simulation results in Part 1 compare to your calculations in Part 0 Can you account for any differences 2 How do the PSpice simulation results compare to your measurements in Laboratory 3 Can you account for any differences Report The problem solutions are due the next period Include a title page and a brief procedure for each circuit Report the results from PSpice and highlight these results on the printouts Include a circuit diagram With each problem Problem statements les and printouts should be included at the end of the report Laboratory 5 Statistical Analysis Introduction The student is already aware of some error introduced by assuming ideal me ters in the measurement process Another uncertainty lies in the use of the circuit components themselves While a component is designed to have a particular Value its nomina value which is marked on the outside covering the case or en capsulation random uctuations in materials and production processes will result in some range of values for the manufactured devices Thus components are usually speci ed by a nominal value and a range called the tolerance in which the actual value is expected to lie For example resistors are speci ed as being within a stated tolerance of the given nominal value This tolerance can be as small as 1 for precision resistors to as large as 20 The tolerance may be indicated by a color band printed on the case of the resistor or by a percentage value printed on the case It is important to know how to identify the tolerance of the resistors used in a particular circuit and to understand what the speci ed nominal value and tolerance for a component means statistically Objective By the end of this lab the student should know how to apply statistical methods to obtain the most accurate estimate of a circuit component Preparation Read Appendix C Fundamentals of Statistical Analysis Become familiar with the concepts of mean standard deviation and variance and the formulas used for calculating these quantities Equipment Needed NLELVIS workstation 26 LABORATORY 5 STATIS TICAL ANALYSIS 27 Procedure 0 The lab instructor will provide each team with a group of ve resistors Each group of resistors should be labeled GROUP A GROUP B GROUP C etc These will be passed in turn from lab team to lab team 1 For each group of resistors measure the resistance of each component us ing the NLELVIS digital multimeter Make a table for each group recording the resistances 2 For the resistors in each group calculate the mean the standard deviation and the variance Make a table which shows these values for each of the groups When you make this table include a row labeled TOTAL and calculate the mean standard deviation and variance for all of the resistance values which you have measured 3 Make another table with rows labeled TEAM 1 TEAM 2 etc and record the values of mean value standard deviation and variance for each resistor group and for the total determined by other lab teams in Part 2 Why do the values differ 4 Using the information in Parts 2 and 3 what would the best estimate of the resistance of each group be What would be the best estimate of the resistance for all groups of resistors combined Why Probing Further 1 Is it reasonable to assume the resistors have a normal distribution Why 2 How does the mean value of resistance obtained by your lab team for the total compare to the nominal value of resistance marked on the case 3 How does the standard deviation obtained by your team compare to the tolerance marked on the case of the resistors Report Your Laboratory Teaching Assistant will inform you when a report is due and on which experiment you will report Make sure that you have recorded all necessary information and data in your laboratory notebook to enable you to prepare a report on this experiment if so directed at some time in the future Laboratory 6 Network Theorems I Introduction An understanding of the basic laws of electrical voltages and currents is essen tial to electrical engineering Circuit analysis is dependent upon knowing the nature of the laws governing voltage and current characteristics This lab studies Kircholf s Voltage Law Kirchoff s Current Law voltage division current division and equiva lent resistance Objective By the end of this lab the student should understand KVL KCL voltage division current division and equivalent resistance combinations Preparation Read the material in the textbook that describes Kirchoff s Voltage Law Kir choff s Current Law voltage division current division and equivalent resistance com binations Be able to perform circuit calculations using these principles Before com ing to class analyze each circuit and determine the theoretical values that should be obtained during the lab Verify your calculations by performing PSpice simula tions for each circuit Record both your calculations and simulation results in your laboratory notebook Equipment Needed NI ELVIS workstation Individual resistors as required Resistance substitution box 28 LABORATORY 6 NETWORK THEOREMS I 29 Procedure 1 Adjust the output of the DC power supply to 10V and verify with the digital multimeter Set up the circuit as shown in Figure 61 Measure and record the total current into the circuit Using the measured current and voltage determine the equivalent resistance of the parallel components in the circuit Replace the resistors with a resistance substitution box set to the equivalent resistance and measure the current as before Compare the experimentally determined equivalent resistance to the theoretical value 10V 3k 2 39kQ 51kQ Figure 61 Determining parallel equivalent resistance 2 Adjust the output of the DC power supply to 10V and verify with the digital multimeter Set up the voltage division circuit as shown in Figure 62 Begin with R5109 and measure the voltage across each resistor Repeat with Rzle ZkQ BkQ 43162 and 51M Compare the measured voltages to those calculated using the voltage divider relation IkQ 10V R Figure 62 Effect of R on the component voltages LABORATORY 6 NETWORK THEOREMS I 30 3 Adjust the output of the DC power supply to 10V and verify with the digital multimeter Set up the current division circuit as shown in Figure 63 Begin with R25lO 2 and measure the currents 11 12 and 13 Repeat with Rgzle 2132 315 43162 and 51kQ Compare the measured currents to those calculated using equivalent circuit resistance and the current divider relation Determine whether or not each set of measurements agrees with Kirchoff s Current Law 1k 2 10V Figure 63 Sum of currents at a node 4 Adjust the output of the DC power supply to 10V and verify with the digital multimeter Set up the circuit as shown in Figure 64 Measure the voltage across each component Compare the measured voltages to those calculated using the voltage divider relation Determine whether or not your measurements agree with Kirchoff s Voltage Law 5109 15kQ 10V 1kQ Figure 64 Sum of voltages around a 100p LABORATORY 6 NETWORK THEOREMS I 31 5 Adjust the output of the DC power supply to 10V and verify with the digital multimeter Set up the circuit as shown in Figure 65 Measure the voltage across each component in loop 1 Repeat for loop 2 and loop 3 Compare your measured values with the terms in the KVL equation written for each loop Determine whether or not your measurements agree with Kirchoff s Voltage Law Explain the reasons for any discrepancies found LOOP 3 Figure 65 Sum of voltages around different loops Probing Further 1 In part 2 what would the value of R have to be so that the voltage across R is 45 of the source voltage Your answer should be quantitative ie a number 2 In part 3 what would the value of R2 have to be so that the current through R2 is 10 times the current through R3 Your answer should be quantitative Report Your Laboratory Teaching Assistant will inform you when a report is due and on which experiment you will report Make sure that you have recorded all necessary information and data in your laboratory notebook to enable you to prepare a report on this experiment if so directed at some time in the future Laboratory 7 Network Theorems II Introduction This lab focuses on the Th venin equivalent and maximum power transfer theorems Complex circuits are often replaced with their Th venin equivalent to simplify analysis For example in the analysis of large industrial power systems the Th venin equivalent is used in short circuit studies Maximum power transfer is also an important concept which allows the designer to determine an optimal design when power is a constraint Objective By the end of this lab the student should be able to verify Th venin s equiv alence theorem and the concept of maximum power transfer 39 Preparation Read the material in the textbook that describes Thevenin s equivalence the orem and maximum power transfer Equipment Needed NI ELVIS workstation Individual resistors as required Resistance substitution box 32 LABORATORY 7 NETWORK TEOREMS II 33 Procedure 1 This part of the lab illustrates the use of Th venin s theorem Adjust the output of the DC power supply to 10V and verify with the digital multimeter Set up the circuit as shown in Figure 71 Measure the open circuit voltage between nodes A and B Measure the short circuit current between nodes A and B ie connect the ammeter between nodes A and B Using these measurements determine the Th venin equivalent circuit Set up the newly determined Th venin equivalent circuit and verify that this circuit has the same open circuit voltage and short circuit current as the previous circuit Save this circuit for Part 2 680 3300 10V 2209 1k 2 Figure 71 Determining the Thevenin equivalent circuit 2 This part of the lab is to illustrate maximum power transfer Use the Th venin equivalent circuit developed in Part 1 For a resistance subsititution box R between nodes A and B measure the current through and voltage across R if R209 Repeat for R21009 1209 5009 in 209 increments Determine the power dissipated by the resistor for each value of R Plot power vs resistance At which value is the power a maximum LABORATORY 7 NETWORK T HEOREMS II 34 Probing Further 1 Use PSpice to determine the Th venin equivalent for the circuit in Part 1 First enter the circuit shown in Figure 71 using node B as the reference or ground node node 0 in PSpice The voltage at node A is then the open circuit voltage Next modify the circuit by placing a voltage source with zero volts a technique used in PSpice to measure current between points A and B The current through this voltage source is the short circuit Current Determine the Th venin equivalent and compare to your experimentally obtained circuit in Part 1 Record your PSpice programs and the data obtained from the simulation in your laboratory notebook by pasting in the printouts Highlight the open circuit voltage value and short circuit current value obtained from the simulation 2 Use PSpice to simulate the circuit of Part 2 Start with the value of R Rmam pm that you determined experimentally to give maximum power transfer and find from the PSpice simulation the power delivered to this resistance Then repeat with 209 increments through 1000 ie for Rm W 1000 g R g Rmam pm 1002 Compare the values of power obtained by simulation with those you obtained experimentally Record your PSpice programs and the data obtained from them in your laboratory notebook Report Your Laboratory Teaching Assistant will inform you when a report is due and on which experiment you will report Make sure that you have recorded all necessary information and data in your laboratory notebook to enable you to prepare a report on this experiment if so directed at some time in the future Laboratory 8 The Oscilloscope Introduction The digital oscilloscope allows the engineer to examine time varying waveforms in order to determine the magnitude frequency phase angle and other waveform characteristics which depend upon the interaction of circuit elements with the sources driving them Objective By the end of the lab the student should be familiar with the controls of a digital oscilloscope and be able to use the instrument to observe periodic waveforms Preparation Review XYZs of Oscilloscopes available at wwwtekcom 60 pages Be familiar with the following voltage scaling Volts division time base seconds division input coupling triggering and measurement probes Equipment Needed NI ELVIS workstation Individual resistors as required Procedure 0 At the beginning of class your LTA will review the virtual oscilloscope software used with the NI ELVIS workstation 1 Basic setup Connect the FUNCOUT out to oscilloscope CH A and ground to oscilloscope CH A Set the function generator to output a lOOHz sine wave with peak amplitude15V and DC offsetzOV Open the oscilloscope window in the Nl ELVIS software Switch the display for Channel A on Set the Channel A display to ON Turn the measure function for Channel A on and record the measured values for RMS voltage peak to peak voltage and waveform frequency Sketch the displayed waveform in your laboratory notebook Compare your measurements with the expected values based on the function generator output 35 LABORATORY 8 THE OSCILLOS COPE 36 2 Source control Using the source pulldown menu for Channel A change the source from BNC Board CH A to FGEN SYNCHOUT Sketch this waveform in your laboratory notebook Return the source setting to BNC Board CH A Change the function generator output to a square wave record the displayed waveform in your laboratory notebook and measure the peak to peak output voltage Repeat this 39measurement for a triangular wave 3 Voltage scaling Reset the function generator to output a sine wave Vary the vertical scale control for Channel A using either the control knob or pull down menu Record the effect that this control has on the displayed waveform Set the control to EOOmV div and measure the peakto peak magnitude of the displayed waveform by counting estimate the number of peakto peak divisions and multi plying by the vertcal scale Compare this result with the measurement given by the oscilloscope 4 Voltage o set manipulation Vary the vertical position control in the oscil loscope and record the effects in your laboratory notebook noting any changes in the measured RMS voltage Return the offset to zero and add a DC offset of 05V to the function generator output Record the effects in your laboratory notebook noting any changes in the measured RMS voltage 5 Time scaling Return the DC offset in the function generator to 0V Using either the timebase dial or pulldown menu adjust the timebase of the oscilloscope display to the fastest setting 5uS div Record theeffect that this setting has on the diSplayed measurements for the waveform Gradually increase the timebase through each available setting until the slowest setting has been reached 200mS div Record the effect that this control has on the measurement of voltage and frequency Return the timebase to a setting where 1 3 full cycles of the output sine wave is viewable Press the SINGLE screen update button to capture a single sweep of the output waveform and measure the period of the waveform by counting estimate the number of time divisions for a single cycle and multiplying by the time scale Compare this measurement to the inverse of the frequency measured by the oscilloscope 6 Triggering synch function Return the screen update to RUN Adjust the triggering pull down menu to SYNCHOUT and record the oscilloscope re sponse Vary the function generator peak amplitude to verify that the oscilloscope is continuing to update the display in this mode of operation 7 Cursor function Set the function generator to output a lOOHz sine wave with peak amplitude15V and DC offset20V Return the triggering function to Immediate Display a single screen update of between 1 3 cycles of the output function Switch the cursors function on and drag the cursors to appropriate points on the waveform to measure the period of the sine wave Then adjust the cursors to measure the peak to peak voltage of the sine wave Compare these measurements to those expected based on the function generator s output settings LABORATORY 8 THE OSCILLOSCOPE 37 8 Connect the voltage divider circuit shown in Figure 81 Set the function generator to output a 1kHz sine wave with peak magnitudele and DC offsetzO Display the function generator output on Channel A of the oscilloscope and the volt age across the 1009 resistor on Channel B Display and measure these voltages simul taneously Measure the period of both waveforms using the cursor function Sketch the waveforms in your laboratory notebook and record your settings for Volts div and seconds div Compare your voltage measurements with theoretical calculations based on the voltage divider equation Compare your waveform period measurement with the theoretical value obtained from the input frequency 1k FUNCTION GENERATOR moo 1kHz SINE Figure 8 1 Voltage divider circuit used in Parts 8 and 9 9 Reverse the polarity for the output voltage measurement on Channel B Repeat your voltage and period measurements as in Part 8 sketch the resulting waveforms in your laboratory notebook and record your settings for Volts div and seconds div Probing Further 1 What purpose does the FGEN SYNCH OUT serve 2 Why does the RMS voltage measurement vary when an offset is added in the function generator but not when changed in the oscilloscope 3 In Part 9 what effect did reversing the polarity of the output voltage mea surement have on the oscilloscope display and the oscilloscope voltage measurements 4 How is the accuracy of your measurements affected by adjustment of the volts division control Seconds division control 5 How would the peak to peak and EMS voltage measurements be affected by the presence of noise in a displayed waveform Would this be a clear representation of the actual signals in a circuit What steps would you take to determine if noise is an issue in your oscilloscope measurements and how would you mitigate it if necessary LABORATORY 8 THE OSCILLOSCOPE 38 Report Your Laboratory Teaching Assistant will inform you When a report is due and on which experiment you will report Make sure that you have recorded all necessary information and data in your laboratory notebook to enable you to prepare a report on this experiment if so directed at some time in the future Laboratory 9 RC and RL Circuits Introduction This lab deals with RC and RL circuits RC and RL circuits are used in many con gurations for a large variety of design purposes In addition real components can be modeled in a given frequency range by a combination of R and C or R and L as appropriate For example a wirewound resistor can be modeled as a pure resistor R at DC or very low frequencies but as the frequency of operation increases the inductive effects of the winding must be taken into account This lab illustrates some of the basic features of the transient response of circuits in which resistance and capacitance or resistance and inductance are both present Objective By the end of this lab the student should know how to measure the time constants of RC and RL circuits Preparation Review the material in the textbook on RC and RL circuits Before coming to the lab determine the theoretical time constants of the circuits used in the lab Be sure to account for the 1509 output impedance of the function generator in your calculations Equipment Needed NI ELVIS workstation Individual resistors and capacitors as needed Inductance substitution box 39 LABORATORY 9 RC AND RL CIRCUITS 40 Procedure 1 Set up the RC circuit shown in Figure 91 Set the function generator to give a square wave output with magnitude equal to SOOmV Measure both the source voltage and the voltage across the capacitor with the digital oscilloscope Adjust the frequency of the function generator so that the waveform shown has de nite fiat sections at the t0p and bottom Using the oscilloscope cursors function determine when the voltage reaches 0632 times its nal value Sketch the waveform for a complete cycle in your notebook recording the voltage scale and time scale values Clearly label the sketched waveforms including initial and nal values Repeat these steps using C0047 uF and CO1 uF For each circuit the frequency of the waveform generator may have to be changed 10kQ FUNCTION GENERATOR so WAVE 39oo1pF Figure 91 RC circuit 2 Now modify the circuit of Figure 91 by swapping the positions of the resistor and capacitor Repeat the measurements in Part 1 using C00luF 0047uF and 01 pF while observing the voltage across the resistor Find the time when the voltage reaches 0368 times its initial value the voltage decays here Compare your measured values of the RC circuit time constant in Parts 1 and 2 with the theoretical values LABORATORY 9 RC AND RL CIRCUITS 41 3 Set up the RL circuit shown in Figure 92 The function generator should be set up as in Part 1 Use an inductance substitution box for the inductor Measure both the source voltage and the voltage across the resistor with the digital oscil loscope Adjust the frequency of the function generator so that the waveform has de nite at sections at the top and bottom Using the oscilloscope cursors function determine when the voltage reaches 0632 times its nal value Sketch the waveforms for a complete cycle in your notebook recording the voltage scale and time scale val ues Clearly label the sketched waveform including initial and nal values Repeat these steps using L400mH 600mH and 800mH For each circuit the frequency of the function generator may have to be changed 200mH FUNCTION GENERATOR 2k 2 so WAVE Figure 92 RL circuit 4 Now modify the circuit of Figure 92 by swapping the positions of the resistor and inductor Repeat the measurements in Part 3 using L200mH 400mH 600mH and 800mH while observing the voltage across the inductor Find the time when the voltage reaches 0368 times its initial value the voltage decays here Compare your measured values of the RL circuit time constant in Parts 3 and 4 with the theoretical values Probing Further 1 Use PSpice to simulate the RC circuit response for the circuit of Figure 91 Repeat for the RL circuit response for the circuit of Figure 92 For both use PROBE to obtain waveforms and compare these waveforms to those you sketched from your experiments 2 Explain the signi cance of the 0632 and 0368 multipliers and why they are used in the lab Report Your LTA will inform you when a lab report is due Laboratory 10 Series RLC Circuits Introduction This lab illustrates some of the properties of RLC circuits Depending on the component values series RLC circuits are either overdamped underdamped or critically damped For the component values which will be used in this experiment the circuit is underdamped and the circuit response is given by 11 iii 2L 4L2 LO where 51 and 32 are the poles of the characteristic equation The circuit current is 8182 2 e o tA cos wdt B sin wdt where a a 2L L i2 27F wd VLO 4L2 T and T 2 period Objective 39 By the end of this lab the student should be able to relate the nature of the physical response of a series RLC circuit to the parameter values a and cud which are determined by the component values Preparation Review the material in the textbook on the RLC circuit response Review the concepts of overdamped underdamped and critically damped response Before coming to the lab calculate the theoretical parameter values of 31 32 04 odd and T for the circuit used in the lab ie do Part 0 of the Procedure Equipment Needed Nl ELVIS workstation Individual resistors and capacitors as required Inductance substitution box 42 LABORATORY 10 SERIES RLC CIRCUITS 43 Procedure 0 Determine the values of 51 32 a wd and T for the circuit shown in Figure 101 1 Set up the circuit as shown in Figure 101 Set the function generator to output a square wave with peak magnitude1V and DC offset0V Since the internal resistance of the function generator is 1509 use 1000 for the resistance R Make sure that the period of the square wave is long enough so that the transient sinusoidal waveforms dampen out ie decrease to a negligible value before the succeeding pulse occurs This will allow for ease of measurement and accuracy Display the voltage across the inductor on the digital osciloscope and adjust the function generator until a good waveform is obtained Draw an accurate representation of this waveform in your laboratory notebook 2500 100mH FUNCTION GENERATOR 001m 30 WAVE Figure 101 Series RLC circuit 2 Using the cursors function in the digital oscilloscope measure the period of the damped sinusoidal waveform Compare this value to T obtained in Part 0 3 Measure the initial value of the damped sinusoidal waveform and the voltage magnitude at the next peak 1 cycle Measure the time differential between these two points Determine the Neper frequency damping coefficient a using your measurements and the equation 0zt lsecondqaeak VinitiaLvaluee Compare your measured a with your calculations in Part 0 4 Observe the voltage drOp across the capacitor using the oscilloscope Sketch this waveform in your laboratory notebook Calculate for the values of L and C used in this circuit the value of total series resistance which gives critical damping Replace the resistance R with this value remember R fcngen 1500 Again observe the voltage drop across the capacitor and sketch the resulting waveform in your laboratory notebook Then replace the resistor with 10 times the value you calculated for critical damping Observe and record the capacitor voltage response LABORATORY 10 SERIES RLC CIRCUITS 44 Probing Further 1 Describe the differences between the underdamped overdamped and crit ically damped responses in the capacitor voltage Vc What are the potential advan tages or disadvantages of an electrical system behaving in each mode of Operation What is the functional form of the transient reSponse be speci c if the circuit is critically damped 2 Explain why the inductor voltage VL is greater than the magnitude of the incoming square wave at the time of a square wave transition initial condition Report Your Laboratory Teaching Assistant will inform you when a report is due and on which experiment you will report Make sure that you have recorded all necessary information and data in your laboratory notebook to enable you to prepare a report on this experiment if so directed at some time in the future Laboratory 11 Design Lab Introduction In this lab you will design and build a circuit to meet certain criteria In calling for a circuit design two fundamental questions must be answered 1 What function is the circuit to perform and 2 How well is the circuit expected to perform this function The detailed answers to these questions are usually called the functional requirement and the speci cations respectively Objective The objective of this laboratory exercise is to introduce you to the nature of the engineering design process ie the process of selecting combinations of components to perform a given function with a given degree of precision Preparation Review the material in your circuits textbook on voltage dividers Before you come to lab you should have your circuit design complete with the design procedure and the resulting circuit recorded in your laboratory notebook This is not a team exercise Each individual will be required to have a design completed in the notebook prior to coming to lab class Equipment Needed you specify Procedure 1 You are to design a voltage regulator circuit which will provide 700V across a load resistance RL which may vary anywhere between 10009 and 15009 The supply voltage source is to be 1000V You are to use resistors R1 and R2 between the 10V source and RL in a voltage divider network You must select the values of these resistors to obtain no more than a l5 variation about 700V as the value of RL ranges from 10009 to 15009 Initially you can assume that the selected values of resistances R1 and R2 are precise is 0 tolerance In your laboratory notebook show your design procedure and your calculations to determine values of R1 and R2 45 LABORATORY 11 DESIGN LAB 46 2 After discussing the circuit design with the lab instructor connect the cir cuit and test it When you have met the speci cations demonstrate the performance of your regulator circuit to your LTA 3 Suppose that the specifications had included the additional requirement that the voltage divider was to dissipate the minimum possible power while meeting the 700V i 5 speci cation How would this have changed your design Probing Further 1 What were the major considerations in making the design Did the avail able components affect your design How would you design the circuit if only 10 tolerance resistors in standard sizes were available What are the standard resistor values available for commercial resistors with a 10 tolerance If R1 and R2 are the nominal standard values of the resistors used in your voltage divider and the extreme values are R1H RlL RzH 11R2 RZL 09R2 then what difference would various combinations make in the performance of your circuit 2 Use PSpice to determine the effect on the output voltage across the load resistor of voltage divider resistor values that vary up to 10 from the nominal values Report Your Laboratory Teaching Assistant will inform you when a report is due and on which experiment you will report Make sure that you have recorded all necessary information and data in your laboratory notebook to enable you to prepare a report on this experiment if so directed at some time in the future Laboratory 12 Lab Review and Presentations Introduction One of the many tasks of an engineer is the presentation of technical infor mation to peers to management and from time to time to members of the general public These presentations may be written or oral You have been given several assignments designed to help you develop your written communication skills This laboratory exercise will help you develop your oral presentation skills You will present your results on a selected laboratory exercise to an audience composed of your peers fellow students and management your LTA Objective One objective is to develOp skills of organizing technical material and compos ing succinct presentation materials transparencies to brie y summarize the essence of your activities and results A further objective is to learn to face your audience make your presentation field questions and possible criticisms while maintaining your poise and self con dence Preparation The student will need to prepare an oral presentation as described below in the Procedure section before coming to lab class Equipment Needed Transparencies for overhead projector or other presentation medium Procedure 1 Your LTA will assign each team a lab to present The team members will prepare appropriate transparencies for an overhead projector to present the lab theory procedure and results The lab class will be divided to accommodate the number of teams in each section Your LTA will inform you at the close of the preceding lab class what time will be allotted to each team for their presentation You should rehearse your presentation and time it to take full advantage of your time allotment but not overrun it 47 LABORATORY 12 LAB REVIEW AND PRESENTATIONS 48 2 Each team will be anonymously reviewed by their peers and these review comments may be considered by the LTA in grading the presentation Review of each group will include 39 Presentation of theory GOOD FAIR POOR Presentation of procedure GOOD FAIR POOR Presentation of results GOOD FAIR POOR Clarity GOOD FAIR POOR Probing Further 39 1 Practically all technical conferences require a short paper as the basis for an oral presentation The paper is published in a Proceedings volume Presentation of such talks and papers can be signi cant in career advancement As you have learned in preparing for this assignment care in planning your talk in preparing your transparencies and in rehearsing to get your timing and wording correct technical conferences have a strict time budget you will be required to adhere if you want to be invited back takes much time and effort But it can pay handsome dividends Report A lab report is not due the next period Please tum in manual evaluation form after class The instructor should return these forms to the faculty coordinatOr Laboratory 13 Final Exam Introduction This has been your first engineering laboratory Although you have been pro vided with a cookbook this manual hopefully you have not just blindly followed instructions in order to get a good grade If you have you have cheated only yourself The theory which you have learned from your textbook and lectures is a way of look ing at reality and thinking about how to organize experience ie dealing with real physical things It is only through combining practical experience with theory that you can begin to develOp the necessary analytical skills to aid you in taking things apart and putting them together in new ways that is the essence of the practice of real engineering Objective This examination is designed to help you and your LTA determine how much you have developed your knowledge skills and self con dence Preparation Review your lab manual Think carefully of the procedures which you have followed and what you have learned from them How do you measure voltage cur rent resistance frequency etc How accurate are your measurements When your measurements do not agree with those of your peers what is the cause How do your measuring instruments affect the circuit in which you are trying to make measure ments Equipment Needed Pencil Calculator Procedure 1 The nal exam will consist of a written exam and a practical exam The practical exam involves performing a lab procedure to obtain a desired result The written exam will cover the application of theory to understand circuit behavior based upon your laboratory experiences in this course 49 LABORATORY 13 FINAL EXAM 50 Probing Further 1 This has been your rst in a series of four 4 laboratory courses What have you done to develop your expertise and skills What Will you do differently in the next course What have you learned from your experience Report No report is due for this experiment Laboratory notebooks are to be turned in to the LTA for evaluation Part III Appendices 51 Appendix A Safety 39 Electricity when improperly used is very dangerous to people and to equip ment This is especially true in an industrial environment where large amounts of power are used and where high voltages are present A l in environments where peo ple are especially susceptible to electric shock such as maintenance of a high voltage system while in operation or in hospitals where electrical equipment is used to test or control physiological functions A Z A 3 and in an experimental or teaching lab oratory where inexperienced personnel may use electrical equipment in experimental or nonstandard con guration Engineers play a vital role in eliminating or alleviating the danger in all three types of environments mentioned above For conditions where standard equipment is used in standard con gurations governmental agencies and insurance underwriters impose strict laws and regulations on the operation and use of electrical equipment including switchgear power lines safety devices etc 39As a result corporations and other organizations in turn impose strict rules and methods of operation on their employees and contractors Engineers who are involved in using electrical equipment in supervising others who use it and in designing such systems have a great respon sibility to learn safety rules and practices to observe them and to see that a safe environment is maintained for those they supervise In any working environment there is always pressure to get the job done and take short cuts The engineer as one who is capable of recognizing hazardous conditions is in a responsible position both as an engineer and as a supervisor or manager and must maintain conditions to protect personnel and avoid damage to equipment Because of their nonstandard activities experimental laboratories are exempt from many of these rules and39 regulations This puts more responsibility on the engineer in this environment to know and enforce the safest working procedures The knowledge and habit forming experience to work safely around electrical equipment and the ability to design safe electrical equipment begins with the rst student laboratory experience and continues through life This includes learning the types of electrical injuries and damage how they can be prevented the physiology of electrical injuries and steps to take when accidents occur 52 APPENDIX A SAFETY 53 Physiology of Electrical Injuries There are three main types of electrical injuries electrical shock electrical burns and falls caused by electrical shock A fourth type sunburned eyes from looking at electric arcs such as arc welding is very painful and may cause loss of work time but is usually of a temporary nature Other injuries may be indirectly caused by electrical accidents eg burns from exploding oil immersed switch gear or transformers Although electric shock is normally associated with high voltage AC contact under some circumstances death can occur from voltages from substantially less than the nominal 120 Volts AC found in residential systems Electric shock is caused by an electric current passing through a part of the human body The human body normally has a high resistance to electric currents so that a high voltage is usually required to cause lethal currents This resistance is almost all in the skin but when the skin is wet its resistance is much lower When a person is hot and sweaty or is standing in water contact with 120 Volts or less is likely to cause a fatal shock Electric shock is not a single phenomenon but is a disturbance of the nerves that is caused by electric current A current through a part of the body such as the arm or leg will cause pain and muscle contraction If a victim receives an electric shock from grasping a live conductor a current of greater than 15 to 30mA through the arm will cause muscle contractions so severe that the victim cannot let go Similar currents through leg muscles may cause sudden contractions causing the victim to jump or fall resulting in possible injuries or death It is also possible for a prolonged period of contact of more than a minute or so to cause chest muscles to be contracted preventing breathing and resulting in suffocation or brain damage from lack of oxygen The predominant cause of death by electric shock is generally attributed to ventricular brillation which is an uncontrolled twitching or beating of the heart that produces no pumping action and therefore no blood circulation Unless corrective action is taken death follows quickly from lack of oxygen to the brain While the amount of current that will cause brillation depends on several variables 05 to 5A through the body will normally cause the very small current through the heart that causes brillation in most peOple Larger currents than this through the heart causes contraction or clamping of the heart muscle and resulting death unless corrective action is taken Burns from electric currents may be caused by electric currents owing in or near parts of the body Such burns if caused by high frequency sources are deeper than burns from other sources and take longer to heal but otherwise are not different from normal heat burns Source of Electric Shock Since electric shock is caused by an electric current through a part of the body it is prevented by not allowing the body to become part of any electric circuit From this viewpoint electric circuits may be classi ed as either grounded or ungrounded APPENDIX A SAFETY 54 Grounded circuits are safer for most conditions since they result in known voltages at other points in the circuit and provide easier and better protection against faulty conditions in the circuit The disadvantage is that a person standing on a non insulated oor can receive a shock by touching only one conductor Almost all electric power generation transmission and distribution systems are grounded to protect people and equipment against fall conditions caused by wind storms lightning etc Residential commercial and industrial systems such as light ing and heating are always grounded for greater safety Communication computer and similar systems are grounded for safety reasons and to prevent or reduce noise crosstalk static etc Many electrOnic equipments or instruments are grounded for safety and noise prevention also Common examples are DC power supplies oscillo scopes oscillators and analog and digital multimeters Ungrounded circuits are used in systems where isolation from other systems is necessary where low voltages and low power are used and in other instances where obtaining a ground connection is dif cult or impractical In the ungrounded circuit contact with two points in the circuit that are at different potentials is required to produce an electrical shock The difficulty is that with no known ground a hidden fault can occur causing some unknown point to be grounded Thus touching a supposedly safe conductor while standing on the ground would result in an electric shock Prevention of Shock in the Laboratory Prevention of electric39shock to individuals and damage to equipment in the laboratory can be done by strict adherence to several common sense rules summarized below 1 When hooking up a circuit connect to the power source last while power is off 2 Before making changes in a circuit turn off or disconnect the power rst if possible 3 Never work alone where the potential of electric shock exists 4 When changing an energized connection use only one hand Never touch two points in the circuit that are at different potentials 5 Know that the circuit and connections are correct before applying power to the circuit 6 Avoid touching capacitors that may haVe a residual charge The stored energy can cause a severe shock even after a long period of time 7 Insulate yourself from ground by standing on an insulating mat where available APPENDIX A SAFETY 55 After Accident Action Since accidents do happen despite all efforts to prevent them plans for appro priate reaction to an accident can save time and lives Such a plan should include immediate availability of rst aid material suitable for minor injuries or for injuries that are likely because of the nature of the work Knowledge of how to obtain trained as39sistance such as Emergency Medical Services EMS should be readily available for everyone Treating victims for electrical shock includes four basic steps that should be taken immediately Step two requires quali cation in CPR and step three requires knowledge of mouthto mouth resuscitation Everyone who works around voltages that can cause dangerous electrical shock should take advantage of the many oppor tunities available to become quali ed in CPR and arti cial respiration Immediate Steps After Electric Shock 1 Shut off all power and remove victim from the electric circuit If the power cannot be shut off immediately use an insulator of some sort such as a wooden pole to remove victim from the circuit Attempts to pull victim from the circuit with your hands will almost always result in your joining the victim in the electric shock 2 If you are a quali ed in CPR check for ventricular brillation or cardiac arrest If either is detected external cardiac massage should be started at once Whether you are quali ed in CPR or not notify EMS and the ECE Department at once using the telephone numbers listed below 3 Check for respiratory failure and take appropriate action This may have resulted from physical paralysis of respiratory muscles or from a head injury Sometimes many hours pass before normal respiration re turns Arti cial respiration should be continued until trained EMS assistance arrives 4 Check for and treat other injuries such as fractures from a fall or burns from current entry and exit sites Investigations are always after accidents As an engineer you will be involved as a part of the investigating team or in providing information to an investigator Information obtained and notes written immediately after the emergency will aid this investigation and assist in preventing future accidents of a similar nature Emergency Numbers Emergency FireEMS 911 or 656 2222 Student Health Center 656 2233 ECE Department Of ce 656 5650 APPENDIX A SAFETY 56 Appendix A References Al WF Cooper Electrical Safety Engineering NewnesButterworth London Boston 1978 AZ WH Buschsbaum and B Goldsmith Electrical Safety in the Hospital Medical Eco nomics Company Oradel NJ 1975 A S J G Wester editor Medical Instrumentation Application and Design Houghton Mif in Company Boston 1978 Appendix B Fundamentals of Electrical Measurements Electrical engineers make and use a wide variety of measurements of electrical circuit variables such as voltage current frequency power and energy as well as of electrical circuit parameters such as resistance capacitance and inductance Even though many instruments which can be used to make such measurements are com mercially available the proper use and interpretation of these measurements depend on a fundamental understanding of how these measuring instruments work their capabilities and their limitations This appendix provides a brief overview of the fundamentals of electrical mea surements There are many textbooks on the subject See References B l through B 6 of this appendix for example As your exposure to various types of electrical measuring instruments increase in this and subsequent laboratory courses you will find some of these books or similar books that you can nd useful in developing your understanding and measurement skills In addition many commercial instru ment manufacturers publish handbooks and applications notes that provide more information on Specific measurement techniques Measurement of Current and Voltage The basic electrical circuit variables of current and voltage are measured with ammeters for current and voltmeters for voltage These instruments may use either analog or continuous or digital or numerical indicators or readouts Analog Meter Instruments The analog instruments were those developed early in the history of electrical science and technology Most are based on the d Arsonval galvanometer movement A brief description of this meter movement and its use in ammeters and voltmeters is given in the textbook ELECTRIC CIRCUITS Fifth Edition by J W Nilsson B7 More information is available in References B 1 and B Z In the d Arsonval gal vanometer current through a coil of ne wire develops a magnetic eld which opposes that of a permanent magnet to rotate a needle across a scale which is calibrated marked off in units of the measured variable 57 APPENDIX B FUNDAWNTALS OF ELECTRICAL MEASUREMENTS 58 Current Measurement The basic meter movement can be modeled as a resistance RM which rep resents the resistance of the ne wire coil in series with an ideal zero resistance meter movement M as shown in Figure B1 The important parameters for this meter are the full scale current I M F3 and RM The full scale current is the value of current that rotates the indicating needle through its maximum arc Note that a larger current may damage the meter movement permanently rendering it useless vM M gt Figure 31 Electrical circuit model for d Arsonval meter movement The voltage drop across the meter movement will be VM IRM B 1 and the maximum rated voltage drop for the meter will be VMFS IMFSRM B 2 When it is necessary to measure currents larger than I M F3 a resistor with a value of resistance Rsh smaller than that of the meter RM is placed in parallel with the meter movement The parallel resistor is usually called the shunt resistor Figure B2 shows a circuit diagram for this arrangement M RM I Wa IM gt RM lsh Figure Bl Electrical circuit model for d Arsonval meter movement with a shunt resistance APPENDIX B FUNDAMENTALS OF ELECTRICAL MEASUREMENTS 59 From Figure B2 it is obvious that VM VM I I Is 2 B 3 M h RM I39 Rsh and that the full scale current I F3 for this metering arrangement will be R IFS IMFS 1 RM 13 4 sh Below in Figure BB is the circuit diagram symbol for an ammeter compared to the equivalent circuit model for the analog ammeter which uses a d Arsonval galvanome ter vM RM T NVc r T I IM EQUIVALENT CIRCUIT MODEL e CD A U CIRCUIT DIAGRAM SYMBOL FOR AN I AMMETER Figure B3 Electrical circuit model for ammeter using a d Arsonval meter movement with a shunt resistance and the circuit diagram symbol for the ammeter Example B Z Consider a d Arsonval meter movement with the following characteristics IMFS 50HA RM 5129 Find the value of shunt resistance that will enable the measurement of up to 1mA of current gtFrom Equation B 4 RM IFS 1mA Rsh IMFS 50MA 1 20 APPENDIX B FUNDAMENTALS OF ELECTRICAL MEASUREMENTS 60 01 RM 5000 T6 T9 263169 Exercise B I What value of R5 is required to make a meter that would measure 100mA full scale Rah Voltage Measurement A d Arsonval galvanometer movement can also be used to make a voltmeter Instead of placing a resistor in shunt a resistor is placed is series with the meter move ment Figure B4 shows the equivalent circuit for such an arrangement connected in series with a voltage source V23 which has an internal resistance R3 associated with it Also shown is the circuit symbol used to indicate the presence of such a voltmeter in a circuit R X EQUIVALENT CIRCUIT MODEL lt SOURCE METER RX gt CIRCUIT DIAGRAM i SYMBOL FOR A VOLTMETER l lt SOURCE METER gt Figure BA Equivalent circuit for voltmeter circuit using a d Arsonval galvanometer with series resistor and the circuit symbol used for a voltmeter APPENDIX B FUNDAMENTALS OF ELECTRICAL MEASUREMENTS 61 The maximum voltage VFS that can be applied will depend upon the size of the series resistor R3 By applying Kirchoff s voltage law to the circuit of Figure B4 we have VFS IMFSRs RM Rm 135 Errample B Q Using the meter movement previously used suppose that it is desired to make a voltmeter with a 100V full scale rating From Equation B 5 VFS Rs 39 RM Rm IMFS 100 R3 12 50106 5000 R and RS 199549 Rm Often R95 is not precisely known or is inconvenient to determine If RC is small enough compared to Rs to be considered negligible it will have a small effect on the accuracy of the voltage measurement If not the loading effect of the voltmeter on the voltage source must be taken into account A procedure for doing this will be used in Laboratory Experi ment Number 2 Exercise B 2 Assuming that Rm is small say 1009 what value of RS would be required to make a voltmeter with 10V full scale rating using the same meter movement as in the example above Analog Electronic Instruments From the circuit diagram of Figure BA it can be seen that if the unknown voltage source VIE has an internal resistance associated with it that approaches or exceeds that of the voltmeter circuit the reading will be in error due to the voltage drop across the internal resistance of the source The analog electronic instruments were developed to provide a high input impedance for the voltmeter in order to minimize such a loading effect of the measuring instrument on the circuit being measured The early versions of such electronic voltmeters used vacuum tube ampli ers and were consequently known as vacuum tube voltmeters or VTVM s More recent instruments use the high impedance offered by field effect transistors in operational ampli ers FET input op amps The output voltage of the op amp can then be used to drive a d Arsonval galvanometer to provide a continuous analog indication How ever the advent of digital circuits and indicating devices has enabled the development of digital meters which will be brie y reviewed in the next section APPENDIX B FUNDAMENTALS OF ELECTRICAL MEASUREMENTS 62 Digital Instruments Figure B5 shows a block diagram of an electronic ammeter with a digital readout Figure B6 shows a block diagram of an electronic voltmeter with a digital readout Except for the input circuitry both instruments are similar in design and construction CURRENT TO AID DISPLAY VOLTAGE CONVERTER DRIVER DIGITAL DISPLAY VOLTAGE TO R AID DISPLAY 39N N VOLTAGE CONVERTER DRIVER DIGITAL DISPLAY Figure B6 Block diagram of electronic voltmeter with a digital display readout The input module of the digital electronic ammeter in Figure 35 and that of the digital electronic voltmeter in Figure B6 is usually built around an operational ampli er and other transistor devices You will learn more about these in ECE 320 and ECE 321 The A D spoken as A to D converter and other digital circuit devices you will learn more about in ECE 371 The input resistance of the digital electronic ammeter is designed to be as low as possible ideally zero whereas the input resistance of the digital electronic voltmeter is designed to be as high as possible ideally in nite HOW Good Are Your Measurements In engineering practice although qualitative words like good and bad are used more often than not quantitative descriptions are used ie how good or bad in numerical terms When making measurements three attributes are of interest accuracy precision and resolution Accuracy is usually expressed as a percentage and respresents the difference in percent of the readout value of the measuring instrument compared to the actual or true value Usually the accuracy speci cation of an instrument is expressed as a percentage of the fullscale value on a given range For example if you are using a voltmeter with an accuracy of 2 of full scale on the O lOV range and you obtain a scale reading of 48V you would expect the actual voltage to be within the range of 48ID2V For digital display instruments the accuracy speci cation is usually given as a percentage of full scale on the range being used plus one digit the lowest order APPENDIX B FUNDAMENTALS OF ELECTRICAL MEASUREMENTS 63 digit Thus for example if a digital voltmeter with 4 digits is being used on the 0 20V range and the accuracy speci cation is i01 1 digit a reading of 1342V will indicate an actual input voltage in the range 1342003V Precision is used to indicate how closely the readout can be read For ex ample an ammeter using a d Arsonval meter movement for readout with a scale calibrated in units of 02 of each major subdivision can provide an estimate doWn to about 01 of a major subdivision Ie an ammeter with a scale subdivided into 10 major subdivisions 01 mA per major subdivision with these major subdivisions further partitioned into 5 minor subdivisions of 002 mA each might allow and es timate of 001 mA or 10 MA if the indicating needle is backed with a mirror that allows the observer to carefully align the needle with the scale For a digital readout the precision is determined by the number of digits available in the display For example a readout that has 4 digits on a voltmeter being used on the 0 20V range would have a precision of 001V and on the 0 200V range would have a precision of 01V Note that the precision of an instrument will usually be better than its accuracy Resolution is used to indicate the smallest change of the input variable to which the instrument sensibly through its readout can respond Just because the original speci cations for a given instrument state a given accuracy doesn t mean that the instrument is guaranteed to retain that accuracy in de nitely Measuring instruments like all manufactured items are subject to wear and tear An instrument which is being used within its ratings ie hasn t been abused electrically or mechanically like being drOpped on a workbench or onto the oor still requires periodic maintenance and recalibration The procedure for re calibration is usually speci ed in the instruction manual for the instrument provided by the manufacturer It is the usual good practice for the person performing the recalibration to place a sticker on the outside of the instrument giving the date of servicing and the name or initials of the person responsible In industrial plants where accurate measurements are important the calibration procedures are usually performed in an instrument shop which is equipped with secondary standards of voltage current resistance etc traceable to the National Institute of Standards and Technology NIST formerly known as NBS or the National Bureau of Standards 39 When using an instrument refer to the instruction manual for correct usage and for speci cations on the accuracy and resolution of the instrument Also check to see what the last date of calibration was The longer the interval between calibration and your date of using the instrument the more suspect is the accuracy of your measurements It is good practice to have a known voltage source such as a fresh 15V battery to quickly check a voltmeter or current source such as a 15V battery in series with a 15kQ resistor to provide 1mA to obtain a quick check of an ammeter If several instruments are available they can be used with the same source to obtain a comparison of readings The best rule of measurement is check it out before you believe the readings APPENDIX B FUNDAMENTALS OF ELECTRICAL MEASUREMENTS 64 How Good Are Their Measurements Compared To Mine Often the measurements made by two engineers or two different groups of engineers under ostensibly similar conditions using ostensibly similar sets of instru ments and equipment will result in different values sometimes what appear to be signi cantly different value The consequences may range from a friendly dispute to lawsuits Various engineering organizations such as the American Society for Testing Materials ASTM go to great lengths to standardize measurement procedures and then to compare the results obtained by various people in different organizations using these procedures One method of doing this is the so called roundrobin method In the roundrobin method a set of objects to be measured for example a set of standard voltage cells will be sent from laboratory to laboratory for mea surements to be made Then each group making the measurements will report them to a central group the referee for comparison Following a statistical analysis of the individual and pooled measurements the results are reported back to all of the participants This method helps to establish the practical limits on repeatability of the measurements made using the procedure In general there are many types of errors that can interfere with an accurate measurement of a variable such as a voltage value These can be partitioned B l B 2 into three categories human errors systematic errors and random errors Human errors are caused by a loss of concentration ignorance or laziness on the part of the human being making and or recording the measurements Such errors include choosing the wrong scale or even instrument misreading the scale transposing numbers when recording etc For this reason it is helpful to have one or more lab partners observing or repeating the measurements in order to catch such errors before they become a problem Systematic errors may due to equipment instrument errors or to environ mental conditions An example of the instrument error is the use of an instrument that is out ofcalibration or one for which the zero has not been properly adjusted Many instruments are temperature sensitive and will give erroneous readings when used out of their speci ed ambient temperature range They mayalso be sensitive 39 to the presence of largemagnetic elds or sensitive to high humidity In some cases the effects of environmental conditions may be accounted for if known but quite often either an accurate measurement of actual instrument temperature or its local humidity are not known so correction is not possible Random errors are those errors which are due to unknown causes and which are operative when the human errors and the systematic errors have been brought under control or otherwise accounted for These random errors can be minimized or accounted for by the use of multiple measurements and statistical analysis Statistical analysis terminology and methodology are discussed in Appendix C of this manual APPENDIX B FUNDAMENTALS OF ELECTRICAL MEASUREMENTS 65 Appendix B References B l F Spitzer and B Howard PRINCIPLES OF MODERN INSTRUMENTATION New York Holt Rinehart and Winston 1972 B2 TL Zapf Accuracy and Precision pp 4 1 through 4 20 in CF Coornbs Jr ed BASIC ELECTRONIC INSTRUMENT HANDBOOK New York NY McGraw Hill 1972 B 3 S Wolf GUIDE TO ELECTRONIC INSTRUMENT AND LABORATORY PRACTICE Englewood Cliffs NJ Prentice Hall 1973 B 4 HV Malmstadt CG Enke SR Crouch and GHor1ick ELECTRONIC MEASUREMENT FOR SCIENTISTS Menlo Park CA WA Benjamin Inc 1974 B5 HH Chiang ELECTRICAL AND ELECTRONIC INSTRUMENTATION New York NY John Wiley and Sons 1984 13 6 EF Mazda ELECTRONIC INSTRUMENTS AND MEASUREMENT TECHNIQUES Cambridge UK Cambridge University Press 1987 B 7 JW Nilsson ELECTRIC CIRCUITS Fifth Edition Reading MA Addison Wesley 1996 Appendix C Fundamentals of Statistical Analysis When the human errors and systematic errors associated with a measurement procedure have been eliminated or brought under control there will still be some variability in the measurements when they are repeated or when some other per son presumably equally skilled makes the measurement These are called random errors They are characterized by irregularity which may appear to be a source of potential disagreement or confusion However the use of statistical analysis meth ods makes possible the extraction of essential information Two characteristics of the measurements are of interest the arithmetic mean average value of a series of measurements and some measure of the spread or dispersion ie the degree of variability about the mean Consider a series of measurements 111 22 3 513 of some quantity For example 6 measurements of a voltage source over a period of an hour repeated for 5 consecutive hours Table 0 provides values for each of these measurements Measurement N0 Hour 1 Hour 2 Hour 3 Hour 4 I Hour 5 5043 5048 5040 5044 5047 5038 5043 5044 5044 5041 5036 5046 5042 5039 5040 5035 5042 5039 5036 5042 5040 5040 5043 5045 5037 5043 5038 5041 5400 5039 manpower 4 Table C1 Voltage measurements of a source over a 5 hour interval For each sample of 77 measurements one can calculate an average or arith metic mean value from 21502 J m 39 C l m n lt gt and a variance 42133939 002 82 3 3 2 77 1 66 APPENDIX C FUNDAMENTALS OF STATISTICAL ANALYSIS 67 More often the square root of the variance called the standard deviation 9 is used rather than the variance As can be seen from Equation 0 2 the variance is the mean square deviation of each measurement from the average value Therefore the standard deviation is the root mean square deviation of the set of measurements Equations C 1 and 3 2 have been used to calculate the average voltage variance and standard deviation for each of the sets of voltage measurements in Table 3 1 The results are shown in Table 0 2 Parameter Hour 1 Hour 2 Hour 3 Hour 4 Hour 5 Vm 5039 5043 5042 5041 5041 92 11810 5 13810 5 03510 5 12710 5 11610 5 3 00034 00037 00019 00036 00034 All Hours VaryCW 5041 3 00015 Table 01 Voltage measurements of a source over a 5 hour interval Also calculated and shown in Table 3 2 is the average of the sample averages and the standard deviation of these averages It can be seen that the sample averages are more tightly grouped about the average of the averages or the grand mean than are the individual measurements are grouped about their respective sample mean values Why would one go through such calculations What is the interpretation of the results From the original data tabulations in Table 3 1 it can be seen that the maximum voltage measured was 5047V and the minimum was 5035V This corresponds to a difference of 12mV which compared to the grand mean value of 5041V is only 024 for the range Max Min and only i012 about the grand mean On a percentage basis these results look very good However there are situations when the ability to resolve a few millivolts might be important so it is important to characterize the ability of the voltmeter to make precise measurements Another item of interest is to estimate how much of a deviation might be expected over a longer period of time Again looking at Table 31 it can be seen that over any of the one hour periods the readings uctuated up and down around the grand mean value Only twice in a 5 hour period was the grand mean value obtained 2 out of 30 readings This wandering back and forth is characteristic of the variation expected from the random interplay of various uncontrolled physical factors The average value of a measurement might be though about as information and the random uctuations as noise which interferes in this case rather slightly with the process of obtaining the information APPENDIX C FUNDAMENTALS OF STATISTICAL ANALYSIS 68 We only took 5 samples one per hour of 6 readings each What would we expect from say 20 readings per hour taken over the period of a month Or perhaps through automation 1200 readings per hour taken over a month What information would we gain by greatly increasing the number of observations and of course greatly increasing the work of analyzing the data In other words how good are our inferences made from a few samples about the true values of mean and variance or standard deviation 39 The theory of measurement errors and their determination is based on the following observations C l 1 The most probable value that can be assigned to a measured magnitude on the basis of equally trustworthy direct measurements is the arithmetic mean 2 In any large number of measurements positive and negative errors of the same magnitude are equally likely to occur 3 Small errors are much more likely to occur than large ones 4 All of the errors of measurement in a given series lie between equal positive and negative limits 39 These observations have led to the use of the normal distribution function to model the expected error distribution of a large approaching in nity series of measurements of the same kind This large in nite reference group is called the population The average value of the population of measurements is given the symbol a Its variance is symbolized as 02 and its standard deviation is or In terms of these parameters the normal distribution function is 1 m p2 x m3 2 72 0 3 Figure 01 shows a plot of this distribution function in terms of a normalized vari able 3 z 39u 0 4 039 or 1 22 Me 2 3 5 to give a universal curve A related curve is that of the integral of 3 5 or fz Pz 32 f 63 2 du 3 6 00 APPENDIX C FUNDAMENTALS OF STATISTICAL ANALYSIS 69 05 045 04 3 E 025 02 015 01 005 Units are c deviations from p Figure 31 Normal distribution function versus 2 0 s 04 02 0 i 5 4 1 2 3 4 5 Units are c deviations from u Figure 02 Cumulative normal distribution function versus 2 APPENDIX C FUNDAMENTALS OF STATISTICAL ANALYSIS 70 The integral in 36 is known as the cumulative normal distribution function Values of this integral for positive values of z ie for it gt p are given in Table 03 Values of the integral for z lt 0 ie for a lt u are obtained from the relation P z 1 Pz o7 Table 03 Values of the cumulative distribution function P z for positive values of z The value of z is determined by z row heading column heading A plot of the cumulative normal distribution function versus Z is given in Figure C2 PrOperties of the normal distribution function and the cumulative nor mal distribution functions are discussed in Reference C 2 Where there are numerous examples of the applications of these functions in statistical analysis From Equations 0 6 and 37 as well as the values of Pz given in Table 03 it can be seen that the limit of Pz as39z gt 00 is 100000 Thus Pz is a function that varies smoothly from 0 at oo to 100000 at 00 It is interpreted as a measure of the probability that 9 the measured variable lies Within a given range about the true value LL Let s explore more of this interpretation Looking at the values in Table 03 at z 100000 Pz 2 0834134 05 034134 the number 05 is subtracted from the value in Table 03 to yield the integral value from 0 to 2 instead of from 00 to APPENDIX C FUNDAMENTALS OF STATISTICAL ANALYSIS 71 2 Therefore for the error a as L 2 lzla ie z i10000 the probability is 1 0 2 1 1000 2 Pac lt1000 e 2 d d l M QT 1000 z m 0 e Z 0139 Plm 0 lt 1000 03413 03413 06826 Thus about 68 of the time we expect the random error to lie within 1c7 of a By the same reasoning we expect the error to lie within 20 about 955 of the time and the error will lie within i3a about 997 of the time This is all well and good but when we really don t know u and 039 how can we make any predictions about the expected error of our readings It happens that both of these statistical parameters that are suf cient to describe the population from Equations C 5 and 3 6 can be estimated from our samples Of course you realize that as we went from 6 samples hour to 20 samples hour to 1200 samples hour to in nity we came into the all encompassing knowledge of how to calculate very precisely the expected error Thus it would seem that the smaller is our sample the less precise will be our estimate of the population statistical parameters a and a It happens that the equations used to calculate the arithmetic mean Equa tion 3 1 and that for the variance Equation 3 2 which are repeated below for convenience of reference Z 93339 37m 31 7 3 1 77 and n 2 33339 93cm2 Eittf v 0 give a maximum likelihood estimator of u and a maximum likelihood estimator of 02 How good are these estimates It depends upon the number of measurements n used to obtain the estimates The goodness of the estimate is usually expressed in terms of con dence intervals The details of doing this are discussed in Reference 3 2 in Chapter 11 Small Sampling Theory Figure 33 shows the effect of sample size on the percentage error in the estimation of a In this gure the di erence between the estimate obtained by using Equation 3 1 and u is compared to the estimate of 039 obtained by using Equation 3 2 or mm mw lt00 to obtain a universal curve Thus for a given value of s and sample size n the value of u is expected to lie within the range cum 3 yn s This expectation is expressed as probability For the example shown in Figure 33 a probability of APPENDIX C FUNDAMENTALS OF STATISTICAL ANALYSIS 72 90 has been selected The resulting limiting values represented by the curves are called the 90 con dence limits and may be interpreted as meaning that 9 times out of 10 the estimate of u lies within the limits shown 90 Percent Confidence Limit yn 0 I z i 0 20 i i 40 80 80 100 120 140 Sample size n Figure 33 90 con dence limit yn on the error in estimating h using may versus sample size n Figure 3 4 shows the effect of sample size on the error in the estimation of 0 using the value of 3 calculated from Equation CI 2 Thus 0 lies between the limits 3 rim lt 039 lt 3 7quot71 3 9 Where 7101 is obtained from the lower curve of Figure 04 and 7quot is ob tained from the upper curve of Figure 34 The curves shown in Figure 04 are calculated using the 90 percent con dence limits Let us now reexamine the data of Table 31 Since the hour to hour aver ages appear to behave randomly we will combine all of the measurements to get a pooled sample size of 30 measurements For the pooled sample V0 5041 and 32 0003262 Using the curve in Figure 33 for n 30 yn 0316 and the population mean value of the measurements V is expected 9 times out of 10 to lie within the limits Vav yn 39 s lt V lt Vav s or 5040V lt V lt 5042V APPENDIX C FUNDAMENTALS OF STATISTICAL ANALYSIS 73 The standard deviation of measurements in the sample population is expected to lie in the range de ned by Equation 0 9 or 000274V lt 0 lt 000424V It can also be seen that if we had used only 6 measurements one hour s worth for our calculations the uncertainty in both the mean value and the standard deviation would have been much larger w N N m f 90 Percent Confidence Limits r n and run RRM irultngt 1 1 quotquotquot39 2 rln 05 i i i o 10 20 30 4o 50 so 70 so 90 100 110 Sample size n Figure CA 90 con dence limits 739 and Tu on the estimate of 75 using 3 versus sample size n APPENDIX C FUNDAMENTALS OF STATISTICAL ANALYSIS 7 4 Appendix C References 0 1 AdeF Palmer THE THEORY OF MEASUREMENTS New York NY McGraW Hill Publishing Co 1912 See Chapter IV 02 MR Spiegel STATISTICS 2nd Edition New York NY Schaum s Outline Series McGraw Hill 1988 Appendix D Resistor Identi cation Table A1 Standard resistor values 10 18 33 56 11T 20 r 36 62 12 22 39 68 13W 24 43 75 15 27 47 82 16 r 30 51 91 T Value unavailable for 10 L multiplier i Value unavailable for 100 multiplier Table A2 Resistor color code Color Value Multiplier Tolerance Black 0 1 Brown 1 10 Red 2 100 Orange 3 1000 Yellow 4 10 000 Green 5 105 Blue 6 106 Violet 7 107 Gray 8 108 White 9 109 Gold 10391 5 Silver 10 2 10 None 20 75

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