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1. c Repeat steps a and b for the following frequencies 10 Hz 100 Hz 1 kHz 10 M Hz Calculate the difference from the original 3 0 Vms Put the results in tabular form Questions Section 3 What trend is evident in the percent error Section 4 Period Measurement with the Frequency Counter a Configure the FG to supply a 3 0 Vms sine wave 1 Hz with no DC offset Try to measure the period of this waveform using the counter Again as in section 2 a you should observe that the multimeter display jumps around at this very low frequency lt 2 Hz b Configure the FG to supply a 3 0 Vims sinewave 100 Hz c Measure and record the period of this waveform using the counter d Repeat steps a and b for frequencies of 10 Hz 100 Hz 1 kHz 10 M Hz Calculate the difference from the original 3 0 Vims Put the results in tabular form Questions Section 4 What trend is evident in the percent error Section 5 Analysis of the Measurement Error In this section we learn to analyze the measurement error Overview of Error Analysis Data collection and interpretation is key to successful laboratory experimentation There are two main types of experimental errors in physical measurements Systematic Errors will cause the entire distribution of data points to be offset with respect to the true value Causes of systematic error include poor measurement technique errors in instrumental calibration software errors o
2. 5 5 00 V triangle wave 2K Hz with no DC offset 6 6 00 V p square wave 2K Hz with no DC offset d Repeat steps a and b for this waveform 7 6 00 V Square wave 2K Hz with 1 0V DC offset e Display the results AC DC and RMS values in tabular form for all seven waveforms Questions Section 1 1 2 3 How closely do waveforms 1 and 2 match each other for the AC reading Why is that What would you mathematically expect waveform numbers 3 4 and 5 to be for RMS values How close are these predictions to the actual scope readings How closely do the measurements for waveforms 6 and 7 match what was calculated in question 3 and 4 of the prelab Section 2 Voltage Measurements at Various Frequencies a b c Configure the FG to supply a 3 0 Vim sine wave Hz with no DC offset Using the AC scale on the Agilent 34401A multimeter measure the voltage output of the FG You should observe that at this very low frequency the multimeter display jumps around Repeat steps a and b for the following frequencies 10 Hz 100 Hz 1 kHz 10 M Hz Calculate the difference from the original 3 0 Vms Put the results in tabular form Question Section 2 What trend is evident in the percent error Section 3 Function Generator Counter Accuracy a Configure the FG to supply a 3 0 Vim sine wave 500 Hz with OV DC offset b Use the Agilent 34401A Counter to measure this frequency and record the results
3. Exp 3 USE OF BASIC ELECTRONIC MEASURING INSTRUMENTS Part Il amp ANALYSIS OF THE MEASUREMENT ERROR PURPOSE e To become familiar with more of the instruments in the laboratory e To become aware of operating limitations of input and output devices e To understand and be able to work with root mean square RMS measurements e To learn the principles of analyzing the measurement error including variance and Gaussian distribution This experiment relates to the following learning objectives of the course 1 Ability to interconnect equipment and devices such as multimeter function generator and oscilloscope to achieve required results Acquire practice in recording data and results and maintaining a proper engineering notebook Ability to analyze and evaluate data AN LAB EQUIPMENT Agilent E3640A DC Power Supply Agilent 34410A Digital Multimeter Timer Counter Agilent 33120A Function Generator FG Agilent 54621A Oscilloscope STUDENT PROVIDED EQUIPMENT 1 BNC Double Banana 1 BNC T Adapter 1 BNC BNC Cable m Experiment Sections 1 Agilent 33120A Function Generator and Agilent 54621 A Oscilloscope 2 Voltage Measurements at Various Frequencies 3 Function Generator Counter Accuracy 4 Period Measurement with the Frequency Counter 5 Analysis of the Measurement Error Section 1 Agilent 33120A Function Generator Function Generator FG Background Note e Inthe section 1 of this experiment you will conf
4. alysis would it be acceptable if you obtain additional measurement values from students on other benches 4 What change do you expect in the difference between the calculated and manually measured values of the standard error as the number of measurements increase explain 5 Do you expect the error distribution look more normal as the number of measurements increase
5. aximum of three in the Agilent 54621A oscilloscope otherwise you can find out how by checking its user s manual also available on line 1 6 00 V sine wave 500 Hz with no DC offset 2 2 12 Vims sine Wave 500Hz with no DC offset 3 2 00 Vims Sine wave 1k Hz with 2 0 V DC offset Note for each waveform enter its parameters into the function generator exactly as stated In particular this will require you to switch back and forth between V peak to peak and V RMS by pressing Enter Number then either A or b Recall each of the three stored waveforms one at a time and perform the following measurements 1 AC Peak to Peak 2 DC Average 3 RMS values Perform these measurements on the oscilloscope only To do so press Quick Measure button then Select and Measure each of the following AC Peak to Peak DC Average and RMS For example here is the results that were obtained for the first waveform c zt Agilent Technologies a gt AvgC1 yo E RMS 19 2 130 Measure Settin noe MS RMS Note how as indicated by the dashed lines the scope automatically performs the calculations over an integer number of periods of the periodic waveform Always be sure the vertical sensitivity of the scope is set so the waveform is not clipped otherwise it might perform the calculations incorrectly Repeat steps a and b for the following three waveforms 4 2 00 Vims sine wave 1k Hz with 2 0 V DC offset
6. be assumed to represent the varying outcomes values of a measurement process RMS value of the FG output waveform d Freeze the waveform by pressing the Stop Run button on the scope panel thereby making it turn red to read and record the digital readout at any moment Once the Sop Run button of the scope has been pressed successive individual traces can be obtained by pressing the Single button Obtain 10 RMS readings and record them in a table e Calculate the mean and the standard deviation for the set of ten measurements f Increase the number of readings to 20 and recalculate the mean and standard deviation Record the differences from the previous values in your table g Increase the number of measurements to 40 and plot the distribution of the error deviation from the mean h Manually fit the above standard error distribution to a Gaussian distribution function as shown in the figure above Estimate the standard deviation from the manually fitted Gaussian curve Remember the curve spans 30 about the mean and record the percent difference from the calculated value Questions section 5 1 How and why do the estimates of the mean and standard deviation change as we increased the number of samples from 10 to 20 2 How close is the mean value based on the 20 point sample set to the theoretical value If we increase the number of measurements to say 1000 will we nearly reach the theoretical value 3 For error an
7. e used in the power formula like a DC value For example assume that we apply a DC voltage Voc toa resistor R causing a DC current lic through it The DC power which of course is a constant and is the same as the average power 1s 2 2 be V po REI R Power c source 7 Yio DC For AC source we use the same formula except we use the RMS value of voltage and current That is 2 2 lous V pus R ame R Power ic source Vue RMS The RMS value of a function is defined as the square root of the average value mean of the square of a function If a function V t is periodic with a period of T then the RMS of this function is mathematically defined as Vans T 1 2 Vit dt 0 To obtain the RMS value of a function we follow these three steps 1 Square the function over a period 2 Calculate the mean of this new function 3 Take the square root Example Assume we have a triangular function with a period of T 4 and an amplitude of A with no At A DC offset Because of symmetry it suffices to integrate v from 0 to 1 1 e to average over a quarter period to find the RMS value Vrms sja The RMS of sine waveforms can be obtained similarly as done in the pre lab Experimental procedure Note throughout this experiment set the FG to High Z output termination Impedance a Configure and store the following waveforms in the FG You should be able to figure out how to store waveforms up to a m
8. gned in order to assist in the construction of Gaussian distribution plots required in parts e through Number of Bins j iow g This is done by categorizing entered values into _ Entet New Value specific bins of a predetermined Size Once all the Entered Yalues Bin Quantities Range 3 Standard Deviation values have been entered the data is exportable to any spreadsheet applications The Gaussian Calculator is also useful in quickly calculating the average value and standard deviation The executable code and the user guide are available either from the EE 241 directory or by clicking the following links Copy to Clipboard ETENEE Fundamental Formulas Standard P Gaussian Calculator User Guide Berton DA Average mean yz i l Reset A Gaussian Calculator Executable Pee Standard deviation standard Errori Clase The Gaussian Calculator application a Configure and store the following waveforms in the Agilent function generator V pp Sine wave 1 KHz with no DC offset b Connect the FG output to the Agilent oscilloscope for this experiment you can use a black coaxial BNC to BNC cable c Set the scope to AC coupled and press the Quick Meas button on the scope panel and then select RMS measurement You will notice that the digital readout for the RMS value of the waveform of the scope is not stable the least significant digit is constantly changing The continually changing readout can
9. igure the FG to supply seven different waveforms Each of these waveforms will be initially stored by the FG which will allow you to recover them quickly There are some instructions on how to operate the FG given below Please use the manual for reference You must first change the Output Termination Impedance of the FG to High Z by following the steps outlined in page 40 of the FG operating manual You can store up to 3 waveforms at a time with the Agilent 33120A Function Generator e The maximum DC Offset that can be generated by the Agilent 33120A Function Generator is given in the operating manual as V lt V __ and V lt 2V_ whichever is less max 2 Offset PP pe Where Vmax 10V for a High Impedance termination and 5V for a 50Q termination RMS Value of a Waveform Sinusoidal signals e g alternating current or voltage waveforms AC signals are periodic and hence change with time For these signals the instantaneous power which 1s proportional to the square of the signal at any given point of time is also periodic and hence changes with time However for these signals we are often interested in the average power and not care for the instantaneous power The root mean square RMS value defined below is a single measure value of a sinusoidal signal that can be used in the average power calculations The RMS value is also called the effective value because it is effectively like a DC value 1 e it can b
10. ment of R For this reason we can only obtain estimates of the true average and standard error If a measurement is carried out M times where m lt n then we can use the above equations to estimate the average and standard error by replacing N with m This means that more measurements translate to smaller errors in the estimate of the true mean and standard error Note that the measurements are assumed to be unbiased and carried out under identical conditions Gaussian distribution Gaussian The standard error does not provide a bate distribution complete representation of the i measurement error spread deviation from the mean To obtain a complete distribution of the error we can divide the range of values between the minimum and maximum into m equal intervals or bins and plot the frequency of occurrence for each range of values within each bin For many physical quantities the distribution 2 has a bell shaped form known as a i Gaussian or normal distribution For a Gaussian distribution 68 3 percent of the measurements are within one standard deviation of the mean As shown almost all measured values fall within 3o0 of the 2 mean The distribution has the f x l x _ X OAD I mathematical form or l Jy frequency fe Ww w i 14 B R2 u 10 9 8 7 6 Measurement intervals bins Experimental Procedure Note An executable program The Gaussian Calculator has fameen nnn gg been desi
11. r failure to correct for external conditions e g temperature Possible sources of systematic error must be considered in all of the stages of the experiment from design to data analysis Random Error is the deviation from the true value which occurs in any physical measurement The assumption is that this error will result in experimental readings which are equally distributed between too high and too low and the mean value reflects the true average value to within some precision Analysis of Random Error Suppose you want to measure a physical property e g the RMS value of a waveform The numerical values that the physical quantity can assume may be represented by a random variable R whose population is the set X X2 X3 Xn Each X represents a possible value or outcome Note that the population of R may include multiple occurrences of the same value Two important statistical properties of this set are the average value mean and the standard deviation standard error defined as Fl i l Average mean Ee Standard deviation standard error o n n li The Standard Deviation also known as Standard Error measures the spread of the data about the mean value It represents the expected error in a single measurement of a physical quantity 1 e the expected deviation from its true mean In laboratory measurements we can only obtain a subset of the entire population 1 e all possible outcomes of a measure

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