However, after testing it was found that the physical circuit was not sufficiently reliable for student use. So after carefully analyzing its behavior, he wrote a program to mimic its behavior as closely as possible. You will be using that program when you obtain your data.
Our College Physics Laboratory project for Winter is to completely analyze that circuit. You will be able to manipulate the switches remotely using the form above, and the program which controls the circuit will report to you the voltages measured (relative to ground) at points VA through VF. You will use that data in your analysis.
It may be useful to read the sections from the text on circuit elements and circuit analysis, since this laboratory project deals extensively with those subjects and you may not have completed them before undertaking your analysis.
We will be using many of the statistical techniques from last quarter in our analysis, as well as some new ones. And we will read several more historically important papers in physics in order to refine our understanding of the structure of effective experimental reporting. Your final report will be a culmination of this entire endeavor.
Only one of the circuits above was constructed, so only one student at a time will be able to use it. But the experiment takes only a few seconds to perform, so if the circuit is in use by another student you will be notified, and as soon as it is free your data will be sent to you.
Each student will have a different set of configurations, numbered 1 through 8, which will be assigned by your instructor the first day of class. For each configuration, check the boxes above for the switches you want to close. Note that switch S1 is always closed during the taking of data.
Switch S2 allows us to include or exclude all of the capacitors at one time, while switches S3 and S4 allow us to exclude capacitors C2 and C3, respectively.
Resistors R1 and R4 are always in the circuit, while switches S5 and S6 allow us to remove resistors R2 and R3, respectively, and switches S7, S8 and S9 allow us to bypass resistors R5, R6 and R7, respectively.
Once you have chosen which switches you want to close for a particular data set, push the button labeled "Perform the experiment" and your data will shortly be returned to you in the form of a table. If all of the capacitors have been excluded from your circuit, a single line will be returned with the time since switch S1 was closed (in seconds), followed by the values of the voltages measured at the points VA through VF. If any capacitors have been included, you will receive 100 lines of data with the values of the voltages at those test points every 10 milliseconds (approximately) after the switch was thrown. Record all data without rounding.
The analog to digital converter which measures the voltages has 12 bit accuracy. This means that every voltage is a multiple of the reference voltage (2.5 V) divided by 212, or .00061 V.
In order to get a sense of the size of these components, the resistors are each about 1/4 of an inch long, not including the wires, or leads, connected to each end. These resistors are rated at 1/4 Watt (meaning that if you push more power through them they will be damaged). C1 and C2 are polyester film capacitors, while C3 is a ceramic disc capacitor. Although it is not evident here, the longer lead on capacitors and LEDs is the positive lead.The resistors are color-coded with their values and tolerances. The tolerance is the percentage deviation that is allowed in the labeled value of the resistor. These resistors all have a gold tolerance color band (on the right hand side), signifying that their actual value may be within 5% of the value described by the other color bands. A silver tolerance band would signify 10%.The switches are electronic CMOS switches, connected as shown in the implementation diagram.
Due to the size of these resistors, it is somewhat difficult to determine the color codes in some cases. Resistors R1 and R7 have brown, black and green bands; R2 and R4 have red, red and green bands; R3 has yellow, violet and green bands; R5 has brown, green and green bands, and R6 has orange, orange and green bands.
The color codes on the resistors begin at the end where the colored band is on the edge of the resistor (on the left hand side in the photo). The first two (which we will call c1 and c2) encode the significant digits of a number between 10 and 99, and the last (c3) encodes the power of ten by which that number is multiplied:
R = ((c1 * 10) + c2) x 10c3 W.The color codes are:
So a code of "yellow, violet, red" would signify a 4700 W resistor (also called "4.7 K").
black = 0 red = 2 yellow = 4 blue = 6 gray = 8 brown = 1 orange = 3 green = 5 violet = 7 white = 9
The measured values of the resistors are
Compute the labeled value of each resistor using the color chart above, and the relative deviation between its measured and labeled values, relative to the labeled value. Are they all within accepted tolerance?
R1 = 980 KW R3 = 4.61 MW R5 = 1.49 MW R7 = 980 KW R2 = 2.21 MW R4 = 2.17 MW R6 = 3.3 MW
but they are labeled as
C1 = .047 mF C2 = .022 mF C3 = .01 mF
In each case, the 3 indicates 103 picoFarads (1000 pF = 1 nanoFarad, and 1000 nF = 1 mF). If there were no third digit, the one or two digit value would be in pF. The K indicates a tolerance of 10% (a J would indicate 5%). In fact, they have the following measured values:
C1 = 473K C2 = 223K C3 = 103K
For each capacitor, compute the relative deviation between its measured and labeled values. Are they all within accepted tolerance?
C1 = .0503 mF C2 = .0229 mF C3 = .00993 mF
Compute the equivalent resistance of the entire circuit, the total current flowing through the circuit and the total power dissipated by it. Check that Kirchhoff's Laws (as formulated at the bottom of the section on circuit analysis) are obeyed. Are they obeyed exactly? If not, compute the relative deviations between the voltages or currents which should be the same.
Compute the equivalent resistance of the entire circuit, the total current flowing through the circuit and the total power dissipated by it. Check that Kirchhoff's Laws are obeyed (as above).
Compute the equivalent resistance of the entire circuit, the total current flowing through the circuit and the total power dissipated by it. Check that Kirchhoff's Laws are obeyed (as above).
We know that there is an absolute error budget of .000305 V in every voltage measurement. We also know that our resistor measurements are only accurate to 3 decimal places; given their magnitude of 1 MW, this translates to an absolute error budget of 5 KW in every resistance measurement. Compute the RMS error budget for each of your total current calculations using the formulaRMS = (DVrel2 + DRrel2)1/2Note that DRrel depends on your configuration; each time you add two resistances, the absolute error doubles, and each time you multiply (or divide) two resistances, the relative error increases by a factor of 21/2.If the RMS error is greater than or equal to the current deviation for each of the Kirchhoff Current Law checks for the total current, those deviations can be attributed to systematic error. If not, what are the sources of the random error?
Plot VA as a function of time. Compute the relative deviation of your last VA value with respect to Vin. What does this tell you about this circuit?
Plot VC on the same graph and describe its relation to the previous plot.
Plot the currents through R1 and R4 as functions of time, on the same graph. Describe the relationships between the various graphs.
Note that the graph of charge vs. time is analogous to that of position vs. time. Therefore the slope of the charge graph has an important relationship to the currents through the resistors. What is it? This is one of the keys to understanding the behavior of the circuit.
V(t) = Vin (1 - e-t / R C),where R is the equivalent total resistance of the circuit and C is the equivalent total capacitance of the circuit; their product is called the time constant. What relationship does that value have to the time it takes to charge the capacitors in the circuit?
If we divide the data for VA by Vin, subtract that quotient from 1 and then take the natural logarithm:
f(t) = ln (1 - VA / Vin),This "linearized" function should be a straight line with slope -1 / R C. In order to compute this slope, we are going to perform a linear regression.
If any of the VA values are equal to 1.53 V, eliminate them from the data set. (Why?) Be sure to eliminate the corresponding values of t. Compute f(t) for the remaining values of VA. Then compute the average of them (F), and the average of the time values (T). If we have N values of t and N values of f(t), the slope (m) and intercept (b) are given by
m = (N * T * F - S (ti fi)) / (N (T2) - S ((ti)2))where the S denotes the sum taken from i = 1 to i = N.b = F - m T
Compute m and b. Compute your experimental value of the time constant from the slope. Compute the relative deviation of this measured time constant with respect to the expected value.
Plot the charge stored on each capacitor as a function of time (all on the same graph).
Plot the current through R1 as a function of time.
Compute the relative deviation of the measured value of the time constant with respect to the expected value.
Plot the charge stored on each capacitor as a function of time (all on the same graph).
Plot the current through R1 as a function of time.
Compute the relative deviation of the measured value of the time constant with respect to the expected value.
Plot the charge stored on each capacitor as a function of time (all on the same graph).
Plot the current through R1 as a function of time.
Compute the relative deviation of the measured value of the time constant with respect to the expected value.
Plot the charge stored on each capacitor as a function of time (all on the same graph).
Plot the current through R1 as a function of time.
Compute the relative deviation of the measured value of the time constant with respect to the expected value.
One of the more interesting facets of this experiment is the dependence of the relative deviations for the time constant on the time constant. How might you account for that? Is there something in the experimental design that might change this situation?
For each paper, do the following:
Do not simply cut and paste definitions from the web; write them in your own words!
This paper reports one of the few real mysteries at large in the world of physics: spacecraft from Earth are not behaving as expected with regard to all known laws of physics. The Pioneer 10 and 11 spacecraft are experiencing an acceleration toward the sun in excess of that predicted by any known model. To a physicist, this paper reads like a good mystery novel.
Look up the terms "Astronomical Unit", "Doppler Shift", "ephemeris" and "dark matter".
Don't let the size of this paper scare you; it will be very easy to outline. Identify the structure of the paper by section rather than by paragraph. Note that it is a review, which means that it is more expository than the other papers we have read.
Redshift can be used as both a distance and a measure of time because of the constant velocity of light as it travels. It is denoted by the letter "z"; z = 0 means here and now, while z = infinity means at the farthest point away in the past, both in space and time: the time and "place" of the Big Bang. H0 denotes the Hubble Constant. "mag" is an abbreviation for magnitude. WM is the fraction of the mass-energy in the universe which is matter, and WL is the fraction of the mass-energy in the universe which is attributable to the cosmological constant. The letters "B", "V", "R" and "I" denote spectral color bands roughly corresponding to blue, green, red and infrared. Extinction refers to the disappearance of spectral information due to absorption by dust and other matter.
Look up the terms "redshift", "supernova", "Hubble Constant", "magnitude", "luminosity" and "cosmological constant".
2007, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.
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