Experiments with Electricity and Magnetism for Physics 336L

Experiments with
Electricity and Magnetism
for
Physics 336L
by
R. Sonnenfeld
New Mexico Tech
Socorro, NM 87801
c
Copyright 2014
by New Mexico Tech
February 22, 2014
i
Contents
Course schedule
v
Course Goals
1
Safety
1
Required Supplies
2
Lab Reports
2
1 Circuits (n=1)
Understand DC series and parallel circuits and light-bulbs . . . . . . . . . . . . . . . . .
8
8
2 Internal Impedance and Error Analysis (n=1)
13
Measuring input impedance of a voltmeter and output impedance of a battery . . . . . 13
3 Complex Impedance (n=1)
16
Measure complex impedance of a resistor and a capacitor . . . . . . . . . . . . . . . . . 16
4 Magnetic Field, Inductance, Mutual Inductance, and Resonance (n=2)
18
Characterizing an LR circuit and determining impedance of an inductor with an oscilloscope and voltmeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5 Hysteresis (n=2)
22
Measure hysteresis loop of a soft iron torus and learn how a transformer works . . . . . 22
6 Diodes and Transistors (n=1)
29
Characterize a diode, then build a one-transistor amplifier . . . . . . . . . . . . . . . . . 29
7 Operational Amplifiers (n=2)
32
Learn basic op-amp theory then build an inverting and non-inverting op-amp . . . . . . 32
8 Index of Refraction of Air (n=1)
37
Compare optical path length in air to vacuum using an interferometer . . . . . . . . . . 37
9 Low-Pass Filters (n=1)
38
Measure frequency response of an RC filter and determine its affect on some recorded
music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
10 Electric Field, Capacitance, Dielectric Constant (n=1)
41
Use a capacitor to measure the dielectric constant of liquid nitrogen . . . . . . . . . . . 41
11 Negative Resistance (n=1)
42
Measure I-V curve of neon lamp. Use negative resistance to create an oscillator . . . . . 42
12 Superconductivity (n=1)
43
Understand Meissner effect with a High Tc superconductor . . . . . . . . . . . . . . . . 43
ii
13 Serial Ports (n=1)
48
Decode the signals that come out of a computer terminal. . . . . . . . . . . . . . . . . . 48
14 Miscellaneous Experiments (n=various!)
Develop an outlined experiment or create your own . . . . . . .
14.1 Design your own! . . . . . . . . . . . . . .
Concieve and develop your own experiment . . . . . . . . . .
14.2 Lock-in Amplifiers . . . . . . . . . . . . .
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Lock-in amplifiers allow you to see a tiny signal in the presence of substantial noise
14.3 TV remote control . . . . . . . . . . . . .
Detect and decode the infra-red control pulses sent by a TV remote .
14.4 Resistance and temperature . . . . . . . .
How does resistivity vary with temperature? . . . . . . . . . .
14.5 Electric Motors . . . . . . . . . . . . . . .
Electric motors can be used in reverse as electric generators . . . .
14.6 Magnetic materials . . . . . . . . . . . . .
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Measure magnetic properties of materials by incorporating them in an inductor.
14.7 Electric and magnetic fields of a power line
Figure out how to measure the E and/or B-field from a power line
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14.8 Flames . . . . . . . . . . . . . . . . . . . . .
What are the electrical properties of flames? . . . . . . . . . . .
14.9 Guitar Pickup . . . . . . . . . . . . . . . .
How do electric guitar pickups work? . . . . . . . . . . . . . .
14.10Build a Geiger Counter . . . . . . . . . . .
Build a Geiger Counter . . . . . . . . . . . . . . . . . . .
14.11Build a rail gun . . . . . . . . . . . . . . . .
Build a rail gun! Group project . . . . . . . . . . . . . . . .
14.12Build an electromagnetic can crusher . . . .
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A pulsed magnetic field can cause a coke-can to crush itself via eddy curret
Appendices
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iii
Acknowledgments
A number of faculty and students at New Mexico Tech have contributed to the experiments
for this junior-level laboratory class. The ones I know about are:
Prof. Marx Brook,
Prof. J. J. (Dan) Jones,
Prof. Paul Krehbiel,
Prof. Charles B. Moore,
Prof. William Rison,
Mr. Robert Rogers, and
Prof. William P. Winn
iv
2014 Course schedule
Date
1/16
1/23
1/30
2/06
2/13
2/20
2/27
3/06
3/13
3/20
3/27
4/03
4/10
4/17
4/24
5/1
5/8
Lab
Introduction
Circuits
Internal Imp.
Complex Imp.
BLMR
BLMR
Hysteresis
Hysteresis
Plan projects
BREAK
Diodes
OpAmps
OpAmps
Option1
Option2
Make-up
Finals
Due
Examples/Writeup
Int. Imp.
Complex
—
BLMR
—
—
—
Hysteresis
Diodes
—
Opamps
Option1
Option2
Make-Up
Table 1: This is the approximate schedule we will follow. Labs are due one week after they are
completed. There is slack in the schedule for illness or other personal needs.
v
Introduction
Course Goals and Time Committment
There are four purposes for this course.
• To give practical examples of concepts learned in Phys333/334
• To let you experience some of the excitement (and confusion) of experimental physics.
• To give you a working comfort with basic electronics as it appears universally in a modern
physics lab.
• To somewhat enhance your abilities with data analysis and error propagation.
The labs vary in length, but will take at least two hours to get through and understand. The course
grade is based on your lab reports and your level of effort in class. The individual measurements
you are asked to do are not particularly time-consuming; that is intentional. The time you are not
spending putting things together and making measurements is meant to be spent understanding
your results and how they fit the basic electromagnetic theory that you have learned to this point
in your career.
Depending on your level of understanding, it will require one to three hours after each class
session to complete the data analysis, perform the needed derivations, and do the background
reading to needed to understand your results.
References
The following references will be invaluable to you throughout your career if you do any kind of
experimental work. They are not required, but if you choose to buy them now, you will likely not
need another reference for basic electronics or error analysis for at least a decade.
Horowitz, Paul, and Hill, Winfield, The Art of Electronics (2nd edition), Cambridge University
Press, New York City, 1989.
Taylor, John R., An Introduction to Error Analysis, 2nd Edition, University Science Books,
Sausalito, California, 1997.
Safety
• Shoes and socks are required for minimal safety.
• Campus police are available at 835-5434 in case of emergency.
• Know the location of the nearest FIRE EXTINGUISHER. Can this fire extinguisher be
used on electrical fires?
• Know the location of the nearest FIRE ALARM pull lever.
• Report defective or damaged equipment to the instructor.
• Construct experiments so they can not fall and so that people will not trip over wires.
• Do not energize any experiment using more than 30 V until the instructor has inspected it.
1
• Conventional lab protocol calls for no food or drink under any circumstances. I have loosened
this ban because I have observed that science runs on caffeine. Thus, black coffee, tea, or
bottled water are the only drinks allowed in the lab. Sugar and sweeteners are destructive
to electronics, and many drinks (e.g. Coca-Cola) are corrosive. In the event of a spill, you
are responsible to promptly clean and dry any lab equipment involved. You may be required
to dissassemble, dry, and retest your equipment if liquid has penetrated the case.
• Food and snacks are not permitted in the lab. If you are hungry, you can take a quick snack
break out in the Workman lobby.
• One reason that labs usually ban food and drink is that hazardous chemicals (e.g. lead)
are associated with electronics. Thus, you shall wash your hands between working with lab
equipment and eating.
Laboratory Protocol
• The Laboratory hours are 2 PM to 4:30 PM. Please arrive on time. Habitual lateness/absence will affect your grade. If you finish an experiment early, begin working on
your analysis or on the next experiment.
• The lab has wireless internet connectivity. You may bring a lap-top to assist in data plotting/analysis or for web research related to your analysis.
• Please limit your conversations in the laboratory to the experiments.
• You are encouraged to help your fellow student learn the material, but all are responsible
for their own understanding.
Required Supplies
Failure to bring the following supplies every time can result in grade reductions. (See section on
grading).
• This lab manual is obviously needed every class.
• A 3-ring binder for lab write-ups should be brought to every class. Some labs refer back to
earlier labs. Even if you do your labs purely electronically, print previous labs out and put
them in the binder.
• A notebook for recording your data that includes data from earlier labs. This notebook may
be the same 3-ring binder mentioned above or a separate laboratory notebook. Do what
works best for you.
• A calculator or computing device with a calculator built in. This allows you to do preliminary data analysis in lab.
Lab Reports & Moodle
You can find this class on Moodle. There is a section for uploading your lab report. I would like
all reports to be submitted via Moodle. This may mean that you need to scan in hand-written
tables, sketches, and graphs include them in your report. To facilitate this, you probably want
to put these all together on a single sheet (or two) of paper. Then you only need to scan one or
2
two pages rather than several. (Cameras are alternatives to scanners, provided the final result
is readable). If you have the technology, and wish to create your graphs, sketches, and tables
directly in electronic form, this is of course acceptable.
I will provide on request LaTeX templates for a lab report if you already know or would like
to learn LaTeX. Inferior programs like Microsoft Word, or less inferior programs like Open-Office
Writer are also acceptable. Open-Office can easily directly generate a .pdf from any file. Please
do not submit .docx or .odt files, only pdfs.
Your lab report shall be clear and legible. It may be typed, or hand-written directly in your
lab book and scanned to pdf. Your report should have the following sections:
1. Heading – In the following order, state
(a) Your name
(b) The name of the experiment
(c) Date the report was written.
(d) Date the laboratory work was finished.
2. Vocabulary – Define each word and state how it relates to the experiment.
3. Apparatus and Procedure –
Describe what you did. Use figures to show how the equipment and circuits were hooked up.
For each piece of equipment you use, record (and report) the following:
(a) Description (e.g. voltmeter, oscilloscope)
(b) Manufacturer Model (e.g. HP 6235A)
(c) Serial number
Recording the Model and Serial numbers will help you get consistent results if you need to
continue the same lab over one or more periods. This information will also be helpful should
the equipment become faulty and in need of repair.
4. Answers to Numbered Questions – Some labs are more formally structured than others.
In labs where I have numbered the measurements you should report or the calculations you
should do, it is efficient for both of us to simply structure the rest of your report around
reporting what you were asked to report and calculating/deriving what you were asked to
calculate or derive. In labs that are more free-form, you will want to continue with the
order as shown below. Even for labs with numbered questions, please read the sections
below suggesting how to present raw data, calculations, and derivations.
5. Error Discussion – The next section of this manual explains formal error analysis. In
includes a fancy expression with partial derivatives. This level of analysis is only required
for the lab called Internal Impedance. HOWEVER, a brief (couple of sentence) discussion of
errors is required in every lab (even those that mostly revolve around answering numbered
questions). Your discussion should specify what the error is in your raw measurements
and how your arrived at that number. You should also estimate the error in your derived
measurements.
3
6. Measured variables and data tables – The column headings of your data table should
list the name of the variable being measured and the units of the numbers contained. In
~ or B
~ or ǫ. In the laboratory,
theoretical courses, variables often represent quantities like E
variables represent the number of divisions on an oscilloscope or the voltage on a voltmeter.
An imprecise table would have a column headed “Voltage (V)”. A more precise table would
indicate the part of the circuit the voltmeter was connected to. A well-defined variable could
be called VAB , where A and B are the two points on your circuit to which the voltmeter
probes were applied. VAB would be further defined by the inclusion of the circuit diagram
with the table with A and B marked on it.
It is not unusual for the raw data taken by a physicist to be somewhat disorganized. (This
varies with the physicist!). For example, when measuring the amplitude of a resonant circuit
vs. the frequency at which it is driven, one may take a number of data points that are not
too interesting (because they are not near the resonance). You might also obtain data that
you later decide are incorrect or ambiguous (because you forgot to note the scale of an
oscilloscope, for example). Your notebook may thus contain a jumble of dubious data and
good data. For the lab report, you can recopy the data table in a well-ordered way. Feel free
to omit any dubious or redundant data points. Alternatively, do enough measurements to
understand what constitutes a valid datum and what columns should be in a table. Then,
draw a neat table and continue your experiment, completing the table as you go. You can
then scan the table and include it in your lab report without rewriting it.
7. Graphs – Show data as points (not lines) and theoretical curves as lines. Use computer
plotting software, or plot by hand in your notebook or on graph paper. Axes should be
labeled with tick-marks and and well-defined variables (described previously in discussion
of tables).
8. Calculations – Calculations should be shown in your report. When numerical values are
plugged into formulae or derived results, they should be explicitly written in your report. I
know that you can plug ten numbers into a formula in your calculator and just write down
the answer, but am explicitly asking that you not do that. You (or I) should be able to look
at your calculations and see exactly what numbers you used at every step. This is good
practice in your career. It will help you find your mistakes and documents exactly what you
did.
9. Derivations – In many laboratory exercises, you will be asked to derive results that are
basic to understand the experiment or doing calculations.
Many derivations require a figure; draw large, clear figures. Every derivation using the
integral form of Maxwell’s equations must have an accompanying figure that shows the
locations of line integrals, surface integrals, and volume integrals.
10. Summary and Conclusions – It is typical in an article or report to have a summary
and conclusions section. I will not require this, nor will your grade depend on it. The
reason is that this manual specifies what data should be acquired and analysis completed
in significant detail. Grades depend on the quality of the data and analysis.
4
Error Analysis
Types of Error
All experimental uncertainty is due to either random errors or systematic errors. Random errors
are statistical fluctuations (in either direction) in the measured data due to the precision limitations of the measurement device. Random errors usually result from the experimenter’s inability
to take the same measurement in exactly the same way to get exact the same number. Systematic errors, by contrast, are reproducible inaccuracies that are consistently in the same direction.
Systematic errors are often due to a problem which persists throughout the entire experiment.
Note that systematic and random errors refer to problems associated with making measurements. Mistakes made in the calculations or in reading the instrument are not considered in error
analysis. It is assumed that the experimenters are careful and correct!
All measured quantities should be recorded with error bars. It is all right to list the error at
the beginning of a data table. (e.g. All measurements are ±0.001V ).
Minimizing Error
Here are examples of how to minimize experimental error.
Example 1: Random errors
You measure the mass of a ring three times using the same balance and get slightly different
values: 17.46 g, 17.42 g, 17.44 g
The solution to this problem is to take more data. Random errors can be evaluated through
statistical analysis and can be reduced by averaging over a large number of observations.
Example 2: Systematic errors The cloth tape measure that you use to measure the length of
an object had been stretched out from years of use. (As a result, all of your length measurements
were too small.)
The Ohm-meter you use reads 1 Ω too high for all your resistance measurements (because if
the resistance of the probe leads).
Systematic errors are difficult to detect and cannot be analyzed statistically, because all of the
data is off in the same direction (either to high or too low). Spotting and correcting for systematic
error takes careful thought into how your equipment works, and cleverness to measure how far off
it is from correct.
* How would you compensate for the incorrect results of using the stretched out tape measure?
* How could you determine your probe resistance, and correct the resistance measurements?
Handling Errors
Note that uncertainties (Random or Systematic) are not obtained by comparing your results with
the “accepted” values you find in the literature. Instead, they are found in the following way:
1. Estimate the uncertainties in the measured quantities (e.g. voltages, resistances, frequencies,
lengths, etc.) and
• All instruments have an inherent accuracy which can only bed determined by comparing
the instrument to a more accurate instrument or to a “standard” for the quantity of
interest. Explain what you compared your instruments to to determine this inherent
accuracy.
5
• Uncertainty in a length may be due to precision with which a scale can be read.
Uncertainty in a time may be due to unknown reaction time for starting/stopping a
timer.
• Random uncertainties may be estimated by making the same measurement 10 times
and calculating the standard deviation. Show such repeated measurements, and the
standard deviation calculation, in your log-book.
• Repeated measurements will not uncover inaccuracies in instruments.
2. Propagate the uncertainties in the measured quantities to find uncertainties in the calculated
results.
• Use the general formula for propagation of errors given on page 75 of Taylor’s Error
Analysis text. (See References section above – An excerpt follows)
• Assume you are measuring the variable q which depends on the measured quantities,
x, . . . , z. If the uncertainties δx, . . . , δz are random, then the uncertainty in q is
δq =
s
∂q
δx
∂x
2
+ ... +
∂q
δz
∂z
2
(1)
Course Grades
If a lab is scheduled to take two class hours, it is given a weighting factor n = 1, if scheduled for
four hours, n = 2. The total number of points for the ith lab is pi ∗ ni .
If you attend every lab, there are 13 sessions in this course. The amount of work required
occupies nominally 12 sessions. However, some of you will be slower. Plan on completing all 13
sessions. Grades for the course will be based on:
1. Attendance and diligence in the laboratory and
2. The number of points P for the course, computed from the following algorithm:
P =
1 X
(pi ∗ ni )
12
Example: A student does labs 1,2,3,4,5,6,7,9 and 11 and gets 8.0 on each,
except gets 10.0 on lab 5 and earns a 6 on lab 7.
P=(8*1+8*1+8*1+8*2+10*2+8*1+6*2+8*1+8*1)/12=8.0=B-
6
Figure 1: Practice problems for circuits lab. (Unspecified resistor values in Problem 2 are 1000 Ω.)
7
Figure 1: Kirchoff’s Current Law: The sum of currents entering a node is equal to the sum of
currents leaving it.
Kirchoff’s Voltage Law: The sum of voltage around any closed loop in a circuit is zero.
1
Circuits (n=1)
Background theory
In Physics 333, you learned the basic principles you need to analyze any circuit.
1. Electrical charge is conserved. This leads to Kirchoff’s current law.
2. The electrostatic force is a conservative force. This leads to Kirchoff’s voltage law.
Kirchoff’s famous laws for circuit analysis are stated above in the figure caption.
Referring to the left panel of Figure 1, we see that the current law means that IA = IB +IC +ID .
This immediately leads to the useful corollary that in a series circuit (which has only one branch)
the current is the same at all points in the circuit.
To understand the origin of the Voltage law shown in the right panel, consider that the
definition of a conservative force is that the Work done is equal over any possible path between
two points. Recalling that Voltage is just the work done per electron, we see that the voltage drop
between any two points in the circuit must be equal no matter what path is taken. An equivalent
way of stating this is that the closed loop voltage is 0 over any possible loop through the circuit.
Because they are derived from fundamental physical principles, Kirchoff’s Laws are valid for
any AC or DC circuit containing any combination of passive components, semiconductors and
active components. For this lab, we will also need Ohm’s law, V = IR. Ohm’s law is an example of
an IV characteristic. IV characteristics depend on the type of device being studied. A resistor fits
Ohm’s law exquisitely, while a light-bulb only approximately fits Ohm’s law. Later in the course
we will study the IV characteristics of diodes and transistors. For a diode, I = IS (eVD /nkT − 1),
which as you can see is not Ohm’s law at all!
Basic Formulae for DC Circuits
V = IR
(1)
W = QV
(2)
P = IV
(3)
8
Ref f = R1 + R2 + R3 + ...
(4)
1/Ref f = 1/R1 + 1/R2 + 1/R3 + ...
(5)
1/Zef f = 1/Z1 + 1/Z2 + 1/Z3 + ...
(6)
Ohm’s Law (eqn 1) is not fundamental. It happens to be true for materials called resistors.
It is a simple case of an I-V curve or constitutive relation.
The Power dissipation formula (eqn 3) is merely a consequence of taking the derivative of
equation 2, which is just the definition of voltage.
The series and parallel circuit laws (eqn 4-5) come directly from Ohm’s law and Kirchoff’s
laws. A beautiful thing about these laws is that they generalize to complex impedances (eqn 6).
Here “R” is replaced with “Z”, which will be discussed in future labs.
In terms of fundamental physics, you need only conservation of charge and energy to derive all
circuit laws. However, the simple formulae above come up so frequently in circuit analysis that
they are worth memorizing.
Philosophy of Experimental Physics
An experimentalist spends a lot of time “thinking” as well. Results are almost never what you
expect them to be, and apparatus is full of quirks that you need to learn via familiarity. In this
experiment you will measure current and voltage in simple circuits with high accuracy. (Three
figures is high accuracy for an experiment). Some obvious questions occur.
1. How does one measure a voltage?
2. How does one measure a current?
3. How do you hook up a power supply?
4. What value of resistor should I use?
5. Why do the light bulbs behave differently than the resistors?
The answers to the first three questions are contained in Kirchoff’s two laws. It requires
thought about what a series and what a parallel circuit is and how current must flow.
The answer to the fourth question depends on Kirchoff’s Laws, Ohm’s Law and the accuracy
of your measuring instruments. If you choose too large a resistor, or too small a resistor, your
measurements will suffer for accuracy. So pick something and begin the experiment and see if you
have chosen wisely. This iterative process is characteristic of much experiment.
Series and Parallel Circuits with Resistors
You will build several simple series and parallel circuits and explicitly measure the voltage and
current to verify Kirchoff’s Laws and build up familiarity and confidence with the use of electrical
equipment. This lab should serve to remind you of things you already know, and because it is
relatively simple, you should be able to predict your results with three significant digits.
Using the power supply and several identical resistors of value R, build and study circuits i-ii.
( 1 ) Set the power supply to 6.00 V (Check it with a multimeter). Before building circuit i,
measure and record the resistor values you have selected (the two resistors will be slightly
different even if nominally the same). Use this and the voltage to predict the current in the
circuit.
9
Figure 2: Circuits i and ii are simple series circuits. Circuit iii illustrates a combination of series
and parallel circuits. Circuit iv is a very useful configuration called a “Voltage Divider”.
( 2 ) Now build circuit i, measure (and record) the current at point A and point C. Should they
be equal? Are they? Do they agree with your calculations? If they do not, you made a
mistake.
( 3 ) Before building the circuit, measure and tabulate the three resistors to be used. You should
now be able to predict the voltages for circuit ii. Make a table for circuit ii in which you
include the predictions and measurements for VBD , VBC , and VBA .
( 4 ) Build circuit iii and test Kirchoff’s current law for node C. Make predictions of values for
all the currents at node C and verify with measurements. It is more interesting to make
predictions before doing the measurements. If you end up surprised, say so, then revise your
predictions (without erasing the original ones).
( 5 ) Circuit iv is called a “Voltage Divider”. This is because VOU T = VF G is always a fixed
1
fraction of VIN = VAB . Design a 10:1 voltage divider (a circuit for which VVout
= 10.0
. You
in
may not find resistors to provide an exact 1/10 ratio; it is sufficient to get between 1/9 and
1/11. Further, predict the division ratio of the circuit you actually build to three signficant
digits based on the value of resistors you do find. Verify your prediction with three different
values of VIN .
( 6** ) Now that you have a working familiarity with Kirchoff’s laws, design a more complicated
circuit with two different parallel legs in series. (Use several different resistor values in your
circuit design.) Write the design in your lab book and lab report and predict Ref f . Measure
10
current through circuit and verify your prediction.
1
Using resistance to measure temperature of a light-bulb
( 7 ) Get two Ecko #46 lightbulbs out of the box and measure and record the resistance through
each of them with your Ohmmeter. With some wire and a solder iron, solder the two
lightbulbs into a series circuit.
( 8 ) Connect the two lightbulbs to the 6 V output of your power supply. Measure the current
through the bulbs and the Voltage of the supply. Use this to calculate the resistance of two
light-bulbs in series, and thus the resistance of one of the bulbs.
( 9 ) Bypass one of the bulbs (just move your alligator clip so it is not in the circuit) so you can
measure the current, voltage, and thus resistance of a single bulb.
( 10 ) Calculate and record the resistance of a single bulb, a single one of the double bulbs and
the bulb that you measured with your Ohmmeter. The values will all be very different.
( 11 ) The following formula allows you to calculate the resistivity of tungsten wire as a function
of temperature. Assuming that room temperature is 300 K, calculate the temperature of
the single bulb, and the double bulb. Light bulbs get very hot!
R = Rref × [1 + α(T − Tref )]
α = 4.4 × 10−3
1/K
(7)
( 12 ) Based on what you have learned, is it right to treat a lightbulb like a resistor? Why or
why not?
Vocabulary
⋄
⋄
⋄
⋄
Series circuit
Parallel circuit
Kirchoff’s Voltage Law
Kirchoff’s Current Law
Equipment
-
HP 6235A Triple output Power Supply
Proto-board or Spring Board
Your Personal Multimeter
Resistors
Tektronix Oscilloscope
Laboratory Time
Two hours
1
** If time is short, I may waive the custom circuit requirement for this first lab, so do this part last.
11
Figure 3: Find the indicated voltages and currents for the given circuit.
12
2
Internal Impedance and Error Analysis (n=1)
Sources of error generally fall into two classes, systematic errors and random errors. Random
errors are due to uncontrollable “noise” or reading accuracy of instruments.
Systematic errors are often due to uncalibrated instruments or innaccuracies introduced by
factors such as internal impedance of the measuring device.
Therefore, an important property to know about a voltmeter (or an oscilloscope) and a power
supply (or a battery) is its internal impedance.
Measurements
1. Calibrate a Fluke multimeter (not your meter) vs. a resistance standard and a voltage
standard. (Calibration involves plotting Vmeas = G × Vstd + B). In a simple calibration
there is both a “gain (G) ” and an “offset” (B) error. Take a half a dozen voltage and
resistance data points using the full range of the resistance and voltage standards. I only
have a single voltage standard, so you will have to share. If you find yourself waiting around,
you can proceed to step 3 and borrow the standards later.
Assume that the standards are perfectly correct. After you have calibrated your meter, use
it to measure the voltage of one of the standard cells around the room. How close does your
result come to the number on the standard cell?
Record GV , BV , BR , and GR in your notebook. You will use them to correct your measurements in the rest of the lab. However, there will still be measurement errors.
2. If your voltmeter read 1.000 V, what is your best estimate of what the voltage really is?
Also, what do you think is the absolute error on this measurement. What if your meter
read 10.00 V? What about 10 mV? What is your best estimate on the error in resistance if
your meter reads 0 Ω? 10 Ω? 1 kΩ? 1 MΩ?
3. Take measurements and do calculations to find the internal impedance of a Fluke 7X multimeter when it is set to measure voltage. Use the circuit shown below in figures “a” and “b”,
with at least three values for the resistance R. Use (at least) the following values: R=22
MΩ, 1.5 MΩ, and 150 kΩ.
(NOTE: You have a choice about what voltage to set your power supply to to do this experiment. Does it matter? Set it to whatever voltage is likely to give you the most precise
result. Please note: Do NOT use the voltage standard as your power supply.)
4. Find the internal impedance of an oscilloscope at the BNC connector on the chassis of the
oscilloscope.
5. Draw a circuit diagram of a circuit to measure the internal impedance of a small battery.
(Indicate the battery itself as an “ideal battery” in series with an internal impedance).
6. Make measurements to find the internal impedance of a 9-V battery.
7. The experiment you just did is probably somewhat destructive to the battery. So let’s finish
the job. Choose a 1/4 Watt resistor of two to four ohms, connect it to your 9 V battery, and
get your hands away quickly. What happens? Estimate how much power you dissipated
through your 1/4 Watt resistor. Do not forget to include the internal impedance of your
battery when doing this calculation..
13
V
δV
∂Ri
∂V
∂Ri
∂V δV
R
δR
∂Ri
∂R
∂Ri
∂R δR
VB
δVB . . .
δRi
Table 2: Table listing each value of resistance and each voltage, the uncertainties in each, and
the partial derivatives of the internal impedance with respect to the resistances measured. These
are the quantites required by the general error analysis formula of Taylor. NOTE: In order to
find the partial derivatives, you need to derive an explicit expression for Ri in terms of R, VB and
V . This derivation should be in your lab report, as should the expressions for partial derivatives
themselves.
8. If you enjoyed this destruction, you can take it up a notch. Find an aluminum plate and
place it on your lab bench because you are going to produce something red-hot and we do
not want scorch marks. Working over the plate, wrap a single strand of steel wool around
your battery terminals. It will glow red hot and melt almost instantly. You have made a
fuse!
Calculations and Analysis
9. Calculate uncertainties in the internal impedance of the voltmeter (not the oscilloscope) for
each value of resistance used. Use the general formula for propagation of errors given in
the introduction to this lab manual. Make a table like Table 1. (See the formula in the
introduction to this manual) What is your best estimate for value of the internal impedance?
Vocabulary
⋄
⋄
⋄
⋄
⋄
⋄
⋄
Random Error
Systematic Error
Input impedance
Output impedance
Equivalent circuit
Open-circuit voltage
Short-circuit current
Equipment
-
Voltage Standard
Resistance Standard
Oscilloscope—one only
Fluke 7X multimeter—one only
Resistors
Power supply
9V Battery and battery clip
Laboratory Time
Two hours
14
Figure 1: Circuits for measuring the internal impedance of a voltmeter. (a) Circuit to find VB .
(b) Circuit to find Ri after VB is known.
References
Taylor, J. R., An Introduction to Error Analysis, University Science Books, Sausalito, California,
1997.
15
3
Complex Impedance (n=1)
Since complex numbers are widely used in physics and electrical engineering, it is useful to see
how a complex number can be measured. Here is an experiment to do that.
The experiment
The impedance of a capacitor and resistor in series is a complex number. Determine the complex
impedance of a resistor (R) and capacitor (C) in series by applying a sinusoidal voltage across
the series combination and then doing the following:
1. Measure and record the resistance and capacitance you will use. (Your multimeter will do
both!).
2. Sketch the circuit you intend to build. It consists of capacitor C connected to the center
pin of the function generator output, followed in series by resistor R. [Note: The style of
connector used on our function generator and oscilloscopes is called a “BNC” connector].
3. Build the circuit and probe the total voltage from the function generator VF G on one ’scope
channel.
4. Measure the current I through the circuit by probing the voltage across resistor R.
5. Measure the relative amplitude and phase of VF G and I for an input frequency
f = 200 Hz.
Hint: Get both signals on the oscilloscope at once. Trigger on the input voltage and set
the trigger level to 0V. This is a way to easily quantify the phase shift between the applied
voltage and current.
6. Copy the two waveforms into your lab book and label them VF G and VR . The graph should
be accurate enough to see the relative phases and amplitudes of the waveforms. You will be
referring to this graph in future labs.
7. Based on your measurements and the theory outlined on the next page, calculate the complex
impedance Ztotal of the circuit as well as the impedance ZC of the capacitor by itself. Assume
that your measurement of the resistor is correct. (You did measure the resistor, didn’t you?).
Finally, calculate the capacitance of the capacitor.
8. Repeat all of the above for f = 1000 Hz (You may omit the sketch. You will not it will be
difficult to do).
9. Using the measured values of R and C, work backwards to predict what phase shifts and
amplitudes you would have expected to see. Comment on the agreement (or lack thereof).
Equipment
-
Oscilloscope
Function generator
Multimeter
R=33 kΩ Resistor
C=0.047µF Capacitor
16
Laboratory Time
Two hours
About Complex Impedance
Complex impedance (generally denoted by the symbol Z) is a useful concept because it allows
inductors (L) and capacitors (C) to be treated by the same rules as ordinary resistors (R). (Note:
These rules were discussed in the lab called “Circuits”.) The complex impedance of a capacitor
1
, where ω is the angular frequency of the applied sine wave. Complex impedance
is ZC = iωC
Z is expressed in units of Ohms just like ordinary resistance R. Likewise, the normal rules for
combining resistors in series and parallel can be used for complex impedances.
Recall that for a voltage divider made of ordinary resistors R1 and R2 in series:
VOU T
= R2 /(R1 + R2 )
VIN
(8)
(This formula assumes that VOU T refers to the voltage drop across resistor R2 ) A complex
voltage divider can be made of any combination of passive components with complex impedance
Z1 and Z2 in series leading to:
VOU T
= Z2 /(Z1 + Z2 )
VIN
(9)
Derivation of Complex Impedance of a Capacitor and an Inductor
First, let’s generalize Ohm’s Law from V = IR to V = IZ. Next, apply a sinusoidal voltage to
the circuit under study:
h
i
V (t) = V0 cos(ωt + φ) = Re V0 ei(ωt+φ) .
(10)
Current is in general defined I = dQ/dt. For a capacitor Q = CV , so I = C dV /dt. Using
I
. Thus we
the form of V(t) above, we obtain I = C iωV (t). Solving for V(t) we get V (t) = iωC
1
identify iωC as the complex impedance of a capacitor (ZC ).
Given the equation V = L dI/dt for an inductor, see if you can apply a similar argument to
derive the complex impedance for an inductor (ZL ).
Going from measured quantities to impedances
In your experiment, you are measuring VF G (t) and I(t). For simplicity, assume that VF G (t) =
ℜ(eiωt ), with φ = 0. Then Ohm’s law becomes V0 eiωt = I0 ei(ωt−φ) Z0 eiφ . Inspection should show
you that you can calculate Z0 from your V0 and I0 measurements. Further, when you measure
−φ in the current in steps 3–5 of the experiment, you are really measuring φ in the impedance.
Refresher on Complex Arithmetic
Recall that for any complex number Z:
Z = x + iy = Z0 eiφ
(11)
where Z0 = x2 + y 2 and φ = atan(y/x). Also Z02 = Z ∗ Z, where Z ∗ ≡ x − iy. Finally, it is
1
1
useful to know that if Z = a+ib
, then Z ∗ = a−ib
p
17
4
Magnetic Field, Inductance, Mutual Inductance, and
Resonance (n=2)
Introduction
In this laboratory exercise you will learn
• how to determine impedance from measurements of an alternating current and voltage,
• how to calculate mutual and self inductance,
• how to measure the amplitude of an oscillating magnetic field,
• how a circuit behaves around a resonance, and
• how a transformer works.
This is a good time to review the concepts listed above in a junior-level textbook on electricity
and magnetism.
Laboratory Work
Figure 1: Circuit for measuring impedance and self-inductance and exploring what happens
around a resonance.
1. Hook up the circuit in the figure below. According to the manufacturer, the coil (called
“Outer Coil”) has 2920 turns of 29 AWG wire (wire diameter φ = 0.29 mm). The resistance
R = 310Ω will be used to measure the current in the circuit.
2. Self inductance:
(a) For a frequency f = 1000 Hz, measure the amplitudes of the voltage VF G (t) from the
function generator and the voltage VR across the resistor. Also measure the phase
difference between VF G and VR . Make sketches of the waveforms and label VF G and
VR . Compare these sketches to those you made in the lab called “Complex Impedance”.
Note how they differ. Estimate uncertainties of the amplitudes and phase (based on
how well you can read them off the ’scope).
(b) From the measurements construct the complex functions V (t) and I1 (t).
18
(c) Find the total impedance ZT ≡ V (t)/I1 (t), which will be a complex number.
(d) Subtract the impedance of the resistor ZR ≡ R + 0i from the total impedance ZT to
find the impedance of the Outer Coil, Z1 .
(e) The impedance of the Outer Coil is Z1 = R1 + iωL1 , where R1 is the resistance of the
Outer Coil due to the resistance of the wire in it. L1 is the self inductance and ω is the
angular frequency (radians/second). R1 can be measured with an ohmmeter, which
does not see the inductance because it applies a steady (DC) voltage. Compare your
two values of R1 (determined from the complex impedance and from direct Ohmmeter
measurement) with the value given by the manufacturer.
(f) Calculate the self inductance of the Outer Coil L1 and estimate its uncertainty. Compare L1 with the value given by the manufacturer, which is 63 mH (after correcting
the typo in the manufacturer’s instruction sheet).
3. Resonance:
(a) Look for a resonance between 35 and 40 kHz. It will show up clearly as a null in your
measurement of current. Measure this frequency to three significant figures.
Compare the relative phases of VF G (t) and I1 (t) at frequencies above and below the
resonant frequency. Sweep back and forth through the resonance and note what you
observe about the phase. This is not a precision measurement. You are merely trying
to observe what happens to the phase as the circuit passes through resonance.
(b) Disconnect the oscilloscope probe that measures V (t) and use it to measure the voltage
V2 (t) across the Search Coil, which is wound on a wooden dowel. Keep the other probe
connected across the resistor R so you can watch the behavior of the current. Place
the Search Coil inside the Outer Coil and observe what happens to V2 (t) and I1 (t)
as the frequency is increased through the resonant frequency. Notice that while I1 (t)
is nearly zero at resonance, V2 (t) is not. But V2 (t) is proportional to dB/dt which is
proportional to the rate of change of current in the Outer Coil. How can this happen
when I1 (t) is nearly zero? (It is likely we will discuss the answer to this question in
class. Please answer it in your lab report as well).
Figure 2: Circuit with Outer Coil and inner Search Coil. The Search Coil can be used to find the
magnetic field from the Outer Coil. There is a mutual inductance between the two coils. As a
unit, the two coils are a transformer.
19
4. Magnetic Field Amplitude and Mutual Inductance.
The Search Coil can be used to find the magnetic field produced by the Outer Coil. For
the calculations of magnetic field amplitude and mutual inductance described below, set
the function generator for a large amplitude and a frequency f = 1000 Hz and make the
following measurements:
(a) Voltage across the resistor in series with the Outer Coil ( for determining the current).
(b) Voltage across the leads from the Search Coil.
5. Transformer.
The two coils together are a transformer. In a transformer, power flows from the primary
winding (Outer Coil in this case) to the secondary winding (Search Coil in this case). Remove
the resistor from the primary circuit so that the function generator is connected directly
to the primary winding (Outer Coil). And then, for calculations described below, do the
following:
(a) Find and note the range of frequencies for which V2 (the search-coil voltage) is nearly
constant. (Hint, set your frequency to roughly 5 kHz and work your way down and
up from there. The answer can be approximate, but you should include in your lab
report how you defined nearly constant.
(b) Having determined the frequency range over which V2 is roughly constant, set the
frequency somewhere in the middle of this range and make the following two measurements.
(c) Measure the voltage V1 across the primary winding (Outer Coil).
(d) Measure the voltage V2 across the secondary winding (Search Coil).
Calculations and analysis
6. Derive an approximate expression for the magnetic field strength B near the center of the
Outer Coil when the current through that coil is I1 , the number of turns of wire is N1 ,
and the length of the coil is ℓ1 . Make the approximation that B = 0 outside the coil. The
quantities I1 , N1 , and ℓ1 have subscript 1 to indicate that they are associated with the
Outer Coil (also called the primary of the transformer). Later, the subscript 2 will be used
for variables associated with the secondary coil of the transformer (Search Coil).
7. Derive an approximate expression for the self-inductance of the coil. Your expression will
involve N1 , A1 , ℓ1 and physical constants. A1 is the area enclosed by one turn of wire in
the coil.
8. Use the results in the previous items to estimate a numerical value for the inductance of the
coil. Also estimate its impedance Z1 = iωL1 at a frequency f = 1 kHz.
9. Compare the inductance L1 of the Outer Coil deduced from your measurements of voltage
amplitude and phase with the value obtained from the equation for L1 above. Can the
discrepancy be explained by the uncertainty in the measured values? If not, can you come
up with a systematic reason for the error, and can you perhaps come up with a way to
calculate L1 more accurately? Explain, and do it.
20
10. Starting with one of Maxwell’s equations, develop the equations for determining B from
the voltage induced in the Search Coil, given the cross-sectional area A2 and number of
turns N2 of the Search Coil. Variables associated with the Search Coil will have subscript 2.
Compare the B value derived from your measurements with that calculated as in (1) above,
using the actual value of the current.
11. The combination of the Outer Coil and Search Coil constitutes a mutual inductor. Derive
the following approximate equation for the mutual inductance between the Outer Coil and
the Search Coil using the approximate expression for B above:
M12 = µ0 N1 N2 A2 /ℓ1 .
(12)
In addition, compute the induced voltage in the Search Coil.
12. Derive the following expression for the ratio of voltages of a transformer:
V2
N2 A2
=
,
V1
N1 A1
(13)
Compare your measured value with that predicted from the above expression.
13. Notice that the above equation predicts that V2 /V1 will be independent of frequency. For
what range of frequencies is this true? For extra credit, you may speculate on why the
equation fails, or at low frequencies.
14. Using the value for L1 and the resonant frequency, calculate the capacitance of the coil.
Vocabulary
⋄
⋄
⋄
⋄
⋄
Search coil
Mutual inductance
Tank circuit
Primary winding
Secondary winding
Equipment
-
Outer Coil (2920 turns)
Search Coil (50 turns wound on wood)
Oscilloscope
Function generator
310 Ω resistor
Laboratory Time
Four hours
21
5
Hysteresis (n=2)
Introduction
A system is said to exhibit hysteresis when the state of the system does not reversibly follow
changes in an external parameter. In other words, the state of system depends on the history
of the system (“history-sis”). The classic examples of hysteresis use Ferromagnetic materials.
The state of the system is given by the magnetic moment per unit volume, M, and the external
parameter is the magnetic field, H. In this experiment, we explore hysteresis in the ferromagnetic
core of a transformer by measuring both the magnetic flux density B and the magnetic intensity
H in the core. The variable M can be deduced from H and B.
The magnetic intensity vector H is defined by the equation
H≡
1
B−M ,
µ0
(14)
where B is the magnetic flux density (or magnetic induction), µ0 is the magnetic permeability of free space and M is the macroscopic magnetization. The macroscopic and microscopic
magnetizations are related by the defining equation
1 X
mi .
∆V →0 ∆V
i
M = lim
(15)
M is the vector sum of the atomic magnetic moments divided by the volume, ∆V , of the sample.
For an isotropic, linear material, M is linear in H, as
M = χm H ,
(16)
where the dimensionless magnetic susceptibility χm is assumed constant. In this case equation (14)
becomes,
H =
so that
and
1
B−M
µ0
µ0 H = B − µ 0 χ m H
B = µ0 (1 + χm )H .
(17)
(18)
(19)
By comparing this last equation with the free space equation,
B = µ0 H ,
(20)
the magnetic permeability, µ, of the material is defined to be
µ = µ0 (1 + χm ) .
22
(21)
The relative permeability Km is defined as
Km =
so that
µ
µ0
Km = 1 + χm .
(22)
(23)
In a ferromagnetic material, the magnetization is produced by cooperative action between
domains of collectively oriented atoms. Ferromagnetic materials are not linear, so that
χm = χm (H) .
(24)
However, the above equations are still applicable if we realize that µ is no longer a constant, that
is
µ(H) = µ0 [1 + χm (H)]
(25)
and
B = µ(H)H .
(26)
This behavior can be explained by examining the microscopic structure of a ferromagnetic
material. The material actually is polycrystaline, consisting of many small crystals of the material.
Each of these crystal grains is divided into groups or ‘domains’ of atoms. Within a domain, the
magnetic moments of the atoms are all aligned parallel to each other. These domains, on the order
of hundreds of Angstroms across, are essentially completely magnetized as long as the temperature
remains below the Curie temperature of the material. In the absence of an externally-applied
magnetic field, the magnetization vectors of the domains in a given grain can be aligned so as
to minimize the net magnetization of that grain. Furthermore, the grains themselves can be
randomly oriented as shown in Figure 1. Thus the macroscopic magnetization M is zero.
Figure 1: Magnetic domains in a ferromagnetic material when the applied field H is small.
If a weak external magnetic field is applied to a speciman with M = 0, within each crystal
grain the domains whose magnetization vectors are oriented more in the direction of the applied
field will grow at the expense of the less favorably oriented domains. That is, the domain walls
move and the material as a whole acquires a net macroscopic magnetization, M.
23
For weak applied fields, movement of the domain walls is reversible and χm is constant. Thus
the macroscopic magnetization M is proportional to the applied field,
M = χm H ,
(27)
B = µH ,
(28)
and
where µ is constant.
For larger fields the domain wall motion is impeded by impurities and imperfections in the
crystal grains. The result is that the domain walls do not move smoothly as the applied field is
steadily increased, but rather in jerks as they snap past these impediments to their motion. This
process dissipates energy because small eddy currents are set in motion by the sudden changes in
the magnetic field and the magnetization is therefore irreversible.
For a sufficiently large applied field the favorably oriented domains dominate the grains. As the
applied field is further increased, the magnetization directions of the less well oriented domains are
forced to become aligned with the applied field. This process proceeds smoothly and irreversibly
until all domains are aligned with H. Beyond this point, no further magnetization will occur.
The magnitude of M = |M| at this point is called the “saturation magnetization.” A graph of
B = |B| versus H = |H| for the above process is shown in Figure 2. This graph, which starts at
B = 0 and H = 0, is called the normal magnetization curve.
Figure 2: The normal magnetization curve of B versus H.
For large H, the domains appear as in Figure 3.
Figure 3: Ferromagnetic domains when the applied H is large.
The alteration of the domains and their direction of magnetization gives rise to a permanent
magnetization which persists even after the applied field is removed.
24
Figure 4: Magnetic hysteresis loop.
If we try to demagnetize the material by decreasing H, the B vs. H curve of Figure 2 is not
followed. Instead, B does not decrease as rapidly as does H. Thus, when H decreases to zero,
there is still a non-zero Br , known as the “remnance.” Only when H reaches the value −Hc does
B become zero. This value Hc is called the “coercive force.” Continuing the cycle H1 , zero, −H1 ,
zero, and back to H1 , the B–H curve looks like that in Figure 4. Since B always “lags” behind
H, the curve in the above graph is called a “hysteresis” loop (from the Greek “to lag”).
The area inside the loop can be shown to be proportional to the energy per unit volume that
is required to change the orientation of the domains over a complete cycle (Warburg’s Law):
W =
Z
I
HdB dτ .
(29)
vol cycle
This energy goes into heating the specimen, and is in joules if H, B and v are measured in A/m,
Tesla and m3 , respectively.
Applying Maxwell’s Equations
Using the apparatus diagrammed in Figure 5, we can observe the hysteresis curve of a ferromagnetic material by the following technique. Suppose we wind two coils of wire on a torus of the
material. Since ferromagnetic materials are good “conductors” of magnetic flux, the coupling
constant between the two coils will be close to one.
Since both the magnetizing field H and the voltage drop across R1 are proportional to the
instantaneous magnetizing current, the horizontal deflection on the scope is proportional to H.
Faraday’s Law (one of Maxwell’s equations),
∇×E=−
∂B
,
∂t
(30)
is what we need to calculate the voltage Vs across the secondary winding:
Vs = N 2 A
Integrating gives
B=
1
N2 A
Z
25
dB
.
dt
(31)
Vs dt .
(32)
Figure 5: Schematic of apparatus to generate and display hysteresis loops.
1
R2 and C across the secondary act as an integrator. If the resistance R2 ≫ ωC
, then the current
in the secondary is determined almost entirely by R2 , so that we may write is = Vs /R2 . In the
capacitor, the current is clearly dq
dt , so
Vs
dq
= is =
.
dt
R2
(33)
Hence the potential difference across C at any instant is
Vy =
1
q
=
C
R2 C
Z
Vs dt =
N2 AB
,
R2 C
(34)
so that the vertical deflection is proportional to B:
B = k B Vy .
(35)
An analysis of the transformer primary circuit begins with Ampere’s Law, another one of
Maxwell’s equations,
∂D
∇×H=J+
,
(36)
∂t
and yields
H = k H Vx
(37)
for the average of H around the iron core. after recognizing that the displacement current can be
neglected at low frequencies.
Laboratory Measurements
1. Set up the apparatus using a 10Ω, 5W resistor for R1 , a 500kΩ resistor for R2 , and a 0.1µF
capacitor for C. To display a hysteresis loop, run the oscilloscope in xy mode and display
Vx on the x axis and Vy on the y axis. Be sure to include the isolation transformer (why?).
2. Starting with the Variac at zero, slowly turn up the voltage in increments such that each
successive increment produces a hysteresis curve that is distinguishable from the previous
one. Trace these curves, continuing until the material reaches saturation.
3. For the Variac setting that gives the largest hysteresis curve, measure the temperature rise
in a one-minute time interval.
26
Calculations and Analysis
~
4. Transformer Primary: Starting with the Maxwell equation that relates the curl of H
~
to J, derive (37) and find an expression for kH . Draw a figure showing the path for the line
integral and the area for the surface integral.
5. Transformer Secondary: Starting with the Maxwell equation that relates the curl of
~ to the time derivative of B,
~ derive (31), (32), (34), and (35). Draw a large, clear figure
E
that shows the path for the line integral and the area for the surface integral superimposed
on the transformer.
6. Choose one of your largest hysteresis loops and, using Warburg’s law, calculate the energy
loss per cycle due to hysteresis, the temperature rise of the core per cycle and the number
of cycles and elapsed time necessary to raise the temperature of the core by 1◦ C. Be careful
doing this calculation; it will take time; be sure your result is reasonable.
7. Plot the normal magnetization curve using the end points of the hysteresis loops.
8. From the normal magnetization curve (B vs. H), plot the relative permeability µ/µ0 of the
material as a function of H.
9. What is the theoretical limiting slope of the magnetization curve B(H) when H → ∞?
(Hint: start with the most fundamental relation between H and B: B = µ0 (H + M).) Estimate the error in your measured value. Compare the theoretical value with your measured
value and explain why they differ.
Vocabulary
⋄
⋄
⋄
⋄
Hysteresis Loop
Soft magnetic material
Saturation Magnetization
Remanence
Reference
D. J. Griffiths, Introduction to Electrodynamics, Prentice Hall, Englewood Cliffs, New Jersey,
1989.
R. G. Lerner and G. L. Trigg, Encyclopedia of Physics, Second Edition, VCH Publishers, Inc.,
New York, 1991, pages 529–530.
Equipment list
-
Isolation transformer
Variac
AC patch cord
Ferromagnetic Torus (Rowland Ring)
Oscilloscope
Resistor R1 (10 Ω, 5 W) and Resistor R2 (470–530 kΩ)
Capacitor C (0.1 µF)
Plastic film & fine-tip “sharpie” marker for tracing oscilloscope pattern
Thermometer and insulation
27
Laboratory Time
Four hours
28
6
Diodes and Transistors (n=1)
Diodes and transistors are two of the fundamental building blocks for modern digital electronics.
The exercises below are a step toward understanding their properties.
Some aspects of the behavior of a transistor can be understood from the following two principles:
1. Current starts to flow in the forward direction (the direction of the arrow in circuit symbols)
through diode junctions when the voltage across the junction reaches a threshold, which is
about 0.6 volts for silicon diodes. The base-emitter and collector-base junctions of transistors
are diodes.
2. Transistors amplify currents. The current that flows from the collector terminal to the
emitter terminal is called Ic . The current that flows from the base terminal to the emitter
terminal is called Ib . The current gain is
hF E =
Ic
.
Ib
(38)
Diode Characteristics
The diode we will use is the emitter-base junction of a NPN transistor. The flat side of the
transistor has labels for the three pins: e, b, c, for emitter, base, collector.
1. Using the circuit of Figure 1 measure the voltage Vbe across the diode and deduce the current
Ib through it. Measure enough values to make a good graph of Ib (Vbe ).
2. Then reverse the polarity of the voltage applied to the series combination and repeat the
measurements. Let both Vbe and Ib be negative numbers after reversing the polarity.
3. Combine the data from both polarities on a single graph of Ib (Vbe ). This function, Ib (Vbe ),
is called the constitutive relation for the diode.
4. Make a few measurements at large voltages using the collector/base junction instead of the
emitter/base junction of the transistor. Comment on the difference between the two diodes.
5. Use the circuit of Figure 2 to rectify an AC signal. Sketch V1 (t) and VR (t) from the oscilloscope; use the same voltage scales and the same zero position for both V1 (t) and VR (t).
Explain what you see using the constitutive relation for the diode and the constitutive
relation for a resistor (Ohm’s law).
Transistor Amplifier
In the circuit of Figure 3, a sine wave V1 (t) from a function generator drives a current Ib through
the base-emitter junction of a transistor; the resistor in series with the junction limits the current.
The purpose of the following measurements is to study amplification by the transistor.
6. Use the two inputs of the oscilloscope to compare Vce (t) and V1 (t). Sketch the waveforms.
Explain the shape of Vce (t).
7. Call τ the time when Vce is a minimum, which is also when the current Ic is a maximum.
For the specific time τ do the following:
29
Figure 1: Circuit for finding I vs. V for a diode.
Figure 2: Circuit to rectify a voltage.
(a) Measure the base current Ib (τ ) and the collector current Ic (τ ) and calculate the ratio
Ic /Ib . This ratio is the current gain of the transistor, which is often called hF E in
manufacturer’s data sheets. Compare the current gain you deduce from your measurements with the current gain you expect from the manufacturer’s specifications for the
transistor.
(b) Find the power delivered to the base-emitter junction and the power dissipated by the
resistor RL , which is called the load resistor. How do you account for the fact that the
energies are different? Where does the extra energy come from?
Vocabulary
⋄
⋄
⋄
⋄
⋄
Forward bias
Reverse bias or back bias
Diode drop
Breakdown voltage
Carrier
30
Figure 3: Circuit for studying transistor amplification.
Equipment list
-
Power Supply
Function generator
NPN Transistor: 2N4401 or MPSW01
Oscilloscope
Resistors
Laboratory Time
Two hours
31
7
Operational Amplifiers (n=2)
Background
Uses of Operational Amplifiers
Measurements in every branch of science and engineering typically begin with a transducer that
changes whatever parameter is being measured into an electrical signal for further amplification
and processing. Because of their versatility, superb performance, ease of use, and low cost,
operational amplifiers (op amps) have become the main building block for amplification and
processing of electrical signals before they are digitized and ingested by a computer. Operational
amplifiers are used in circuits to perform the following functions:
• Amplifying.
• Adding, subtracting, and multiplying.
• Integrating and differentiating.
• Detecting peaks and holding them.
• Filtering (low pass, high pass, and band pass).
• Modulating and demodulating.
Books listed under References below show many useful circuits using operational amplifiers
and explain how they work in more detail than we give in the next section.
What Operational Amplifiers Do
As you would expect, an amplifier in general increases some electrical quantity, usually voltage. In
general, VOutput = A × VInput where the factor A is called the Gain of the amplifier. An integrated
circuit operational amplifier such as we will be studying in this lab amplifies the voltage difference
v+ − v− between its ‘+’ and ‘−’ inputs (see Figure 1). The output voltage is
V2 = AOL (v+ − v− ) ,
(39)
where AOL is the open-loop voltage gain of the amplifier.
Two properties of operational amplifiers make them particularly useful:
• The open-loop gain AOL is large (typically 105 at low frequencies).
• The resistance Rin between the + and − inputs is large (typically 10 MΩ for operational
amplifiers which use bipolar junction transistors (BJT) amplifiers, and 1010 Ω or higher for
operational amplifiers which use field effect transistors (FET’s).
Operational amplifiers are used by providing negative feedback (usually resistance) from the
output to the − input, which substantially reduces their gain from its open-loop value. This
new gain is called the closed-loop gain and we will denote it as G. When the open-loop gain
AOL is much greater than R2 /R1 (see Figure 1), then the voltages v+ and v− are very nearly
equal. Furthermore, when the resistance used in the feedback loop is small compared with the
internal resistance of the operational amplifier, then the currents into the + and − inputs can be
neglected. When these conditions are true the operational amplifier is said to be ideal.
32
The Golden Rules
An ideal operational amplifier can be analyzed by adding the following two simple rules to the
usual rules for circuit analysis:
1. The + and − inputs are at the same potential.
2. No current flows into either input.
The above rules are so generally useful that they are sometimes call The Golden Rules of Op-Amp
behavior. One implication of these rules, for an ideal operational amplifier is that the closed loop
gain G depends only on the feedback elements and not on its open-loop gain AOL nor on its
internal resistance.
Figure 1: Inverting and non-inverting configurations.
Experiments
You will do five mini-experiments on op-amp circuits. In the first, you will see an op-amp work
properly (ideally). You will then learn some of the limitations of op-amps and see some non-ideal
behavior by observing clipping, slew-rate limit, and band-width limit. In the fifth mini-experiment,
you will quantitatively test the OP-AMP GAIN HYPOTHESIS for DC voltages where op-amps
behave almost ideally. Your write-up should have an entry corresponding to each experiment and
the numbered measurements or calculations below.
ALSO – Before proceeding further, measure the resistances R1 and R2 with your multimeter
and report those values. You will need them for section V of this experiment.
I. Proper operation of an Op-Amp
Build an inverting amplifier. Refer to the left-panel of figure 1 and build that circuit. You
should choose 1000 Ω <= R1 <= 100 kΩ and R2 ∼ 10 × R1 .
33
Set proper Power and Signal voltages Up until now, your circuits have been “passive”, which
means there is no need to power them. Op-amps are “active”, they need power, as well as
inputs and outputs. We will call the power voltages VP S (PS for power-supply). We will call
the input and output voltages (“signal voltages”) V1 and V2 . In Figure 1, VP S = ± 10 V. On
your function generator, set V1 to roughly 0.1 V and the frequency to roughly f = 100 Hz.
Determine approximate gain (G) Apply sine wave, square wave, and triangle waves at the
inputs (V1 ) and sketch V1 and V2 to scale in your write-up. Make the sketches large (page
width). V2 should be amplified relative to V1 . What is the approximate amplification factor
you got? Why do you think this is called an inverting amplifier ?
II. Clipping of an Op-Amp
Set VP S = ± 10 V. Set function generator to f = 100 Hz and an amplitude of V1 = 1.5 V . Sketch
the appearance of the sine, triangle, and square waves under these conditions. Draw the new
curves on top of the old ones (perhaps with a dotted line or a different color pencil).
Next set VP S = ± 13 V and add another layer to the sketches of the sine, square, and triangle
waves.
Finally set VP S = ± 18 V and note what you see on the ’scope (no need to sketch).
Why do you think this phenomenon is called Clipping? Can you explain the approximate
relationship between VP S and the waveforms V2 ? Look up Maximum Output Voltage Swing on
the data sheet for these Op-Amps. Does this help you understand what is going on?
III. Slew rate of an Op-Amp
Set function generator to f = 100 Hz, square wave and an amplitude of V1 = 1.5 V . Set VP S
high enough to avoid clipping. Now slowly increase f until you begin to notice that the square
wave is not so square. Sketch the appearance of the square wave at three different frequencies as
it gets more and more distorted. Indicate what frequency you are drawing (or put a proper time
scale on the x-axis of your sketch).
Return your function generator to 100 Hz, change to sine-wave, and repeat this observation
and your sketches.
Finally do this with a triangle wave. You might think at first that the triangle wave is immune
to this distortion. Make sure that the input wave and the output wave both show on the ’scope.
You will then be able to tell what is happening to the triangle wave as the slew-rate limit is
reached. For the triangle wave, make a single sketch including both input and output that shows
the effect.
What’s a slew rate? Look up slew rate in the data sheet. Can you understand it? Can you
tell from some of your measurements what the measured slew rate limit is? (It will probably be
higher than the data sheet says). It is easiest to make this measurement of slew rate on the square
wave.
IV. Bandwidth of an Op-Amp
The bandwidth of an amplifier is defined as the range of frequencies over which the amplifier
performs correctly. Your Op-Amps operate all the way down to zero frequency, therefor their
bandwidth is defined as the maximum frequency at which they operate correctly.
The word correctly can be defined in several ways. The typical definition is that the bandwidth
of an amplifier is the frequency at which the output is 3 dB lower than it is at low frequencies
34
(beyond the bandwidth, the output generally keeps decreasing). The location of the 3 dB point
is often called the “knee” of the amplifier frequency response. A loss of 3 dB is equivalent to a
reduction in Amplitude of roughly a factor of 2. Therefor, if you determined that your amplifier
had a gain of 10 at low frequencies, the bandwidth would be the frequency at which the gain of
the op-amp was 5.
One of the reasons you learned about slew-rate, is, for larger inputs, the slew-rate distorts
the waveform at lower frequencies than the bandwidth does. Thus, for larger inputs, you cannot
correctly measure the bandwidth. To be sure of getting a proper bandwidth measurement, you
should input quite a small signal, such as V1 = 5 mV pp. (If your function generator cannot put
out such a small signal, build a voltage divider to get down to this level.)
Once you have set V1 properly, increase the frequency to measure the bandwidth using a
sinewave. Record the bandwidth and check it against the Gain Bandwidth Product (GBWP)
listed in your data sheet. After you do this, try putting in triangle waves and square waves at
different frequencies. Draw three sketches of each showing increasing distortion as you reach the
bandwidth. Can you explain these distortions? Why didn’t the sine wave distort?
V. Theoretical Gain of an Op-Amp
In the last sections you got the approximate gains of the Op-Amps at low frequency, however it
is hard to measure the Gain precisely using an oscilloscope, so we will do it instead using DC
voltages and a voltmeter.
Hook up the op-amp input (V1 ) to the 6 V output of your power supply and vary (V1 ) and
measure V2 . Use two multimeters to get measurements of V1 and V2 to three significant figures.
Make a table of your results for V1 and V2 and calculate the precise op-amp gain.
After doing this, rebuild your circuit to make a non-inverting op-amp (Figure 1, right-hand
panel). Repeat the measurements from the previous paragraph and tabulate these as well. Now
use your two tables (and the measured values of R1 and R2 to evaluate the truth of the following
statements:
OP-AMP GAIN HYPOTHESIS: “The closed-loop gain of an inverting amplifier is
G = −R2 /R1 and the gain of an non-inverting amplifier is G = R2 /R1 .”
Does the inverting amplifier agree with this hypothesis? To what precision?
Does the non-inverting amplifer agree? If it doesn’t agree, is this measurement error, or is the
hypothesis wrong?
Analysis
Use the Golden Rules of Op-Amps to arrive at a formula for the gain of inverting and non-inverting
op-amps. Show your work. Feel free to consult other sources to help you in your analysis.
Compare your new analysis with the measured gains.
Vocabulary
⋄
⋄
⋄
⋄
Open loop gain
Closed loop gain
Bandwidth
Corner frequency or Roll-off frequency
35
⋄ Slew Rate
⋄ Clipping
References
Powers, Thomas R., The Integrated Circuit Hobbyist’s Handbook, High-Text Publications, Solano
Beach, California, 1995.
Horowitz, Paul, and Hill, Winfield, The Art of Electronics, Cambridge University Press, New
York City, 1989.
Equipment
-
Operational Amplifier (LT1001)
Oscilloscope
±15 volt power supply
Resistors: R and 10R (R > 2000Ω)
Ohmmeter
Laboratory Time
Four hours
36
8
Index of Refraction of Air (n=1)
Even though the wavelength of light in air is only slightly smaller than the wavelength of light
in a vacuum, it can be measured.
Measurements
1. Use the equipment listed below to determine the index of refraction of air.
2. Extrapolate your results to predict the index of refraction at 1 atmosphere (1013.25 millibars) and 0o C.
3. What measurement uncertainty leads to the largest uncertainty in the index of refraction?
What is the resulting uncertainty in the index of refraction?
Calculations
4. From your data and the ideal gas law, find out how the ratio of pressure to absolute temperature, p/T , depends on the fringe number Nf . The fringe number Nf is defined in the
following way: fringe number 1 is the fringe you see at the beginning. As new fringes appear
as you decrease (or increase) the pressure, number them 2, 3, 4, etc.
5. From theory derive the function Nf (n) where n is the index of refraction.
6. From the above, find the function p(n) relating pressure and index of refraction. Then find
a relation between 1 − n and p(1) − p(n). Put in numbers to get 1 − n. Then find n.
Equipment list
-
Hand vacuum pump
Chamber with flat windows
Pressure sensor
Laser
Michelson Interferometer
Thermometer
Laboratory Time
Two hours
37
9
Low-Pass Filters (n=1)
We will learn about and apply the concept of “filtering”. We will predict and measure the time
constant of an RC circuit in an AC environment. Finally, we will use what we know to predict
and then measure the capacitance of a home-made parallel-plate capacitor.
The experiment
We revisit the series circuit of resistor and capacitor to understand, time-constants, Bode plots,
and low-pass filters.
Filters and Frequency response
1. Sketch the circuit you intend to build. It consists of resistor R1 connected to the center pin
of the BNC of a function generator, followed in series by capacitor C1 . (N.B. The circuit is
slightly different than the one you built for the Complex Impedance lab. There the resistor
was connected directly to ground. Here the capacitor is connected directly to ground).
2. Build the circuit and probe the total voltage from the function generator VF G on one ’scope
channel.
3. Measure the Voltage across the capacitor VC at about a dozen different frequencies between
1 Hz and 3 MHz. The frequencies should be roughly uniformly spaced in a logarithmic sense,
but more tightly focused on the range of frequencies where there is a noticeable change in
the behavior of the circuit. The savvy physicist surveys the frequencies of interest before
taking the time to do detailed measurements.
4. Measure the relative amplitude and phase of VF G and VC for each frequency. Hint: Get
both signals on the oscilloscope at once. Trigger on the input voltage and set the trigger
level to 0V. This is a way to easily quantify the phase shift between the applied voltage and
current.
Make a table with columns (1) fF G , (2) VF G , (3) VC , (4) VVFCG ,and (5) φC . Make a plot on
log-log paper of column 4 vs. column 1 and a plot on semilog paper of column 5 vs. column
1. Such paper is provided for you on next page. This type of plot (log-log and semilog
for amplitude and phase vs. frequency) is called a “Bode Plot”. Note: Measuring φC is
somewhat time-consuming. You can first survey the frequency range of interest and decide
where to concentrate your effort measuring φ accurately. At other frequencies, you can just
approximate it.
5. Using what you already know about complex impedance, write down the complex voltage
divider equation for this new circuit in which VF G is the input and VC is the output. Solve
for the ratio VVFCG . If you do this correctly, the expression R1 C1 should appear somewhere in
your ratio. Define the variable ω1 ≡ R11C1 and calculate its value. ω1 and the corresponding
f1 are often called the “corner frequencies”. (Note the “corner” in your amplitude Bode
plot at about this frequency). Note that the corner frequency ω1 is also the reciprocal of
the time constant of the circuit, RC.
6. Using your voltage divider equation, what is the amplitude ratio for ω = R11C1 ? Finding a
frequency that gives such an amplitude ratio is a quick way to find the corner frequency.
38
You have just created and analyzed a “low-pass filter”, so called because it attenuates high
frequencies, without much affecting the low frequencies (thus “passing them”). Circuits like these
were (until recently with all-digital technology) attached to the bass and treble knobs of your
stereo system. Filters are still important throughout science and technology. Turn your scope up
to maximum sensitivity and then push the “BW limit” switch to see a low-pass filter in operation.
Filters and Music
Before the advent of digital technology, the base and treble knobs on stereo systems merely
changed the resistance of an RC filter like the one you just created. It is worth listening to the
effect your filter has on music. Bring an iPod or similar player and design a filter that cuts out
frequencies above 100 Hz. Play your iPod through a speaker (we have some you can use) and
note how the low-pass filter affects it.
Can you design a high-pass filter that only plays music above 1000 Hz? Try it and see how it
sounds. (You get credit for this part of the lab just by demonstrating it to your instructor.)
You should know in your designs that the output impedance of an iPod is about 30 Ω. You
must include this series resistor when designing your filter.
Equipment
-
Oscilloscope
Function generator
Capacitance meter
Ohmmeter
R1 =3.3 kΩ Resistor
R2 =You select it Ω Resistor
C1 =0.047µF Capacitor
Laboratory Time
Two hours
39
100
80
Phase (φ in degrees)
60
40
20
0
−20
−40
−60
−80
−100
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
Frequency Response of an RC filter
1
10
0
Amplitude Ratio
10
−1
10
−2
10
−3
10
−4
10
0
10
1
10
2
10
3
4
10
10
Frequency (Hz)
40
5
10
6
10
7
10
10
Electric Field, Capacitance, Dielectric Constant (n=1)
Laboratory Work
Measure the dielectric constant of liquid nitrogen.
Some of the effort in this laboratory project will be designing the experiment. You will
need to make calculations to see what kinds of equipment the experiment will require. The first
measurements should be done quickly and simply to uncover unsuspected problems.
Report
In addition to describing your laboratory work and the results, consider the subtle effect leading
to the Clausius-Mosotti formula in Problem 4.38, Page 200 of D. J. Griffiths, Introduction to
Electrodynamics, Third Edition, Prentice Hall, Upper Saddle, NJ, 1999. Can the Clausius-Mosotti
formula explain the descrepancy between your measurement and the accepted value of liquid
nitrogen? (You should be able to find the polarizability of one nitrogen molecule from the dielectric
constant of gaseous nitrogen.)
Equipment
-
Liquid Nitrogen
Air-variable capacitor
Styrofoam containers
Miscellaneous electronic equipment
Laboratory Time
Two hours
41
11
Negative Resistance (n=1)
Introduction
In linear devices such as ordinary resistors, “resistance” could be defined either as dV /dI or as
V /I; in either case the resistance of the device would have the same value. What definition
should be used for non-linear devices such as diodes, neon lamps, and spark gaps? There is some
advantage in defining resistance for these kinds of devices as dV /dI. Because of this definition,
these devices exhibit negative resistance, and because of the negative resistance they can be made
to oscillate in special circuits.
Laboratory Work
1. This experiment will use a high voltage power supply; ask the instructor to look at your
circuit before turning on the supply.
2. Find the relation between voltage V and current I for a neon lamp. This relation is called
the “constitutive relation” for the neon lamp.
3. Use the constitutive relation to design and construct an oscillator. A capacitor and a resistor
will be required in the circuit. Draw a diagram of your circuit. Measure the frequency of
the oscillator. (HINT: The capacitor needs to be put in parallel with the lamp, while the
resistor needs to be in series with the combination. Based on your measurementsso far, you
should be able to pick a resistor and capacitor that will work well.)
4. Now that you have seen the oscillator work once, you are more likely to understand what is
going on. Select a different resistor or capacitor with the goal of either doubling or halving
the oscillator frequency (you decide which you want to do. If your first frequency came out
small, you might want to double it, but if the flashing is pretty rapid, you might aim to
halve the frequency.)
Calculations and Analysis
5. From the values of the capacitor and the resistor and the voltage at which the neon
lamp becomes conducting, calculate the frequency of the oscillator. Compare your calculated frequency with the measured frequency. Repeat the calculation for the second
resistor/capacitor set you choose.
Equipment list
-
High voltage supply
Voltmeters
Neon lamp
Assorted parts
Laboratory Time
Two hours
42
12
Superconductivity (n=1)
Introduction
The discovery of ceramic compounds that are superconductors when immersed in liquid nitrogen
has made it possible to investigate some properties of superconductivity using very simple apparatus. In this lab we will use a commercial high-temperature-superconductivity experiment kit to
study some of aspects of superconductors. The kit contains a disk of ceramic superconductor, a
module consisting of a smaller ceramic superconductor disk with an attached thermocouple, and
two very strong, small magnets.
A superconductor is a substance which acts like a perfect conductor when its temperature is
below some critical value, the critical temperature Tc . Thus when T < Tc a superconductor acts
as if its resistance were zero, so that if a current were to be established in a ring of superconductor
it would persist indefinitely.
In 1933, Meissner discovered that a magnet was repulsed by a superconductor, an effect now
known by his name. The following are some considerations of how the Meissner effect might come
about.
Suppose that a strong magnet is brought up to a superconductor so that the superconductor
experiences a finite dB/dt. Then an emf would be induced in the superconductor such that a
current would begin to flow. This induced current would continue to flow after the magnet stops
moving (dB/dt = 0) because zero voltage will maintain a current when the resistance is zero.
According to Lenz’s law the induced current will be in a direction such that the B field due to it
would be opposed to the B field of the magnet. Thus there would be a repulsive force between
the small magnet and the superconductor. If the small magnet had a sufficiently strong field then
the repulsion could result in the magnet being levitated above the superconductor.
However, if the magnet were placed on the superconductor while the superconductor was
warmer than the critical temperature, and then the superconductor was cooled to below the critical
temperature, the magnet would not be levitated since dB/dt would never have been different from
zero. Nevertheless, if one were then to try to remove the magnet from the superconductor, there
would be an attractive force between the two of them resisting the removal of the magnet due to
the B field of the current induced in the superconductor as the magnet was moved. Since there is
zero resistance in the superconductor we might expect that the induced current would be entirely
on the surface and not in the interior of the superconductor.
Another way that a superconductor might repulse a magnet is if microscopic magnetic dipoles
are induced in the superconductor which then oppose the applied field. Then, inside the superconductor there would be zero B field, and the external field of the dipoles induced in the
superconductor would repel the source of the applied field, that is, the magnet. The superconductor would then be described as diamagnetic. This implies that the effect would be a volume, or
bulk effect, becoming stronger when the volume of the superconductor is increased. In this case,
if the magnet were placed on the superconductor before it was cooled, as the superconductor became superconducting it would exclude the magnetic field from its interior and the magnet would
be levitated. Of course, this assumes that the strength of the magnet is sufficient to support its
own weight.
This experiment is divided into two parts. In the first part you will study the Meissner Effect.
In the second part you will determine the critical temperature of the ceramic superconductor.
43
Preparation Before Lab
Read the article on Superconductivity by Philip B. Allen in Encyclopedia of Physics, Second
Edition, edited by Lerner and Trigg, VCH Publishers, New York, 1991, pages 1198–1203.
Safety Procedures
Liquid Nitrogen
Liquid nitrogen (LN2) is COLD, it will kill skin cells by freezing them.
• Wear safety glasses when using LN2.
• Do not allow any LN2 to contact your body.
• Do not touch anything that has been immersed in LN2 before it has warmed to room
temperature. Use the plastic tweezers provided.
• Be very careful to avoid splashing of the LN2.
• Avoid overfilling any vessel or spilling the LN2.
• Do not put LN2 in any vessel with a tight-fitting lid.
• The extreme cold of LN2 will damage many surfaces, so do not pour LN2 on or in anything
not intended for LN2 contact.
• Let the lab instructor dispose of any unused LN2.
Ceramic Superconductor Handling Precautions
The ceramic superconducting material is made of oxides of various metals. The salts of these
metals are toxic if ingested.
• Never handle the ceramic disks with your hands. Use tweezers provided.
• Do not under any circumstances ingest the materials.
• Do not expose the superconductor disk to water. After use in LN2, carefully wipe the
superconductor to remove water or frost. Then warm it under a lamp to completely dry it.
• The ceramic disks are brittle. Handle gently!
• Do not expose the ceramic disks to temperatures greater than 100◦ F.
Things to Do
Meissner Effect
The two possible causes of levitation (the Meissner effect) may be summarized as:
1. Induced currents (due to change of B with time, which by Lenz’s law oppose the change in
magnetic flux).
2. Diamagnetism (induced magnetic dipoles which oppose an applied B field).
44
Write clear and concise descriptions of your observations and answer the questions below.
1. Place the 1 inch diameter ceramic superconductor disk in an empty styrofoam cup. Place
the cubic magnet on the uncooled superconductor, then cool to superconducting state with
liquid nitrogen. If (1), then no levitation (no change in B with time). If (2), levitation.
2. If necessary, lift the magnet above the superconductor. If (1), a force will oppose removal of
magnet (of about same magnitude as force which opposes moving magnet closer to superconductor). If (2), force always repulsive. Also, try to move the levitated magnet toward
the edge of the superconductor disk. What happens?
3. Surface current vs. bulk effect. Repeat the previous two steps using two superconductor
disks stacked one on top of the other. If (1), the current is only on surface, and there is no
dependance on the thickness of the superconductor. If (2), double thickness of superconducting disk (two disks), increases height of levitation. Also, as superconductor warms up
to non-superconducting state, the magnet will abruptly fall if (1), but gently descend if (2).
Magnetic Field studies
4. What is the orientation of the magnetic field of the suspended cube magnet? To test if there
is a preferred orientation, cause the magnet to rotate by blowing on the cube’s edge with a
straw. Can you get it to rotate about another axis? From this you should be able to infer
the axis of the magnet. (This also demonstrates the possibility of a frictionless magnetic
bearing.)
5. Try to change the orientation of suspension by using a second magnet (whose poles can be
determined using the compass provided) to attract the levitated magnet into an upright
position (while spinning, say). Is the upright position stable? Why or why not?
Measuring Superconducting transition temperature
6. Measure the critical temperature Tc of the ceramic superconductor using the module provided consisting of encapsulated superconductor disk and attached thermocouple. The
module has six wires. The red and blue wires are the thermocouple wires. Since the voltages produced by the thermoucouple are just a few millivolts, you need to set your voltmeter
to its most sensitive range. (Some meters have a special 300 mV setting, others just autorange.) In order to determine the temperature of the superconductor you need to know the
relation between temperature and thermocouple voltage. See the table for the particular
thermocouple you are using. The table below is appropriate for the type K thermocouple
included in the Colorado Superconductor kits.
TABLE OBTAINED FROM http://srdata.nist.gov/its90/download/all.tab
May 2009
ITS-90 Table for type K thermocouple
C
0
-1
-2
-3
-4
-5
-6
-7
Thermoelectric Voltage in mV
45
-8
-9
-10
-270 -6.458
-260 -6.441 -6.444 -6.446 -6.448 -6.450 -6.452 -6.453 -6.455 -6.456 -6.457 -6.458
-250 -6.404 -6.408 -6.413 -6.417 -6.421 -6.425 -6.429 -6.432 -6.435 -6.438 -6.441
-240
-230
-220
-210
-200
-6.344
-6.262
-6.158
-6.035
-5.891
-6.351
-6.271
-6.170
-6.048
-5.907
-6.358
-6.280
-6.181
-6.061
-5.922
-6.364
-6.289
-6.192
-6.074
-5.936
-6.370
-6.297
-6.202
-6.087
-5.951
-6.377
-6.306
-6.213
-6.099
-5.965
-6.382
-6.314
-6.223
-6.111
-5.980
-6.388
-6.322
-6.233
-6.123
-5.994
-6.393
-6.329
-6.243
-6.135
-6.007
-6.399
-6.337
-6.252
-6.147
-6.021
-6.404
-6.344
-6.262
-6.158
-6.035
-190
-180
-170
-160
-150
-5.730
-5.550
-5.354
-5.141
-4.913
-5.747
-5.569
-5.374
-5.163
-4.936
-5.763
-5.588
-5.395
-5.185
-4.960
-5.780
-5.606
-5.415
-5.207
-4.983
-5.797
-5.624
-5.435
-5.228
-5.006
-5.813
-5.642
-5.454
-5.250
-5.029
-5.829
-5.660
-5.474
-5.271
-5.052
-5.845
-5.678
-5.493
-5.292
-5.074
-5.861
-5.695
-5.512
-5.313
-5.097
-5.876
-5.713
-5.531
-5.333
-5.119
-5.891
-5.730
-5.550
-5.354
-5.141
-140
-130
-120
-110
-100
-4.669
-4.411
-4.138
-3.852
-3.554
-4.694
-4.437
-4.166
-3.882
-3.584
-4.719
-4.463
-4.194
-3.911
-3.614
-4.744
-4.490
-4.221
-3.939
-3.645
-4.768
-4.516
-4.249
-3.968
-3.675
-4.793
-4.542
-4.276
-3.997
-3.705
-4.817
-4.567
-4.303
-4.025
-3.734
-4.841
-4.593
-4.330
-4.054
-3.764
-4.865
-4.618
-4.357
-4.082
-3.794
-4.889
-4.644
-4.384
-4.110
-3.823
-4.913
-4.669
-4.411
-4.138
-3.852
-9
-10
C
0
-1
77 Kelvin is -196 C
-2
-3
-4
-5
-6
-7
-8
Type K thermocouple should read -5.829 mv
7. After setting up your circuit cool the superconductor/thermocouple module with LN2. Once
it has come to equilibrium at the temperature of LN2 (77 K) the thermocouple voltage should
be 6.43 mV.
8. Remove the module from the LN2 and place it on an insulating surface (an empty styrofoam
cup will do). Now balance the cylindrical Samarium magnet so that it is levitated over the
center of the superconducting disk.
9. Observe the magnet carefully, measuring its levitation height, and simultaneously recording
the thermocouple voltage every 10 s. If the levitation is due to a surface effect the levitation
should cease abruptly, whereas if it is a bulk effect, the levitation should weaken over a finite
time interval.
10. Redo this experiment by first placing the magnet on the encapsulated superconductor and
then cooling it down to LN2 temperature, while observing the magnet and recording the
thermocouple voltage.
11. Describe your observations and measurements.
Four point probes and resistance measurements
You will see text on the superconducting module that says ”500 mA” maximum. Here’s what
it means. The module has two wires connected to a thermocouple. You have already used
46
those. The other four wires comprise something called a four-point probe, a sketch of which
is shown below. Four point probes are the established way to measure small resistances.
You apply a current betweenn the two out wires, and currect cause a small DV
Other questions
12. Why does the LN2 boil when you pour it into the “dish?” and when you put the superconductor disk into it?
13. When the LN2 has evaporated, the magnet stays levitated for many more minutes. Why is
this so?
14. Describe the way in which the magnet falls as the superconductor warms to above the
critical temperature and explain this behavior.
15. Under some circumstances, the magnet may suddenly scoot to one side of the brass cup as
the assembly warms. Why is this?
16. Why does the assembly develop a layer of frost only after the LN2 has boiled away?
17. If the two different determinations of Tc yielded different values of Tc , explain why and
determine which is the correct value.
Equipment list
-
1 inch diameter ceramic superconductor disk.
Ceramic superconductor and thermocouple module (“device”).
Cubic neodymium rare-earth alloy magnet.
Cylindrical samarium rare-earth alloy magnet.
Plastic tweezers.
Insulated dish (styrofoam cup).
CSI Instruction Manual.
Safety glasses.
Liquid Nitrogen (LN2)
Digital Multimeter.
Magnetic Compass.
Plastic straw.
Laboratory Time
Two hours
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13
Serial Ports (n=1)
Background
Serial data communications is important in many scientific instruments. If you build an instrument
yourself, or need to automate a commercial instrument, you will likely need to understand it.
Like so much else in electronics, serial communications have advanced and proliferated since
the PC revolution of the early 1980’s. You have probably heard of USB (Universal serial bus),
and SATA (Serial ATA, for hard drives), which are high performance interfaces for commercial
computers. There are also many serial interfaces used at the chip or subsystem level, among these
are I2C (Inter-IC bus) and SPI (Serial Peripheral Interface).
The grandparent of all serial-protocols is the RS-232 protocol, often simply called ”the serial
port protocol”. Despite its antiquity (a variant was first used in teletypes for telegraphy in
the early 20th century), it is still used by GPS devices and various scientific instruments and
is available on your laptop computer via a “terminal program”. In particular, because if its
simplicity, it is a good protocol to learn. The other modern protocols previously mentioned have
much in common with it, and you can understand their more sophisticated features more easily
if you already understand RS-232.
Connectors and pin definitions
The RS-232 protocol was used most heavily for PC’s to communicate with modems. Because
modems interfaced to telephones, they had a lot of complexity. They had to detect dial tones and
know when the line was clear for data. In fact the full RS-232 interface defined a 25-pin connector.
Early on PC-manufacturers cut this down to 9-pins, and used a “9-pin D connector” (see Figure
below). This is still more complicated than needed for most applications. In practice, one needs
only the “RX” (Receive) and “TX” (Transmit) wires, plus ground, and sometimes power. When
you read descriptions of RS-232 and serial ports on line, you will run into references to arcane
signals like request to send (RTS), clear to send (CTS) and data terminal ready (DTR) that were
important in modem and telephone days. Fortunately, you need none of these things for this lab.
You only need to understand RX and TX.
In this lab, you will be using only RX, TX and ground. The RS-232 specification says that
pin 2 is RX (receive). Pin 2 is sometimes called RD (Receive data). Pin 3 is TX (transmit),
also known as TD (Transmit data). Pin 5 is ground (by now you know that you always need a
ground!). Looking at the diagram below, you will see that the “male” and “female” connectors
are mirror images. Thus what you think is Pin 2 may be Pin 4, while Pin 1 may be Pin 5. To
make matters worse, there are two types of cables in common use for serial ports.
Experiment
Connect an oscilloscope to the serial port of a laptop computer and analyze the waveforms that
come out of MINICOM or HYPERTERM. Figure out how to translate oscilloscope waveforms
into letters. This article will be extremely useful. (http://en.wikipedia.org/wiki/RS-232 ).
It is your job to figure out the connectors and how to connect to the appropriate pins. After
that, the waveform analysis is straightforward.
Your report should include a sketch of the waveforms of your initials shown at 19200 baud and
at 2400 baud. Indicate what changes as you change the baud rate of MINICOM or HYPERTERM.
48
Figure 1: 9-Pin “D” connectors: They are called “D” because of their shape. This type of
connector is also referred to as a “DB9”. They are the same size and shape as the video (SVGA)
connector on the back of your computer, but they have fewer pins. Notice that pin 2 on the
“male” connector becomes pin 4 on the “female” connector. This causes endless confusion.
You should figure out how to decode all the capital letters and all the numbers on a computer
keyboard. When you are ready, I will type you a 10 character or less secret message. Decode this
and include it in your report.
1. A “straight” cable connects every pin on one end to the same numbered pin on the other
end. You would think this is obvious, but there is another common option.
2. A “cross-over” cable connects pin 2 on one end to pin 3 on the other end.
If you undertand why cross-over cables exist, you will understand more about serial ports.
In principle, you only need one wire (plus ground) to send data. In fact this is correct. For a
computer to send serial data, it merely triggers a line The concept of a cross-over vs. a straight
through cable also applies to Ethernet, where it also causes much confusion among beginning
engineers and scientists.
Equipment
- Laptop computer with working serial ports
- Tektronix TDS-1002 Digital Oscilloscope
- Serial port cables
Laboratory Time
Two hours
49
Figure 2: Why you need a crossover cable: The computer on the right is “talking” into its TX
line. The computer in the left is “listening” to its RX line. If the left computer wants to “talk”,
it wiggles it’s TX line, which must obviously be connected to the RX line of the computer on the
right.
Equipment
- Laptop computer with working serial ports
- Tektronix TDS-1002 Digital Oscilloscope
- Serial port cables
Laboratory Time
Two hours
50
14
Miscellaneous Experiments (n=various!)
Some of these experiments require advance notice because I will have to procure needed
supplies. The amount of notice required is indicated on each.
The following experiments are not completely described. Part of each exercise is to identify
the core scientific objective. Most are n=2 or more because of the work of figuring out what to
do and perhaps assemblying supplies. I will help you with these and can decide on their value in
advance.
14.1
Design your own!
This takes one month advance notice because we have to discuss what you will do and obtain
materials you might need.
Students in this class have tried to design their own transmitter, stereo amplifier, spark gap,
or magnetic levitation device. I can help you turn your idea into something that might work.
14.2
Lock-in Amplifiers
This project has a lot of physics and is not lethal. We have both a lock-in amplifier and a
mechanical chopper. The ”chopper” when applied to a laser beam or LED, can allow you to see
a weak light from an LED in an otherwise brightly lit room. Perhaps you can even make a laser
motion detector out of this apparatus.
14.3
TV remote control
This requires only one week advance notice.
Using a phototransistor or photodiode, look at the infrared light signal emitted by a remote
control unit for a television set. See if you can determine the control codes.
14.4
Resistance and temperature
Requires one week advance notice.
Experiment with the following:
1. Thermistor
2. Metal film resistor
3. Carbon resistor
4. Tungsten filament
5. Copper wire
Low temperatures can be produced with liquid nitrogen. You could get some ideas on experimental techniques from an article by K. G. Vandervoort, J. M. Willingham, and C. H. Morris
(American Journal of Physics, 63(8), 759-760, 1995.
51
14.5
Electric Motors
Requires NO advance notice.
If you rotate a motor shaft (say, with another motor) it will generate a voltage. Connect the
shaft of one motor to the shaft of another motor and make measurements to observe the generated
voltage in both the driving motor and the driven motor.
14.6
Magnetic materials
Requires one month advance notice to order the materials. It should be possible to study the
magnetic properties of different types of materials by seeing how they affect the inductance of a
coil in an oscillating circuit.
14.7
Electric and magnetic fields of a power line
This takes three weeks advance notice because additional materials must be obtained.
An article in Physics Today asks, “Do the all-pervasive low-frequency electromagnetic fields of
modern life threaten our health?” (W. R. Bennett, Jr., Cancer and Power Lines, Physics Today,
April 1994, page 23-29.) It might be interesting to measure the electric and/or magnetic fields of
the biggest power line you can find in the vicinity of Socorro.
14.8
Flames
There is a striking photograph of a flame being pulled toward a Van de Graaff generator in The
Physics Teacher, vol. 33, page 44, January 1995. It could be interesting to study the electrical
properties of flames. Flames can be used as speakers as well.
14.9
Guitar Pickup
This requires three weeks notice to order the pickups.
Electric guitar pickups are beautiful examples of electromagnetic induction. Working through
all the details of the transducer is exciting and educational.
14.10
Build a Geiger Counter
This is a cool project that can be done in 3-4 weeks by two people. There are several articles on
how to do this, including one from Make magazine that is fairly complete. We should talk by
mid-February so I can order parts for you. Unlike the rail gun and can-crusher, this project is
not dangerous. I will provide some Uranium ore that can be used to test the final result.
14.11
Build a rail gun
Because it is both dangerous and complex, this project may be undertaken as a paper study.
Research rail-guns and write a report that outlines how you would go about building one and
what physics (B-fields, currents, stresses on the system) you can work out on paper. A paper
study requires no advance notice and can be good for n=2 for a single person. To actually even
begin to build one is probably n=5 and we should discuss this by mid-February.
Electromagnetic rail guns can reach very high speeds. This is a wonderful and educational
engineering project in pulsed power and mechanical design. It is also dangerous! I will help you.
This should be a group project.
52
14.12
Build an electromagnetic can crusher
Another good group project that is potentially fatal ... but not until energized. I can guide you
through the steps needed to charge large capacitors and fire them safely.
Because it is both dangerous and complex, this project may be undertaken as a paper study.
Research can-crushers and write a report that outlines how you would go about building one and
what physics (B-fields, currents, stresses on the system, forces on the can) you can work out on
paper. A paper study requires no advance notice and can be good for n=2 for a single person.
To actually even begin to build one is probably n=5 and we should discuss this by mid-February.
53
Appendices – Data Sheets / Circuits Concept
Test
54