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PHYSICS LAB II
LABORATORY MANUAL
Number:
Name:
Department:
2017-2018
2
Contents
İçindekiler Contents ................................................................................................................................................. 2
Report Format ......................................................................................................................................... 5
UNIT SYSTEMS ......................................................................................................................................... 7
Measurements and Uncertainty ............................................................................................................. 9
Graphical Representation of Data ......................................................................................................... 11
Testing Electrical Circuits with the M ultimeter .................................................................................... 14
DIGITAL MULTIMETER ........................................................................................................................... 15
Multimeter as an Ampermeter ...................................................................................................... 16
Experiment 1 : Ohm's Law and Not Ohm's Law .................................................................................. 18
Experiment 2 : Emf and Internal Resistance ......................................................................................... 26
Experiment 3: Kirchoff’s Law ................................................................................................................. 30
Experiment 4: Current-Balance Experiment (F = ILBsin) .................................................................... 34
Experiment 5: Transformer Basics I ....................................................................................................... 39
Experiment 6: Ohm’s Law, RC and RL Circuits in DC ............................................................................. 45
Experiment 18/2: Ohm’s Law, RC and RL Circuits ................................................................................. 50
EXPERIMENT 7. RL, RC, RLC Circuts (AC analysis) .................................................................................. 54
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GENERAL INSTRUCTIONS
1. You must arrive on time since instructions are given and announcements are made at
the start of class.
2. You will do experiments in a group but you are expected to bear your share of
responsibility in doing the experiments. You must actively participate in obtaining the
data and not merely watch your partners do it for you.
3. The assigned work station must be kept neat and clean at all times. Coats/jackets
must be hung at the appropriate place, and all personal possessions other than those
needed for the lab should be kept in the table drawers or under the table.
4. The data must be recorded neatly with a sharp pencil and presented in a logical way.
You may want to record the data values, with units, in columns and identify the quantity
that is being measured at the top of each column.
5. If a mistake is made in recording a datum item, cancel the wrong value by drawing a
fine line through it and record the correct value legibly.
6. Get your data sheet, with your name, ID number and date printed on the right corner,
signed by the instructor before you leave the laboratory. This will be the only valid
proof that you actually did the experiment.
7. Each student, even though working in a group, will have his or her own data sheet
and submit his or her own written report, typed, for grading to the instructor by the next
scheduled lab session. No late reports will be accepted.
8. Actual data must be used in preparing the report. Use of fabricated, altered, and other
students’ data in your report will be considered as cheating.
9. Be honest and report your results truthfully. If there is an unreasonable discrepancy
from the expected results, give the best possible explanation.
10. If you must be absent, let your instructor know as soon as possible. Amissed lab can
be made up only if a written valid excuse is brought to the attention of your instructor
within a week of the missed lab.
11. You should bring your calculator, a straight-edge scale and other accessories to
class. It might be advantageous to do some quick calculations on your data to make sure
that there are no gross errors.
12. Eating, drinking, and smoking in the laboratory are not permitted.
13. Refrain from making undue noise and disturbance.
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Semester II Student’s
NAME - SURNAME :
NO :
DEPARTMENT :
DATE: EXP NAME.: INSTRUCTOR’S SIGNATURE
1st Week
2nd
Week
3rd
Week
4th
Week
5th
Week
6th
Week
7th
Week
8th
Week
9th
Week
10th
Week
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Report Format
The laboratory report must include the following:
1. Title Page: This page should show only the student’s name, ID number, the name of
the experiment, and the names of the student’s partners.
2. Objective: This is a statement giving the purpose of the experiment.
3. Theory: You should summarize the equations used in the calculations to arrive at the
results for each part of the experiment.
4. Apparatus: List the equipment used to do the experiment.
5. Procedure: Describe how the experiment was carried out.
6. Calculations and Results: Provide one sample calculation to show the use of the
equations. Present your results in tabular form that is understandable and can be easily
followed by the grader. Use graphs and diagrams, whenever they are required.
It may also include the comparison of the computed results with the accepted values
together with the pertinent percentage errors. Give a brief discussion for the origin of
the errors.
7. Conclusions: Relate the results of your experiment to the stated objective.
8. Data Sheet: Attach the data sheet for the experiment that has been signed by your
instructor.
6
Introduction
The aim of the laboratory exercise is to give the student an insight into the significance
of the physical ideas through actual manipulation of apparatus, and to bring him or her
into contact with the methods and instruments of physical investigation. Each exercise
is designed to teach or reinforce an important law of physics which, in most cases, has
already been introduced in the lecture and textbook. Thus the student is expected to be
acquainted with the basic ideas and terminology of an experiment before coming to the
laboratory.
The exercises in general involve measurements, graphical representation of the data,
and calculation of a final result. The student should bear in mind that equipment can
malfunction and final results may differ from expected values by what may seem to be
large amounts. This does not mean that the exercise is a failure. The success of an
experiment lies rather in the degree to which a student
has:
• mastered the physical principles involved,
• understood the theory and operation of the instruments used, and
• realized the significance of the final conclusions.
The student should know well in advance which exercise is to be done during a specific
laboratory period. The laboratory instructions and the relevant section of the text
should be read before coming to the laboratory. All of the apparatus at a laboratory
place is entrusted to the care of the student working at that place, and he or she is
responsible for it. At the beginning of each laboratory period it is the duty of the
student to check over the apparatus and be sure that all of the items listed in the
instructions are present and in good condition. Any deficiencies should be reported to
the instructor immediately.
The procedure in each of these exercises has been planned so that it is possible for the
prepared student to perform the experiment in the scheduled laboratory period. Data
sheets should be initialed by your instructor or TA. Each student is required to submit a
written report which presents the student’s own data, results and the discussion
requested in the instructions. Questions that appear in the instructions should be thought
about and answered at the corresponding position in the report. Answers should be
written as complete sentences.
If possible, reports should be handed in at the end of the laboratory period. However, if
this is not possible, they must be submitted no later than the beginning of the next
exercise OR the deadline set by your instructor.
Reports will be graded, and when possible, discussed with the student.
7
UNIT SYSTEMS
SI UNITS
Sciences are built upon measurements. Measurements are expressed with numbers. This
allows the logic, precision and power of mathematics to be brought to bear on our study
of nature.
Units of measurement are names which characterize the kind of measurement and the
standard of comparison to which each is related. So, when we see a measurement
expressed as "7.5 feet" we immediately recognize it as a measurement of length,
expressed in the unit "foot" (rather than other possible length units such as yard, mile,
meter, etc.) Since many possible units are available for any measurement it is essential
that every measurement include the unit name. A statement such as "the length is 7.5" is
ambiguous, and therefore meaningless.
8
THE METRIC SYSTEM
As early as 1670, European scientists were recommending reform of the chaotic unit
systems then in use: systems which differed from country to country. The reformers
urged (l) uniformity and universality, (2) simple ratios of sizes of units, (3) rational
relations between units, and (4) units referenced to constants of nature (such as the
circumference of the earth, boiling point of water, etc.)
In 1791, in the aftermath of the Revolution, the French National Assembly adopted a
more rational system based upon decimal ratios. This came to be known as the metric
system. In the United States, at this time, there was also interest in reform of units and
standards. In 1786 Congress approved a decimal system of coinage.
In 1790 Congress considered a report on units which Secretary of State Thomas
Jefferson had prepared at the urging of George Washington. In the report Jefferson
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proposed, as one alternative, a decimal system of weights and measures. His system had
several unfortunate features, (1) it retained some of the old unit names (pound, foot,
inch, furlong, mile, etc.) but assigned them new sizes (1 foot contained 10 inches, for
example), and (2) his system was not fully compatible with the metric system then
being developed in France. Congress, confused and ill-informed (as usual) took no
action on the proposal.
John Quincy Adams' 1821 Report Upon Weights and Measures was an exhaustive
study, presenting pros and cons of unit reform. Though praising the virtues of the
French Metric system (and noting some shortcomings) he concluded that the U. S. had
not attained sufficient maturity to require adoption of the system. Further he noted that
the states had laws of weights and measures which were substantially uniform. To
impose a new system on all states would raise sticky questions of states' rights.
So our best opportunity for adoption of a sensible unit system slipped by. While other
countries, one by one, adopted the metric system, the U. S. arrogantly went its own way,
feeling no need nor desire to adopt a "foreign" system of units.
In 1890, metric units were established as the legal basis of all weights and measures in
the United States, but this did little to establish the use of the metric units in industry,
commerce, and everyday life. Today the revised and standardized metric system, called
the international system (SI, for System‚ International) is used in nearly all countries.
The United States, South Africa and less than a dozen non-industrialized countries have
not made a commitment to convert fully to the metric system.
But change is coming"slowly. Several states have marked their highway distance signs
in both miles and kilometers. Some radio stations report daily temperatures in both
degrees Fahrenheit and Celsius. U. S. automakers use metric parts in auto engines.
Science and medicine have been almost exclusively metric in all countries for many
years. Other industries use metric standards because of international trade and
competition.
The relative simplicity of the metric system is well illustrated by comparing
measurements of length in the United States System with those of the metric system.
Measurements and Uncertainty
A measurement result is complete only when accompanied by a quantitative statement
of its uncertainty. The uncertainty is required in order to decide if the result is adequate
for its intended purpose and to ascertain if it is consistent with other similar results."
National Institute of Standards and Technology
No measuring device can be read to an unlimited number of digits. In addition when we
repeat a measurement we often obtain a different value because of changes in conditions
that we cannot control. We are therefore uncertain as to the exact values of
measurements. These uncertainties make quantities calculated from such measurements
uncertain as well.
10
Finally we will be trying to compare our calculated values with a value from the text in
order to verify that the physical principles we are studying are correct. Such
comparisons come down to the question "Is the difference between our value and that in
the text consistent with the uncertainty in our measurements?".
The topic of measurement involves many ideas. We shall introduce some of them by
means of definitions of the corresponding terms and examples.
Sensitivity - The smallest difference that can be read or estimated on a measuring
instrument.
Generally a fraction of the smallest division appearing on a scale. About 0.5 mm on our
rulers. This results in readings being uncertain by at least this much.
Variability - Differences in the value of a measured quantity between repeated
measurements. Generally due to uncontrollable changes in conditions such as
temperature or initial conditions.
Range - The difference between largest and smallest repeated measurements. Range is
a rough measure of variability provided the number of repetitions is large enough. Six
repetitions are reasonable. Since range increases with repetitions, we must note the
number used.
Uncertainty - How far from the correct value our result might be. Probability theory is
needed to make this definition precise, so we use a simplified approach.
We will take the larger of range and sensitivity as our measure of uncertainty.
Example: In measuring the width of a piece of paper torn from a book, we might use a
cm ruler with a sensitivity of 0.5 mm (0.05 cm), but find upon 6 repetitions that our
measurements range from 15.5 cm to 15.9 cm. Our uncertainty would therefore be 0.4
cm.
Precision - How tightly repeated measurements cluster around their average value. The
uncertainty described above is really a measure of our precision.
Accuracy - How far the average value might be from the "true" value. A precise value
might not be accurate. For example: a stopped clock gives a precise reading, but is
rarely accurate. Factors that affect accuracy include how well our instruments are
calibrated (the correctness of the marked values) and how well the constants in our
calculations are known. Accuracy is affected by systematic errors, that is, mistakes that
are repeated with each measurement.
Example: Measuring from the end of a ruler where the zero position is 1 mm in from
the end.
Blunders - These are actual mistakes, such as reading an instrument pointer on the
wrong scale. They often show up when measurements are repeated and differences are
larger than the known uncertainty. For example: recording an 8 for a 3, or reading the
wrong scale on a meter..
Comparison - In order to confirm the physical principles we are learning, we calculate
the value of a constant whose value appears in our text. Since our calculated result has
an uncertainty, we will also calculate a Uncertainty Ratio, UR, which is defined as
UR = |experimental value − text value|
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Uncertainty
A value less than 1 indicates very good agreement, while values greater than 3 indicate
disagreement. Intermediate values need more examination. The uncertainty is not a
limit, but a measure of when the measured value begins to be less likely. There is
always some chance that the many effects that cause the variability will all affect the
measurement in the same way.
Example: Do the values 900 and 980 agree?
If the uncertainty is 100 , then U R = 80/100 = 0.8 and they agree,
but if the uncertainty is 20 then U R = 80/20 = 4 and they do not agree.
Graphical Representation of Data
Graphs are an important technique for presenting scientific data. Graphs can be used to
suggest physical relationships, compare relationships with data, and determine
parameters such as the slope of a straight line.
There is a specific sequence of steps to follow in preparing a graph. (See Figure 1 )
1. Arrange the data to be plotted in a table.
2. Decide which quantity is to be plotted on the x-axis (the abscissa), usually the
independent variable, and which on the y-axis (the ordinate), usually the dependent
variable.
3. Decide whether or not the origin is to appear on the graph. Some uses of graphs
require the origin to appear, even though it is not actually part of the data, for example,
if an intercept is to be determined.
4. Choose a scale for each axis, that is, how many units on each axis represent a
convenient number of the units of the variable represented on that axis. (Example: 5
divisions = 25 cm)
Scales should be chosen so that the data span almost all of the graph paper, and also
make it easy to locate arbitrary quantities on the graph. (Example: 5 divisions = 23 cm
is a poor choice.) Label the major divisions on each axis.
5. Write a label in the margin next to each axis which indicates the quantity being
represented and its units.Write a label in the margin at the top of the graph that indicates
the nature of the graph, and the date the data were collected. (Example: "Air track:
Acceleration vs. Number of blocks,
12/13/05")
6. Plot each point. The recommended style is a dot surrounded by a small circle. A
small cross or plus sign may also be used.
7. Draw a smooth curve that comes reasonably close to all of the points. Whenever
possible we plot the data or simple functions of the data so that a straight line is
expected. A transparent ruler or the edge of a clear plastic sheet can be used to "eyeball"
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a reasonable fitting straight line, with equal numbers of points on each side of the line.
Draw a single line all the way across the page. Do not simply connect the dots.
8. If the slope of the line is to be determined, choose two points on the line whose
values are easily read and that span almost the full width of the graph. These points
should not be original data points. Remember that the slope has units that are the ratio
of the units on the two axes.
9. The uncertainty of the slope may be estimated as the larger uncertainty of the
two end points,
𝑚 = 𝑠𝑙𝑜𝑝𝑒 =(𝑦2 − 𝑦1)
(𝑥2− 𝑥1)
13
General Graphic Types
I- Linear Graps:
y = mx + A
II-Power Graphs
y = K.xm
Logy = LogK + m.logx
III- Exponential Graphs
Y = K.amx
a = e specific case
y = K.emx
Lny = LnK + mx
x
y=mx - A
y=mx
y=mx + A
y
m < 0
y=k.xm
m>1
0<m<1
x
y
14
Testing Electrical Circuits with the M ultimeter
A multimeter is an electrical instrument capable of measuring voltage, current, and resistance. Digital multimeters have numerical displays, like digital clocks, for indicating the quantity of voltage, current, or resistance. Analog multimeters indicate these quantities by means of a moving pointer over a printed scale.
Analog multimeters tend to be less expensive than digital multimeters, and more beneficial as learning tools for the first time student of electricity.
Connect the black lead to the common (-) hole. This may well already be done, depending on the type of meter you use.
Connect the red lead to the volts (+) hole. Be careful, if this lead Is placed in the wrong hole (ohm or amp) it can cause permanent damage to the meter. This may well already be done, depending on the type of meter you use.
Turn the selector switch to AC volts (usually red). If there are Multiple selections, use the highest setting possible (if power unknown) or go one level higher than the estimated power.
Some digital meters have "Auto-range" and don't require any selection apart from AC volts.
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DIGITAL MULTIMETER
16
Multimeter as an Ampermeter
Turn Power Off before connecting multimeter.
Break Circuit.
Move multimeter leads (if needed).
Place multimeter in series with circuit.
Select highest current setting, turn power on, and work your way down.
Turn power off.
Disconnect multimeter.
Reconnect Circuit
Mutimeter as a Voltmeter
Select the DC or AC Volts
Start at the highest volts scale and work your way down.
Probe leads are connected in parallel.
Be very careful to not touch any other electronic components within the equipment
and do not touch the metal tips.
Mutimeter as an Ohmmeter
Resistance is the measure of electrical "friction" as electrons move through a
conductor. It is measured in the unit of the "Ohm," that unit symbolized by the capital Greek
letter omega (Ω).
Set your multimeter to the highest resistance range available. The resistance function
is usually denoted by the unit symbol for resistance: the Greek letter omega (Ω), or
sometimes by the word "ohms."
Touch the two test probes of your meter together. When you do, the meter should
register 0 ohms of resistance. If you are using an analog meter, you will notice the
needle deflect full-scale when the probes are touched together, and return to its resting
position when the probes are pulled apart.
It is essential that electricians be able to use clamp-on ammeters, or multi-meters, for in-place troubleshooting of motors and controllers as well as other circuits. Many clamp-on ammeters can be used to measure voltage and resistance as well as current. They come in various sizes with selections of scales that make them extremely versatile tools.
17
The resistance scale on an analog multimeter is reverse-printed from the other scales:
zero resistance in indicated at the far right-hand side of the scale, and infinite
resistance is indicated at the far left-hand side. There should also be a small
adjustment knob or "wheel" on the analog multimeter to calibrate it for "zero" ohms of
resistance. Touch the test probes together and move this adjustment until the needle
exactly points to zero at the right-hand end of the scale.
Digital multimeters set to the "resistance" mode indicate non-continuity by displaying
some non-numerical indication on the display. Some models say "OL" (Open-Loop),
while others display dashed lines. If you are using a digital multimeter, you should see
a numerical figure close to 10 shown on the display, with a small "k" symbol on the
right-hand side denoting the metric prefix for "kilo" (thousand). Some digital meters
are manually-ranged, and require appropriate range selection just as the analog meter.
If yours is like this, experiment with different range switch positions and see which
one gives you the best indication
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Experiment 1 : Ohm's Law and Not Ohm's Law
I. Purpose of this Experiment
The main purpose of this experiment is to review the measurement of voltage (V), current (I),
and resistance (R) in dc circuits. In the first part, you will measure the internal resistance of a
battery and examine the relationship between V and I in a resistor which obeys Ohm's law. In
second part of the lab, you will measure the resistance of some electrical devices that do not
obey Ohm’s law.
II. References
Halliday, Resnick and Krane, Physics, Vol. 2, 4th Ed., Chapters 32, 33
Purcell, Electricity and Magnetism, Chapter 4
Taylor, An Introduction to Error Analysis, Second Edition
II. Equipment
Digital multimeters
Resistor board with 4, 5 10, and 20 ohm resistors
6-volt battery
knife switch
10 V Power Supply
diode board with switching diodes, LED and 40 ohm resistor
III. Introduction
Voltage
When an electric charge moves between two points that have an electric potential
difference between them, work is done on the charge by the source that is creating the
potential. The amount of work that is done is equal to the decrease in the potential energy of
the charge. The difference in potential energy is equal to the product of the difference in the
electrical potential between the points and the magnitude of the electric charge. In the SI
system of units, the unit of electrical potential difference is the volt (written as V). For this
reason, almost everyone who work with circuits say "voltage difference" instead of "electrical
potential difference". Voltages are measured using a voltmeter. Voltage differences are
always measured between two points, with one lead of the voltmeter connected to one point
and the second lead connected to another point. On the other hand, diagrams of circuits almost
always show the voltage at individual locations in the circuit. If the voltage is given at one
point, then this means that the second point was at "ground" potential or "zero volts" and this
ground point is labeled on a circuit schematic using a special symbol.
Current
The rate at which charge passes through a surface is called the electrical current.
Current is measured in Ampères, commonly called amps, with units written as A. One amp
of current is defined as one Coulomb of charge passing through a cross-sectional area per
second. Since an electron has a charge of -1.609x10-19
C, This is equivalent to about 6×1018
electrons passing per second. Current is measured using an ampermeter which is placed in a
circuit so that the current flows into the positive terminal of the ammeter and out the negative
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terminal. Since the current flows through the ammeter, and we do not want the ammeter to
disrupt the current that is ordinarily flowing through a circuit, an ammeter has a low
resistance. Never connect an ammeter directly across a battery (or other voltage source), since
this will result in a large current flowing through the ammeter, possibly damaging it or the
battery. Note that in contrast a voltmeter has a high resistance.
Resistance and Ohm's Law
When current is driven through an ordinary electrically conducting material, such as a
metal or semiconductor at room temperature, it encounters resistance. You can think of
resistance as a sort of frictional drag. In a sample made of a good conductor, the current is
directly proportional to the potential difference, i.e.
VR
I1
This relationship is called Ohm's Law and is usually written:
V=IR
In this relationship, I is the current flowing through the sample. The potential difference V is
the difference in voltage between one end of the sample (where the current enters) to the other
end (where the current leaves). Finally, R is the resistance of the sample. In the SI system of
units, resistance has unit of ohms, which is written as .
In many materials the resistance does not change with the amount of voltage applied
or the current passing through it, over a large range of both parameters, so it is a constant to a
very good approximation. The resistors used in this lab are made of thin metal films or carbon
(a semiconductor). You should find that they obey Ohm’s Law very well. Metals are
examples of good conductors. They have a high density of electrons that are relatively free to
move around, so that connections made with metal tend to have a low resistance. In an
electrical insulator, the electrons are more tightly bound and cannot move freely. In a
semiconductor, most of the electrons are tightly bound, but there is a small fraction
(compared to a metal) that are free to conduct current. The small density of carriers in
semiconductors makes them more resistive than metals, and much more conducting than
insulators. It also gives them many other unusual properties, some of which we will see in this
lab.
Batteries and EMF
There are a variety of ways to generate a voltage difference. Batteries produce an
electrical potential difference through chemical reactions. If the plus (+) and minus (-) leads
of a battery are connected across a resistor, a current will flow out of the positive terminal of
the battery (which has a more positive potential than the negative terminal), through the
resistor and into the negative terminal. In other words, the positive current flows from the
positive to the negative terminal of the battery. Inside the battery, chemical reactions drive a
current flow from the more negative region to the more positive region. As a result, a battery
can be thought of as a charge pump that is trying to push positive charge out of the + terminal
and suck positive charge into the - terminal.
In physics and EE textbooks, one also encounters the terms electromotive force or
EMF. The term EMF comes from the idea that a force needs to be exerted on charges to
move them through a wire (to overcome the resistance of the piece of wire to the flow of the
current). The battery can be thought of as the source of this force. However, the EMF of a
battery is just the voltage difference generated across the terminals of the battery and is
measured in volts. So EMF is not actually a force, despite its name. In Physics 2, we will not
make distinctions between the EMF, the voltage difference, and the electrical potential
difference, but use these terms interchangeably.
20
Part of this experiment is to measure the EMF and internal resistance of a battery.
When a current flows inside a battery it is also encounters resistance and the battery is said to
have an internal resistance. Batteries with low internal resistance, such as the 12 V lead-acid
batteries commonly found in cars, can deliver a lot of current. They need to be treated with
caution; shorting together the terminals of a battery (or other voltage source) with a low
internal resistance could lead to melted wires, a fire, or the battery exploding. On the other
hand, batteries with high internal resistance cannot deliver much current and show significant
loss of voltage when current is supplied.
Electrical Symbols
Components used in electrical experiments have standard symbols. Those required in
this experiment are shown in Fig. (1.2). You should understand what each symbol represents
and use them when drawing schematics of your own circuits.
Figure 1.2. Some common symbols used in electrical circuits.
Figure 1.3 Simple circuit with a battery and two resistors showing direction of positive current
flow I.
Electrical Circuits
An electrical circuit is formed by using wires to connect together resistors, batteries,
switches, or other electrical components into one or more connected closed loops. Where
three or more wires meet, the current will split between the different paths. However each
new path for current flow that is created at these junctions must rejoin another channel at
some other point, so that all loops close. All loops that are created must be closed so that
current can flow.
electrical
ground
( V = 0 )
21
Kirchhoff's Rules
There are two very useful rules for analyzing electrical circuits and finding the
currents and voltages at different points in a circuit.
Rule 1: In going round a closed loop the total change in voltage must be zero.
Rule 2: The current flowing into any junction where wires meet is equal to the
current flowing out of the junction.
For example, applying the first rule to Fig.1.3 and assuming that the conductors
joining the components have zero resistance, we find the potential differences between the
lettered points in the circuit are given by:
FGEF
DECD
BCABBA
VV
IrVV
IRVVVV
0
0
0
Summing all the differences we get:
0 IrIRVAA
which can be rewritten:
)( rRI .
As another example, we can apply rule 2 to Fig. 1.3. Considering the nodes at points P
and Q in the circuit, we get
At P: Current in = I At Q: Current in = I1 and I2
Current out = I1 + I2 Current out = I
Both points yield the equation I = I1 + I2.
Figure I-3
Q P
22
Figure I-4: (a) Connecting resistors R1 and R2 in series produces a resistance21 RRR . (b)
Connecting resistors R1 and R2 in series produces a resistance21
21
RR
RRR
.
Series and Parallel Resistors
In Physics 174, you measured the resistance of two resistors when they were
connected in series (see Fig. 1.4 a) and in parallel (see Fig. 1.4 b). For the series connection,
one finds 21 RRR , i.e. the resistance adds. For the parallel connection, one finds
21
21
RR
RRR
.
These elementary results can be derived by applying Kirchoff's rules. For example,
consider the series connected resistors. Since current is conserved, the current I in R1 must be
the same as the current I in R2. Hence the voltage drop across R1 is 11 IRV and the voltage
drop across R2 is 22 IRV . Thus we can write )( 2121 RRIVV . This is equivalent to
writing IRV where 21 VVV and 21 RRR , i.e. two resistors connected in series
are equivalent to one resistor whose value is equal to their sum. This argument can be
generalized to n resistors in series, and one finds i iRR .
Next, consider the parallel connected resistors. The potential difference V between O
and O ’ must be the same whether we go along OABO ’ or OCDO ’. Also conservation of
current requires that 21 III , where:
11 RVI
is the current through R1 and
22 RVI
is the current through R2. Substituting these expressions for I1 and I2 into our equation for I
gives:
21
11
RRVI .
This is equivalent to writing IRV where )( 2121 RRRRR . This argument can be
generalized to n resistors connected in parallel and one finds
n
i iRR 1
11.
Diodes
Not everything obeys Ohm's law, i.e. current is not necessarily proportional to voltage.
In this lab you will also measure the characteristics of a common type of electrical device
(a) (b)
O'
O
23
called a diode. A diode consists of a junction of an “n-type” semiconductor and a "p-type"
semiconductor. The current in n-type semiconductors is carried by negative charges (the
electrons), while in p-type semiconductors the current is best thought of as being carried by
positive charges (called "holes" that are due to missing electrons). When n and p materials are
brought together, a few electrons will drift from n to p and some holes will drift from p to n.
This charge transfer between n and p regions generates an internal electrical potential at the
junction which opposes further transfer of electrons and holes between the two sides. It is
possible to drive current from p to n (i.e. holes from the p region to n and electrons from the n
region to the p) only if this potential “barrier” is overcome by applying a sufficiently large
voltage difference across the diode. For current to flow, the p region must be positive with
respect to the n. Applying a positive voltage to n and a negative voltage to p produces only a
very small “leakage current”. Thus the diode acts like a one-way valve with low resistance to
current flowing in the direction of the arrow, and high resistance to current flowing in the
opposite direction. If too much voltage is applied in either direction, the diode will be
destroyed.
The symbol for a diode is shown in Fig. 1.1 (a). The tip of the triangle points in the
direction that current can flow with low resistance. The characteristics of an IN914 switching
diode are shown in Table 1.1. This is one of the diodes that you can measure in the lab. A
light emitting diode (LED) has also been provided. In an LED, the current flow causes
emission of light with a fairly well-defined wavelength or color. LEDs are efficient, reliable
and long-lived, provided you don't apply too much voltage across them. A red, yellow or
green LED can typically withstand about 3 V and about 5V for a blue LED.
Figure 1.1: (a) Electrical symbol for a diode. (b) When Vb > Vg+Va, current flows through
the diode, from b (the p-type region or anode) to a (the n-type region or cathode). Here Vg is
the threshold voltage that needs to be reached before significant forward conduction occurs.
When Vb < Vg+Va the flow of current is blocked. In particular, when Vb<Va, the device is
said to be "reverse biased" and only a very small leakage current will flow. (c) Sketch of the
physical layout of an 1N914 switching diode. The dark black band is on the cathode.
b a
(a)
(c)
(b)
I
anode cathode
24
Table 1.1 Some electrical characteristics of the IN914 switching diode.
Peak Reverse Voltage 75 V
Average Forward Rectified Current 75 mA
Peak Surge Current, 1 Second 500 mA
Continuous Power Dissipation at 25°C 250 mW
Operating Temperature Range -65 to 175°C
Reverse Breakdown Voltage 100 V
Static Reverse Current 25 nA
Static Forward Voltage 1 V at 10 mA
Capacitance 4 pF
Typical threshold voltage 0.6 V
25
NOTES
26
Experiment 2 : Emf and Internal Resistance
Purpose: to find internal resistance of a battery.
Table1. Resistor Color codes
Teory: Real batteries are constructed from materials which possess non-zero resistivities. It
follows that real batteries are not just pure voltage sources. They also possess internal
resistances.
Incidentally, a pure voltage source is usually referred to as an emf (which stands for
electromotive force). Of course, emf is measured in units of volts. A battery can be modeled
as an emf connected in series with a resistor , which represents its internal resistance.
Suppose that such a battery is used to drive a current through an external load resistor ,
as shown in Fig. 1. Note that in circuit diagrams an emf is represented as two closely
spaced parallel lines of unequal length. The electric potential of the longer line is greater than
that of the shorter one by volts. A resistor is represented as a zig-zag line.
27
Figure 1: A battery of emf and internal resistance connected to a load resistor of
resistance .
Thus, the voltage of the battery is related to its emf and internal resistance via
R = V/I
(ohm)
R (ohm)
I (A)
V (volt)
28
Tablo 2:
R (color code)
()
R (multimeter)
()
%
Differenc
e
R
1
R
2
R
3
R
4
R
5
29
30
Experiment 3: Kirchoff’s Law
Objectives
To calculate expected voltages and currents for each component using Kirchhoff’s Laws.
To measure the actual voltage and current for each component.
To compare the expected and actual values of the voltages and currents
To determine if the circuit obeys Kirchhoff’s Laws.
Kirchhoff's current law (KCL)
This law is also called Kirchhoff's first law, Kirchhoff's point rule, or Kirchhoff's junction
rule (or nodal rule).
The principle of conservation of electric charge implies that:
At any node (junction) in an electrical circuit, the sum of currents flowing into that node is
equal to the sum of currents flowing out of that node or equivalently
The algebraic sum of currents in a network of conductors meeting at a point is zero.
Recalling that current is a signed (positive or negative) quantity reflecting direction towards
or away from a node, this principle can be stated as:
Kirchhoffs Voltage Law or KVL,
states that “in any closed loop network, the total voltage around the loop is equal to the sum of
all the voltage drops within the same loop” which is also equal to zero. In other words the
algebraic sum of all voltages within the loop must be equal to zero.
31
V1 V2 V3 V4 Vtotal
Circuit
EWB Sim.
32
I = E/Rtot I1 I2 I3 I4 I2+I3+I4
Circuit
EWB Sim.
33
NOTES
34
Experiment 4: Current-Balance Experiment (F = ILBsin)
INTRODUCTION
Procedure
If you're using a quadruple-beam balance:
Set up the apparatus as shown in figure 1.1.
Determine the mass of the magnet holder and magnets with no current flowing.
Record this value in the column under “Mass” in Table 1.1.
Set the current to 0.5 amp. Determine the new “mass” of the magnet assembly.
Record this value under “Mass” in Table 1.1.
Subtract the mass value with the current flowing from the value with no current
flowing. Record this difference as the “Force.”
Increase the current in 0.5 amp increments to a maximum of 5.0 amp, each
time repeating steps 2-4.
If you're using an electronic balance:
Set up the apparatus as shown in figure 1.1.
Place the magnet assembly on the pan of the balance. With no current flowing,
press the TARE button, bringing the reading to 0.00 grams.
Now turn the current on to 0.5 amp, and record the mass value in the “Force”
column of Table 1.1.
Increase the current in 0.5 amp increments to a maximum of 5.0 amp, each
time recording the new “Force” value.
Data Processing
Plot a graph of Force (vertical axis) versus Current (horizontal axis).
Analysis
What is the nature of the relationship between these two variables? What does this tell
us about how changes in the current will affect the force acting on a wire that is inside a
magnetic field?
35
36
Experiment 4/B: Force versus Length of Wire
Procedure
Set up the apparatus as in Figure 2.1.
Determine the length of the conductive foil on the Current Loop. Record this value
under “Length” in Table 2.1.
If you are using a quadruple-beam balance:
 With no current flowing, determine the mass of the Magnet Assembly. Record this
value on the line at the top of Table 2.1.
Set the current to 2.0 amps. Determine the new “mass” of the Magnet Assembly.
Record this value under “Mass” in Table 2.1.
Subtract the mass that you measured with no current flowing from the mass that you
measured with the current flowing. Record this difference as the “Force.”
Turn the current off. Remove the Current Loop and replace it with another. Repeat steps
2-5.
If you are using an electronic balance:
Place the magnet assembly on the pan of the balance. With no current flowing, press the
TARE button, bringing the reading to 0.00 grams.
Now turn the current on, and adjust it to 2.0 amps. Record the mass value in the “Force”
column of Table 2.1.
Turn the current off, remove the Current Loop, and replace it with another. Repeat steps
2-4.
Data Processing
Plot a graph of Force (vertical axis) versus Length (horizontal axis).
Analysis
What is the nature of the relationship between these two variables? What does this tell
us about how changes in the length of a current-carrying wire will affect the force that it
feels when it is in a magnetic field?
37
Mass with I=0; ________________
Length
L(cm)
Mass (g) Force F(N) Length
L(cm)
Mass (g) Force F(N)
38
39
Experiment 5: Transformer Basics I
Introduction
When an alternating current passes through a coil of wire, it produces an alternating
magnetic field. This is precisely the condition needed for the electromagnetic induction
to take place in a second coil of wire. In this lab you will investigate several of the
factors influencing the operation of a transformer.
Equipment Needed - Supplied
1. The four coils from the PASCO SF-8616 Basic Coils Set
2. The U-shaped Core from the PASCO SF-8616 Basic Coils Set
3. Optional: the additional coils from the PASCO SF-8617 Complete Coils Set
Equipment Needed - Not Supplied
1. Low voltage ac power supply 0-6 VAC, 0-1 amp such as PASCO Model SF-9582
2. AC voltmeter 0-6 VAC
3. Banana connecting leads for electrical connections
Procedure
1. Set up the coils and core as shown in Figure 1. In the diagram, the coil to the left will
be referred to as the primary coil, and the one to the right will be the secondary coil.
Note that we are putting in an alternating current to the primary at one voltage level, and
reading the output at the secondary.
2. With the 400-turn coil as the primary and the 400-turn coil as the secondary, adjust
the input voltage to 6 volts a.c. Measure the output voltage and record your results in
Table 1.1.
3. Repeat step 2 after inserting the straight cross piece from the top of the U-shaped
core. Record your results. (See Figure 2.)
4. Repeat step 2 after placing the coils on the sides of the open U-shaped core. Record
your results.
40
5. Finally, repeat step 2 after placing the cross piece over the U-shaped core. Record
your results.
6. Using the core configuration which gives the best output voltage compared to input
voltage, try all combinations of primary and secondary coils. Use a constant input
voltage of 6.0 volts a.c. Record your data in Table 1.2.
41
Analysis
1. Which core configuration gives the maximum transfer of electromagnetic effect to
the secondary coil? Develop a theory to explain the differences between configurations.
2. From your data in table 1.2, for a primary having a constant number of turns, graph
the resulting output voltage versus the number of turns in the secondary. What type of
mathematical relationship exists between numbers of turns of wire and the resulting
output voltage? Is the data ideal? Why or why not?
3. Consider further improvements to your transformer. What additional changes might
you make to increase the transfer from one coil to the other?
42
43
NOTES
44
45
Experiment 6: Ohm’s Law, RC and RL Circuits in DC
OBJECTIVES 1. To explore the measurement of voltage & current in circuits
2. To see Ohm’s law in action for resistors
3. To explore the time dependent behavior of RC and RL Circuits
PRE-LAB READING
INTRODUCTION When a battery is connected to a circuit consisting of wires and other circuit elements
like resistors and capacitors, voltages can develop across those elements and currents
can flow through them. In this lab we will investigate three types of circuits: those with
only resistors in them and those with resistors and either capacitors (RC circuits) or
inductors (RL circuits). We will confirm that there is a linear relationship between
current through and potential difference across resistors (Ohm’s law: V = IR). We will
also measure the very different relationship between current and voltage in a capacitor
and an inductor, and study the time dependent behavior of RC and RL circuits.
The Details: Measuring Voltage and Current Imagine you wish to measure the voltage drop across and current through a resistor in a
circuit. To do so, you would use a voltmeter and an ammeter – similar devices that
measure the amount of current flowing in one lead, through the device, and out the other
lead. But they have an important difference. An ammeter has a very low resistance, so
when placed in series with the resistor, the current measured is not significantly affected
(Fig. 1a). A voltmeter, on the other hand, has a very high resistance, so when placed in
parallel with the resistor (thus seeing the same voltage drop) it will draw only a very
small amount of current (which it can convert to voltage using Ohm’s Law VR
= Vmeter
=
Imeter
Rmeter
), and again will not appreciably change the circuit (Fig. 1b).
Figure 1: Measuring current and voltage in a simple circuit. To measure current
through the resistor (a) the ammeter is placed in series with it. To measure the voltage
drop across the resistor (b) the voltmeter is placed in parallel with it.
The Details: Capacitors Capacitors store charge, and develop a voltage drop V across them proportional to the
amount of charge Q that they have stored: V = Q/C. The constant of proportionality C is
the capacitance (measured in Farads = Coulombs/Volt), and determines how easily the
capacitor can store charge. Typical circuit capacitors range from picofarads (1 pF = 10-
12
F) to millifarads (1 mF = 10-3
F). In this lab we will use microfarad capacitors (1 μF =
10-6
F).
RC Circuits
46
Consider the circuit shown in Figure 2. The capacitor (initially uncharged) is connected
to a voltage source of constant emf . At t = 0, the switch S is closed.
Figure 2 (a) RC circuit (b) Circuit diagram for t > 0
In class we derived expressions for the time-dependent charge on, voltage across, and
current through the capacitor, but even without solving differential equations a little
thought should allow us to get a good idea of what happens. Initially the capacitor is
uncharged and hence has no voltage drop across it (it acts like a wire or “short circuit”).
This means that the full voltage rise of the battery is dropped across the resistor, and
hence current must be flowing in the circuit (VR
= IR). As time goes on, this current will
“charge up” the capacitor – the charge on it and the voltage drop across it will increase,
and hence the voltage drop across the resistor and the current in the circuit will
decrease. This idea is captured in the graphs of Fig. 3.
Figure 3 (a) Voltage across and charge on the capacitor increase as a function of time
while (b) the voltage across the resistor and hence current in the circuit decrease.
After the capacitor is “fully charged,” with its voltage essentially equal to the voltage of
the battery, the capacitor acts like a break in the wire or “open circuit,” and the current
is essentially zero. Now we “shut off” the battery (replace it with a wire). The capacitor
will then release its charge, driving current through the circuit. In this case, the voltage
across the capacitor and across the resistor are equal, and hence charge, voltage and
current all do the same thing, decreasing with time. As you saw in class, this decay is
exponential, characterized by a time constant t, as pictured in fig. 4.
47
Figure 4 Once (a) the battery is “turned off,” the voltages across the capacitor and
resistor, and hence the charge on the capacitor and current in the circuit all (b) decay
exponentially. The time constant τ is how long it takes for a value to drop by e.
The Details: Inductors Inductors store energy in the form of an internal magnetic field, and find their behavior
dominated by Faraday’s Law. In any circuit in which they are placed they create an
EMF ε proportional to the time rate of change of current I through them: ε = L dI/dt.
The constant of proportionality L is the inductance (measured in Henries = Ohm s), and
determines how strongly the inductor reacts to current changes (and how large a self
energy it contains for a given current). Typical circuit inductors range from nanohenries
to hundreds of millihenries. The direction of the induced EMF can be determined by
Lenz’s Law: it will always oppose the change (inductors try to keep the current
constant)
RL Circuits If we replace the capacitor of figure 2 with an inductor we arrive at figure 5. The
inductor is connected to a voltage source of constant emf . At t = 0, the switch S is
closed.
Figure 5 RL circuit. For t<0 the switch S is open and no current flows in the circuit. At
t=0 the switch is closed and current I can begin to flow, as indicated by the arrow.
As we saw in class, before the switch is closed there is no current in the circuit. When
the switch is closed the inductor wants to keep the same current as an instant ago –
none. Thus it will set up an EMF that opposes the current flow. At first the EMF is
identical to that of the battery (but in the opposite direction) and no current will flow.
Then, as time passes, the inductor will gradually relent and current will begin to flow.
After a long time a constant current (I = V/R) will flow through the inductor, and it will
be content (no changing current means no changing B field means no changing
magnetic flux means no EMF). The resulting EMF and current are pictured in Fig. 6.
48
Figure 6 (a) “EMF generated by the inductor” decreases with time (this is what a
voltmeter hooked in parallel with the inductor would show) (b) the current and hence
the voltage across the resistor increase with time, as the inductor ‘relaxes.’
After the inductor is “fully charged,” with the current essentially constant, we can shut
off the battery (replace it with a wire). Without an inductor in the circuit the current
would instantly drop to zero, but the inductor does not want this rapid change, and
hence generates an EMF that will, for a moment, keep the current exactly the same as it
was before the battery was shut off. In this case, the EMF generated by the inductor and
voltage across the resistor are equal, and hence EMF, voltage and current all do the
same thing, decreasing exponentially with time as pictured in fig. 7.
Figure 7 Once (a) the battery is turned off, the EMF induced by the inductor and hence
the voltage across the resistor and current in the circuit all (b) decay exponentially. The
time constant τ is how long it takes for a value to drop by e.
The Details: Non-Ideal Inductors So far we have always assumed that circuit elements are ideal, for example, that
inductors only have inductance and not capacitance or resistance. This is generally a
decent assumption, but in reality no circuit element is truly ideal, and today we will
need to consider this. In particular, today’s “inductor” has both inductance and
resistance (real inductor = ideal inductor in series with resistor). Although there is no
way to physically separate the inductor from the resistor in this circuit element, with a
little thought (which you will do in the pre-lab) you will be able to measure both the
resistance and inductance.
APPARATUS
1. Science Workshop 750 Interface In this lab we will again use the Science Workshop 750 interface to create a “variable
battery” which we can turn on and off, whose voltage we can change and whose current
we can measure.
49
2. AC/DC Electronics Lab Circuit Board
We will also use, for the first of several times, the circuit board pictured in Fig. 8. This
is a general purpose board, with (A) battery holders, (B) light bulbs, (C) a push button
switch, (D) a variable resistor called a potentiometer, and (E) an inductor. It also has (F)
a set of 8 isolated pads with spring connectors that circuit components like resistors and
capacitors can easily be pushed into. Each pad has two spring connectors connected by
a wire (as indicated by the white lines). The right-most pads also have banana plug
receptacles, which we will use to connect to the output of the 750.
Figure 8 The AC/DC Electronics Lab Circuit Board, with (A) Battery holders, (B) light
bulbs, (C) push button switch, (D) potentiometer, (E) inductor and (F) connector pads
4. Resistors & Capacitors We will work with resistors and capacitors in this lab. Resistors (Fig. 8a) have color
bands that indicate their value (see appendix A if you are interested in learning to read
this code), whereas capacitors (Fig. 8b) are typically stamped with a numerical value.
Figure 10 Examples of a (a) resistor and (b) capacitor. Aside from their size, most
resistors look the same, with 4 or 5 colored bands indicating the resistance. Capacitors
on the other hand come in a wide variety of packages and are typically stamped both
with their capacitance and with a maximum working voltage.
GENERALIZED PROCEDURE This lab consists of five main parts. In each you will set up a circuit and measure
voltage and current while the battery periodically turns on and off. In the last two parts
you are encouraged to develop your own methodology for measuring the resistance and
inductance of the coil on the AC/DC Electronics Lab Circuit Board both with and
without a core inserted. The core is a metal cylinder which is designed to slide into the
coil and affect its properties in some way that you will measure.
Part 1: Measure Voltage Across & Current Through a Resistor
50
Here you will measure the voltage drop across and current through a single resistor
attached to the output of the 750.
Part 2: Resistors in Parallel Now attach a second resistor in parallel to the first and see what happens to the voltage
drop across and current through the first.
Part 3: Measuring Voltage and Current in an RC Circuit In this part you will create a series RC (resistor/capacitor) circuit with the battery
turning on and off so that the capacitor charges then discharges. You will measure the
time constant in two different ways (see Pre-Lab #5) and use this measurement to
determine the capacitance of the capacitor.
Part 4: Measure Resistance and Inductance Without a Core The battery will alternately turn on and turn off. You will need to hook up this source to
the coil and, by measuring the voltage supplied by and current through the battery,
determine the resistance and inductance of the coil.
Part 5: Measure Resistance and Inductance With a Core In this section you will insert a core into the coil and repeat your measurements from
part 1 (or choose a different way to make the measurements).
Experiment 18/2: Ohm’s Law, RC and RL Circuits
Answer these questions on a separate sheet of paper and turn them in before the
lab
1. Measuring Voltage and Current In Part 1 of this experiment you will measure the potential drop across and current
through a single resistor attached to the “variable battery.” On a diagram similar to the
one below, indicate where you will attach the leads to the resistor, the battery, the
voltage sensor , and the current sensor . For the battery and sensors make sure that you
indicate which color lead goes where, using the convention that red is “high” (or the
positive input) and black is “ground.” Reread the pre-lab description of this board
carefully to understand the various parts. When you draw a resistor or other circuit
element it should go between two pads (dark green areas) with each end touching one of
the spring clips (the metal coils). Do NOT just draw a typical circuit diagram. You need
to think about how you will actually wire this board during the lab. RECALL: ammeters
must be in series with the element they are measuring current through, while voltmeters
must be in parallel.
2. Resistors in Parallel In Part 2 you will add a second resistor in parallel with the first. Show where you would
attach this second resistor in the diagram you drew for question 1, making sure that the
ammeter continues to measure the current through the first resistor and the voltmeter
measures the voltage across the first resistor.
3. Measuring the Time Constant τ
As you have seen, current always decays exponentially in RC circuits with a time
constant τ: I = I0
exp(-t/τ).
51
We will measure this time constant in two different ways.
(a) After measuring the current as a function of time we choose two points on the curve
(t1,I
1) and (t
2, I
2). What relationship must we choose between I
2 and I
1 in order to
determine the time constant by subtraction: τ = t2 – t
1? Should we be able to find a t
2 that
satisfies this for any choice of t1?
(b) We can also plot the natural log of the current vs. time, as shown at right. If we fit a
line to this curve we will obtain a slope m and a y-intercept b. From these fitting
parameters, how can we calculate the time constant?
(c) Which of these two methods is more likely to help us obtain an accurate
measurement of the time constant? Why?
Part 3: Measuring Voltage and Current in an RC Circuit
3A: Using a Single Resistor
1. Create a circuit with the first resistor and the capacitor in series with the battery
2. Connect the voltage sensor (still in channel A) across the capacitor
3. Record the voltage across the capacitor V and the current sourced by the battery I (Press
the green “Go” button above the graph). During this time the battery will switch
between putting out 1 Volt and 0 Volts.
Question 4:
Using the two-point method (which you calculated in Pre-Lab #3a), what is the time
constant of this circuit? Using this time constant, the resistance you measured in
Question 1 and the typical expression for an RC time constant, what is the capacitance
of the capacitor?
52
t(s)
0 10 20 30 40 50 60 70 80 90
V1
(volt)
V2
(volt)
V3
(volt)
53
54
EXPERIMENT 7. RL, RC, RLC Circuts (AC analysis)
INTRODUCTION
In this experiment the impedance Z, inductance L and capacitance C in alternating current
circuits will be studied.
The parameters of the circuit will be varied to produce the condition called resonance.
The inductive and capacitive reactance are defined as follows:
Inductive Reactance = XL = 2pifL
Capacitive Reactance = XC = fC2
1
.
The impedance in a series AC
circuit is found by adding the
individual reactances and
resistance as vectors as shown
in Figure 1.
Z
R
XL
XC
XL XC
Fig. 1
Voltages are all equal to the
current I, times the individual
or combined reactances. They
can be calculated from a
diagram which has the same
form as that shown in Figure
2.
Vtotal = IZ
VR = IR
VL = IXL
VC = IXC
VL VC
55
Fig. 2
As the frequency is varied from low to high, a minimum value of total impedance Z is found
when XL = XC, or f = LC2
1
. The value of Z at this resonance frequency is Z = R. If the
applied voltage is kept constant, then when Z is a minimum, I will be at a maximum, so both
Z and I have the general form as shown in Figure 3.
Z
f
I
f
Fig. 3
The width of the curves in the
above is of great importance in
such devices as radio and TV
receivers (we only want one
channel at a time), and is
measured by the ratio of the
width to the center frequency, as
shown in Figure 4.
Small R 0.707 Imax
Larger R
I
f1 fo f2 f
Imax
Fig. 4
56
When the peak is narrow, the
circuit is said to have a high Q,
where the quality Q is defined as:
Q = 12
o
ff
f
. A high Q corresponds
to a small value of the total series
resistance (coil resistance plus any
other resistance). Q can also be
shown to be given by Q = R
XL ,
where XL = oL, with fo being
the resonant frequency. Figure 5
indicates the relationship between
Z, R, XL and XC as f varies from
f1 to fo to f2.
45o
45o
Z for f = f2
Z for f = f1
Z for f = fo (Z = R)
XC XL = R
Locus of points as f varies
XL XC = R
Fig. 5
L (H)
R
(ohm)
XL
teoric VR (volt) I=VR/R
Z
den=Vtop/I Z teo=22
RX L
1
2
3
57
C
(farad) R (ohm)
Xc
teoric VR (volt) I=VR/R Z den=Vtop/I
Z teo=22
RX C
1
2
3
58