Physics 241 Lab: RC Circuits
–
DC Source
http://bohr.physics.arizona.edu/~leone/ua/ua_spring_2010/phys241lab.html
Name:____________________________
Section 1
:
1.1.
To
day you will investigate two similar RC circuits. The first circuit is the
charging up
the
capacitor circuit. In this circuit (shown below) the capacitor begins without any charge on it and is
wired in series with a resistor and a constant voltage source
. The voltage source begins charging the
capacitor until the capacitor is fully charged. The
charging up equation
that describes the time
dependence of the charge on the capacitor is
. The final charge on the
capacitor, Q
max
is det
ermined by the internal structure of the capacitor (i.e. its capacitance):
.
Use a graphing calculator (or mad graphing skills) and make a quick sketch of
on the
axes. Assume that the source voltage is 9 V, t
he resistance is 1.0x10
3
Ω
and the capacitance is 1.0x10

3
F. The amount of time that equals the resistance times the capacitance is called the time constant:
. Create your sketch so that Q(t=
τ
) is sketched above the delineated tic
mark.
Your sketch
below:
1.2.
The second circuit is the
discharging
the capacitor circuit. In this circuit (shown below) the
capacitor begins with some initial charge and is wired in series with a resistor. The capacitor begins
discharging through th
e resistor until no charge remains on the capacitor plates. The
discharging
equation
that describes the time dependence of the charge on the capacitor is
.
(Also think of a switch:
Use a graphing calculator (or mad graphing ski
lls) and sketch a graph of
on the axes
below. Assume that the resistance is 1.0x10
3
Ω
and the capacitance is 1.0x10

3
F. Find the initial
charge on the capacitor by assuming the capacitor had been charged to 9 volts by a battery be
fore
being discharged through the resistor. Create your sketch so that Q(
τ
) is sketched above the delineated
tic mark. Be sure to include charge values along the y

axis.
Your sketch below:
What is the decimal value of e

1
to 3 decimal places? _______
__ Engineers usually approximate this
number as 1/3 (.333) in order to think quickly about exponential decay. For example, if you plug in
t=
τ
(one decay time constant), the amount of charge left on the capacitor has decayed to approximately
1/3 of its in
itial value. Approximately how much of the initial charge is left on the capacitor after the
circuit has operated for t=3
τ
seconds?
Your work and answer:
1.3.
Now examine the time dependence of the voltage across the capacitor for the same dischargin
g
capacitor in part c. As the charge on the capacitor changes, the voltage difference across the capacitor
plates also changes. In fact, the definition of capacitance easily relates
by a
constant:
. Therefore
, the equation describing the time dependent decay of the voltage
across the capacitor is simply
, where
.
You will experimentally test this
equation later in this lab.
Sketch a graph of
on the axes below using your answer to the
previous question (graph of Q
CAP
). Be sure to include voltage values along the y

axis.
Your sketch
below:
As the capacitor discharges, it causes a current to flow through the resistor. Because energy must be
conserved, the magnitude of the voltage across the resistor is the same as the voltage across the
capacitor (they are the only circuit components!). Because the resistor is Ohmic, the current through
the resistor can be related to its voltage and resista
nce. This gives a time dependent equation for the
current through the resistor of
. You should notice that the time dependence of the
charge on the capacitor
, the
voltage across the capacitor
, and the
current through the resistor
al
l
exhibit the
same exponential decay function
, and are simply related to each other using properties you
already know. Relate this equation for resistor current to the others by using Ohm’s law to determine
I
o
in terms of
R
,
C
and
Q
o
.
Your work and answe
r:
Section 2
:
2.1.
A
differential equation
is an equation that involves derivatives. Most all equations designed to
model reality in the physical sciences make use of differential equations so a good working knowledge
of this type of mathematics is es
sential to any working physical scientist or engineer. The following
table compares an algebraic equations to a differential equations using two examples:
Examine the differential equation
. One solution to this differential equa
tion is
. Check the solution by plugging it into the differential equation to see if it works.
Your work and answer:
2.2.
When analyzing circuits, you often must write a differential equation describing the behavior of
t
he circuit. This is most easily done by using conservation of energy to write a voltage equation. Then
use fundamental concepts to relate voltage to charge on the capacitor to create a differential equation
for Q(t). Examine the discharging circuit for
today’s lab and the construction of the differential
equation that describes it:
DISCHARGING WITHOUT SOURCE
SOLUTION:
Check the solution function by substituting it into
both sides
of the differential eq
uation for Q
cap
(t).
Differentiate where appropriate to prove the left

hand side of the equation equals the right

hand side
with the solution substituted in for Q
cap
(t).
Your work and answer:
What trivially happens to the initial charge Q
o
in
this checking process? Your answer to this should
allow you to see that the initial amount of charge on the capacitor Q
o
is not determined by the
differential equation. Basically you choose the initial amount of charge to put on the capacitor plates
and
the differential equations determines how quickly that charge discharges through the resistor.
Your answer:
2.3.
The other circuit for today’s lab, charging the capacitor, also has a differential equation to
describe its behavior in time:
CHARGING WITH CO
NSTANT SOURCE
SOLUTION:
(Not a question)
Section 3
:
3.1.
A large capacitance and large resistance translate into a slow time constant so that you may
easily measure the rate of decay with a stopwatch. Y
ou are supplied with a 1000
F electrolytic
capacitor. Electrolytic capacitors are “one

way” capacitors.
Be careful to only apply voltage
correctly to the electrolytic capacitor or you will damage it (the negative terminal is marked on
the capacitor).
Y
ou will discharge your capacitor in an RC circuit with approximately 10 k
Ω
Remember
.
What time constant should you expect with R = 10 k
Ω
and C = 1000
F?
Your work and answer:
Since approximately 4 time constants allows the circuit to discharge to 2% of its initial value, how long
should you measur
e the decay of the capacitor’s charge?
Your answer:
3.2.
Charge an electrolytic
capacitor without resistance
in the correct direction
using the 9

Volt
battery (this happens quickly since there is very little resistance). Then switch to discharge the
c
apacitor through a ~
10 k
Ω
resistor (if the resistance is too small, the capacitor will discharge too
rapidly to measure). Collect (voltage, time) data by having the DMM measure voltage across the
capacitor while it discharges through the resistor using a
stopwatch. You should collect more data at
the beginning when there is rapid voltage change.
[Collect your data now.]
Make a “raw graph” of your data by plotting V
cap
(t) vs. t.
[Make your graph now.]
Next you will linearize your data by taking the na
tural logarithm of your voltages. Since
, taking the natural logarithm of the function cancels the exponential:
.
The function
is the equation of a line with slope

1/(RC) and y

intercept
ln(V
o
).
Thus, if you make the graph of
on regular (Cartesian) graph paper, you will obtain a
line with a slope equal to

1/RC if your data is exponentially related.
Graph your linearized data by
taking the natural logarithm of your
measured voltages ln(V) and plot these vs. t on regular graph
paper. This should give you a line with slope equal to

1/RC.
Make your graph now. Then find C
from the slope and R:
Section 4
:
The following picture shows the digital oscillosc
ope and labels its most common features.
You now need to practice using the digital oscilloscope so that you are prepared to make
measurements with it. Keep in mind that the oscilloscope is simply a tool that allows you to analyze
the details of a rap
idly changing voltage. With that in mind, you will now practice the more common
measurements that are made as well as their uses.
Create a sinusoidal wave with your function generator with a
very
small
voltage (i.e. use a
special feature of the function g
enerator and a frequency in the 1

100 kHz range. Also add a small
negative
DC offset. Use the autoset button to quickly get your signal on the screen so you can adjust
your function generator DC offset correctly. Be sure that your channel is on “1x prob
e” and that your
trigger is set to the correct source.
Do this now.
Input the sine wave voltage source on channel 1 and determine the
average voltage
of your sine wave.
Use the measurement feature set to measure the mean
Your observations:
Get the d
igital oscilloscope to tell you on its screen the wave’s period and frequency. Use the
measurement feature set to measure the period.
Your observations:
Get the digital oscilloscope to tell you on its screen the wave’s amplitude. Use the measurement
feature set to measure the peak

to

peak voltage.
Your observations:
Adjust the amplitude of the wave on the function generator until you see that the wave spends more of
its time being negative than positive (with some positive). This will change y
our answer to the last
question
(see this)
. Use a two cursor measurement
of time
and get the oscilloscope to tell you on its
screen how much time the sine wave spends being positive. Then do the same thing to find out how
much time the wave spends being
negative.
Your observations and answers:
Now use a two

cursor measurement
in voltage
and get the oscilloscope to tell you on its screen the
voltage drop of the wave from its maximum positive value to zero.
Your observations:
Change your sin
e wave to a triangle wave of 500,000 Hz and use the DC offset so that the minimum of
the triangle wave is zero volts. Examine a part of the triangle wave that is decreasing. Use a two
cursor measurement
in time
and space
to find how long it takes for the
triangle wave to decrease from
its highest value to one half of that value.
Your observations and answer:
Section 5
:
5.1.
Most digital electronics make extensive use of capacitors. However, the decay rates are
typically much too rapid to measure w
ith a DMM. In this part of the lab you will create an RC circuit
using a 0.1
F capacitor and a 1 k
Ω
resistor and you will rapidly charge and discharge the capacitor
with an oscillating square wave. What time constant
τ
will this produce?
Your answer:
You should choose a frequency of 1/(20
τ
) Hz so that there is plenty of time for the capacitor to
discharge fully. What is this frequency?
Your answer:
Use your function generator to create a
square wave
with a voltage alternating between V
MIN
= 0
Vo
lts and V
MAX
= 3 volts and the correct frequency. Do this by 1
st
setting the frequency. Then set the
wave to oscillate between +1.5 V and

1.5V and use the DC offset to shift your signal to have V
MIN
= 0
volts. The voltage across the capacitor should l
ook like shark fins on your oscilloscope as the
capacitor exponentially charges and then discharges.
Do this now.
5.2.
Now set up an RC circuit with
R=1 k
Ω
and
C=0.1
F
in series with the same square wave
from the previous question. Then answer the f
ollowing questions:
During the time interval that the square wave is at +3 volts, is the capacitor being charged or
discharged?
Your answer:
During the time interval that the square wave is at +0 Volts, is the capacitor being charged or
discharged?
Your answer:
The following is an important reminder that you won’t need in today’s lab. Many times you may need
to find the current in a circuit. Which component must you measure if you want to determine the
current of the circuit and why?
Your ans
wer and explanation:
Observe the voltage across the capacitor and the total circuit voltage simultaneously using a bottom
ground configuration. You should see the “shark fin” pattern that is modulated by the alternating
square wave source voltage (turning
on then off). Use a double cursor measurement to find the time it
takes for your charged capacitor to decrease by half.
Your observation:
When a physical quantity decays exponentially, the time it takes for it to decay to ½ its original value
is call
ed the half

life t
½
. Solve the
half

life equation
for t
½
to find what t
½
should be in this circuit in
terms of R and C:
.
Your work and answer:
Combine the results of the previous questions and calculate the experimentally
determined capacitance
C of your capacitor using your half

life measurement.
Your work and answer:
Now use the double cursor method to find the time it takes for your capacitor to discharge from ½ of
its initial value to ¼ of its initial value.
Your
observation:
The decaying exponential function has the unique property that each consecutive halving of its value
occurs in the same amount of time. Using this knowledge, predict how long it should take for your
capacitor to discharge to 1/128 of its i
nitial value.
Your work and answer:
Now use the cursors to collect voltage vs. time data for your
decaying
capacitor. Then linearize your
data, graph it on regular graph paper, and compute to the capacitance C from the slope. Be sure your
value i
s close to the labeled value.
Collect your data and make your graph now. Then show your
work and answer for C below:
Report Guidelines:
Write a separate section using the labels and instructions provided below. You
may add diagrams and equation
s by hand to your final printout. However, images, text or equations
plagiarized from the internet are not allowed!
Title
–
A catchy title worth zero points so make it fun.
Goals
–
Write a 3

4 sentence paragraph stating the experimental goals of the lab
(the big
picture). Do NOT state the learning goals (keep it scientific). [~1

point]
Concepts & Equations
–
[~5

points] Be sure to write a separate paragraph to explain each of
the following concepts.
o
Describe the operation and features of the digital o
scilloscope.
o
Discuss what differential equations are and how to check their solutions. You should
give an example (keep it simple).
o
Discuss the construction process of the two differential equations that model RC
circuits with DC source. Discuss the solu
tions to these differential equations.
o
Discuss all you know about capacitors (should be a lot from lecture).
o
Discuss how to find the capacitance of a capacitor using an RC circuit with a DC
source.
o
Discuss how to determine if data has an exponential relati
onship.
o
Discuss what the half

life is of an exponential relationship is and how it works.
Procedure & Results
–
Write a 2

4 sentence paragraph for each section of the lab describing
what you did and what you found. Save any interpretation of your results
for the conclusion
.
[~4

points]
Conclusion
–
Write at least three paragraphs where you analyze and interpret the results you
observed or measured based upon your previous discussion of concepts and equations. It is all
right to sound repetitive since it
is important to get your scientific points across to your reader.
Write a separate paragraph analyzing and interpreting your results from your open

ended
experiment. Do NOT write personal statements or feeling about the learning process (keep it
scientif
ic). [~5

points]
Graphs
–
All graphs must be neatly hand

drawn during class, fill an entire sheet of graph
paper, include a title, labeled axes, units on the axes, and the calculated line of best fit if
applicable. [~5

points]
o
The two graphs from secti
on 3.
o
The graph from section 5.
Worksheet
–
thoroughly completed in class and signed by your TA. [~5

points.]
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