Description
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RC Circuits
Purpose
a. To study the transient behavior of voltage and current in RC circuits.
b. To measure an RC circuit time constant (), = RC.
c. To determine the capacitance of an unknown capacitor from the time constant.
d. To verify the equivalent capacitance of capacitors in parallel and series combinations.
Theory
The voltage drop across a capacitor of capacitance C is given by
VC
Q
C
(1)
The voltage drop across a resistor of resistance R is given by
VR I R
(2)
In this laboratory you will measure the voltage across the resistor and capacitor in two
situations (1) when the capacitor is being charged
and (2) when the capacitor is being discharged.
Figure 1 shows the circuit of charging and
discharging of a capacitor C. When the single pole
double throw switch (SPDT) is toggled to position
1, it is a charging circuit as it completes the circuit
with the battery, resistor and capacitor. When the
switch is toggled to position 2, it disconnects with
battery and makes a discharging circuit.
Draw separate circuit diagrams for
charging and discharging processes on your note
book. Indicate clearly the direction of current and
the polarity of voltage across the capacitor.
Figure 1: Circuit diagram for charging
and discharging of a capacitor.
Charging
Referring to the Fig. 1, when the capacitor of capacitance C is being charged by the power
supply of emf, , through the resistance R, the voltages around the loop satisfy:
VC (t ) VR (t ) 0
VC (t ) R I (t ) 0
(3)
The solutions to the above expression for VC(t) and I(t) are the time-dependent (transient)
behavior of voltage across the capacitor and current in the circuit. If the capacitor is uncharged in the
beginning, the solutions to the above expression for VC(t) and I(t) given by
VC (t ) [1 e t / ]
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(Inverse exponent)
(4)
1
I (t )
[e t / ] I [e t / ]
0
R
(Natural exponent)
(5)
where is called the time constant of the circuit and given by
= RC
(6)
and Io = /R is the maximum current in the circuit. When the capacitor is fully charged, the charge on
it Q = C/.
Discharging
When the capacitor is being discharged through the resistance R, by toggling the SPDT switch
to position 2, the transient behavior of VC (t) and I(t) are given by
VC (t ) V0 [e t / ]
I ( t ) I 0 [e
t /
(7)
(8)
]
In the above expressions Vo= Q/C, and = RC.
Apparatus
Two capacitors, power supply, decade resistor box, wires, differential voltage probe, current probe,
LabQuest interface, and a single pole double throw switch (SPDT).
Note that the capacitors are polarity sensitive and you should wire the capacitors carefully (if you are
unsure about how to wire the capacitors ask your instructor for help).
Description of Apparatus
In order to realize the circuit shown in the Fig. 1, we will use a power supply instead of the
battery. We will use a differential voltage probe for voltage measurement and current probe for current
measurement. We will use a decade resistance box so that we can vary the resistance in the circuit.
Schematic of the circuit connection that you will construct will look like in Fig. 2.
Decade
resistance box
LabQuest
red
Current probe
Power supply
Voltage
probe
Charge
Discharge
SPDT switch
black
Fig. 2. Apparatus and schematic of circuit connection.
Procedure
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1. Make sure the power supply off. Wire the circuit as shown in the figure above using one of the
capacitors given. Note that the capacitor has a polarity – connect it properly! Connect voltage
probe across the capacitor and connect the current probe in series with the resistor and capacitor
using the proper polarities. The sensors should be connected to a LabQuest which is then
connected to a computer.
2. Open Logger Pro in the computer. Before taking the measurements, zero the sensors by
clicking
on menu bar (or from set up sensor) with the power supply off. Set the sampling
rate to 100/ sec {Experiment Data collection Sampleing Rate = 100 Done}. Now slowly
rotate the voltage and current knobs and set the voltage to 5 V.
3. Make sure that the capacitor is initially discharged by placing the switch in the discharge
position, with R set to zero. Then set the resistance, using the decade box, to a suitable value
(compute an approximate value of R to give τ ~ 0.5 sec.). Note down the value of resistance in
table 1.
4. Now, click “Collect” on the computer screen and then, shortly after the data starts collecting,
throw the switch to the CHARGE position. Wait until the capacitor is fully charged and throw
the switch back to the DISCHARGE position. You will now have, in one graph, both the
charging and discharging curves for the voltage and current. Save the data in a temporary file
before continuing!. Change the scales in both graphs to clearly see the plots on the screen. You
may use Auto Scale option by clicking
5. Curve Fitting: For both the charging and discharging portions of the curves for the capacitor,
you will select an appropriate region and fit the data with the appropriate function to determine
the time constant () for the circuit. First, select a portion of a graph in charging portion from
voltage versus time graph for fitting. The region you select should begin and end on the portion
of the curve that is changing significantly. After selecting the region, now click on “Analyze”
and select “Curve Fit”, and then choose the function “Inverse Exponent” or “Natural
Exponent” depending upon the nature of the curve. Then click on “Try Fit” and finally on
“OK” to show the fitting parameters in a box on the graph. Note that is the reciprocal of the
fitting constant C (this C is not the capacitance!). You will also see bracket symbols ([ ]) in
the beginning and end of the graph covering the data used to make fitting. You can move those
brackets to modify the portion of the graph.
Fit the curve for the discharging portions too. If the two values of C are in good agreement and
if the “Root MSE” for each is less than ~ 0.01, record the average values of Otherwise repeat
the analysis or entire measurement.
Complete the fitting for the graph from current versus time too. Record the values in table 2.
You should include a graph from ONE observation in your report. Print ONLY ONE of your
favorite graph.
6. For this first measurement only, examine the table of values for VC(t) and I(t) for five different
values of t spanning the time during which (i) the capacitor is charging and (ii) the capacitor is
discharging, compute the value of [VC (t)+VR(t)] and record these values. Here, VR(t) = I(t). R.
[Physics 2150 only] Using the “Integrate” function in the analysis software, integrate I(t)
separately for both the charging and the discharging processes. Record the values of these
integrals and also record the voltage VC on the fully charged capacitor.
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7. Repeat steps 4 – 5 above for at least three other significantly different values of R. These should
span the entire range for which you can measure the time constants, i.e. from τ ~ 0.1 sec to τ ~ 2
sec. You will now have at least four values of τ, corresponding to four different values of R,
which can be used to construct a graph of τ vs. R.
8. Repeat the procedure described in steps 4 – 6 for another capacitor, thereby obtaining at least
four values of τ as a function of R for the second capacitor.
9. Now, connect the two capacitors in series and measure the time constant using the method
described in 3 – 5 above, for one value of R only.
10. Connect the two capacitors in parallel and measure the time constant using the method
described in 3 – 5 above, for one value of R only.
Computations
In this laboratory you have observed the charging and discharging behavior of a RC circuit.
And measured the time constant (= RC) for the circuit.
Compare the nature of the voltage versus time graph and current versus time graph while
charging and discharging.
For capacitor 1, plot a graph of τ vs. R from the data in table 1 and 2. Determine the
experimental value of the capacitance (C1) from the graph and record in table 3. Compare your result
with the capacitance value written on the capacitor.
Similarly, for capacitor 2, plot a graph of τ vs. R from the data in table 1 and 2. Determine the
experimental value of the capacitance (C2) from the graph and tabulate in table 3. Compare your result
with the capacitance value written on the capacitor.
Calculate the experimental value of capacitance from the value of τ for series combination of
the capacitors and tabulate in table 3.
Similarly, calculate the experimental value of capacitance from the value of τ for parallel
combination of the capacitors and tabulate in table 2.
Calculate the theoretical values of capacitance for series and parallel combinations. Theoretical
values are computed using the appropriate formulas for series and parallel combinations of capacitors
and the experimentally determined values of C for the individual capacitors. What are the formulae for
calculating equivalent capacitance in series and parallel combination?
[Physics 2150 only] Compute and compare the values of Q determined by (i) Q I dt (ii) Q = C VC
for both the charging and discharging processes in step 6.
Questions to be answered in your report:
1. Does your graph of τ vs. R pass through the origin? Should the graph necessarily pass through the
origin? What does a non-zero intercept signify?
2. Discuss your observations of [VC(t) + VR(t)] for both the charging and discharging processes.
3. In one of the current versus time graph, choose a current (near the time ~ 0 sec), say I1. Now find
the current which is 1/e (= 0.37) of I1. How long does it take to change from I1 to 0.37 I1? Compare
it with time constant.
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4. [Physics 2150 only] Compare and discuss the values of Q determined by the two different methods
in part 5).
5. Do your experimental values for the series and parallel combinations of capacitors agree with the
theoretically calculated values? Discuss in terms of the experimental errors involved.
6. Your capacitors were charged to a maximum voltage of 5 volts. In each of the four cases (that is for
the individual capacitors and for the series and parallel combinations), when the capacitors were
fully charged, what was:
A. The charge (in microcoulombs) stored on the plates of each capacitor.
B. The energy (in microjoules) stored in each capacitor.
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Data Sheet
Date experiment performed:
Name of the group members:
Table 1. For voltage versus time graph
Capacitance, C
Resistance ()
Charging ()
Discharging ()
Average ()
Charging ()
Discharging ()
Average ()
C1
C1
C1
C1
C2
C2
C2
C2
Ceq – Series
Ceq – Parallel
Table 2. For current versus time graph
Capacitance, C
Resistance ()
C1
C1
C1
C1
C2
C2
C2
C2
Ceq – Series
Ceq – Parallel
Table 3
C1
C2
—
—
Ceq – Series
Ceq – Parallel
Experimental
Theoretical
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Capicitance, C
C1
C1
C1
C1
C2
C2
C2
C2
Resistance (Ω)
Charging, τ
Discharging ,τ Average (τ )
0.438
0.448
0.504
0.473
0.537
0.511
0.598
0.727
0.406
0.333
0.274
0.272
0.460
0.401
0.392
0.376
195
200
227
270
195
200
227
270
0.457
0.443
0.486
0.857
0.261
0.270
0.341
0.361
Ceq-series
230
0.198
0.206
0.202
Ceq- Parallel
230
0.824
0.817
0.820
Table 2. For current versus time graph
Capicitance, C
C1
C1
C1
C1
C2
C2
C2
C2
Resistance (Ω)
Charging, τ
195
200
227
270
195
200
227
270
Discharging ,τ Average (τ )
0.406
0.462
0.434
0.424
0.467
0.446
0.534
0.475
0.505
0.564
0.610
0.587
0.257
0.406
0.331
0.257
0.272
0.264
0.327
0.542
0.434
0.401
0.405
0.403
Ceq-series
230
0.187
0.200
0.193
Ceq- Parallel
230
0.803
0.833
0.818
Table3.
C1
Experimental
Theoretical
Ceq-series
C2
–
–
Ceq- Parallel
C1=2200 μF
C2=1000 μF
Title
Name 1:
Course number
LAI:
Date
Name 2:
Section:
Name 3:
Experiment: Measurement Example
Name 4:
Synthesis Question 1:
Purpose: The aim of this experiment was to confirm that the mass of the machine key
falls within the tolerance given by its manufacturer.
IV: NA
Guest User 5/23/2018 9:27 AM
Comment [1]: Always start the report with
a purpose that matches the Synthesis Question.
Avoid copying the Synthesis Question.
DV: NA
CV: NA
Guest User 5/23/2018 9:28 AM
Comment [2]: Identify independent
variable (IV), dependent variable (DV), and
Controlled variables (constants – CV) when it
applies.
Materials:
•
Machine key
•
Postal scale
•
Digital balance
Procedure: Several measurements of the key mass were made using two different devices
and compared it to the tolerance.
First, a postal scale with tick marks 6 grams apart was used to measure the mass of the
machine key.
Guest User 4/5/2018 2:14 PM
Comment [3]: The procedure should be
written in the way that anyone following your
steps should be able to produce the same thing
without confusion.
Do not include numbers in the procedure.
Next, a digital balance was used to measure the key’s mass. According to the manual of
the device, its uncertainty of the balance is ± 0.1 grams.
Data:
The measurements are organized in Table 1 below.
Table 1: Measurements and Given Tolerance of Machine Key Mass
Source
Measurement (g)
Postal Scale
93 ± 3
Digital Scale
95.1 ± 0.1
Manufacturer
93 ≤ x ≤ 96
Guest User 4/26/2018 12:24 PM
Comment [4]: Data should always be
organized in a data table. Include all trials and
the average value of the dependent variable.
The units should be entered only in the heading
of the table. If the uncertainty has a constant
value for all measurements (for example the
uncertainty given by the manifacture), include
it in the heading. If you calculate the standart
deviation, then create a separate column.
In some labs you do not measure quantities,
but observe what is happening. In these kind of
labs, the “Data” section is replaced by the
“Observation” section.
The third data (manifacturer) refers to the manufacturer statement that the mass should be
95 g, but could be as low as 93 g or as high as 96 grams.
1
Evaluation of Data (Analysis):
The measurements in the data table were also shown graphically in the Figure 1 below
for an easier comparison.
Guest User 4/5/2018 2:14 PM
Comment [5]: Evaluation of data (a.k.a.
Analysis) include calculations, graphs, analysis
of the graph, how the uncertainty has been
selected, etc.
Figure 1: Comparing Measurements and Tolerance
The uncertainty of the postal scale was chosen 3 grams as one half of the smallest
devision of the scale, which is 6 grams.
According to the graphical representation, the mass given by the postal scale is consistent
with the manufacturer’s claim because the green and red lines overlap. However, the
postal scale measurement, m = 93 ± 3 g, can not be used to conclude that the
manufacturer’s value is accurate because this measurement also includes masses outside
also answer questions
asked in lab manual.
the tolerance. The measurement from the digital balance, m = 95.1 ± 0.1 g, on the other
hand, confirms that the key’s mass is within the manufacturer tolerance, 93 ≤ x ≤ 96 g.
The possible masses given by that measurement fall within tolerance.
Conclusion:
The experimenter can conclude that the mass of the key measured with the digital
balance, m = 95.1 ± 0.1 g, does indeed fall within the range claimed by the manufacturer,
93 ≤ x ≤ 96 g.
2
Guest User 4/5/2018 2:14 PM
Comment [6]: The conclusion should
answer the question raised in the purpose.
Synthesis Question 2:
Purpose: The goal of this experiment is to measure the mass of a single sheet of paper
and evaluate the success of a clever measurement trick.
IV: NA
DV: NA
CV: NA
Materials:
•
Print papers
•
Postal scale
•
Digital scale
Procedure:
The total mass of 20 sheets of paper was measured using the mechanical postal
•
scale.
•
The mass of the 20 sheets was divided by 20 to find the mass of one sheet.
•
The mass of one sheet was measured using the digital scale.
Data:
The measurements are organized in Table 2 below.
Table 2: Measurements and Given Tolerance of Papers Mass
Measurement (g)
Measurement (g)
(for 20 sheets)
(for 1 sheets)
Postal Scale
96 ± 3
4.8 ± 0.2
Digital Scale
NA
4.5 ± 0.1
Source
3
Guest User 4/5/2018 2:13 PM
Comment [7]: The trick is successful if the
value obtained by using it is within the same
range as the value given by the digital scale.
This is a quick way of measuring the mass of
one page with small uncertainty when
measuring tools with high precision are not
present.
Evaluation of Data (Analysis):
The measurements in the data table were also shown graphically in the Figure 2 below.
Figure 2: Comparing Direct and Calculated Measurements of Paper Mass
As seen by the graphical representation of the measurements, the uncertainties do not
overlap. The discrepancy between the two measured values is 0.3 g. This discrepancy is
the same as the sum of the uncertainties. This means that the only possible value that
satisfies both measurements is a mass of exactly 4.60 grams, a claim that is not supported
by the measurements, therefore it can not be made without further data collections. Since
the digital scale has a smaller uncertainty, the direct measurement from it would be more
acceptable. However, the value of the mass given by the measurement trick is within the
range of the value given by the digital scale. Therefore, the trick is considered to be a
successful one.
The percentage difference between the two values of the mass of one sheet is calculated
below.
!! − !!
4.8 − 4.5
% !”##$%$&’$ = ! + ! × 100% =
× 100% = 6%
4.8 + 4.5
!
!
2
2
The value of the percentage difference of 6% would be another good reason to state that
the trick gave a useful approximation of the actual value of the mass of one page.
Conclusion:
The mass of one page found by using the clever measurement trick (measuring the mass
of 20 pages and dividing it by 20 to find the mass of one page) is 4.8 ± 0.2 gr. Since this
value is within the same range as the value from digital scale, 4.5 ± 0.1 gr, the trick is
considered to be successful.
Guest User 4/26/2018 12:34 PM
Comment [8]: The conclusion addresses
whether the trick is successful or not, stated
in the purpose.
4
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