D.E. Shaw
.
.
Resistance and Capacitance
Spring 2009
ramp up wave function. Set the amplitude for
4.0 volts and set the frequency for 0.1 Hz.
J
Click on the Measurements and Sample Rate
I
button and choose both the output voltage
330
10
33
100
and current to measure. Finally select a
100
Ω
Ω
Ω
μf
μf
sample rate of 200 Hz and leave the generator
lamp
on Auto. Minimize the generator window.
• In the setup window use the Sampling
A B C
D E F G
Options tab to select a delay start time of 0.1
seconds and an automatic stop time of 9
Equipment: Pasco RLC Circuit Board, one
RLC Board
seconds. Minimize the setup window.
Pasco voltage probe, four connecting wires
Fig. 1
• We use the Calculator to obtain the
with banana plugs that allow two connections at
resistance for each applied voltage from Ohm's Law (Eq. 1).
each end, a separate 100 ohm resistor and a multimeter.
Click on Calculator. In the definition window replace the
default values by: R = x/y. Then click on the please define
Part A: Resistance
variable "x" button and select the data measurement "output
voltage". At this point a please define variable "y" button should
(i) Ohm's Law
appear. Click and select the data measurement "output current".
Finally click the Accept button and then minimize this window.
Theory: If a material obeys Ohm's Law the current density "J"
• Create a voltage versus current plot by dragging the output
is directly proportional to the applied electric field "E". The
voltage to a plot icon. Then drag the current icon to this plot to
proportionality factor is the conductivity of the material. Ohm's
replace the time axis with the current. (Recall that you must
Law can also be expressed as:
drag the current icon directly over the axis and only drop when a
box only surrounds the axis.)
ΔV = IR ...(1)
• Create a resistance versus time plot by dragging the calculated
resistance values to the plot icon to create a different plot of the
where ΔV is the potential difference applied to the material, "R"
resistance versus time.
is its resistance and "I" is the current.
• Click on start to obtain a data set.
Experimental:
Analysis:
• Fig. 1 shows the Pasco RLC circuit board. Use a wire to
First examine the voltage versus current curve. Does the resistor
connect the signal terminal (marked by a sine wave) of the 750
obey Ohm's Law? Explain. If the resistor obeys Ohm's Law
Pasco Interface Box to terminal "H" of the circuit board.
what does the slope of this curve represent? Do a linear fit and
Connect the ground terminal of the Interface Box (located
obtain the slope.
beside the signal terminal) to terminal "C" of the circuit board.
Is the resistance versus time curve consistent with your first
With these connections the output voltage produced by the
plot? Neglecting the first few points obtained for very small
interface box is applied directly across the 100-ohm resistor.
currents and potentials find the average resistance.
• The interface box has a signal generator that can produce
Compare the resistance obtained from both plots with the
several different types of output signals. You can select the type
nominal value of 100 ohms given by the manufacturer.
of signal, the peak voltage and the frequency. You can also
choose to measure the output voltage and/or the
(ii) Resistors in Series and Parallel:
current produced by the generator. For this part of the
Theory: When resistors are in series the same current
experiment we will apply a positive ramp voltage
flows through each and the potential difference applied
having a peak voltage amplitude of 4.0 volts and a
10
33
100
to the combination must be the sum of the individual
frequency of 0.1 Hz. The output voltage applied to
Ω
Ω
Ω
potential differences across each resistor. The equivalent
the resistor will start at zero and increase linearly to
resistance for series resistors is the sum of the individual
4.0 volts in a time of 10 seconds.
B
A
C
resistances. When resistors are in parallel the potential
• In the Data Studio program, click on the signal
difference across each is the same and the sum of the
V
output icon button (which is on the extreme right
currents must equal the current flowing into the resistor
side) to open the signal generator control window.
Resistor network
combination. The reciprocal of the equivalent resistance
Select the positive up ramp wave function not the
Fig. 2
Introduction: The objectives of this
experiment are to study some properties of
resistors and capacitors including Ohm's Law,
series and parallel connection of resistors,
temperature dependence of resistance, charge
and discharge of a capacitor through a resistor
and series and parallel combinations of
capacitors.
H
for parallel resistors is the sum of the reciprocals of the
individual resistances.
Experimental:
• Use a wire to connect the ground terminal of the 750 Pasco
Interface Box to terminal "C" of the circuit board and connect a
second wire from the signal terminal of the box to terminal "A"
of the circuit board. Finally connect a wire between terminals
"B" and "C". With these connections the output voltage
produced by the interface box is applied across the network of
resistors shown in Fig. 2.
• Use all of the same Data Studio settings from part (i). Obtain a
plot of voltage versus current and resistance versus time.
Analysis: Does Ohm's Law appear to apply this network of
resistors? Repeat the analysis done in Part A (i) to find the
equivalent resistance of this network and obtain the mean of the
resistance values obtained from each graph.
• Use the theory of parallel and series resistors to compute the
theoretical equivalent resistance of this network. Compare the
theoretical result with the mean experimental value and find the
percent difference.
• For Ohm's Law (in the form of Eq. 1) to be valid, the plot of
ΔV versus I must be linear and pass through the origin. Does
Ohm's Law apply over the complete range of data obtained? Is
there a limited region where Ohm's law applies fairly well?
• If your voltage versus current curve is linear over a small
range of low currents and voltages, do a linear fit and obtain the
resistance from the slope. Use the smart tool to find the initial
resistance (R0) from the resistance versus time plot. How do
these values compare? From the same curve find the maximum
resistance (R) when the filament is at its maximum temperature.
• Compute the ratio (R/R0) of the high temperature maximum
resistance to the room temperature resistance (R0). Use Equation
(2) and your computed resistance ratio to estimate the maximum
temperature of your filament.
(iv) Power Dissipated in the Lamp:
Change the signal to a positive square wave having the
amplitude of 5 V and a frequency of 0.1 Hz. Select the sample
rate of 500 Hz. With this signal the potential difference applied
to the lamp will be zero and then at some later time it will
almost instantaneously become 5 volts
(iii) Resistance of an Incandescent
which is equivalent to a simple circuit
T (K)
tungsten
Lamp:
with a battery, lamp and switch in series.
2000
2000
Theory: The resistance of materials
Click the Sampling Options tab and
depends on their temperature. For good
under Delay Start select then Output
metallic conductors the resistance
Voltage then Rise Above and finally 0.1
increases with temperature as a result of
volts. Under Automatic Stop, set the data
increased collisions between the moving
collection time for 0.3 seconds. Use the
1000
1000
charge carriers and the fixed atoms. We
Calculator to calculate the power by
let “R” and “R0” be the resistances at the
taking the product of the voltage and
temperature “T” and room temperature
current. Drag the power data icon to the
(R/R0)
respectively of the tungsten lamp
plot icon. Collect a set of data. If the
0
filament. The relation between the
light is on initially, the data collection
1212
00
44
88
temperature “T” and the ratio (R/R0) is
will not begin until the lamp goes off
Fig. 3
shown in Fig. 3. The data in this graph
and then comes back on again. Using
was calculated from data measured by
some of the results found earlier explain
Jones and Langmuir and published in the Handbook of Physics
the basic shape of the curve obtained for power as a function of
and Chemistry. The relation connecting the temperature and the
the time. Why do light bulbs often blow out when they are first
resistance ratio is:
turned on?
⎛ R
T (K ) = 88 + 213⎜⎜
⎝ R0
⎞
⎛ R
⎟⎟ − 2.407⎜⎜
⎠
⎝ R0
2
⎞
⎟⎟ ...(2 )
⎠
Experimental: Before making the electrical connections, use
the multimeter to measure the resistance of the lamp filament at
room temperature (R0). Use a wire to connect the ground
terminal of the 750 Pasco Interface Box to terminal "D" of the
circuit board and connect a second wire from the signal terminal
of the box to terminal "I" of the circuit board. The applied signal
is applied directly across the lamp filament.
Using all of the same settings used in part (i) and (ii), obtain a
plot of voltage versus current and resistance versus time.
Analysis:
Part B: Capacitance
(i) Charging of a Capacitor
Theory: When an uncharged capacitor with capacitance "C" is
charged through a resistance "R" by connecting it to a battery or
power supply the charge, q(t), on the capacitor at any time "t" is:
−t
⎞
⎛
q(t ) = q f ⎜⎜1 − e RC ⎟⎟ ...(3)
⎠
⎝
where qf is the final charge on the capacitor when it is fully
charged and the charging begins at the time zero. Since
q = CΔV the potential difference, ΔV(t), across the capacitor at
any time is:
−t
⎛
⎞
ΔV (t ) = ΔV f ⎜⎜1 − e RC ⎟⎟ ...(4 )
⎝
⎠
Since no current exists in the circuit when the capacitor is fully
charged, ΔVf is the same as the battery or power supply
potential.
The current in the circuit at any time "t" is:
I (t ) = I 0 e
−t
RC
...(5)
where I0 is the initial current (V/R).
Experimental:
Analysis:
• Observe the curves obtained for the potential difference across
the capacitor and the current in the circuit. Do these plots seem
to be in qualitative agreement with the theoretical expectations?
• Use the inverse exponent fit and the natural exponent fit for
the potential difference and current plots respectively. The Data
Studio functions for the inverse exponent and natural exponent
fits are:
(
)
A 1 − e − C ′x + B
Ae
−C ′x
and
+B
respectively. Data Studio uses C rather than C’. We will
• The standard circuit for studying the charging of a capacitor
represent the value found by Solver to be C’ to avoid possible
uses a capacitor, resistor, battery and switch in series. In this
confusion with the capacitance C. By
experiment we replace the battery and
comparing these fitting functions with equations
switch with a positive square wave potential.
H
(4) and (5) we see that the time constant “RC” is
This potential is applied to a capacitor and
J
the reciprocal of the value C’ returned by the
resistor placed in series with it. Since the
I
Data Studio program. The capacitance C is
applied potential will suddenly rise from
330
100
100
related to C’ by:
Ω
μf
zero to 5 volts it is not necessary to use a
μf
mechanical switch in the circuit. The period
1
1
C′ =
⇒ C=
of the square wave will be long enough to
RC
RC ′
C
G
allow the capacitor to fully charge before the
A B
D
E
F
The fitting procedures for exponential and
V
applied potential returns to zero.
inverse
exponential fits occasionally produce
RC Circuit
• Use a wire to connect the ground terminal
erratic
results.
Before you accept the fitted
Fig. 4
of the 750 Pasco Interface Box to terminal
coefficients as being satisfactory make sure
"G" as shown in Fig. 4. Connect a second
the fitted curve passes close to the data points. The important
wire from the signal terminal to terminal "C". Finally connect a
term is "C (the exponent)". If you are unable to get a good fit
wire from "H" to "J" to place the 100 ohm resistor in series with
double click on the data box that displays the fit. This will open
the 330 μf capacitor.
a curve fit window that allows you to enter initial estimates of
• Connect a Pasco voltage probe to channel "A" and the red
the fitting parameters. Enter a trial value of 28 for "C" (which is
wire of this probe to terminal "J" and the black wire to terminal
the approximate value of 1/RC). You will probably find that the
"G" to measure the potential across the capacitor.
program is now able to obtain a good fit using the trial value as
• In the Setup window click on the signal generator and choose
a starting point for the fitting procedure. Click on the Accept
the positive square wave function. Choose an amplitude of 5.0
button. For both curves compare your fitted values of "C" with
volts and a frequency of 0.1 hz. Using the Measurement and
the expected value of "1/RC" calculated from the known values
Sample Rate button select both the output voltage and current to
of the resistance and capacitance. Find the percent difference. Is
measure and choose a sample rate of 1000 hz. Leave the signal
your experimental data consistent with the theoretical
generator on Auto then minimize the generator window.
predictions? Our capacitor model neglects any charge that may
• Click on the channel "A" icon and select the voltage sensor.
leak directly through the capacitor from one plate to the other
• Use the Sampling Options tab to select the following start and
and any charge that passes through the voltage probes to
stop conditions. Under Delay Start select Data Measurement.
measure the potential difference ΔV. These possible sources of
Select Output Voltage then Rise Above 0.1 volts. Select Start
error are examined in the Optional Project Section. However,
Signal Generator prior to start condition. Under Automatic
the typical internal resistance of the voltage probe is about 106
Stop select Data Measurement and stop the data collection when
ohms and the leakage resistance of a typical capacitor used in
the Voltage, ChA(V), rises above 4.8 volts. Note that we do not
this experiment is even larger. Therefore the effect of leakage
use the Output Voltage here since it will rise above 4.8 volts
resistance and internal resistance of the meter should be very
immediately. With these settings data collection will begin
small unless the capacitor is defective which could create a
automatically when the voltage on the capacitor just begins to
much smaller internal resistance.
rise above 0.1 volts and will stop automatically when it rises
(ii) Charge on a the Capacitor
above 4.8 volts, when it is almost fully charged. Minimize the
Theory: The objective is to find the charge on the fully charged
setup window.
capacitor studied in part (i) by using the fact that the current
• Drag the Voltage, Cha A(V) data to the plot icon. Also drag
during the charging process can be measured.
the Output Current to the plot icon to make a separate graph.
The current into the capacitor is:
• Collect a data set.
dq
I=
dt
The total charge “Q” that accumulates on the capacitor during
the charging process is:
t
t
0
0
Q = ∫ dq = ∫ Idt ...(5a )
The charge on the capacitor when it is fully charged is the
integral of the current over the charging time interval.
Experimental:
•Use the same set up as in the (i) except for the start and stop
conditions. Set the Start Condition so that data begins when the
output voltage rises above 3 (V) and select Start Generator
before Start Condition. Finally select the Stop Condition so that
data stops when the output voltage falls below 3 (V). With these
settings all of the current will be displayed.
•Collect a data set. You should observe the current to the
capacitor decreasing exponentially with time. Click anywhere
on the plot to display a Graph Settings window. Select Axis
Settings and for the X axis select a maximum of 0.4 (s) so that
the current curve will be clearly displayed.
•Click on the Σ icon and select area. Draw a measurement box
and select the area under the current curve from its initial value
until it is almost zero. This area is the integral of the current
over the charging time and is, therefore, the charge on the
capacitor according to Eq. (5a).
•Calculate the charge on the capacitor using: Q = CΔV. Use
your measured value of “C” from part (i) and 5.00 (V) for ΔV.
•Find the percent difference between the charges on the
capacitor obtained in the two previous steps.
(iii) Discharging of the Capacitor
Theory: A charged capacitor with capacitance "C" can be
discharged through a resistance "R" by connecting one side of
the capacitor to a resistor and then to the other side of the
capacitor. This situation occurs, in this experiment, when the
signal generator's output drops to zero. The charge, q(t), on the
capacitor at any time "t" after the discharge begins is:
q(t ) = q0 e
−t
RC
...(6)
where q0 is the initial charge on the capacitor when it is fully
charged and the discharging begins at the time zero. Since q =
CΔV the potential difference, ΔV(t), across the capacitor at any
time is:
ΔV (t ) = ΔV0 e
−t
RC
...(7 )
where ΔV0 is the potential across the capacitor before it begins
to discharge.
The magnitude of the current in the circuit at any time "t" is:
I (t ) = I 0 e
−t
RC
...(8)
where I0 is the initial current (ΔV0/R).
Experimental:
• In this case the discharge current will flow in the opposite
direction and Data Studio will indicate this by displaying the
current as a negative quantity.
• The same settings are used for the signal generator. The only
changes required are for the start and stop conditions. Use the
Options button to select the following start and stop conditions.
Under Delay Start select Data Measurement. Select Output
Voltage then Fall Below 4.8 volts. Also select Start Signal
Generator prior to start condition. Also choose in the same way
to stop the data collection when the Voltage, ChA(V), falls
below 0.1 volts. With these settings data collection will begin
automatically when the voltage on the capacitor just begins to
decrease (from its fully charged state) and will stop
automatically when it gets close to zero, when it is almost fully
discharged. Minimize the setup window.
• Collect a data set.
Analysis:
• Observe the curves obtained for the potential across the
capacitor and the current in the circuit. Do these plots seem to
be in qualitative agreement with the theoretical expectations?
• Fit both of your data sets using the Data Studio natural
exponent fit. The important term is "C (exponent)". If you are
unable to get a good fit double click on the data box that
displays the fit. This will open a curve fit window that allows
you to enter initial estimates of the fitting parameters. Enter a
trial value of 28 for "C" (which is the approximate value of
1/RC). You will probably find that the program is now able to
obtain a good fit using the trial value as a starting point for the
fitting procedure. Click on the Accept button. For both curves
compare (find % difference) your fitted values of "C" with the
expected value of "1/RC" calculated from the known values of
the resistance and capacitance. Is your experimental data
consistent with the theoretical predictions?
(iv) Capacitors in Parallel
Theory: When capacitors are in parallel the potential across
each must be the same while the charges can be different. Two
capacitors in parallel can be replaced by a single equivalent
capacitance which is the sum of the individual capacitances.
Experimental: Use the same set-up (Fig. 4) used in collecting
your last data set but add a wire between the terminals "F" and
"G". This connection places the 100 and 330 μf capacitors in
parallel. The two capacitors will be discharged through the 100
ohm resistor and the discharge data will be analyzed to find the
equivalent capacitance. Collect a set of data for the common
potential across both capacitors as they discharge through the
resistor.
Analysis:
Fit the curve of potential versus time to obtain an experimental
value of RC' where C' is the equivalent capacitance. Compute
outlined that uses the same equipment and methods used in this
experiment.
Measurement of R’: Use the same 5 volt square wave used in
the capacitor experiment but remove the automatic start and stop
(v) Capacitors in Series
conditions. Connect the output of the generator directly to the
330 mF capacitor and connect the voltage probes directly across
Theory: Capacitors in series have the same charges but
the capacitor. Create a graph to display the voltage measured
different potentials. Two capacitors in series have an equivalent
across the capacitor.
capacitance whose reciprocal is the sum of the reciprocals of the
Click on Start and soon as the measured voltage is at 5 volts pull
separate capacitors.
the signal plug out of the Pasco box and collect data for
approximately 400 seconds. What you are observing is the
Experimental: Use the meter to measure the resistance of the
discharge of the capacitor through the equivalent
separate resistor. Connect the ground
resistance, R’. Do a natural exponent fit and
terminal of the 750 Pasco Interface Box
determine R’ from the measured time constant and
(the second terminal from the right end) to
J
the known value of C. How does this resistance
terminal "G". Connect one end of the
compare with the 100 ohm resistor used to
Pasco
separate resistor to terminal "F" and the
330
100
discharge the capacitor in Part B?
μf
μf
other end of the resistor to the signal
Measurement of Rc: Use the same method as in
terminal of the 750 Pasco Interface Box
the previous step but as soon as you pull out the
(The terminal at the right end). An external
R
G
signal plug also disconnect the voltage probe from
F
resistor must be used since the coil is
terminal “J”. Since this will remove “Rm” we will
connected to terminals "J" and "H" and
have the discharge of the capacitor through “Rc”
consequently the resistors on the board can
only. Unfortunately while the voltage probe is
Series
Capacitors
not be used to discharge the series
disconnected we will not be measuring the
Fig. 5
capacitors. Connect the red wire from the
potential across the capacitor. As a result, every
channel "A" voltage probe to terminal "F"
ten seconds reconnect the voltage probe to “J” for just long
and connect the black wire to terminal "G". The voltage
enough for the potential difference across the capacitor to be
measured by channel "A" is the potential across the two series
recorded. Immediately disconnect and wait ten more seconds
capacitors.
and repeat until a total of about 400 seconds have elapsed.
Collect a data set of the potential across the series capacitors as
Using this approach the effect of the meter resistance is almost
they discharge.
eliminated since the voltage probes were connected for a very
short time.
Analysis:
Use the smart tool to measure the capacitor potential differences
Fit the curve of potential versus time to obtain an experimental
and corresponding times at the short time intervals when the
value of RC' where C' is the equivalent capacitance. Be sure to
voltage probe was connected. Transfer these measurements to
use the correct value of "R" in computing C'. Compute the
an Excel worksheet and plot the measured potential differences
theoretical equivalent capacitance and find the percent
versus the times and do a natural exponential fit. Find the time
difference between the experimental and theoretical values.
constant and then determine Rc. Finally use Eq. 8 to obtain the
meter resistance, Rm.
Optional Project: Measurement of the Leakage Resistance
How important are the capacitor leakage resistance and the
of the Capacitor.
meter resistance in Part B of this experiment?
A more detailed model of the
capacitor used in this experiment is
shown in Fig. 6. The parallel
resistors Rc and Rm are the capacitor
leakage and Pasco meter resistances respectively. The
equivalent resistance is R’:
the theoretical equivalent capacitance and find the percent
difference between the experimental and theoretical values.
Rc
1
1
1
=
+
′
R Rc Rm
...(9)
We can measure R’ and Rc separately
and then use Eq. (9) to find the meter
resistance, Rm.
A method to obtain R’ and Rc is briefly
Rm
Pasco voltage probe
Fig. 6
Check List: Minimal Requirements for Lab Notebook Report
The significance of each graph must be discussed and the fitted values (such as the intercept and slope) must be compared with model values
when possible.
Part A(i) Ohm’s Law:
9 Data Studio plots of potential versus current and resistance versus time.
9 The resistance obtained from both plots and the percent difference between both of these values and the nominal value of 100 Ω.
Part A(ii) Series and Parallel:
9 The calculated theoretical resistance of the resistor network.
9 The resistances obtained from both the potential versus current and resistance versus time plots and the percent difference from the calculated
value.
Part A (iii) Lamp:
9 The resistance of the filament measured at room temperature by the multimeter.
9 Data Studio plots of the potential of the filament versus the current and the resistance versus time. Discussion of these plots.
9 The resistance of the filament obtained at low temperatures from both graphs.
9 Comparison of these resistances with the value measured with the meter.
9 Calculation of the maximum filament temperature.
Part A (iv) Lamp Power
9 Data Studio plot of the power dissipated as heat in the resistor versus time. Discussion of this plot.
Part B(i) Capacitor Charging
9 Data Studio plots of the potential versus the time and the current versus time.
9 Calculation of the capacitance from both plots. The percent differences between the measured capacitances and the nominal (marked) value of
the capacitance.
Part B(ii) Charge on Capacitor
Comparison of the charge on the capacitor obtained using two different methods.
Part B(iii) Capacitor Discharging:
9 Data Studio plots of the potential versus the time and the current versus time.
9 Calculation of the capacitance from both plots. The percent differences between the measured capacitances and the nominal (marked) value of
the capacitance.
Part B (iv) Parallel Capacitors:
9 Calculated value of the equivalent parallel capacitance.
9 Data Studio plot of the potential versus the time.
9 Calculation of the capacitance from the plot. The percent differences between the measured capacitance and the calculated parallel equivalent
capacitance.
Part B (v) Series Capacitors:
9 Calculated value of the equivalent sewries capacitance.
9 Data Studio plot of the potential versus the time.
9 Calculation of the capacitance from the plot. The percent differences between the measured capacitance and the calculated series equivalent
capacitance.