Download RC Circuits - Chabot College

,Chabot College Physics Lab
RC Circuits
Scott Hildreth
Goals:
 Continue to advance your understanding of circuits, measuring resistances, currents, and voltages
across multiple components.
 Extend your skills in making “breadboard” connections of electrical components, tracing connection
errors, diagnosing problems, and resolving them logically.
 Verify relationships between voltage, current, capacitance, and time for charging and discharging RC
circuits in series and parallel configurations.
Background:
By now you should be comfortable making connections on a circuit board, and measuring current and
voltage digitally using the Vernier equipment and standalone multimeters. This lab activity will challenge
you to verify the equations for charging and discharging capacitors in simple DC circuit arrangements. As
with the previous lab, you will:
a) Theoretically calculate the rate of charging and discharging simple single, series, and parallel
capacitors in an RC circuit, using 10 resistors, a 9V battery, and 0.1 Farad capacitors.
b) Construct models of these circuits using the PhET Circuit Construction Kit online, and test your
hypotheses about the currents and voltages using this model.
c) Experimentally use the Vernier and Pasco circuit boards to actually build an RC circuit, and
measure the actual values with the multimeter, computer, and oscilloscope.
Write up:
This experiment does not require a formal lab write up; each group should be able to record data and
answer all questions, and submit one organized summary with all questions neatly answered and the data
clearly recorded. If you do an organized job of taking your data and recording answers, you should be able
to submit the summary at the end of the lab period. If not, you must submit it no later than one week from
the date of the lab (but remember the exam is coming the next week! Spend your time studying for that,
not making this look pretty.)
MATERIALS
Computer running Logger Pro software
LabPro interface
Vernier Current & Voltage Probe System
Additional Vernier Voltage Probes
Oscilloscope (optional)
6-8 sets of alligator clip wires
Batteries
adjustable low-voltage DC power supply
Vernier Circuit Board
Digital Multimeter w/ (2) voltage probes
BACKGROUND
The charge q on a capacitor’s plate is proportional to the potential difference V across the capacitor:
q
V  ,
C
where C is the capacitance, measured in the unit of the farad, F, (1 farad = 1 coulomb/volt).
If a capacitor of capacitance C (in farads), initially uncharged, is connected across a resistor R (in ohms) to
a potential V0 (volts), a time-dependent current will flow according to Ohm’s law. This situation is shown
by the RC (resistor-capacitor) circuit below when the switch to the battery is closed:
R
Red
C
Black
Figure 1
As the current flows, the charge q on the capacitor builds up over time, increasing the potential across the
capacitor, which in turn reduces the current. This process creates an exponentially decreasing current
flowing through the resistor. If you measure the voltage drop across the resistor, that voltage will decrease
in time, modeled by:
Charging RC Circuit Voltage across Resistor R = Vr (t )  V0e

t
RC
The rate of the decrease is determined by the product RC, known as the time constant of the circuit. A large
time constant means that the capacitor will discharge slowly. Similarly, the Voltage across the capacitor C
builds up exponentially in time at a rate determined by the same time constant, modeled by:
Charging RC Circuit Voltage across Capacitor C = Vc (t )  V0 1  e

t
 RC


The same time constant RC describes the rate of discharging as well as the rate of charging. The
relationships for discharging RC circuits are similar to those above, although both of the voltage drops
across the resistor and capacitor decrease (they are in parallel!), and direction of the current and the sign of
the voltage drop will be reversed:
Discharging RC Circuit Voltage across Resistor R = Vr (t )  V0e
t
 RC
Discharging RC Circuit Voltage across Capacitor C = Vc (t )  V0 e
t
 RC
Important Note! Record your team’s sketches, equations, and answers to the questions below, clearly and
neatly, as part of your overall data for the experiment. If this is done effectively from the start, you should
NOT have to copy them over for the report.
PART A: THEORETICAL CALCULATIONS OF RC CIRCUITS
1. For an RC circuit with a single capacitor of 0.10 Farads, in series with a single resistor of 10 , and
a voltage source of 9 V, calculate the time constant for the circuit, and develop the charging and
discharging equations for the voltage readings across the resistor and capacitor.
time
Capacitor Voltage
Resistor Voltage
2. Sketch the graphs of Voltage vs. Time for the resistor and the capacitor, for both cases of charging and
discharging (four sketches in all!)
time
3. If you add a second capacitor of equivalent value in series with the first, how will this affect the rates of
charging and discharging? Develop the new charging and discharging equations for this situation.
4. If you add a second capacitor of equivalent value in parallel with the first, how will this affect the rates of
charging and discharging?
PART B: BUILD A MODEL OF YOUR CIRCUIT NETWORK ONLINE
5. Locate, click on, and SAVE the pre-made PhET files RC Circuits 1/2/3 onto the desktop of your
computer, from my website (http://www.chabotcollege.edu/faculty/shildreth/physics/EMapps/ )
Now locate and run the PhET simulator “Circuit Construction Kit – AC/DC” available online at
http://phet.colorado.edu/en/simulation/circuit-construction-kit-ac. We’ll use this simulator to model
your circuits.
In the upper right corner, click LOAD, and browse your
computer’s files to locate the RC Circuit 1 simulation file.
It should look like the image to the right; you might have to
add the voltage charts from the menu, or adjust the locations
of the voltage probes across the components to correctly read
the values.
6. Charge the capacitor by closing the switch on the left. Run the simulation, noting the resulting graphs
of the voltages across the resistor and capacitor. Pause at any time using the pause button below.
Do the graphs match the expected theoretical results? Estimate the voltage value at a specific time
from the graphs (like 1/RC, the time constant!), and verify your equations predicted that value.
Record your results. Do this for both graphs.
7. Discharge the capacitor by opening the switch on the left, and closing the switch on the right. Run the
simulation, noting the resulting graphs of the voltages across the resistor and capacitor. Pause at
any time using the pause button below. Do the graphs match the expected theoretical results?
Estimate the voltage value at some specific time from the graphs, and verify your equations
predicted that value. Record your results. Do this for both graphs.
8. Now model the situation with two capacitors in series
connected to the same voltage and resistor combination.
LOAD the RC Circuit 2 simulation. It should look like
the screen picture shown. You may have to reconnect the
voltmeter chart leads.
9. Close the left hand switch to charge the capacitors.
Examine the resulting voltage graphs. Do they match your
theoretical predictions? How does the voltage drop across
each capacitor compare to the voltage across the battery?
You can add another chart to monitor the voltage across the resistor.
10. Discharge the capacitors by opening the left switch and closing the right switch. Examine the resulting
graphs once more, and verify that they match your theoretical predictions.
11. Does it appear that placing two capacitors in a circuit with one pathway for charge increases or decreases the
amount of charge stored? You may need to return to the original circuit from part I to decide.
12. Now model the situation with two capacitors in parallel
connected to the same voltage and resistor combination.
LOAD the RC Circuit 3 simulation. It should look like the
screen picture shown. You may have to reconnect the
voltmeter chart leads.
Repeat steps 9 – 11 above with this configuration, and again
verify that the graphs match your predictions. Does it appear
that placing two capacitors in a circuit with multiple pathways
for charge increases or decreases the amount of charge stored?
PART C: TEST YOUR MODEL EXPERIMENTALLY
13. Connect the RC circuit as shown in Figure 1 at the start of the lab with the 10-F capacitor and the
100-k resistor. Record the values of your resistor and capacitor in your data table, as well as any
tolerance values marked on them.
14. Connect a Differential Voltage Probe to Channel 1 of the LabPro, as well as across the capacitor,
with the red (positive lead) to the side of the capacitor connected to the resistor. Connect the black
lead to the other side of the capacitor.
15. Open the Capacitor file in the Physics with Vernier folder. A graph will be displayed. The vertical
axis of the graph has potential scaled from 0 to 4 V. The horizontal axis has time from 0 to 10 s.
16. Charge the capacitor for 30 s or so with the switch in the position as illustrated in Figure 1. You can
watch the voltage reading at the bottom of the screen to see if the potential is still increasing. Wait
until the potential is constant.
17. Click
to begin data collection. As soon as graphing starts, throw the switch to its other
position to discharge the capacitor. Your data should show a constant value initially, then
decreasing function.
18. To compare your data to the model, select only the data after the potential has started to decrease by
dragging across the graph; that is, omit the constant portion. Click the curve fit tool , and from the
function selection box, choose the Natural Exponential function, A*exp(–C*x ) + B. Click
,
and inspect the fit. Click
to return to the main graph window.
19. Record the value of the fit parameters in your data table. Notice that the “C” used in the curve fit is
not the same as the C used to stand for capacitance. Compare the fit equation to the mathematical
model for a capacitor discharge proposed in the introduction,
V (t )  V0 e
 t
RC
20. How is fit constant C related to the time constant of the circuit?
21. Make a sketch the graph of potential vs. time. Choose Store Latest Run from the Data menu to store
your data. You will need this data for later analysis.
22. The capacitor is now discharged. To monitor the charging process, click
. As soon as data
collection begins, throw the switch the other way. Allow the data collection to run to completion.
This time you will compare your data to the mathematical model for a capacitor charging,

 t 
V (t )  V0 1  e RC 


Select the data beginning after the potential has started to increase by dragging across the graph.
Click the curve fit tool, , and from the function selection box, choose the Inverse Exponential
function, A*(1 – exp(–C*x)) + B. Click
and inspect the fit. Click
to return to the
main graph window.
23. Record the value of the fit parameters in your data table. Compare the fit equation to the
mathematical model for a charging capacitor.
24. Hide your first runs by choosing Hide Run Run 1 from the Data menu. Remove any remaining fit
information by clicking the gray close box in the floating boxes. Repeat the experiment with a
resistor of lower value. How do you think this change will affect the way the capacitor discharges?
Rebuild your circuit using the 47-k resistor and repeat the steps above.
SAMPLE DATA TABLE
Fit parameters
Trial
A
B
C
1/C
Resistor
Capacitor
Time
constant
R
()
C
(F)
RC
(s)
Discharge 1
Charge 1
Discharge 2
Charge 2
ANALYSIS
1. In the data table, calculate the time constant of the circuit used; that is, the product of resistance in ohms
and capacitance in farads. (Note that 1F = 1 s).
2. Calculate and enter in the data table the inverse of the fit constant “C” for each trial. Now compare each
of these values to the time constant of your circuit.
3. Note that resistors and capacitors are not marked with their exact values, but only approximate values
with a tolerance. If there is a discrepancy between the two quantities compared in Question 2, can the
tolerance values explain the difference?
4. What was the effect of reducing the resistance of the resistor on the way the capacitor discharged?
OPTIONAL EXTENSIONS

Try different value resistors and capacitors and see how the capacitor discharge curves change.

Try two capacitors in parallel. Predict what will happen to the time constant. Repeat the discharge
measurement and determine the time constant of the new circuit using a curve fit.

Try two capacitors in series. Predict what will happen to the time constant. Repeat the discharge
measurement and determine the time constant for the new circuit using a curve fit.