Download Combinations of Capacitors

PHYSICS 201
FALL 99
LAB 5
RC Circuits. Charging a capacitor:
Preliminaries.
Geometry (size and shape) is a factor in determining capacitance. We will be working
with three different sized capacitors (small, medium and large) and hence three different
capacitances: C1, C2 and C3.
Initially we want the capacitors to be discharged. To discharge a capacitor, attach a wire
across its terminals. Why does this discharge a capacitor?
We will be using resistor A from the resistor board. Use the multimeter as an ohmmeter
to measure its resistance.
RA =
The RC circuit.
Connect your Science Workshop interface to a power amplifier and set it up as a DC
voltage source. (See Part 4 of Lab 1 for more detailed instructions.) Connect a voltage
sensor to analog channel B. Then in Science Workshop drag the analog plug icon into
the analog channel B icon and choose "voltage sensor." Drag a graph into analog channel
B so we can see the data as it comes in. Set the voltage to 5.00 V but make sure the
signal generator is off.
Make a circuit with resistor RA and the small capacitor CS in series. The circuit is
represented diagrammatically below. (Note that the values of the voltage, resistance and
capacitance shown in the diagram do NOT correspond to those in your experiment.) The
power amplifier will serve as our battery; the on/off button on the signal generator will
serve as our switch; and the voltage sensor will serve as our voltmeter.
Put the leads of the voltage sensor across the resistor. Start recording data and THEN
switch on the signal generator. On the graph, you should see the voltage jump to some
value close to 5.00 V when the generator is switched on, and then begin to decay away.
When the voltage is close to zero, stop recording.
Repeat these measurements for the circuit with CM replacing CS and then again with CL
replacing CM.
Next place CS and CM in parallel (as shown in the diagram below) and repeat the
measurements.
Next place CM and CL in series (as shown in the diagram below) and repeat the
measurements.
Finally, return to the RC circuit with only CM, place the voltage sensor across the
capacitor this time and record its voltage while the capacitor is being charged.
Remember to discharge your capacitor first. You need only do this measurement for CM.
Analysis.
Drag a Table into Analog Channel B, click on the clock so that each cell contains a time
and a voltage, copy the data and paste it into Excel. Since we started recording data
before we switched on the function generator, we must first eliminate the unwanted initial
data. Highlight the unwanted rows, then go to Edit/Delete. Next, insert a column
between your times and voltages. We would like the data we have kept to start with t=0.
Suppose the first time is in cell A1, then in cell B1, enter the formula =A1-$A$1. Copy
the formula down the rest of the column. Column B should contain the time shifted to
correspond to when you switched on the function generator. Make a plot of Voltage
versus Time. Make similar plots for the other circuits you made measurements on.
Include these plots in your report.
Theoretically, the voltage across the resistor in an RC circuit is given by
V (t )  V 0 e  t /( RC )
where V0 is the initial voltage. Fit your Voltage versus Time data to an exponential.
(Exponential is one of the Types available when you Add a Trendline.) Don’t forget to
Display the equation (found under Options). The number multiplying x in your fit is
supposed to be 1/RC. Use this fact to find the capacitance CS, we will take this value to
be the measured capacitance. Fill in your measured capacitance values below. Don’t
forget units.
Capacitor
CS
CM
CL
CS and CM in parallel
CM and CL in series
Measured Capacitance
Theoretical Capacitance
Calculate the theoretical equivalent capacitance for the parallel and series combinations
and enter them into the table above.
The voltage across the capacitor: a three-way comparison
The first method for obtaining the voltage across the capacitor is to measure it directly.
We did this for the RC circuit with RA and CM.
The second method for obtaining the voltage across the capacitor is to note that since the
capacitor and the resistor are in series. Therefore, ideally whatever voltage is not
dropped over the resistor is dropped over the capacitor, i.e.
Vcapacitor  Vbattery  Vresistor
We measured the voltage across the resistor and know the voltage supplied by the battery
(signal generator), so we can obtain the voltage across the capacitor by subtracting.
The third method involves integration. Recall that the voltage across the resistor is given
by Ohm’s law, V=IRA, which is the case even when the voltage is changing in time.
Since we measured V and RA, we can find I. Next recall that
dQ(t )
I (t ) 
dt
or
Q(t )   I (t )dt
Finally, the voltage across the capacitor is Q/C.
Suppose you have your time and voltage across the resistor data (with the times already
shifted as described above) starting in cells A1 and B1 respectively. In cell C1 enter the
formula =B1/RA, where RA is the number you measured at the start of the lab. Copy it
down the column. Column C should now contain the current. In column D we want the
charge on the capacitor. The first row corresponds to t=0 when the capacitor has no
charge on it. Thus enter 0 into cell D1. In cell D2, we want the charge at the time in cell
A2. The time in cell A2 is probably 0.1 sec. Let us assume that the average current
during the first 0.1 sec is the number in cell C1. Thus the charge at the end of that period
would be D1+C1*0.1. D1 is the charge at the beginning of that period (in this case
zero), C1 is the current and 0.1 the time interval. Enter into cell D2 the formula = D1 +
C1*0.1. Copy it down. Column D should now contain the charge on the capacitor. To
obtain the voltage across the capacitor we need to divide by the capacitance. Enter into
cell E1 the formula =D1/CM, where CM is the capacitance you obtained above.
Make a plot in Excel with all three calculations of the voltage across the capacitor. They
should be fairly similar curves. Include the plot in your report. You should also submit
your Excel file as part of your report.
Generic question on series and parallel.
Computers have parallel and serial ports. What is a port? Distinguish between serial
ports and parallel ports. Discuss briefly in what sense this notion of series and parallel
corresponds to the notion of series and parallel we have been using in simple circuits.