Last updated 10/4/06
Lab 6: RC Circuits
Concepts
• RC time constants
Format
Cookbook/quantitative
Objective
You will measure the voltage across a capacitor as a function of time in a RC
circuit. You will determine the time constant and compare it with the theoretical
value. You will learn how to use an oscilloscope to measure a voltage as a
function of time.
Introduction
Previously we’ve learned about how capacitors store charge on their plates and
have seen how they can serve as a temporary voltage source in a circuit. In our
case we hooked up a capacitor to a light bulb and watched as it lit up. The
problem with using a capacitor as a voltage source, as seen by our rapidly
dimming light bulb, is that it decayed quite quickly compared to battery. How
fast a capacitor can be charged up or discharged is governed by the capacitance
and the resistance of the circuit, the time constant is equal to RC.
! = RC
(Eq 6.1)
A full development of the temporal properties of RC circuits is located in your
textbook. The important results for this lab are as follows:
V (t) = V0e
"t
#
,Q(t) = CVo
"t
#
Equation 6.2 : Voltage for Capacitors discharging
!
where Vo is the initial
voltage across the capacitor (i.e. CVo = Qo is the initial
charge stored in the capacitor (eq 13)
!
"t '
"t '
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V (t) = V f &1" e # ) ,Q(t) = CV f &1" e # )
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!
Equation 6.3: Voltage for Capacitors charging
! have assumed the capacitor is initially uncharged, V is the final
where we
f
voltage of the charged capacitor. (eq 14)
!
From these equations, one can see that, at one time constant (i.e. t= τ), a
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discharging capacitor has only about one-third of its initial charge left &e
"1
1'
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while a charging capacitor has reached two-thirds of the final charge
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2'
"1
&(1 " e ) # ) .
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!
Procedure
Measure the voltage across a charging and discharging capacitor as a function of
time and determine the time constant
!
If you try to measure the voltage using a normal voltmeter, you may not be
successful, because the voltage may change very rapidly with time. For example,
a 1µF capacitor in series with a 1k" resistor will have a time constant of 1ms ,
which is too fast for a normal voltmeter and your eye!
!
To measure voltages which change rapidly with time, one can use a device called
!
!
the oscilloscope which can response quickly to the rapid change of voltage and
display the voltage level on a fluorescence screen (vertical axis is voltage and
horizontal axis is time). Unfortunately, your eye still cannot register to such
rapid variation, the voltage trace on the scope will appear for a few ms and then
disappear. The trick is to repeat the same measurement over and over and let the
oscilloscope retrace the same voltage vs. time long enough for your eye to see it.
Therefore, instead of charging the capacitor once, you charge it over and over
again (you let it discharge in between). The repeat charging and discharging can
be accomplished using a voltage source which turns on and off repeatly, such as a
“square wave”.
Figure 1: Square Wave
Since the oscilloscopes are still a little unfamiliar, if you have trouble with your
oscilloscope at any time please ask the TA for assistance… well, after you’ve
given your problem a bit of thought.
1. Your TA will give you two resistors and a capacitor of known values. Be
sure to check the resistor value with your multimeter.
2. Connect your signal generator (voltage source) into your oscilloscope and
check that there is a square wave signal. Adjust the DC bias knob so that
the voltage switches between 0 volts and some positive value Vo. Record
Vo.
3. Make the following circuit with your capacitor and one of your resistors:
Figure 2
4. Calculate τ for this circuit.
5. Change the Time/Div knob on the oscilloscope so that the x axis of the
screen is scaled to accommodate AT LEAST τ seconds1. In practice you
will likely find that extending to a few times τ is preferred.
6. Set the frequency of the signal generator so that it is ~1/(10τ), that is the
period is 10τ. The square wave is on for 5τ and off for 5τ. This is
enough time for the capacitor to reach almost full charge and enough time
to almost fully discharge. (Use the formulae above to find the voltages of
the charging and discharging capacitor at t=5τ and see how close they are
to the final value at t= " .)
!
The horizontral (x) axis of an oscilloscope
represents “time”, so the oscilloscope “plots” voltage as a
function of time. The x axis is divided into large segments and the amount of time represented by each
segment is controlled by the Time/Div knob. Suppose there were 10 segments along the x axis and you
wanted the screen to be two seconds “wide”. If you adjust the Time/Div knob to .2s/Div, then the whole
screen is 2 seconds.
1
7. Using the oscilloscope, view the voltage across the capacitor. Be sure that
the maximum magnitude of the voltage on-screen is close to Vo. Record
the voltages at several time, enough data points for you to trace out a
smooth curve and for you to fit a function to the data. Sketch the curve in
your lab book and include it in your report. Now use the oscilloscope to
measure the time constant. Obtain, the time constant from both the
charging portion and the discharging portion and see if you get the same
value. (Hint: Which equation, 6.2 or 6.3, matches what you see on screen?
What is the voltage when t = τ? If you know how long it takes to get to
this voltage from Vo then you know the time constant. To get a more
precise value of the time constant, extract it from your fitting function).
8. Replace the large resistor with the small resistor. How do the voltage
curves compare? How do the τ’s compare?