AC Circuits
1
Physics 364
Objective
In this lab we will get a basic introduction to capacitors and the role they play when combined
with resistors in ac circuits. In first-year physics you saw RC dc circuits in which capacitors
charge and discharge exponentially in time. And you also saw that a capacitor in a dc circuit
couldn’t sustain a steady state dc current; such currents always damp out with a characteristic
time of RC. After all, a capacitor is a break in the circuit. However, circuits with capacitors can
sustain a steady state ac current, and capacitors are of utmost importance throughout analog
electronics and also in digital electronics. Capacitors allow us to construct timing circuits that
allow an event to occur at a predetermined time after some other event. Such circuits include
oscillators. Capacitors are also used to build analog circuits that perform calculus operations
(differentiation or integration) on an input signal. Most important of all, RC circuits are used
as filters to attenuate high or low frequency “noise” from an input signal or even completely
block the dc part of an input signal.
2
AC Voltage Divider
Read the handout on oscilloscopes. Play with the controls as suggested in the handout and
record your observations and conclusions in your lab book.
Build an unloaded voltage divider with two 10k resistors, similar to the one you built in
the DC circuits lab. Instead of a DC power supply use a function generator to supply the input
voltage. Set the amplitude of the function generator to 10 volts and the frequency to 5 kHz.
Use the oscilloscope to measure the output voltage. To connect the signal, use a simple
BNC-to-Banana cable. Measure both the amplitude and the peak-to-peak value. Be sure to
estimate uncertainties in your measurements. What is the corresponding RMS value of the
voltage? Also measure the frequency of the signal.
Replace the resistors in your voltage divider with 1 M resistors. Measure the output voltage
now, and compare to the value measured earlier. What do these measurements tell you about
the internal resistance of the oscilloscope?
Replace the BNC-to-Banana cable with a x10 probe and measure the output voltage again.
Explain why you measure what you do.
3
Filters
Build the basic RC circuit shown in Fig. 1 which can be used as both an integrator or low pass
filter. We will explore this circuit’s properties. Before doing so, you should read the attached
handout on capacitors. Make sure to use a capacitor that is not polarized.
A. First let’s explore the response of the circuit to a 500 Hz square wave. Compare the input
to the output signal using a scope. Measure the fall time for the output and check that
this equals the product RC. Remember the fall time is the time it takes for the output to
drop to 37% of its maximum value. This measurement is easiest to make if you use the
percent markings on the scope screen. Put the foot of the square wave on 0% and the
Physics 364
AC Circuits
Figure 1: Lowpass filter
top on 100% (by adjusting the CAL button–be sure to turn it back to “zero” when you’re
done!) Adjust the sweep rate (the time dial) so that most of screen is covered by the fall
from 100% to 37%. Now repeat the procedure to measure the rise time, i.e. the time to
climb from 0% to 63%. Does the rise time also equal RC?
B. Measure the frequency response of the filter by measuring the output voltage at a variety
of frequencies. Apply a sine-wave signal and vary the frequency from 1 Hz to 1 MHz.
Since you will be plotting your data versus log( f ), make at least 2 measurements per
decade of frequency. At each frequency, measure the gain A( f ) = Vout /Vin . Note that
the gain may be less than 1 (indicating the filter is attenuating the signal). Be sure to
measure the input amplitude from the function generator at each frequency, since the
combination of your circuit and limitations of the generator will lead to a signal that
generally decreases in amplitude with frequency. Determine the applied frequency by
measuring the period with the oscilloscope. Also measure the phase difference between
the input and output waveforms. You can measure the time difference between the
output and input signal with the oscilloscope; you need to think about how that relates
to a phase shift. (A good point on the waveforms to use is the point at which the trace
crosses 0 Volts.)
C. Plot A( f ) vs. f , A( f ) vs. log( f ), and log(A) vs. log( f ). The “3 dB point” is the frequency
at which the filter output is 70.7% of the input voltage ( f3dB = 1/2πRC). Confirm that
this corresponds to -3 dB. Use your graphs to determine your experimental value of f3dB .
Does this roughly agree with the predicted value? Add to your graphs the theoretical
prediction for your circuit and discuss the agreement with your data.
Filters of this sort are categorized as either “low pass” or “high pass.” Which type of filter
is this one?
It is very important to be able to estimate the input and output impedance of a circuit in
the limiting cases of very high or very low frequencies. This is actually usually quite easy
to do and involves no calculations (think about how a capacitor behaves at high and low
frequencies.). What is the input impedance of the filter (i.e., the impedance presented
to the function generator) as f → ∞? As f → 0? How does your estimate of the input
impedance agree with your measured values of attenuation?
D. Repeat part 1B & C for the filter circuit shown in Fig. 2.
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Physics 364
AC Circuits
Figure 2: Highpass filter
4
Integrator and Differentiator
A. Now we will see that the circuit in 1A can act as an integrator. In other words, the output
signal is the integral of the input signal. Long before MAPLE and other digital computer
software, analog computers were used to perform mathematical operations. Now such
analog computers are largely obsolete. But simple RC circuits remain incredibly useful.
Drive the integrator with a 100 kHz square wave from the function generator with the
output of the generator at maximum amplitude. What is the output waveform? Why
does this circuit turn a square wave into this waveform? If you are not sure, discuss this
point with your instructor. Now drive the integrator with a triangle wave. Again what
is the output waveform? You should be able to name the mathematical function that
describes this curve. Also try a sine wave.
What range of frequencies does the circuit operate as a filter, and what range as an
integrator?
B. Repeat part A using the circuit shown in Fig. 3. (This circuit is a differentiator instead of
integrator.)
Figure 3: Differentiator
5
References
Significant parts of this lab, including diagrams, are taken from Hayes and Horowitz, Student
Manual for the Art of Electronics, Cambridge University Press (1989).
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Physics 364
AC Circuits
Propagation of Error
The rules for propagating an uncertainty depend on whether the quantity that is being computed is the result of a sum, different, product, or quotient operations. The following rules
guide your calculations.
If q = x + · · · + z − (u + · · · + w), then the uncertainty in q is given by
p
δq = (δx)2 + · · · + (δ y)2 + (δu)2 + · · · + (δw)2 .
If q =
x×···×z
,
(u×···×w)
then
δq
|q|
È
=
δx
x
2
+ ··· +
δz
z
2
+
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δu
u
2
+ ··· +
δw
w
2
.