The Oscilloscope and AC Circuits (Lab 09)

Goals:
The Oscilloscope and AC Circuits
To learn to use an oscilloscope as a DC voltmeter.
To learn to use an oscilloscope to study periodic signals.
To see how the oscillation amplitude of a series RLC circuit depends upon the driving
frequency and exhibits resonance at a particular frequency.
Equipment:
Science Workshop Interface & voltage probes
Phys212 LabKit Battery module
Oscilloscope w/ probes
RLC circuit board
Software: DataStudio, Microsoft Excel
Introduction:
The oscilloscope is an extremely useful instrument for measuring electrical signals that vary with
time. The conventional oscilloscope is essentially like a TV set: it consists of an electron gun that
accelerates a beam of electrons towards a phosphor screen (say, along the z-direction). The beam
is deflected in the x-y plane by two sets of plates that provide an electric field. To measure a
voltage V that varies in a regular manner with time, we provide a periodic voltage to the x-plates
that "sweeps" the electron beam from one side of the phosphor screen to the other — when it
reaches the far right side, the beam snaps back to its starting position. Then, to measure the
waveform of the signal V, we provide the voltage to the y-plates, so that the electron beam goes
up and down, depending on the supplied voltage. When these two motions are combined, the net
result is a "trace" on the phosphor screen that shows the voltage of the signal as a function of
time, much as we might draw on a sheet of graph paper. We will spent the first portion of today’s
lab learning how to use the oscilloscope and how to measure timing and voltage information
directly from the screen.
The oscilloscope is a powerful tool, but our primary concern is to use it to help us visualize and
understand alternating current (AC) circuits. This will involve some concepts that you are
probably still covering concurrently in the lectures and recitations, so we will only cover very
basic ideas in this lab. You are already familiar with Ohm's Law: for an ohmic resistor, V = iR.
This relationship is valid for the voltage V and current i at any given instant in time, and could
tell us the current which flows through a known resistance R when connected to a given voltage
supply V. In direct current (DC) circuits, the current and voltage do not change with time. In an
AC circuit, however, we typically apply a periodic signal, the simplest of which is sinusoidal:
V V 0
sin( t)
.
Note that ω is the angular frequency (measured in radians per second), and is related to the
oscillation frequency f by ω = 2π f.

If we applied this sinusoidal voltage source to a resistor, then Ohm’s law would tell us that the
current through the resistor at any instant of time would be given by
V V0
i sin( t)
I
0
sin( t)
.
R R
Note that every time the voltage V reaches a maximum, the current i reaches a maximum. The
same is true for the voltage and current reaching zero or a minimum. We then say that, for a
resistor, the voltage and current are "in phase." Note also that Ohm’s law relates the amplitudes
V 0 and I 0 .
This behavior is not the same for a capacitor or an inductor when connected to a sinusoidal
voltage source. This is because the voltage and current in these devices are related to each other
through a time derivative. For instance, the voltage across an inductor is proportional to the rate
of change of current (di/dt) and not simply to i. For inductors and capacitors, the voltage and
current are not in phase. You will learn, in fact, that for a single inductor or a single capacitor
connected to a sinusoidal source, the current and voltage are 90° out of phase. In other words,
when V reaches a maximum or minimum, i is zero and vice-versa. For a capacitor, the current i is
90° ahead of the voltage, and in an inductor, the current is 90° behind the voltage. The
amplitudes I 0 and V 0 are related to each other by something that sort of looks like Ohm's Law:
V
0
I
0
V0
I
0
X
C
C
I X I ( L)
0
L
0
for a capacitor
for an inductor
(Eq. 1)
Note that the quantity that looks like a "resistance" (technically called a "reactance") changes
with frequency! From Eq. 1, we can see that a capacitor acts like it has a high reactance at low
frequencies and a low reactance at high frequencies, while for an inductor it's the other way
around.
Today's lab will focus on a "series RLC" circuit in which we connect a resistor, a capacitor and
an inductor in series with a sinusoidal voltage. (We will use the oscilloscope to observe what
happens as we vary the frequency of the driving voltage). The circuit is as shown below:
R
V = V 0 sin(ωt)
C
Figure 1 A series RLC circuit, in which a sinusoidal voltage source V, drives a current through a resistor R, an
inductor L and a capacitor C.
L

For a series RLC circuit, the amplitude of the current is given by
V
0
, (Eq. 2)
Z
I
0
where Z is known as the impedance of the circuit (measured in units of ohms), and is given by
Z R ( X X ) , (Eq. 3)
2 2
L C
At any frequency, the resistance R will always remain constant. The reactances due to the
inductor and the capacitor (X L and X C , respectively), will however vary according to the
frequency of the driving voltage. In Eq. 1, we can see that the reactances of the inductor and the
capacitor vary in different ways with respect to the driving frequency. It stands to reason, then,
that there must be some frequency at which the two reactances exactly balance. At this special
frequency, known as the resonant frequency, we can see that the circuit’s impedance would
reach a minimum, and thus the current amplitude in the circuit would reach a maximum.

Name: ____________________________
Name: ____________________________
Name: ____________________________
Date: ______________
Lab Sect.: __________
Lab Instructor: ______________________
Directions:
In this lab, you are provided with a Tektronix 2213 oscilloscope. The front panel of this
instrument is divided up into six areas:
To the far left is the cathode ray tube (CRT) display.
The next vertical strip contains 6 items, including the power button.
The next two areas contain controls related to the vertical deflection of the electron beam
(Channel 1 and Channel 2).
The last two areas on the right contain controls for the horizontal deflection of the
electron beam, including a time base control for determining the rate at which the
electron beam will sweep across the display and the trigger controls for timing this
sweep.
Before turning the power on, check that the following knobs are in the indicated positions:
1. Auto intensity: turn counterclockwise fully.
2. The three "position" knobs — Channel 1, channel 2 and horizontal -- pointing up.
3. Vertical mode switches: Channel 1
4. Horizontal mode switch: no dly ("no delay")
5. sec/div: 2 ms (2 millisecond/cm)
6. Trigger mode: Auto
Now, push the power switch on. Turn the "Auto Intensity" switch slowly clockwise. You should
see a line appear on the screen. Keep the intensity at the minimum level required for you to
clearly see the line. Prolonged operation at very high intensities can damage the phosphor screen.
If you do not see a line:
Check whether the scope is plugged in.
If the scope is plugged in, push the "beam find" button briefly. This should show you that
the beam is being deflected in a particular direction but is off the screen. You can bring
the beam back by using the "position" knobs.

Lab Activity 1: Using the oscilloscope as a DC voltmenter
Your scope should already have a co-axial cable connected to the "Ch 1" input. The red
connecting wire will be connected to the positive end of any voltage you want to
measure, while the black connecting wire will be connected to the negative (ground)
terminal. In general, it is important to keep to this convention to avoid potentially serious
short circuits!
Note the red knob that is part of the "CH 1 Volts/div" control knob. Gently turn this red
knob back and forth to get a feel for how it works. Then, turn it clockwise as far as it will
go and make sure that it clicks in place. The Channel 1 input is now "calibrated" so that
the numbers on the "volts/div" scale are meaningful! Anytime you use a scope, always
check that the different scale knobs are in the calibrated position!
Now calibrate the time control knob (labeled sec/div). Some of the time dials on the
scopes are mislabeled; if your scope’s time knob has more than one position labeled the
same thing, consult your TAs for what the labels should be.
Connect the red and black ends of the cable to each other to provide a clear input of 0 V.
Adjust the Ch 1 controls as follows:
o Position: adjust this so that the horizontal line is in the center of the display.
o Volts/div: set this to "1" on the 1× side (left side) of the dial. This means that
every vertical displacement of 1 cm (major division on the display scale)
corresponds to 1 V. (Throughout this lab you should adjust this setting as
necessary to obtain more accurate measurements)
o AC-GND-DC switch for CH 1: set this at DC.
Now connect a 1.5-V battery across the scope input, with the black connector to the
"negative" terminal and the red connector to the "positive" terminal.
Note down the vertical deflection of the display line. Try to set the "volts/div" dial to a
value that will give you the most accurate reading.
Repeat the above with the two batteries in series.
Q1. Write down the readings you obtained above.
volts/div
Single battery
vertical deflection [cm]
voltage [V]
volts/div
Two batteries in series
vertical deflection [cm]
voltage [V]

Lab Activity 2: Using the scope to measure the amplitude and frequency of a periodic
voltage signal
Now, it's time to learn how the scope can be used to measure a voltage signal that varies with
time. We'll be using a signal that is provided by the Science Workshop interface box.
First, make sure that there is a pair of wires that are connected to the Science Workshop
interface box at the "output" banana plug outlets (extreme right).
Connect the scope leads to the output leads from the interface box, making sure you
connect the black lead to ground and the red one to the positive output.
Set the scope "CH 1 VOLTS/DIV" controls to measure 1 volt/div.
Set the "sec/div" knob to "5 ms." Make sure the central red knob is in the calibrated
position.
Open DataStudio (start → Programs → Physics → DataStudio), and click on Create
Experiment.
Click on the Output icon (yellow circle) located on the right side of the picture of the
Science Workshop interface box in the main Experiment Setup window.
o From the pull-down menu, select the Sine Wave output.
o Set the Amplitude to 1.000 V.
o Set the Frequency to 100 Hz.
o Click the On button. (If you cannot click On because it is not selectable, make
sure Auto is not selected.)
Observe the oscilloscope display. You should see a sine wave. Discuss amongst your
group how the pattern on the screen is quantitatively related to the frequency of the input
signal.
Try turning the "sec/div" knob to different values and observe how the display changes.
Try varying the frequency of the DataStudio output signal from 30 Hz to 300 Hz.
Observe how the oscilloscope display changes and use the "sec/div" knob to keep the
display at a convenient scale. Does changing this setting change the frequency of the
signal?
Q2. Suppose you want to measure the frequency of a periodic input signal using the scope.
Describe how you would go about doing this, i.e., write down a relationship between the
frequency f of the signal, the setting on the "sec/div" scale, and the size of the pattern on the
scope’s grid (“size of the pattern” means either the horizontal or vertical properties of the
waveform, it is up to you to decide which).

Lab Activity 3: Using the oscilloscope to analyze a series RLC circuit
Now for a real circuit measurement! You have been provided with a circuit board that contains a
number of different components connected together. The large white coil is the inductor. To
increase its inductance, remove the metal rod that is held on the circuit board and place it
vertically in the inductance coil. Notice that on this pre-fabricated circuit board, a white line
between components indicates the presence of a conducting wire connecting those components.
Select the output of the Science Workshop interface box to have an amplitude of 1 V and
a frequency of 300 Hz.
Then, connect the output of the interface box across a series combination of the 10-Ω
resistor, the inductor and the 100-μF capacitor. (Essentially, one lead goes to the free end
of the resistor and the other goes to the free end of the capacitor, with the other
connections already taken care of on the board itself.)
Making sure that you keep the polarities right, connect the input leads of the oscilloscope
across the 10-Ω resistor.
Note: Both the black lead of the oscilloscope and the black lead of the output from the Science
Workshop interface are hooked internally to a common “ground”. If these leads are not
connected to the same point in your circuit, then the circuit is connected to ground (V = 0) at two
different points and you will not be measuring what is intended. If this is the case, rearrange your
circuit so as to have the black leads connected together, with the output of the interface box
applied across the series combination and the oscilloscope arranged to measure the voltage
across the resistor.
Recall that the resistor is an ohmic device, and so the voltage across the resistor is always
proportional to the current through the resistor. We will use this to determine at what frequency
the current amplitude reaches its maximum. So, your goal will be to measure the amplitude of
the voltage across the resistor as a function of frequency of the output voltage.
Vary the frequency down from 300 Hz to 30 Hz, measuring the amplitude of the voltage
at every frequency in Table 1. You will have to adjust both the "sec/div" horizontal scale
as well as the "volts/div" vertical scale as you go through this measurement.
Q3. Using an Excel spreadsheet, plot the voltage amplitude versus frequency. Include a printout
of the graph with your lab report.
Q4. At approximately what frequency is the voltage amplitude a maximum? (Again, this is
known as the "resonant frequency" of the circuit.)
Resonant frequency: _______________________

Driving
Frequency f
[Hz]
300
250
225
200
175
150
125
100
90
80
70
60
50
45
40
35
30
Resistor Voltage
Amplitude
[V]
Table 1 Measured voltage amplitude across resistor in a series RLC circuit as a function of driving frequency
Q5. We saw in the introduction that the impedance Z is a minimum at this particular "resonant"
frequency. Based on the expression for the impedance Z in Eq. 3 and the condition described
above, derive a relationship between the resonant angular frequency ω 0 , the capacitance C, and
the inductance L. Be sure to show how you arrived at your result.
Use this relationship and the value of C = 100 μF to deduce the value of the inductance L.

Q6. Now, we will investigate the effect of the iron bar on has on the circuit. Set the frequency
to one much lower than the resonance frequency you found above. Observe the pattern with the
iron bar in the inductor and then remove the bar. Next, set the frequency to one much higher
than the resonance frequency you found above. Again, observe the pattern with the iron bar in
the inductor and then remove the bar. What change occurred at each frequency when you
removed the iron bar? For which frequency was the change more pronounced? Why would the
change be more pronounced for this frequency?