Name:
Grade:
/26
Section Number:
Laboratory 3
Sinusoids in Engineering: Measurement and Analysis
of Harmonic Signals
3.1
Laboratory Objective
The objective of this laboratory is to understand the basic properties of sinusoids and sinusoid
measurements.
3.2
Educational Objectives
After performing this experiment, students should be able to:
1. Understand the properties of sinusoids.
2. Understand sinusoidal addition.
3. Obtain measurements using an oscilloscope.
3.3
Background
Sinusoids are sine or cosine waveforms that can describe many engineering phenomena. Any
oscillatory motion can be described using sinusoids. Many types of electrical signals such as
square, triangle, and saw-tooth waves are modeled using sinusoids. Their manipulation incurs the
understanding of certain quantities that describe sinusoidal behavior. These quantities are
described below.
3.3.1 Sinusoid Characteristics
Amplitude The amplitude A of a sine wave describes the height of the hills and valleys of a
sinusoid. It carries the physical units of what the sinusoid is describing (volts, amps, meters, etc).
Frequency There are two types of frequencies that can describe a sinusoid. The normal
frequency f is how many times the sinusoid repeats per unit time. It has units of cycles per
second or Hertz (Hz). The angular frequency ω is how many radians pass per second.
Consequently, ω has units of radians per second.
Period The period T is the time it takes a sinusoid takes to complete one cycle. The period is
measured in seconds.
Phase The phase
radians.
of a sinusoid causes a horizontal shift along the t-axis. The phase has units of
1
Time Shift The time shift ts of a sinusoid is a horizontal shift along the t-axis and is a time
measurement of the phase. The time shift has units of seconds.
NOTE: A sine wave and cosine wave only differ by a phase shift of 90° or
they are the same waveform but with a different .
radians. In reality,
3.3.2 Oscilloscope
An oscilloscope is a device used to measure different parameters from an electrical signal (see
characteristics listed Section 3.3.1). It displays voltage waveforms versus time as a result of
connecting the oscilloscope to an external signal generator.
3.3.3 Controls
Volts/Division: Vertical grid that controls the amplitude (sets number of volts per box in the
vertical direction).
Time/Division: Horizontal grid that controls the time scale (sets how much time each box in the
horizontal direction is valued at).
3.3.4 Sinusoidal Relationships
Figure3.1:Sinusoid
The general equation of a sinusoid is given below and refers to Figure 3.1.
(3.1)
The angular frequency is related to the normal frequency by Equation 3.2.
(3.2)
The angular frequency is also related to the period by Equation 3.3.
(3.3)
2
By inspection, the normal frequency is related to the period by Equation 3.4.
(3.4)
The time shift is related to the phase (radians) and the frequency by Equation 3.5.
(3.5)
3.3.5 Lab 3 Procedure
**NOTE: If you are working with the HP Oscilloscopes, proceed to Group A of the
procedures. If you are working with the Tenma Oscilloscopes, proceed to Group B
of the procedures.
GROUP A
1. Connect the output channel of the Function Generator to Ch1 of the oscilloscope.
2. The following steps will guide the user in setting up the function generator and
oscilloscope.
a. Turn on the function generator and set frequency to those found in Table 3.1.
b. Within the function generator, set the amplitude to 1V. Make sure to select the
sine wave for wave type by clicking the button with a sine wave pictured on it.
c. Turn on the oscilloscope and locate the Ch1 volts/div knob. Turn this knob until
the number in the top left corner of the oscilloscope reads 500 mV.
d. Turn the time/div knob until the number to the right of the center reads 500microseconds.
e. To complete the table: enter the frequency value and measure the period by
counting the boxes/tic marks on the oscilloscope (your Lab TA will explain this in
more detail). In the calculated period and angular frequency columns, use the
equations found above to calculate those values.
GROUP B
1. Connect the output channel of the Function Generator to Ch1 of the oscilloscope.
2. The following steps will guide the user in setting up the function generator and
oscilloscope.
a. Turn on the function generator and press the button labeled 5K, which can be
found in the top row of buttons. Then make sure the sine wave is selected from
the wave shape options in the top right.
b. Turn on the oscilloscope and set the Volts/Div, using the Ch1 corresponding
knob, to 0.5 Volts. Set the Time/Div as 0.5 milliseconds.
c. After entering the first frequency value in Table 3.1, rotate the output knob on the
function generator until the distance between the peak and the trough is the length
of two blocks on the oscilloscope.
d. In the calculated period and angular frequency columns, use the equations found
above to calculate those values.
3
3. Complete Table 3.1 using the given values for frequency. Attach hand calculations at the
end of the lab.
Table 3.1 Sinusoid Measurements
Function Generator
Frequency (Hz)
Oscilloscope (Measured)
T (sec)
Calculated
(rad/sec)
T (sec)
1000
1500
2000
2500
Amplitude? (Hint: Use Volts/Division value to determine amplitude):
Figure3.4:SignalsComparedtotheFunctionGenerator
Figure3.2:SignalsComparedtotheFunctionGenerator
4. Using an oscilloscope, make measurements across the two separate resistors and
complete Table 3.2 below. Use f = 2500 Hz. (For Group B: if waves disappear when
setting 2500Hz frequency, turn output knob on Function Generator slightly until waves
appear again).
5. Connect Channel 1 and Channel 2 of the oscilloscope as shown in Fig. 3.2 above and
measure the time shift (ts) by counting the tic marks separating two corresponding points
on each sinusoid. Convert the time shift ts with the following equation:
4
Table 3.2 Sinusoid Measurements
Signal
VC
VL
ts (sec)
(rad)
(degrees)
6. Attach hand calculations for Tables 3.1 and 3.2 at the end of the table.
3.4
Lab Requirements
1. Complete Tables 3.1 and 3.2 (2 points each, 4 total)
2. Write an abstract for this lab and submit it to the Lab 3 folder in your lab section’s
abstract folder found in the Pilot Dropbox. (Required to pass course)
3. Show hand calculations for Tables 3.1 and 3.2. (2 points each, 4 total)
4. Explain the difference between Volts/Division and Time/Division. (4 points)
5. Generate a single MATLAB plot that includes all of the following equations and insert
after this page. (2 points each, 6 total)
Vs(t) =
Vc(t) =
Vl(t) =
Where
, f=2500Hz, and time is 0 < t < 2*T with an increment of 0.00001.
(Note: T=1/f, w=2*pi*f)
6. Using the measured values in Table 3.1, sketch one cycle of a sinusoid for the
frequencies 1500Hz and 2500Hz on a separate sheet of paper and insert after this page. (2
points each, 4 total)
7. Through circuit analysis,
together. Assume that
,
(4 points)
5
Verify this by adding sinusoid equations
, and