Lissajous patterns are formed by displaying two periodic waves at right angles to each other, such as by applying different signals to the horizontal and vertical inputs of an oscilloscope. This technique allows measurement of phase and frequency relationships. When the frequencies are equal and in phase, a diagonal line is produced. Experiments were conducted to measure unknown frequencies and phase differences using Lissajous figures. Frequency was determined by counting intersections with reference lines. Phase was calculated using the ellipse parameters and equation 1.1. Problems with unstable patterns and parallax error were addressed.

1. LISSAJOUS PATTERNS (INTRODUCTION)
OBJECTIVE
1. To use Lissajous figures to take phase measurements.
2. To use Lissajous figures to take frequency measurements.
LIST OF REQUIREMENTS
Equipment
1. General purpose oscilloscope (10MHz)
2. Function generators (1 Hz to 1 MHz)
3. Digital multimeter
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2. THEORY
Lissajous patterns are formed when you combined periodic waves moving back and
forth with periodic waves moving up and down. This exhibit does this electronically allowing
the visitor to control the frequency of the X and Y motions independently. The resulting
pattern can be observed on an oscilloscope and viewed on moving speakers. If the
frequencies are high enough, the speakers produce sound. The combination of tones can be
correlated with the scope pictures.
The oscilloscope can measure voltage, frequency, and phase angle. Two-channel
measurements are very useful and are presented.We also can get lissajous figures in
oscilloscope other than to get the waveform figures.
In electronic applications we can generate Lissajous patterns by applying different
signals to the horizontal and vertical inputs of an oscilloscope. In facts, this techniques was
often to used to measure frequencies in the days before frequency meters. A signal of a
known frequency was applied to the vertical input. The resulting pattern was a function of the
ratio of the two frequencies.
Lissajous Figures
A lissajous figure is produced by taking two sine waves and displaying them at right
angles to each other. This easily done on an oscilloscope in XY mode.If the oscilloscope has
the x-versus-y capability, one can apply one signal to the vertical deflection plates while
applying a second signal to the horizontal deflection plates. The horizontal sweep section is
automatically disengaged at this time. The resulting waveform is called Lissajous figure. This
mode can be used to measure phase or frequency relationships between two signals.
When the two sine waves are of equal frequency and in-phase, diagonal line to the
right will produced. When the two sine waves are of equal frequency and 180 degrees out-
of-phase a diagonal line to the left will produced. When the two sine waves are of equal
frequency and 90 degrees out-of-phase a circle will produced. When the horizontal and
vertical sine wave frequencies differ by a fixed amount, this is equivalent to constantly
rotating the phase between them
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3. Phase Measurements Using Lissajous Figures
Lissajous figures are sometimes used for the measurement of phase. They are
produced in an oscilloscope by connecting one signal to the vertical trace and the other to
the horizontal trace. If the ratio of the first frequency to the second is a rational number, then
a closed curve will be observed on the CRO. If the two frequencies are unrelated, then there
will be only a patch of light observed because of the persistence of the oscilloscope screen.
If the two signals have the same frequency, then the Lissajous figure will assume the
shape of an ellipse. The ellipse’s shape will vary according to the phase difference between
the two signals, and according to the ratio of the amplitude of the two signals. Below shown
some figures for two signals with synchronized frequency and equal amplitudes, but different
phase relationships. The formula used for determining the phase is:
sin(θ)=± (equation 1.1)
Where Ymax is half the maximum vertical height of the ellipse and Yo is the intercept
on the y-axis. Two signals that are identical in frequency and have a phase difference of 450,
but with different amplitude ratios. Note that is necessary to know the direction that the
Lissajous trace is moving in order to determine the sign of the phase difference. In practice,
if this is not known a priori, then it can be determined by testing with a variable frequency
signal generator. In the case, one of the signals under consideration is replaced with the
variable frequency signal. The signal generator is adjusted until its frequency and phase
equal that of the other signal input the CRO. When this happens, a straight line will exist.
The signal generator frequency is then increased a little with the relative phase thus being
effectively changed in a known direction. This can be used to determine the correct sign in
the equation above.
Lissajous figure methods are a little more robust to noise than direct oscilloscope
methods. This is because there are no triggering problems due to random noise fluctuations.
Direct methods are, however, much easier to interpret when harmonics are present. The
accuracy of oscilloscope methods is comparatively poor. The uncertainly of the
measurement is typically in excess of 10.
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4. Frequency Measurements Using Lissajous Figures
Lissajous pattern also helps to measure frequency. The signal whose frequency is to
be measured is given 1 F plates or vertical plates and the signal whose frequency is given to
X-plates or horizontal plates. Now the know frequency or standard frequency is a adjusted
so Lissajous patterns can be obtained on the screen which depends on the ratio of two
frequencies. In the given figure,
Let, fy – Unknown frequency signal applied to vertical plates. Fh – Known frequency signal
applied to horizontal plates.
Two lines are drawn, one vertical and one horizontal so that they do not pass through
any intersection on Lissajous pattern. Then the number of intersections of the horizontal and
vertical lines with the Lissajous patterns and counted separately. So after finding the
tangencies if we know we can easily calculate the unknown frequency applied to vertical
plate.
All electronic circuits in the oscilloscope like attenuators, time base generators,
amplifiers cause some amount of time delay while transmitting signal voltage to deflection
plates. We also know that horizontal signal is initiated or triggered by some portion of output
signal applied to vertical plates of CRT. So the delay line is used to delay the signal for some
time in the vertical section of CRT.
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5. PROCEDURES
PART A : FREQUENCY MEASUREMENT USING LISSAJOUS FIGURES
1. Two signal generators was used and the first generator (generator 1) was considered
as the known or standard frequency source. Assumed that it has correct dial
markings. Generator 2 will be the variable (unknown) source.
2. Generator 1 was set at 1 kHz. The signal voltage was set at 4V peak-to-peak. Vary
the frequency of generator 2 until a circular pattern occurs on the scope.
3. This pattern was recorded in table 3.2. The number of vertical and horizontal
tangencies was recorded and the frequency of the generator 2 was calculated(A
vertical tangencies is a point where the figure is tangent to the vertical axis).
4. These procedures was repeated for other convenient ratios and complete Table 3.2.
Used at least one other standard frequency for the last three entries ( e.g., 3 kHz).
PART B : PHASE MEASUREMENTS USING OSCILLOSCOPE AND LISSAJOUS
FIGURES
1. The circuit was connected as in Figure 3.8.
2. The frequency of Ein was set at 1 kHz. R was set at 0 Ω. The signal voltage was set
at 4 V peak-to-peak. The display was centered. R was changed to 10 kΩ and the
pattern in was recorded in Table 3.3. The measured and calculated values was
recorded in Table 3.3, for different values of R.
Figure 3.8
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6. RESULT(Part A)
Lissajous No.Of Vertical No. Of Frequency of Frequency of
Figure Tangencies Horizontal Generator Generator
Tangencies 1(Hz) 2(Hz)
1 2 500Hz 1kHz
2 1 3kHz 1.5kHz
1 3 500Hz 1.5kHz
2 3 1kHz 1.5kHz
3 2 3kHz 2kHz
1 2 1kHz 2kHz
1 4 500 2kHz
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7. RESULT(Part B)
Resistance Lissajous Figure Measure Values Calculated
(kΩ) Values
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2.4 2.8 58.99 57.86
15
2.8 3.8 47.46 46.7
22
1.2 2.0 36.87 35.88
30
2.2 4.8 27.28 27.95
50
0.8 2.6 17.92 17.66
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8. Discussion
In this experiment we was exposed to another function of oscilloscope which are to
measure frequency and to measure phase angle and also to display lissajous pattern.
From the experiment part A.We can tell that the lissajous pattern are formed by two
frequency that was given.Based on our result we can tell that the No. Of Horizontal
Tangencies and No.of Vertical Tangencies are the ratio of the two input frequency.So we
can tell here that Lissajous Figures are affected by the ratio and the value of the two input
frequency.We also know when high frequency given the pattern will move faster.
From experiment part B.we conducted an experiment to calculated phase angle by
the help of figure in oscilloscope.We can measure the phase angle using the equation 1.1.In
this experiment we also get the info how the pattern when we want to find the phase angle
which is in a circle type.The circle become thinner when it nearly to the 0° angle.
The problem that we get in this experiment is when we get unstable lissahous
pattern.After we tracking the source of this problem,we found that the problem is on the
connecting wire probe which maybe already broken .So we replace it with another ,then we
get a stable result.Another problems is when we calculated the phase angle for the 1st
calculation is not same as the 2nd calculation.This is because when we want measure using
equation 1.1.Our Eyes does not perpendicular with the scale on the oscilloscope which
know as parallax error.So we corrected it by making our eyes perpendicular with the scale
on the oscilloscope.Then we get same calculated value.Other problem is when we want to
take the displayed lissajous pattern .It moving too fast .To take it we need to press stop
button at the oscilloscope to see the clear lissajous pattern.
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9. CONCLUSION
As our conclusion .We can tell that oscilloscope can measure the phase and the
frequency of two frequency.From here we know that oscilloscope has very many function not
limited only to display waveform.The oscilloscope will help us to calculate the phase or the
frequency by his formed lissajous figure.We can conclude that experiment part A show us
that lissajous patern are affected by the ratio of two input frequency while for the experiment
part B that the phase angle below 90° will produce a thinner curve . When achieve to 0° , the
curve become straight line.The calculated phase angle and the calculated phase angle by
the help of oscilloscope is slightly difference.This is because when we calculated phase
angle roughly we round off the number to have a easier measurement.While we taking the
result we need to press stop button in oscilloscope because the figures is moving too
fast.From here we can conclude that the more the frequency the faster the figures move.
.
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10. REFERENCE
1. Rusnani Ariffin & Mohd Aminudin Murad (2011).Laboratory Manual Electrical
Engineering Laboratory 1:University Publication Center (UPENA).
2. Charles K.Alexander & Mattgew N.O.Sadiku (2009).Fundamentals Of Electrical
Circuit Fourth Edition:McGRAW HILL.
3. http://www.jmargolin.com/mtest/LJfigs.htm
4. http://www.allaboutcircuits.com/vol_2/chpt_12/2.html
5. http://ei-notes.blogspot.com/2012/03/method-of-finding-phase-frequency.html
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