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Table of Contents
Background: lissajous figures, create a lissajous pattern, subtracting two signals, activity: the lissajous pattern, a classic phase measurement.
All periodic signals can be described in terms of amplitude and phase. To perform amplitude, frequency, and phase measurements using an oscilloscope and to make use of Lissajous figures for phase and frequency measurements.
We all learn that in basic circuit theory classes. You surely have needed to calculate a signal's phase change when it passes through a circuit. Fortunately, you can measure phase on the lab bench with oscilloscope hardware such as the ADALM1000 and its accompanying ALICE desktop software using several methods.
As in all the ALM labs we use the following terminology when referring to the connections to the M1000 connector and configuring the hardware. The green shaded rectangles indicate connections to the M1000 analog I/O connector. The analog I/O channel pins are referred to as CA and CB. When configured to force voltage / measure current –V is added as in CA- V or when configured to force current / measure voltage –I is added as in CA-I. When a channel is configured in the high impedance mode to only measure voltage –H is added as CA-H. Scope traces are similarly referred to by channel and voltage / current. Such as CA- V , CB- V for the voltage waveforms and CA-I , CB-I for the current waveforms.
Lissajous (pronounced LEE-suh-zhoo) figures were discovered by the French physicist Jules Antoine Lissajous. He would use sounds of different frequencies to vibrate a mirror. A beam of light reflected from the mirror would trace patterns which depended on the frequencies of the sounds. Lissajous' setup was similar to the apparatus which is used today to project laser light shows.
Before the days of digital frequency meters and phase-locked loops, Lissajous figures were used to determine the frequencies of sounds or radio signals. A signal of known frequency was applied to the horizontal axis of an oscilloscope, and the signal to be measured was applied to the vertical axis. The resulting pattern was a function of the ratio of the two frequencies.
Lissajous figures often appeared as props in science fiction movies made during the 1950's. One of the best examples can be found in the opening sequence of The Outer Limits TV series. (“Do not attempt to adjust your picture--we are controlling the transmission.”) The pattern of criss-cross lines is actually a Lissajous figure.
The phase difference, or phase angle, is the difference in phase between the same points, say a zero crossing, in two different waveforms with the same frequency. A common example is the phase difference between the input signal and output signal after it passes through a circuit, cable, or PC board trace. A waveform with a leading phase has a specific point occurring earlier in time than the same point on the other signal. That would be the case of when a signal passes through, a capacitor: the current in the capacitor will lead the voltage across the capacitor by 90º. Conversely, a waveform with lagging phase has a specific point occurring later in time than the other paired signal waveform. Two signals are in opposition if they are 180º out of phase. Signals that differ in phase by ±90º are in phase quadrature (one quarter of 360º).
The time (phase) relationship between two sine waves can of course be measured from a time domain plot such as figure 1. The Time measurement capabilities of most any Oscilloscope, the ADALM1000 and the ALICE software included, can display the relative phase between channel A and channel B in degrees and/or the time delay between A and B. The software scans the waveforms looking for the time points where they cross their average value (zero crossing with DC offset removed). It then uses those time points to report frequency, period, phase, delay, duty-cycle etc. Noise and jitter will introduce errors in the results.
Figure 1, Time plot of two sine waves.
Old timers who have started out their careers using an analog oscilloscope probably remember using the classic Lissajous pattern to measure the phase difference of two sine waves. It can be measured by cross plotting the two sine waveforms on the X-Y display in ALICE as shown in Figure 2. In this figure, the voltage waveform on channel A provides the horizontal or X displacement. Channel B provides the vertical or Y deflection. The Lissajous pattern indicates the phase difference by the shape of the X-Y plot. A straight line indicates a 0º or 180º phase difference. The angle of the line depends on the difference in amplitude between the two signals, a line at 45º to the horizontal means the amplitudes are equal. While a circle indicates a 90º difference. It will only be a true circle if the amplitudes are equal. Phase differences between 0º and 90º appear as tilted ellipses and phase is determined by measuring the maximum vertical deflection (Ymax) and the vertical deflection at zero horizontal deflection (Yx=0). In Figure 2, cursors mark these two locations on the X-Y plot. Note that this is only valid if the X-Y plot is centered on 0,0. Any DC offset in the two waveforms must be removed first.
Figure 2 Using a classic Lissajous display lets you measure the phase difference between two sine waves.
Marker readouts in upper left of the plot show the required values for computing the phase difference.
Φ2 - Φ1 = ± sin−1 (Yx=0/Ymax) for when the top of the ellipse is located in quadrant 1 (Q1)
Φ2 - Φ1 = ± 180-sin−1 (Yx=0/Ymax) for when the top of the ellipse is located in quadrant 2 (Q2)
The sign of the phase difference is determined by inspecting the channel time traces.
In the figure 2 example, the Ymax value is 1.538, YX=0 is 1.064, and the top of the ellipse is in Q1: Φ2 - Φ1 = ± sin−1 (1.064/1.538) = ± sin−1 (0.692) = 44º
The accuracy of this method is dependent on the placement of the cursors but it produces reasonable results with certain artistic panache.
Hardware like the ADALM1000 and ALICE desktop software offer multiple techniques to measure phase. Direct measurement in the time domain supports both static and dynamic measurements of phase. Frequency domain based calculation provides somewhat more accurate results for static phase measurements but requires you to take the difference of the FFT phase data at the fundamental frequency.
Materials: ADALM1000 hardware module
The default function of an oscilloscope is to display voltage signals on the Y axis vs time on the X axis. The ALICE software has a special function that allows the user to plot one signal on the X axis and the other on the Y axis.
Begin with both AWG generators set for 1.0 Min and 4.0 Max values and a frequency of 1 kHz . Use the Time display to ensure that both waveform generators are producing the same signal.
You should see two nearly identical sine waves, both in phase with the other. Remember that the sine wave is defined by three parameters – amplitude, frequency and phase. Check to see that the vertical range and position settings for both channels are the same. (Use the V /Div to adjust this if needed.) If the amplitudes are not the same, the sine waves will not be the same amplitude. If the phases are not the same, the sine waves will not line up horizontally. If the frequencies are not the same, the wave the oscilloscope is triggered on will be stationary, while the other wave will move (slowly we hope) to the left or right. Figure 3 shows signals that have different amplitude and frequency (and DC offset).
Figure 3, Two sine waves that vary in amplitude and frequency.
We can now generate a Lissajous pattern by opening the X-Y Plotter tool. Select CA- V for the X Axis and CB- V for the Y Axis. Compare what you see to the examples of Lissajous figures. Use your favorite screen capture software tool to take a picture of the image and save it. Your TA or instructor can help you with this. Include this picture with your report. [Hint: Use the STOP button to freeze your figure at the point you want to take a picture.]
You should now play with the AWG settings and produce several other representative patterns. Create one pattern that you find particularly interesting. Take a picture of it with the screen capture software.
Next, we will do a different kind of comparison of the two sine waves, one that will prove to be very important in the development of measurement techniques. In this measurement, we will compare two signals to see how close they are to one another by subtracting one from the other.
Go back to the Time display. This should the two sine waves plotted vs time. Adjust them again so that they are as identical as possible. (Try to get them displayed on top of one another by using the Min, Max, Freq and Phase controls in the AWG generator controls.)
Click on the Math Button to open the Math function controls. Click on the Built In Expression list and select CAV-CBV, which will produce a third trace that is the difference between the two channels. You will need to increase the V /Div to 1.0 on both channels to see the difference signal. The two AWG channel outputs have DC offset which should also sum to zero if they are the same. Also adjust the vertical position such that the traces do not go off the grid.
If you have adjusted the two sine waves to be identical, their difference should be zero. How well did you do? Note that the amplitude, the frequency, and the phase must be identical to make the difference zero.
Now make the two signals as identical as possible by adjusting the difference signal away. What did you have to do?
From this activity, you should see the value of comparing one signal against some kind of known reference signal. It is possible to tune a guitar, for example, by comparing the tone a string makes with an electronic reference tone. This results in a perfectly tuned instrument, even when the player has less than perfect pitch. You have also seen how we make what are called differential measurements. There are many advantages to making differential over absolute measurements. You have seen one of the key reasons since differential measurements allow you to focus on smaller quantities since you are working with the difference between two signals.
For Further Reading:
Lissajou Curves Lissajou Curves
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| The optical production of the curves was first demonstrated in 1857 by Jules Antoine Lissajous (1833-1880), using apparatus similar to that at the left. Today we can do the same experiment more easily with a laser beam that reflects from the two mirrors vibrating at right angles to each other and then traces the Lissajous figure on the wall. |
| On the left is a pair of tuning forks permanently mounted at right angles to each other. The apparatus is shown in the 1900 catalogue of Max Kohl at a price of 66 Marks. It is in the collection at St. Mary's College in Notre Dame Indiana. |
| The very heavy piece of cast-iron apparatus at the right sits in my living room, where it menaces the unwary toe. Both tuning forks can be driven, allowing steady-state Lissajous figures to be displayed. At the back is a device once used to hold a light source, and a previous owner has mounted a lens on the base of the device. The light source system can be moved right and left with a crank system. This apparatus, without the light-source holder, is listed at 195 Marks in the 1900 catalogue of Max Kohl of Chemnitz, Germany. Another device that relies on Lissajous Figures is the made by Newton & Co. of London. |
| Another Max Kohl device for producing Lissjous figures is at the left. Both tuning forks are electrically driven, and two additional forks of different frequencies are hanging on the rear side of the apparatus, This demonstration is in the collection of Hobart and William Smith Colleges of Geneva, New York, and was bought in the late 1920s. |
Thomas B. Greenslade, Jr., "Nineteenth Century Textbook Illustrations, XXVII...Harmonographs, Phys. Teach. , 17 , 256-8 (1979) Thomas B. Greenslade, Jr. "Apparatus for Natural Philosophy: Nineteenth Century Wave Machines", Phys. Teach., 18 , 510-517 (1980) George M. Hopkins, Experimental Science , Vol. II, Twenty-Seventh Edition (Munn & Company, New York, 1911), pp 127-129 
Lissajous Figures
Lissajous figure is the pattern which is displayed on the screen, when sinusoidal signals are applied to both horizontal & vertical deflection plates of CRO. These patterns will vary based on the amplitudes, frequencies and phase differences of the sinusoidal signals, which are applied to both horizontal & vertical deflection plates of CRO. The following figure shows an example of Lissajous figure.  The above Lissajous figure is in elliptical shape and its major axis has some inclination angle with positive x-axis. Measurements using Lissajous FiguresWe can do the following two measurements from a Lissajous figure.
Now, let us discuss about these two measurements one by one. Measurement of FrequencyLissajous figure will be displayed on the screen, when the sinusoidal signals are applied to both horizontal & vertical deflection plates of CRO. Hence, apply the sinusoidal signal, which has standard known frequency to the horizontal deflection plates of CRO. Similarly, apply the sinusoidal signal, whose frequency is unknown to the vertical deflection plates of CRO Let, $f_{H}$ and $f_{V}$ are the frequencies of sinusoidal signals, which are applied to the horizontal & vertical deflection plates of CRO respectively. The relationship between $f_{H}$ and $f_{V}$ can be mathematically represented as below. $$\frac{f_{V}}{f_{H}}=\frac{n_{H}}{n_{V}}$$ From above relation, we will get the frequency of sinusoidal signal, which is applied to the vertical deflection plates of CRO as $f_{V}=\left ( \frac{n_{H}}{n_{V}} \right )f_{H}$ (Equation 1) $n_{H}$ is the number of horizontal tangencies $n_{V}$ is the number of vertical tangencies We can find the values of $n_{H}$ and $n_{V}$ from Lissajous figure. So, by substituting the values of $n_{H}$, $n_{V}$ and $f_{H}$ in Equation 1, we will get the value of $f_{V}$ , i.e. the frequency of sinusoidal signal that is applied to the vertical deflection plates of CRO. Measurement of Phase DifferenceA Lissajous figure is displayed on the screen when sinusoidal signals are applied to both horizontal & vertical deflection plates of CRO. Hence, apply the sinusoidal signals, which have same amplitude and frequency to both horizontal and vertical deflection plates of CRO. For few Lissajous figures based on their shape, we can directly tell the phase difference between the two sinusoidal signals. If the Lissajous figure is a straight line with an inclination of $45^{\circ}$ with positive x-axis, then the phase difference between the two sinusoidal signals will be $0^{\circ}$. That means, there is no phase difference between those two sinusoidal signals. If the Lissajous figure is a straight line with an inclination of $135^{\circ}$ with positive x-axis, then the phase difference between the two sinusoidal signals will be $180^{\circ}$. That means, those two sinusoidal signals are out of phase. If the Lissajous figure is in circular shape , then the phase difference between the two sinusoidal signals will be $90^{\circ}$ or $270^{\circ}$. We can calculate the phase difference between the two sinusoidal signals by using formulae, when the Lissajous figures are of elliptical shape . If the major axis of an elliptical shape Lissajous figure having an inclination angle lies between $0^{\circ}$ and $90^{\circ}$ with positive x-axis, then the phase difference between the two sinusoidal signals will be. $$\phi =\sin ^{-1}\left ( \frac{x_{1}}{x_{2}} \right )=\sin ^{-1}\left ( \frac{y_{1}}{y_{2}} \right )$$ If the major axis of an elliptical shape Lissajous figure having an inclination angle lies between $90^{\circ}$ and $180^{\circ}$ with positive x-axis, then the phase difference between the two sinusoidal signals will be. $$\phi =180 - \sin ^{-1}\left ( \frac{x_{1}}{x_{2}} \right )=180 - \sin ^{-1}\left ( \frac{y_{1}}{y_{2}} \right )$$ $x_{1}$ is the distance from the origin to the point on x-axis, where the elliptical shape Lissajous figure intersects $x_{2}$ is the distance from the origin to the vertical tangent of elliptical shape Lissajous figure $y_{1}$ is the distance from the origin to the point on y-axis, where the elliptical shape Lissajous figure intersects $y_{2}$ is the distance from the origin to the horizontal tangent of elliptical shape Lissajous figure In this chapter, welearnt how to find the frequency of unknown sinusoidal signal and the phase difference between two sinusoidal signals from Lissajous figures by using formulae.
/prod01/prodbucket01/media/durham-university/departments-/physics/teaching-labs/VT2A9034-1998X733.jpeg "to study lissajous figures experiment") Lissajous FiguresIntroduction. When using an oscilloscope, we can plot one sinusoidal signal along the x-axis against another sinusoidal signal along the y-axis. The result is a Lissajous figure. Lissajous figures tells us about the phase difference between the two signals and the ratio of their frequencies. Five such figures are shown in figures 1-5 below. /prod01/prodbucket01/media/durham-university/departments-/physics/labs/lissajous1.jpg "to study lissajous figures experiment") Fig. 1: Two sine waves of equal frequency, in phase. /prod01/prodbucket01/media/durham-university/departments-/physics/labs/lissajous2.jpg "to study lissajous figures experiment") Fig. 2: Two sine waves of equal frequency, 180 degrees out of phase. /prod01/prodbucket01/media/durham-university/departments-/physics/labs/lissajous3.jpg "to study lissajous figures experiment") Fig. 3: Two sine waves of equal frequency, 90 degrees out of phase. /prod01/prodbucket01/media/durham-university/departments-/physics/labs/lissajous4.jpg "to study lissajous figures experiment") Fig. 4: Two sine waves, in phase, frequency of horizontal wave twice frequency of vertical wave. /prod01/prodbucket01/media/durham-university/departments-/physics/labs/lissajous5.jpg "to study lissajous figures experiment") Fig. 5: Two sine waves, in phase, frequency of horizontal wave three times frequency of vertical wave.
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1. The Lissajous pattern shown in figure.4 is observed on the CRT screen. Find the phase shift between the signals applied to the X and Y inputs of the scope. 4 2 2 4 y(cm.) x(cm.) Figure.4 2. Figure 5 shows a Lissajous pattern observed on the CRT screen. Determine the frequency relationship between the signals applied to the X and Y inputs of ...
The Lissajous pattern indicates the phase difference by the shape of the X-Y plot. A straight line indicates a 0º or 180º phase difference. The angle of the line depends on the difference in amplitude between the two signals, a line at 45º to the horizontal means the amplitudes are equal. While a circle indicates a 90º difference.
The optical production of the curves was first demonstrated in 1857 by Jules Antoine Lissajous (1833-1880), using apparatus similar to that at the left. Today we can do the same experiment more easily with a laser beam that reflects from the two mirrors vibrating at right angles to each other and then traces the Lissajous figure on the wall.
Advertisements. Lissajous Figures - Lissajous figure is the pattern which is displayed on the screen, when sinusoidal signals are applied to both horizontal & vertical deflection plates of CRO. These patterns will vary based on the amplitudes, frequencies and phase differences of the sinusoidal signals, which are applied to both horiz.
Lissajous figures, named after the French physicist Jules Antoine Lissajous, are complex patterns formed by the interaction of two perpendicular harmonic oscillations. These intriguing patterns are not just aesthetically pleasing but also hold significant value in the study of physics and engineering. This article delves into the world of ...
Colorado State University via OpenStax CNX. Lissajous figures are figures that are turned out on the face of an oscilloscope when sinusoidal signals with different amplitudes and different phases are applied to the time base (real axis) and deflection plate (imaginary axis) of the scope. The electron beam that strikes the phosphorous face then ...
EXPERIMENT 9: LISSAJOUS FIGURES PREPARATION Read the attached Notes on Lissajous Figures. EP-9: Find the amplitude ratio and phase shift for each of the three Lissajous figures in Fig. E9-4. EXPERIMENT Part A -- Phase-Shift Network The RC phase-shift network in Fig. E9-1 is prewired in a box, with R = 1 kΩ and C = 110 nF. Your task is to measure the amplitude and phase of vB while ...
The result is a Lissajous figure. Lissajous figures tells us about the phase difference between the two signals and the ratio of their frequencies. Five such figures are shown in figures 1-5 below. Fig. 1: Two sine waves of equal frequency, in phase. Fig. 2: Two sine waves of equal frequency, 180 degrees out of phase.
Lissajous figures often appeared as props in science fiction movies made during the 1950's. One of the best examples can be found in the opening sequence of The Outer Limits TV series. ("Do not attempt to adjust your picture--we are controlling the transmission.") The pattern of criss-cross lines is actually a Lissajous figure.
Equipment: Tektronix 2235 Oscilloscope, Wave Generators (2) (Preferably the Wavetek digitals), and Video Camera with Power Supply. Procedure: Plug one wave generator into each input of a dual trace scope. Set the oscilloscope to X - Y plotting. If each generator is at the same frequency the figure will be in the form of a circle.
LISSAJOUS FIGURES A Lissajous figure is a figure that is displayed on an oscilloscope in the 'xy' mode. We can simulate these figures by graphing the equation z(t) =A1 cos(ωt +φ1)+jA2 cos(ωt +φ2) on the complex plane. END NOTES: The following m-files are located on the web page under 'Lab 2'. These files are for your benefit. Please do
Here is the practical file of the above mentioned experiment in readable pdf format phase-and-freq-by-lissajous-fig Also find other free study material on our website.
Additional Info. ID Code: H1-27 Purpose: Measurement of the speed of sound in air using Lissajous figures. Description: The signal to the loudspeaker is used as the horizontal input of an oscilloscope, and the signal picked up by the microphone is used as the vertical input, forming Lissajous figures. When they are in phase a diagonal line is produced, running from the lower left to the upper ...
This pdf file covers the basic CRO experiment to determine the frequency of a wave given through a function generator. and the making of Lissajous figures using two function generators. CRO-n-lissajous-figures
To create Lissajous figures, we will use the parameterization. x = cos 2π t a. sin 2π t bwhere a, b are always taken to be posi. ive real numbers. The equations are chosen so that the period of the horizontal component x is a and the period of the vertical. component y is b. To begin, define the paramete.
Area of Study: Acoustics . Disclaimer. Equipment: Spirograph unit, Laser. Procedure: The Spirograph will give Lissajous figures if you use only two of the motors. Adjusting the motor speeds will give a variety of figures. ... S-le: Wallace A. Hilton, "Lissajous Figures", Physics Demonstration Experiments. C. L. Stong, "Water Droplets That Float ...
SEM:2 Lissajous Figures/ 21.4.20 ð If you try to t race the trajectory of the motion of the particle using analytical method find y in terms of x, eliminating t, calculation difference δ is nonzero . Below is a table showing lissajous figure for in the ratio 1:1(both equal), 1:2 ( ω Here amplitudes are taken equal
Investigation: Lissajous Figures. Jules Antoine Lissajous was a French physicist who lived from 1822 to 1880. Like many physicists of his time, Lissajous was interested in being able to see vibrations. He started off standing tuning forks in water and watching the ripple patterns, but his most famous experiments involved tuning forks and ...
allinn, Ehitajate tee 5, Estoniae-mail: [email protected] - The article describes in detail a device for demonstration of Liss. jous figures on a large screen using a document camera and projector. Carry. ng out the specified experiment using this device does not take long. The device itself, containing two sine-w.
The aim of this study is to develop an experimental setup to produce Lissajous curves. The setup was made using a smartphone, a powered speaker (computer speaker), a balloon, a laser pointer and a piece of mirror. Lissajous curves are formed as follows: a piece of mirror is attached to a balloon. The balloon is vibrated with the sound signal ...
01 Lissajous Curves Experiment Manual - Free download as PDF File (.pdf), Text File (.txt) or read online for free. P191 Manual
Lissajous Figures. In the old days, whenever they showed an engineer working, there was usually an oscilloscope nearby with a pattern on the screen. Most often, the pattern was a Lissajous Figure. Jules Antoine Lissajous (1822-1880) was a French physicist who was interested in waves, and around 1855 developed a method for displaying them ...