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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 had position

z ( t ) = A x cos ( ω t + φ x ) + j A y cos ( ω t + φ y ) .

In this representation, A x cos ( ω t + φ x ) is the “x-coordinate of the point,” and A y cos ( ω t + φ ) is the “y-coordinate of the point.” As time runs from 0 to infinity, the point z ( t ) turns out a trajectory like that of [link] . The figure keeps overwriting itself because z ( t ) repeats itself every 2 π ω seconds. Do you see why?

Figure one is a cartesian graph with horizontal axis labeled Re and vertical axis labeled Im. On the graph is an oval-shaped curve centered about the origin diagonally with its longer portions sticking into the third and first quadrant. In the first quadrant, a point on the oval indicates movement in the counter-clockwise direction with an arrow pointing towards the bottom-left of the screen, and is labeled z(t). Figure one is a cartesian graph with horizontal axis labeled Re and vertical axis labeled Im. On the graph is an oval-shaped curve centered about the origin diagonally with its longer portions sticking into the third and first quadrant. In the first quadrant, a point on the oval indicates movement in the counter-clockwise direction with an arrow pointing towards the bottom-left of the screen, and is labeled z(t).
Lissajous Figure on Oscilloscope Screen

Two-Phasor Representation. We gain insight into the shape of the Lissajous figure if we use Euler's formulas to write z ( t ) as follows:

z ( t ) = A x 2 [ e j ( ω t + φ x ) + e - j ( ω t + φ x ) ] + j A y 2 [ e j ( ω t + φ y ) + e - j ( ω t + φ y ) ] = [ A x e j φ x + j A y e j φ y 2 ] e j ω t + [ A x e - j φ x + j A y e - j φ y 2 ] e - j ω t .

This representation is illustrated in [link] . It consists of two rotating phasors, with respective phasors B 1 and B 2 :

z ( t ) = B 1 e j ω t + B 2 e - j ω t B 1 = A x e j φ x + j A y e j φ y 2 B 2 = A x e - j φ x + j A y e - j φ y 2
Figure two is a cartesian graph with six arrows beginning at the origin and pointing away in various directions. The first points with a shallow negative slope into the fourth quadrant, and is labeled A_x/2 e^(-jΦ_x). The second extends into the first quadrant with a shallow positive slope and is labeled A_x/2 e^(jΦ_x). The third arrow extends into the first quadrant with a slightly stronger positive slope than the second arrow, and is labeled B_2. Across this arrow is a small direction arrow indicating movement in the clockwise direction. The fourth arrow extends into the first quadrant with a much stronger positive slope than the third and second arrows, and is labeled B_1. Across this arrow is a direction arrow indicating movement in the counter-clockwise direction. The fifth arrow extends into the first quadrant with an even stronger positive slope than the fourth, and is labeled j(A_y/2) e^(-jΦ_y). The sixth and final arrow extends into the second quadrant with a very sharp negative slope, and is labeled  j(A_y/2) e^(jΦ_y) Figure two is a cartesian graph with six arrows beginning at the origin and pointing away in various directions. The first points with a shallow negative slope into the fourth quadrant, and is labeled A_x/2 e^(-jΦ_x). The second extends into the first quadrant with a shallow positive slope and is labeled A_x/2 e^(jΦ_x). The third arrow extends into the first quadrant with a slightly stronger positive slope than the second arrow, and is labeled B_2. Across this arrow is a small direction arrow indicating movement in the clockwise direction. The fourth arrow extends into the first quadrant with a much stronger positive slope than the third and second arrows, and is labeled B_1. Across this arrow is a direction arrow indicating movement in the counter-clockwise direction. The fifth arrow extends into the first quadrant with an even stronger positive slope than the fourth, and is labeled j(A_y/2) e^(-jΦ_y). The sixth and final arrow extends into the second quadrant with a very sharp negative slope, and is labeled  j(A_y/2) e^(jΦ_y)
Two-Phasor Representation of a Lissajous Figure

As t increases, the phasors rotate past each other where they constructively add to produce large excursions of z ( t ) from the origin, and then they rotate to antipodal positions where they destructively add to produce near approachesof z ( t ) to the origin.

In electromagnetics and optics, the representations of z ( t ) given in [link] and [link] are called, respectively, linear and circular representations of elliptical polarization. In the linear representation, the x - and y -components of z vary along the horizontal and vertical lines. In the circular representation, two phasors rotate in opposite directions to turn out circular trajectories whose sum produces the same effect.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, A first course in electrical and computer engineering. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10685/1.2
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