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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

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
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what is the actual application of fullerenes nowadays?
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That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
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for screen printed electrodes ?
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of graphene you mean?
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or in general
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in general
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Graphene has a hexagonal structure
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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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