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8/3/2019 Superposition Principle...Cuevas and Delacruz
http://slidepdf.com/reader/full/superposition-principlecuevas-and-delacruz 1/12
In physics and systems theory , the superpositionprinciple [1], also known as superposition property ,
states that, for all linear systems, the net response at agiven place and time caused by two or more stimuli isthe sum of the responses which would have been
caused by each stimulus individually. So that if input Aproduces response X and input B produces response Y
then input ( A + B) produces response ( X + Y ).
8/3/2019 Superposition Principle...Cuevas and Delacruz
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y The superposition principle holds because, by
definition, a linear system must be additive.
Superposition may sometimes imply linearity,
depending on whether homogeneity is included or
implied in the definition of superposition.
y In the field of electrical engineering, where the x and y
signals are allowed to be complex-valued (as is
common in signal processing), a linear system must
satisfy the superposition property, which requiresthe system to be additive and homogeneous.[2][3] An
additive system satisfies F ( x1 + x2) = F ( x1) + F ( x2). A
homogeneous system satisfies F (a x) = aF ( x), where a
is a scalar. Often the additivity and homogeneity
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Superposition of
almost plane
waves (diagonallines) from a
distant source
and waves from
the wake of theducks. Linearity
holds only
approximately
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y This principle has many applications in physics and
engineering because many physical systems can bemodeled as linear systems. For example, a beam can
be modeled as a linear system where the input
stimulus is the load on the beam and the output
response is the deflection of the beam. Becausephysical systems are generally only approximately
linear, the superposition principle is only an
approximation of the true physical behavior; it
provides insight for typical operational regions forthese systems.
y The superposition principle applies to an y linear
system, including algebraic equations, linear
differential equations, and systems of equations of
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similar methods
y By writing a very general stimulus (in a linear system)
as the superposition of stimuli of a specific, simple
form, often the response becomes easier to compute,
y
For example, in Fourier analysis, the stimulus iswritten as the superposition of infinitely many
sinusoids. Due to the superposition principle, each of
these sinusoids can be analyzed separately, and its
individual response can be computed. (The responseis itself a sinusoid, with the same frequency as the
stimulus, but generally a different amplitude and
phase.) According to the superposition principle, the
response to the original stimulus is the sum (or
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yFourier analysis is particularly common
for waves. For example, in
electromagnetic theory, ordinary light isdescribed as a superposition of plane
waves (waves of fixed frequency,
polarization, and direction). As long as
the superposition principle holds (which
is often but not always; see nonlinear
optics), the behavior of any light wave
can be understood as a su er osition of
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Application to waves
y W aves are usually described by variations in some parameter throughspace and timefor example, height in a water wave, pressure in asound wave, or the electromagnetic field in a light wave. The value of this parameter is called the amplitude of the wave, and the wave itself
is a function specifying the amplitude at each point.y In any system with waves, the waveform at a given time is a function of
the sources (i.e., external forces, if any, that create or affect the wave)and initial conditions of the system. In many cases (for example, in theclassic wave equation), the equation describing the wave is linear. W hen this is true, the superposition principle can be applied. Thatmeans that the net amplitude caused by two or more waves traversingthe same space, is the sum of the amplitudes which would have beenproduced by the individual waves separately. For example, two wavestraveling towards each other will pass right through each other withoutany distortion on the other side.
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Wave interference
Combined
wavef orm
wave 1
wave 2
Two waves in phase
Two waves 180° out
of phase
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yThe phenomenon of interference between
waves is based on this idea. When two or
more waves traverse the same space, the
net amplitude at each point is the sum of
the amplitudes of the individual waves. In
some cases, such as in noise-cancelling
headphones, the summed variation has a
smaller amplitude than the componentvariations; this is called dest ruct ive
inter fer ence. In other cases, such as in
Line Array, the summed variation will
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Departures from linearity
yIn most realistic physical situations,the equation governing the wave is
only approximately linear. In these
situations, the superposition principle
only approximately holds. As a rule,
the accuracy of the approximationtends to improve as the amplitude of
the wave gets smaller. For examples
of phenomena that arise when the
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Quantum superposition
y In quantum mechanics, a principal task is to
compute how a certain type of wave propagates
and behaves. The wave is called a
wavefunction, and the equation governing thebehavior of the wave is called Schrödinger's
wave equation. A primary approach to
computing the behavior of a wavefunction is to
write that wavefunction as a superposition
(called "quantum superposition") of (possibly
infinitely many) other wavefunctions of a
certain typestationary states whose behavior
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Boundary value problems
y A common type of boundary value problem is (to put it
abstractly) finding a function y that satisfies some equation
y F ( y) = 0 with some boundary specification
y G( y) = z For example, in Laplace's equation with Dirichletboundary conditions, F would be the Laplacian operator in a
region R, G would be an operator that restricts y to the
boundary of R, and z would be the function that y is required to
equal on the boundary of R.
y In the case that F and G are both linear operators, then the
superposition principle says that a superposition of solutions to
the first equation is another solution to the first equation:
y while the boundary values superpose:
y G( y1) + G( y2) = G( y1 + y2) Using these facts, if a list can be