What Is the Principle of Superposition?

This phenomenon often occurs in mathematical physics: the combined effect of several different causes is equal to the cumulative effect of these different reasons alone.

Chinese name: Superposition principle

Foreign name: superposition principle
Alias: Superimposed nature

Basic Introduction to Superposition Principle

Related explanation: superposition principle; superposition principle

For example, in physics, the acceleration generated by several external forces acting on an object is equal to the sum of the accelerations caused by each external force acting on the object alone. This principle is called the superposition principle. The superposition principle is applicable to a very wide range. In the study of linear equations and linear problems in mathematics, the superposition principle is often used.

In physics and systems theory, the superposition principle , also called superposition property , says that for any linear system "at a given place and time, the synthetic response produced by two or more stimuli is The sum of the responses produced by each stimulus individually. "

Thus if input A produces a response X and input B produces a Y, then input A + B produces a response (X + Y).

In mathematical terms, for all linear systems F (x) = y , where x is a certain degree of stimulus (input) and y is some kind of response (output). The superposition of stimuli (ie, "and") yields respectively Superposition of reactions

In mathematics, this property is more often called additivity. In most practical cases, the additivity of F indicates that it is a linear mapping, also known as a linear function or a linear operator.

The principle of superposition is applicable to any linear system, including algebraic equations, linear differential equations, and these forms of equations. Inputs and reactions can be numbers, functions, vectors, vector fields, time-varying signals, or any other object that satisfies a certain axiom. Note that when it comes to vectors and vector fields, superposition is understood as vector sum.

1.If several charges exist at the same time, their electric fields are superimposed on each other to form a combined electric field. At this time, the field strength at a point is equal to the vector sum of the field strengths generated at that point when each charge exists alone. This is called the superposition principle of electric fields.

2.The point potential is the potential of a point in the electric field equal to the algebraic sum of the potentials generated at that point when each point charge exists alone, which is called the principle of potential superposition.

Thus if input A produces a response X and input B produces a Y, then input A + B produces a response (X + Y).

In mathematical terms, for all linear systems F (x) = y, where x is a certain degree of stimulus (input) and y is some kind of response (output). The superposition of stimuli (ie, "and") yields respectively Overlay of responses:

This principle has many applications in physics and engineering, because many physical systems can be modeled as linear systems. For example, a beam can be used as a linear system where the input stimulus is the structural load on the beam and the output response is the deflection of the beam. Because physical systems are usually only approximately linear, the superposition principle is only an approximation of real physical phenomena; from here, the operating area of these systems can be known.

Similar method

Relationship with Fourier analysis and similar methods

By writing a very general stimulus in a linear system as the superposition of some specific simple form of stimulus, the principle of superposition is usually used to make the response easy to calculate.

For example, in Fourier analysis, the stimulus is written as the superposition of an infinite number of sine waves. Due to the superposition principle, each such sine wave can be analyzed separately and the respective response can be calculated. (The response itself is also a sine wave, which has the same frequency as the stimulus, but generally has different amplitudes and phases.) According to the superposition principle, the original stimulus response is the sum (or integral) of all individual sine wave responses.

Another common example is that in Green's function analysis, the stimulus is written as the superposition of an infinite number of impulse functions, and the response is the superposition of the impulse response.

Fourier analysis is commonly used for waves. For example, in electromagnetic theory, ordinary light is described as a superposition of plane waves (waves of fixed frequency, polarization, and direction). As long as the principle of superposition holds (usually but not necessarily; see nonlinear optics), the behavior of any light wave can be understood as the superposition of the behavior of these simple plane waves.

Application of Superposition Principle

Application in wave theory

Main article: Wave and wave equations

Waves are often described as changes in certain parameters through space and time, such as height in water waves, pressure in sound waves, or electromagnetic fields in light waves. The value of this parameter is called the amplitude of the wave, and the wave itself is a function that determines the amplitude at each point.

In any system with waves, the wave form at a given time is a function of the system's source (that is, the external force that may exist or generate waves) and the initial conditions. In many cases (such as the classical wave equation), the equation describing the wave is linear. If this condition holds, then the superposition principle can be used. This means that the combined amplitude of two or more waves propagating in the same space is the sum of the amplitudes generated by each wave individually. For example, two waves traveling opposite each other will pass straight through each other without any deformation on the other side (see the top picture).

Superposition principle wave interference

Main article: Interference

The interference between waves is based on this idea. When two or more waves propagate in the same space, the combined amplitude at each point is the sum of the amplitudes of the individual waves. In some cases, such as noise-cancelling headphone, the amplitude of the synthetic variable is smaller than the individual subvariables; this is called negative interference. In another case, such as a line array, the amplitude of the composite variable is greater than the individual subvariables; this becomes active interference.

synthesis

Wave form

Wave 1

Wave 2

Same phase, 180 ° difference

Loss of linearity

It is worth noting that in most practical physical situations, the equations governing waves are only approximately linear. In these cases, the superposition principle is only approximate. As a rule, when the amplitude of the wave is smaller, the accuracy of the approximation is higher. The phenomenon when the superposition principle does not hold true can be seen in nonlinear optics and nonlinear acoustics.

Quantum superposition

Main article: Quantum superposition

A major problem in quantum mechanics is how to calculate the propagation and behavior of a particular type of wave. This wave is called the wave function, and the equation that governs the behavior of the wave is called the Schrodinger wave equation. One of the main ways to calculate the behavior of a wave function is to write the wave function as (possibly infinite) the superposition of some particularly simple steady state wave functions (called quantum superposition). Because the Schrödinger wave equation is linear, the behavior of the original wave function can be calculated by the superposition principle [1], see quantum superposition.

Superposition principle boundary value

Boundary value problem

Main article: Boundary value problem

A common type of boundary value problem is abstractly looking for a function y to satisfy a certain equation

F (y) = 0

And boundary conditions

G (y) = z

For example, in the Dirichlet boundary condition pull-down Plasma equation, F is a Laplace operator on a region R, G is an operator that limits y to the boundary of R, and z is y on R. Functions to be satisfied on the boundary.

In this case, F and G are both linear operators. The superposition principle says that the superposition of some solutions of the first equation is another solution of the first equation:

If then

And the boundary value is:

G (y1) + G (y2) = G (y1 + y2)

Taking advantage of this fact, if a set of solutions can form the solution of the first equation, these solutions are carefully superimposed to make it satisfy the second equation. This is a common method for solving boundary value problems.

Application example of superposition principle

Other application examples

In a linear circuit of electrical engineering, the input (an applied time-varying voltage signal) and the output (current or voltage anywhere in the loop) are related by a linear transformation. Thus the superposition of the digital signals (ie, the sum) will give the superposition of the response. Fourier analysis is particularly common on this basis. For another related technique in circuit analysis, see Superposition theorem.

In physics, Maxwell's equation implies (possibly changing over time) that charge and current and electric and magnetic fields are related by a linear transformation. Thus the superposition principle can be used to simplify the calculation of the physical field caused by a given charge and current distribution. This principle is also used in other linear differential equations in physics, such as the heat equation.

In mechanical engineering, the deformation of the beam and structure used to superimpose the combined load, if the effect is linear (ie, each load does not affect the results of other loads and the effect of each load does not significantly change the geometry of the structural system [2] ).

In hydrogeology, the drawdown of a water well that was originally used to pump water in an ideal aquifer is superimposed.

In process control, the superposition principle is used for model predictive control.

The superposition principle can be used to analyze the small derivative of a known solution of a nonlinear system using linearization.

In music, theorist Joseph Schillinger uses a form of the superposition principle as the "tempo theory" in his "Music Composition Schlinger System."

Attention problem of superposition principle

Attention should be paid when applying the superposition principle:

(1) Only linear circuits have superposition, and the principle of superposition cannot be applied to non-linear circuits.

(2) Only the independent power supply can perform the zeroing process. For circuits containing controlled sources, do not force the controlled source to take the zero value when using the superposition principle. This is because once the controlled source is forced to take a zero value, it is equivalent to removing the physical component represented by the controlled source in the circuit, which leads to erroneous results.

(3) The calculation of power cannot use the superposition principle. ^[1]

(4) When a power supply does not work temporarily, it is set to zero. For an independent voltage source, short both ends when it is temporarily inactive, and for an independent current source, open both ends. ^[1]