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Illustration of the phase difference of two oscillations of the same frequency

Oscillation phase- a physical quantity used primarily to describe harmonic or close to harmonic oscillations, changing with time (most often uniformly growing with time), at a given amplitude (for damped oscillations - at a given initial amplitude and damping coefficient) determining the state of the oscillatory system in ( any) this moment time. It is also used to describe waves, mainly monochromatic or close to monochromatic.

Oscillation phase(in telecommunications for a periodic signal f(t) with period T) is the fractional part t/T of period T by which t is shifted from an arbitrary origin. The origin of coordinates is usually considered the moment of the previous transition of the function through zero in the direction from negative values to the positive.

In most cases, phase is spoken of in relation to harmonic (sinusoidal or imaginary exponential) oscillations (or monochromatic waves, also sinusoidal or imaginary exponential).

For such fluctuations:

, , ,

or the waves

For example, waves propagating in one-dimensional space: , , , or waves propagating in three-dimensional space (or space of any dimension): , , ,

the oscillation phase is defined as an argument of this function(one of the listed, in each case it is clear from the context which one), which describes a harmonic oscillatory process or a monochromatic wave.

That is, for phase oscillation

,

for a wave in one-dimensional space

,

for a wave in three-dimensional space or space of any other dimension:

,

where is the angular frequency (the higher the value, the faster the phase grows over time), t- time , - phase at t=0 - initial phase; k- wave number, x- coordinate, k- wave vector , x- a set of (Cartesian) coordinates characterizing a point in space (radius vector).

The phase is expressed in angular units (radians, degrees) or in cycles (fractions of a period):

1 cycle = 2 radians = 360 degrees.

  • In physics, especially when writing formulas, the radian representation of the phase is predominantly (and by default), measuring it in cycles or periods (with the exception of verbal formulations) is generally quite rare, but measuring in degrees is quite common (apparently, as explicit and not leading to confusion, since it is customary to never omit the sign of the degree in any oral speech, nor in writing), especially often in engineering applications (such as electrical engineering).

Sometimes (in the semiclassical approximation, where waves close to monochromatic but not strictly monochromatic are used, as well as in the path integral formalism, where waves can be far from monochromatic, although still similar to monochromatic), the phase is considered as depending on time and space coordinates not like linear function, but as, in principle, an arbitrary function of coordinates and time:

Related terms

If two waves (two oscillations) completely coincide with each other, the waves are said to be in phase. In the event that the moments of the maximum of one oscillation coincide with the moments of the minimum of another oscillation (or the maxima of one wave coincide with the minima of the other), they say that the oscillations (waves) are in antiphase. In this case, if the waves are the same (in amplitude), as a result of the addition, their mutual annihilation occurs (exactly, completely - only if the waves are monochromatic or at least symmetrical, assuming the propagation medium is linear, etc.).

Action

One of the most fundamental physical quantities on which the modern description practically any fairly fundamental physical system - action - in its meaning is a phase.

Notes


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See what the "Phase of Oscillations" is in other dictionaries:

    The periodically changing argument of the function describing the oscillations. or waves. process. In harmonic. oscillation u(х,t)=Acos(wt+j0), where wt+j0=j F. c., А amplitude, w circular frequency, t time, j0 initial (fixed) F. c. (at time t =0,… … Physical Encyclopedia

    oscillation phase- (φ) Argument of a function describing a value that changes according to the law of harmonic oscillation. [GOST 7601 78] Topics optics, optical instruments and measurements General terms oscillations and waves EN phase of oscillation DE Schwingungsphase FR… … Technical Translator's Handbook Phase - Phase. Oscillations of pendulums in the same phase (a) and antiphase (b); f is the angle of deviation of the pendulum from the equilibrium position. PHASE (from the Greek phasis appearance), 1) certain moment in the course of the development of any process (social, ... ... Illustrated encyclopedic Dictionary

    - (from the Greek phasis appearance), 1) a certain moment in the course of the development of any process (social, geological, physical, etc.). In physics and technology, the phase of oscillations is especially important, the state of an oscillatory process in a certain ... ... Modern Encyclopedia

    - (from the Greek phasis appearance) ..1) a certain moment in the course of the development of any process (social, geological, physical, etc.). In physics and technology, the phase of oscillations is especially important, the state of an oscillatory process in a certain ... ... Big Encyclopedic Dictionary

    Phase (from the Greek phasis - appearance), period, stage in the development of a phenomenon; see also Phase, Oscillation phase… Great Soviet Encyclopedia

    s; and. [from Greek. phasis appearance] 1. A separate stage, period, stage of development of what l. phenomena, processes, etc. The main phases of the development of society. The phases of the process of interaction between the animal and flora. Enter your new, decisive, ... ... encyclopedic Dictionary

But since the turns are shifted in space, then the EMF induced in them will not reach the amplitude and zero values ​​simultaneously.

At the initial moment of time, the EMF of the loop will be:

In these expressions, the angles are called phase , or phase . The corners and are called initial phase . The phase angle determines the value of the EMF at any moment of time, and the initial phase determines the value of the EMF at the initial moment of time.

The difference between the initial phases of two sinusoidal quantities of the same frequency and amplitude is called phase angle

Dividing the phase shift angle by the angular frequency, we get the time elapsed since the beginning of the period:

Graphic representation of sinusoidal quantities

U \u003d (U 2 a + (U L - U c) 2)

Thus, due to the presence of the phase angle, the voltage U is always less than the algebraic sum U a + U L + U C . The difference U L - U C = U p is called reactive voltage component.

Consider how current and voltage change in a series AC circuit.

Impedance and phase angle. If we substitute into formula (71) the values ​​U a = IR; U L \u003d lL and U C \u003d I / (C), then we will have: U \u003d ((IR) 2 + 2), from which we obtain the formula for Ohm's law for a series alternating current circuit:

I \u003d U / ((R 2 + 2)) \u003d U / Z (72)

Where Z \u003d (R 2 + 2) \u003d (R 2 + (X L - X c) 2)

The value of Z is called circuit impedance, it is measured in ohms. The difference L - l/(C) is called circuit reactance and denoted by the letter X. Therefore, the impedance of the circuit

Z = (R 2 + X 2)

The relationship between the active, reactive and impedances of the AC circuit can also be obtained using the Pythagorean theorem from the resistance triangle (Fig. 193). The resistance triangle A'B'C' can be obtained from the voltage triangle ABC (see Fig. 192,b), if all its sides are divided by the current I.

The phase angle is determined by the ratio between the individual resistances included in a given circuit. From the triangle A'B'C (see Fig. 193) we have:

sin? =X/Z; cos? =R/Z; tg? =X/R

For example, if the active resistance R is much greater than the reactance X, the angle is relatively small. If there is a large inductive or large capacitive resistance in the circuit, then the phase shift angle increases and approaches 90 °. Wherein, if the inductive resistance is greater than the capacitive one, the voltage and leads the current i by an angle; if the capacitive resistance is greater than the inductive one, then the voltage lags behind the current i by an angle.

An ideal inductor, a real coil and a capacitor in an alternating current circuit.

A real coil, unlike an ideal coil, has not only inductance, but also active resistance, therefore, when an alternating current flows in it, it is accompanied not only by a change in energy in a magnetic field, but also by a transformation electrical energy into a different kind. In particular, in the wire of a coil, electrical energy is converted into heat in accordance with the Lenz-Joule law.

It was previously found that in an alternating current circuit the process of converting electrical energy into another form is characterized by circuit active power P , and the change in energy in a magnetic field is reactive power Q .

In a real coil, both processes take place, i.e., its active and reactive powers are different from zero. Therefore, one real coil in the equivalent circuit must be represented by active and reactive elements.

fluctuations called movements or processes that are characterized by a certain repetition in time. Fluctuations are widespread in the surrounding world and can have a very different nature. These can be mechanical (pendulum), electromagnetic (oscillatory circuit) and other types of oscillations. free, or own oscillations are called oscillations that occur in a system left to itself, after it has been brought out of equilibrium by an external influence. An example is the oscillation of a ball suspended on a thread. Harmonic vibrations such oscillations are called, in which the oscillating value varies with time according to the law sinus or cosine . Harmonic vibration equation looks like:, where A - oscillation amplitude (the value of the greatest deviation of the system from the equilibrium position); - circular (cyclic) frequency. Periodically changing cosine argument - called oscillation phase . The oscillation phase determines the displacement of the oscillating quantity from the equilibrium position at a given time t. The constant φ is the value of the phase at time t = 0 and is called the initial phase of the oscillation .. This period of time T is called the period of harmonic oscillations. The period of harmonic oscillations is : T = 2π/. Mathematical pendulum- an oscillator, which is a mechanical system consisting of a material point located on a weightless inextensible thread or on a weightless rod in a uniform field of gravitational forces. The period of small natural oscillations of a mathematical pendulum of length L motionless suspended in a uniform gravitational field with free fall acceleration g equals

and does not depend on the amplitude of oscillations and the mass of the pendulum. physical pendulum- An oscillator, which is a rigid body that oscillates in the field of any forces about a point that is not the center of mass of this body, or a fixed axis perpendicular to the direction of the forces and not passing through the center of mass of this body.

24. Electromagnetic oscillations. Oscillatory circuit. Thomson formula.

Electromagnetic vibrations- These are fluctuations in electric and magnetic fields, which are accompanied by a periodic change in charge, current and voltage. The simplest system where free electromagnetic oscillations can arise and exist is an oscillatory circuit. Oscillatory circuit- this is a circuit consisting of an inductor and a capacitor (Fig. 29, a). If the capacitor is charged and closed to the coil, then current will flow through the coil (Fig. 29, b). When the capacitor is discharged, the current in the circuit will not stop due to self-induction in the coil. The induction current, in accordance with the Lenz rule, will have the same direction and recharge the capacitor (Fig. 29, c). The process will be repeated (Fig. 29, d) by analogy with pendulum oscillations. Thus, electromagnetic oscillations will occur in the oscillatory circuit due to the conversion of energy electric field capacitor() into energy magnetic field coils with current (), and vice versa. The period of electromagnetic oscillations in an ideal oscillatory circuit depends on the inductance of the coil and the capacitance of the capacitor and is found using the Thomson formula. Frequency is inversely related to period.

Definition

Initial phase of oscillation is a parameter that, together with the amplitude of oscillations, determines the initial state of the oscillatory system. The value of the initial phase is set in the initial conditions, that is, at $t=0$ c.

Consider harmonic oscillations of some parameter $\xi $. Harmonic oscillations are described by the equation:

\[\xi =A(\cos ((\omega )_0t+\varphi)\ )\ \left(1\right),\]

where $A=(\xi )_(max)$ - oscillation amplitude; $(\omega )_0$ - cyclic (circular) oscillation frequency. The parameter $\xi $ lies within $-A\le \xi \le $+A.

Determination of the oscillation phase

The entire argument of the periodic function (in this case, the cosine: $\ ((\omega )_0t+\varphi)$), which describes the oscillatory process, is called the oscillation phase. The magnitude of the oscillation phase at the initial moment of time, that is, at $t=0$, ($\varphi $) is called the initial phase. There is no established designation of the phase, we have the initial phase denoted by $\varphi $. Sometimes, to emphasize that the initial phase refers to the time $t=0$, the index 0 is added to the letter denoting the initial phase, for example, $(\varphi )_0.$

The initial phase unit is the angle unit, radian (rad) or degree.

The initial phase of oscillations and the method of excitation of oscillations

Let us assume that for $t=0$ the displacement of the system from the equilibrium position is equal to $(\xi )_0$, and starting speed$(\dot(\xi ))_0$. Then equation (1) takes the form:

\[\xi \left(0\right)=A(\cos \varphi =\ )(\xi )_0\left(2\right);;\] \[\ \frac(d\xi )(dt) =-A(\omega )_0(\sin \varphi =\ )(\dot(\xi ))_0\to -A(\sin \varphi =\frac((\dot(\xi ))_0)(( \omega )_0)\ )\ \left(3\right).\]

We square both equations (2) and add them:

\[(\xi )^2_0+(\left(\frac((\dot(\xi ))_0)((\omega )_0)\right))^2=A^2\left(4\right). \]

From expression (4) we have:

Divide equation (3) by (2), we get:

Expressions (5) and (6) show that the initial phase and amplitude depend on the initial conditions of oscillations. This means that the amplitude and the initial phase depend on the way the oscillations are excited. For example, if the weight of a spring pendulum is deflected from the equilibrium position and by a distance $x_0$ and released without a push, then the equation of motion of the pendulum is the equation:

with initial conditions:

With this excitation of vibration spring pendulum can be described by the expression:

Addition of oscillations and initial phase

An oscillating body is capable of taking part in several oscillatory processes simultaneously. In this case, it becomes necessary to find out what the resulting oscillation will be.

Let us assume that two oscillations with equal frequencies occur along one straight line. The equation of the resulting oscillations will be the expression:

\[\xi =(\xi )_1+(\xi )_2=A(\cos \left((\omega )_0t+\varphi \right),\ )\]

then the amplitude of the total oscillation is equal to:

where $A_1$; $A_2$ - amplitudes of folding oscillations; $(\varphi )_2;;(\varphi )_1$ - initial phases cumulative vibrations. In this case, the initial phase of the resulting oscillation ($\varphi $) is calculated using the formula:

The equation for the trajectory of a point that takes part in two mutually perpendicular oscillations with amplitudes $A_1$ and $A_2$ and initial phases $(\varphi )_2and(\varphi )_1$:

\[\frac(x^2)(A^2_1)+\frac(y^2)(A^2_2)-\frac(2xy)(A_1A_2)(\cos \left((\varphi )_2-(\ varphi )_1\right)\ )=(sin)^2\left((\varphi )_2-(\varphi )_1\right)\left(12\right).\]

In the case of equality of the initial phases of the oscillation components, the trajectory equation has the form:

which indicates the movement of a point in a straight line.

If the difference between the initial phases of the added oscillations is $\Delta \varphi =(\varphi )_2-(\varphi )_1=\frac(\pi )(2),$, the following formula becomes the trajectory equation:

\[\frac(x^2)(A^2_1)+\frac(y^2)(A^2_2)=1\left(14\right),\]

which means the motion path is an ellipse.

Examples of problems with a solution

Example 1

Exercise. Oscillations of the spring oscillator are excited by a push from the equilibrium position, while the load is given an instantaneous speed equal to $v_0$. write down initial conditions for such an oscillation and a function $x(t)$ describing these oscillations.

Solution. Message to the load of the spring pendulum instantaneous speed equal to $v_0$ means that when describing its oscillations using the equation:

initial conditions will be:

Substitute into expression (1.1) $t=0$, we have:

Since $A\ne 0$, then $(\cos \left(\varphi \right)\ )=0\to \varphi =\pm \frac(\pi )(2).$

Let's take the first derivative $\frac(dx)(dt)$ and substitute the time $t=0$:

\[\dot(x)\left(0\right)=-A(\omega )_(0\ )(\sin \left(\varphi \right)\ )=v_0\to A=\frac(v_0) ((\omega )_(0\ ))\ \left(1.4\right).\]

It follows from (1.4) that the initial phase is $\varphi =-\frac(\pi )(2).$ Let us substitute the resulting initial phase and amplitude into equation (1.1):

Answer.$x(t)=\frac(v_0)((\omega )_(0\ ))(\sin (\ )(\omega )_0t)$

Example 2

Exercise. Two vibrations in the same direction add up. The equations for these oscillations are: $x_1=(\cos \pi (t+\frac(1)(6))\ );;\ x_2=2(\cos \pi (t+\frac(1)(2))\ )$. What is the initial phase of the resulting oscillation?

Solution. Let us write the equation of harmonic oscillations along the X axis:

Let's transform the equations given in the condition of the problem to the same form:

\;;\ x_2=2(\cos \left[\pi t+\frac(\pi )(2)\right](2.2).\ )\]

Comparing equations (2.2) with (2.1), we obtain that the initial phases of the oscillations are:

\[(\varphi )_1=\frac(\pi )(6);;\ (\varphi )_2=\frac(\pi )(2).\]

We depict in Fig. 1 a vector diagram of oscillations.

$tg\ \varphi $ total fluctuations can be found from Fig.1:

\ \[\varphi =arctg\ \left(2.87\right)\approx 70.9()^\circ \]

Answer.$\varphi =70.9()^\circ $