Another characteristic of harmonic oscillations is the phase of oscillations.
As we already know, for a given amplitude of oscillations, at any moment in time we can determine the coordinates of the body. It will be uniquely specified by the argument of the trigonometric function φ = ω0*t. The quantity φ, which is under the sign of the trigonometric function, called the oscillation phase.
The units for phase are radians. The phase uniquely determines not only the coordinate of the body at any time, but also the speed or acceleration. Therefore, it is believed that the oscillation phase determines the state of the oscillatory system at any time.
Of course, provided that the amplitude of oscillations is specified. Two oscillations that have the same frequency and period of oscillation may differ from each other in phase.
- φ = ω0*t = 2*pi*t/T.
If we express time t in the number of periods that have passed since the beginning of the oscillations, then any value of time t corresponds to a phase value expressed in radians. For example, if we take time t = T/4, then this value will correspond to the phase value pi/2.
Thus, we can plot the dependence of the coordinate not on time, but on phase, and we will get exactly the same dependence. The following figure shows such a graph.
Initial phase of oscillation
When describing the coordinates of oscillatory motion, we used the sine and cosine functions. For cosine we wrote the following formula:
- x = Xm*cos(ω0*t).
But we can describe the same trajectory of motion using a sine. In this case, we need to shift the argument by pi/2, that is, the difference between sine and cosine is pi/2 or a quarter of the period.
- x=Xm*sin(ω0*t+pi/2).
The value pi/2 is called the initial phase of the oscillation. The initial phase of oscillation is the position of the body at the initial moment of time t = 0. In order to make the pendulum oscillate, we must remove it from its equilibrium position. We can do this in two ways:
- Take him aside and let him go.
- Hit it.
In the first case, we immediately change the coordinate of the body, that is, at the initial moment of time the coordinate will be equal to the amplitude value. To describe such an oscillation, it is more convenient to use the cosine function and the form
- x = Xm*cos(ω0*t),
or the formula
- x = Xm*sin(ω0*t+&phi),
where φ is the initial phase of oscillation.
If we hit the body, then at the initial moment of time its coordinate is equal to zero, and in this case it is more convenient to use the form:
- x = Xm*sin(ω0*t).
Two oscillations that differ only in the initial phase are called phase-shifted.
For example, for vibrations described by the following formulas:
- x = Xm*sin(ω0*t),
- x = Xm*sin(ω0*t+pi/2),
the phase shift is pi/2.
Phase shift is also sometimes called phase difference.
>> Oscillation phase
§ 23 PHASE OF OSCILLATIONS
Let us introduce another quantity characterizing harmonic oscillations - the phase of oscillations.
For a given amplitude of oscillations, the coordinate of the oscillating body at any time is uniquely determined by the cosine or sine argument:
The quantity under the sign of the cosine or sine function is called the phase of oscillation described by this function. The phase is expressed in angular units of radians.
The phase determines not only the value of the coordinate, but also the value of other physical quantities, such as speed and acceleration, which also vary according to a harmonic law. Therefore, we can say that the phase determines, for a given amplitude, the state of the oscillatory system at any time. This is the meaning of the concept of phase.
Oscillations with the same amplitudes and frequencies may differ in phase.
The ratio indicates how many periods have passed since the start of the oscillation. Any time value t, expressed in the number of periods T, corresponds to a phase value expressed in radians. So, after time t = (a quarter of a period), after half a period =, after a whole period = 2, etc.
You can depict on a graph the dependence of the coordinates of an oscillating point not on time, but on phase. Figure 3.7 shows the same cosine wave as in Figure 3.6, but different phase values are plotted on the horizontal axis instead of time.
Representation of harmonic vibrations using cosine and sine. You already know that during harmonic vibrations the coordinate of a body changes over time according to the law of cosine or sine. After introducing the concept of phase, we will dwell on this in more detail.
The sine differs from the cosine by shifting the argument by , which corresponds, as can be seen from equation (3.21), to a time period equal to a quarter of the period:
But in this case, the initial phase, i.e., the phase value at time t = 0, is not equal to zero, but .
Usually we excite oscillations of a body attached to a spring, or oscillations of a pendulum, by removing the body of the pendulum from its equilibrium position and then releasing it. The displacement from equilibrium is maximum at the initial moment. Therefore, to describe oscillations, it is more convenient to use formula (3.14) using a cosine than formula (3.23) using a sine.
But if we excited oscillations of a body at rest with a short-term push, then the coordinate of the body at the initial moment would be equal to zero, and it would be more convenient to describe changes in the coordinate over time using the sine, i.e., by the formula
x = x m sin t (3.24)
since in this case the initial phase is zero.
If at the initial moment of time (at t = 0) the phase of oscillations is equal to , then the equation of oscillations can be written in the form
x = x m sin(t + )
Phase shift. The oscillations described by formulas (3.23) and (3.24) differ from each other only in phases. The phase difference, or, as is often said, the phase shift, of these oscillations is . Figure 3.8 shows graphs of coordinates versus time of oscillations shifted in phase by . Graph 1 corresponds to oscillations that occur according to the sinusoidal law: x = x m sin t and graph 2 corresponds to oscillations that occur according to the cosine law:
To determine the phase difference between two oscillations, in both cases the oscillating quantity must be expressed through the same trigonometric function - cosine or sine.
1. What vibrations are called harmonic!
2. How are acceleration and coordinate related during harmonic oscillations!
3. How are the cyclic frequency of oscillations and the period of oscillation related?
4. Why does the frequency of oscillation of a body attached to a spring depend on its mass, but the frequency of oscillation of a mathematical pendulum does not depend on mass!
5. What are the amplitudes and periods of three different harmonic oscillations, the graphs of which are presented in Figures 3.8, 3.9!
Oscillatory processes are an important element of modern science and technology, therefore their study has always been given attention as one of the “eternal” problems. The goal of any knowledge is not simple curiosity, but its use in everyday life. And for this purpose, new technical systems and mechanisms exist and appear every day. They are in motion, manifest their essence by performing some kind of work, or, being motionless, retain the potential, under certain conditions, to move into a state of movement. What is movement? Without delving into the wilds, we will accept the simplest interpretation: a change in the position of a material body relative to any coordinate system, which is conventionally considered motionless.
Among the huge number of possible options for motion, oscillatory motion is of particular interest, which differs in that the system repeats the change in its coordinates (or physical quantities) at certain intervals - cycles. Such oscillations are called periodic or cyclic. Among them, there is a separate class whose characteristic features (speed, acceleration, position in space, etc.) change in time according to a harmonic law, i.e. having a sinusoidal shape. A remarkable property of harmonic vibrations is that their combination represents any other options, incl. and non-harmonic. A very important concept in physics is the “oscillation phase,” which means fixing the position of an oscillating body at a certain point in time. The phase is measured in angular units - radians, quite conventionally, simply as a convenient technique for explaining periodic processes. In other words, the phase determines the value of the current state of the oscillatory system. It cannot be otherwise - after all, the phase of oscillations is an argument of the function that describes these oscillations. The true value of the phase for an oscillatory motion can mean coordinates, speed and other physical parameters that change according to a harmonic law, but what they have in common is a time dependence.
Demonstrating vibrations is not at all difficult - for this you will need the simplest mechanical system - a thread of length r, and a “material point” suspended on it - a weight. Let's fix the thread in the center of the rectangular coordinate system and make our “pendulum” spin. Let us assume that he willingly does this with angular velocity w. Then during time t the angle of rotation of the load will be φ = wt. Additionally, this expression must take into account the initial phase of oscillations in the form of angle φ0 - the position of the system before the start of movement. So, the total angle of rotation, phase, is calculated from the relation φ = wt+ φ0. Then the expression for the harmonic function, which is the projection of the coordinates of the load onto the X axis, can be written:
x = A * cos(wt + φ0), where A is the vibration amplitude, in our case equal to r - the radius of the thread.
Similarly, the same projection on the Y axis will be written as follows:
y = A * sin(wt + φ0).
It should be understood that the phase of oscillations in this case does not mean the measure of rotation “angle”, but the angular measure of time, which expresses time in units of angle. During this time, the load rotates through a certain angle, which can be uniquely determined based on the fact that for cyclic oscillation w = 2 * π /T, where T is the period of oscillation. Therefore, if one period corresponds to a rotation of 2π radians, then part of the period, time, can be proportionally expressed by an angle as a fraction of the total rotation of 2π.
Vibrations do not exist on their own - sounds, light, vibration are always a superposition, an imposition, of a large number of vibrations from different sources. Of course, the result of the superposition of two or more oscillations is influenced by their parameters, incl. and the oscillation phase. The formula for the total oscillation, usually non-harmonic, can have a very complex form, but this only makes it more interesting. As stated above, any non-harmonic oscillation can be represented as a large number of harmonic oscillations with different amplitude, frequency and phase. In mathematics, this operation is called “series expansion of a function” and is widely used in calculations, for example, of the strength of structures and structures. The basis of such calculations is the study of harmonic oscillations, taking into account all parameters, including phase.
Oscillation phase complete - argument of a periodic function describing an oscillatory or wave process.
Oscillation phase initial - the value of the oscillation phase (total) at the initial moment of time, i.e. at t= 0 (for an oscillatory process), as well as at the initial moment of time at the origin of the coordinate system, i.e. at t= 0 at point ( x, y, z) = 0 (for the wave process).
Oscillation phase(in electrical engineering) - the argument of a sinusoidal function (voltage, current), counted from the point where the value passes through zero to a positive value.
Oscillation phase- harmonic oscillation ( φ ) .
Size φ, standing under the sign of the cosine or sine function is called oscillation phase described by this function.
φ = ω៰ t
As a rule, phase is spoken of in relation to harmonic oscillations or monochromatic waves. When describing a quantity experiencing harmonic oscillations, for example, one of the expressions is used:
A cos (ω t + φ 0) (\displaystyle A\cos(\omega t+\varphi _(0))), A sin (ω t + φ 0) (\displaystyle A\sin(\omega t+\varphi _(0))), A e i (ω t + φ 0) (\displaystyle Ae^(i(\omega t+\varphi _(0)))).Similarly, when describing a wave propagating in one-dimensional space, for example, expressions of the form are used:
A cos (k x − ω t + φ 0) (\displaystyle A\cos(kx-\omega t+\varphi _(0))), A sin (k x − ω t + φ 0) (\displaystyle A\sin(kx-\omega t+\varphi _(0))), A e i (k x − ω t + φ 0) (\displaystyle Ae^(i(kx-\omega t+\varphi _(0)))),for a wave in space of any dimension (for example, in three-dimensional space):
A cos (k ⋅ r − ω t + φ 0) (\displaystyle A\cos(\mathbf (k) \cdot \mathbf (r) -\omega t+\varphi _(0))), A sin (k ⋅ r − ω t + φ 0) (\displaystyle A\sin(\mathbf (k) \cdot \mathbf (r) -\omega t+\varphi _(0))), A e i (k ⋅ r − ω t + φ 0) (\displaystyle Ae^(i(\mathbf (k) \cdot \mathbf (r) -\omega t+\varphi _(0)))).The oscillation phase (total) in these expressions is argument functions, i.e. expression written in parentheses; initial oscillation phase - value φ 0, which is one of the terms of the total phase. Speaking of the full phase, the word full often omitted.
Oscillations with the same amplitudes and frequencies may differ in phase. Because ω៰ =2π/T, That φ = ω៰t = 2π t/T.
Attitude t/T indicates how many periods have passed since the start of the oscillations. Any time value t , expressed in the number of periods T , corresponds to the phase value φ , expressed in radians. So, as time passes t=T/4 (quarter period) φ=π/2, after half the period φ =π/2, after a whole period φ=2 π etc.
Since the functions sin(...) and cos(...) coincide with each other when the argument (i.e. phase) is shifted by π / 2 , (\displaystyle \pi /2,) then, in order to avoid confusion, it is better to use only one of these two functions to determine the phase, and not both at the same time. According to the usual convention, a phase is considered the argument is cosine, not sine.
That is, for the oscillatory process (see above) the phase (full)
φ = ω t + φ 0 (\displaystyle \varphi =\omega t+\varphi _(0)),for a wave in one-dimensional space
φ = k x − ω t + φ 0 (\displaystyle \varphi =kx-\omega t+\varphi _(0)),for a wave in three-dimensional space or space of any other dimension:
φ = k r − ω t + φ 0 (\displaystyle \varphi =\mathbf (k) \mathbf (r) -\omega t+\varphi _(0)),Where ω (\displaystyle \omega )- angular frequency (a value indicating how many radians or degrees the phase will change in 1 s; the higher the value, the faster the phase grows over time); t- time ; φ 0 (\displaystyle \varphi _(0))- initial phase (that is, the phase at t = 0); k- wave number; x- coordinate of the observation point of the wave process in one-dimensional space; k- wave vector; r- radius vector of a point in space (a set of coordinates, for example, Cartesian).
In the above expressions, the phase has the dimension of angular units (radians, degrees). The phase of the oscillatory process, by analogy with the mechanical rotational process, is also expressed in cycles, that is, fractions of the period of the repeating process:
1 cycle = 2 π (\displaystyle \pi ) radian = 360 degrees.
In analytical expressions (in formulas), the phase representation in radians is predominantly (and by default) used; the representation in degrees is also found quite often (apparently, as extremely obvious and not leading to confusion, since it is never customary to omit the degree sign either in oral speech or in recordings). Indicating the phase in cycles or periods (except for verbal formulations) is relatively rare in technology.
Sometimes (in the quasi-classical approximation, where quasi-monochromatic waves are used, i.e. close to monochromatic, but not strictly monochromatic), as well as in the path integral formalism, where waves can be far from monochromatic, although still similar to monochromatic) the phase is considered, which is a nonlinear function of time t and spatial coordinates r, in principle, an arbitrary function.
But because the turns are shifted in space, then the EMF induced in them will not reach amplitude and zero values at the same time.
At the initial moment of time, the EMF of the turn will be:
In these expressions the angles are called phase , or phase . The angles are called initial phase . The phase angle determines the value of the emf at any time, and the initial phase determines the value of the emf at the initial time.
The difference in the initial phases of two sinusoidal quantities of the same frequency and amplitude is called phase angle
Dividing the phase angle by the angular frequency, we obtain the time elapsed since the beginning of the period:
Graphic representation of sinusoidal quantities
U = (U 2 a + (U L - U c) 2)
Thus, due to the presence of a 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.
Let's consider how current and voltage change in a series alternating current circuit.
Impedance and phase angle. If we substitute the values U a = IR into formula (71); U L = lL and U C =I/(C), then we will have: U = ((IR) 2 + 2), from which we obtain the formula for Ohm’s law for a series alternating current circuit:
I = U / ((R 2 + 2)) = U / Z (72)
Where Z = (R 2 + 2) = (R 2 + (X L - X c) 2)
The Z value is called circuit impedance, it is measured in ohms. The difference L - l/(C) is called circuit reactance and is denoted by the letter X. Therefore, the total resistance of the circuit
Z = (R 2 + X 2)
The relationship between active, reactive and impedance of an alternating current 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 we divide all its sides by the current I.
The phase shift angle is determined by the relationship between the individual resistances included in a given circuit. From 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 significantly greater than the reactance X, the angle is relatively small. If the circuit has a large inductive or large capacitive reactance, then the phase shift angle increases and approaches 90°. Wherein, if the inductive reactance is greater than the capacitive reactance, the voltage and leads the current i by an angle; if the capacitive reactance is greater than the inductive reactance, 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 one, has not only inductance, but also active resistance, therefore, when alternating current flows in it, it is accompanied not only by a change in energy in the magnetic field, but also by the conversion of electrical energy into another form. Specifically, in the coil wire, 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 active power of the circuit P , and the change in energy in the 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.