Harmonic oscillations in an electric oscillatory circuit. An equation describing processes in an oscillatory circuit

Free vibrations in the circuit.

The AC circuits considered in the previous sections suggest that a pair of elements - a capacitor and an inductor form a kind of oscillatory system. Now we will show that this is indeed the case, in a circuit consisting only of these elements (Fig. 669), even free oscillations are possible, that is, without an external source of EMF.

rice. 669
Therefore, a circuit (or part of another circuit) consisting of a capacitor and an inductor is called oscillatory circuit.
Let the capacitor be charged to a charge qo and then an inductor connected to it. Such a procedure can be easily carried out using the circuit, the scheme of which is shown in Fig. 670: first the key is closed in position 1 , while the capacitor is charged to a voltage equal to the EMF of the source, after which the key is transferred to the positions 2 , after which the discharge of the capacitor through the coil begins.

rice. 670
To determine the dependence of the charge of the capacitor on time q(t) apply Ohm's law, according to which the voltage across the capacitor U C = q/C equal to the EMF of self-induction that occurs in the coil

here, "prime" means derivative with respect to time.
Thus, the equation turns out to be valid

This equation contains two unknown functions - the dependence on the charge time q(t) and current I(t), so it cannot be solved. However, the current strength is a derivative of the charge of the capacitor q / (t) = I(t), so the derivative of the current strength is the second derivative of the charge

Taking into account this relation, we rewrite equation (1) in the form

Surprisingly, this equation completely coincides with the well-studied equation of harmonic oscillations (the second derivative of the unknown function is proportional to this function itself with a negative coefficient of proportionality x // = −ω o 2 x)! Therefore, the solution to this equation will be the harmonic function

with circular frequency

This formula defines natural frequency of the oscillatory circuit. Accordingly, the oscillation period of the capacitor charge (and the current strength in the circuit) is equal to

The resulting expression for the oscillation period is called J. Thompson's formula.
As usual, to define arbitrary parameters A, φ in the general solution (4), it is necessary to set the initial conditions - the charge and the current strength at the initial moment of time. In particular, for the considered example of the circuit in Fig. 670, the initial conditions have the form: at t = 0, q = qo, I=0, so the dependence of the capacitor charge on time will be described by the function

and the current strength changes with time according to the law

The above consideration of the oscillatory circuit is approximate - any real circuit has active resistance (connecting wires and coil windings).

rice. 671
Therefore, in equation (1), the voltage drop across this active resistance should be taken into account, so this equation will take the form

which, taking into account the relationship between charge and current strength, is converted to the form

This equation is also familiar to us - this is the equation of damped oscillations

and the attenuation coefficient, as expected, is proportional to the active resistance of the circuit β = R/L.
The processes occurring in an oscillatory circuit can also be described using the law of conservation of energy. If we neglect the active resistance of the circuit, then the sum of the energies of the electric field of the capacitor and the magnetic field of the coil remains constant, which is expressed by the equation

which is also an equation of harmonic oscillations with a frequency determined by formula (5). In its form, this equation also coincides with the equations following from the law of conservation of energy during mechanical vibrations. Since the equations describing the oscillations of the electric charge of a capacitor are similar to the equations describing mechanical oscillations, it is possible to draw an analogy between the processes occurring in an oscillatory circuit and the processes in any mechanical system. On fig. 672 such an analogy was drawn for the oscillations of a mathematical pendulum. In this case, the analogs are "capacitor charge q(t)− pendulum deflection angle φ(t)” and “current I(t) = q / (t)− pendulum speed V(t)».


rice. 672
Using this analogy, we qualitatively describe the process of charge and electric current oscillations in the circuit. At the initial moment of time, the capacitor is charged, the strength of the electric current is zero, all the energy is contained in the energy of the electric field of the capacitor (which is similar to the maximum deviation of the pendulum from the equilibrium position). Then the capacitor begins to discharge, the current strength increases, while the self-induction EMF occurs in the coil, which prevents the current from increasing; the energy of the capacitor decreases, turning into the energy of the magnetic field of the coil (an analogy - the pendulum moves to its lower point with increasing speed). When the charge on the capacitor becomes equal to zero, the current strength reaches its maximum value, while all the energy is converted into the energy of the magnetic field (the pendulum has reached its lowest point, its speed is maximum). Then the magnetic field begins to decrease, while the self-induction EMF maintains the current in the same direction, while the capacitor begins to charge, and the signs of the charges on the capacitor plates are opposite to the initial distribution (analogue - the pendulum moves to the opposite initial maximum deviation). Then the current in the circuit stops, while the charge of the capacitor again becomes maximum, but opposite in sign (the pendulum has reached its maximum deviation), after which the process will be repeated in the opposite direction.

ELECTROMAGNETIC OSCILLATIONS AND WAVES

§1 Oscillatory circuit.

Natural vibrations in the oscillatory circuit.

Thomson formula.

Damped and forced oscillations in the c.c.

1. Free vibrations in c.c.

An oscillatory circuit (c.c.) is a circuit consisting of a capacitor and an inductor. Under certain conditions in the c.c. electromagnetic fluctuations in charge, current, voltage and energy can occur.

Consider the circuit shown in Figure 2. If you put the key in position 1, then the capacitor will be charged and a charge will appear on its platesQ and tension U C. If you then turn the key to position 2, then the capacitor will begin to discharge, a current will flow in the circuit, while the energy of the electric field enclosed between the plates of the capacitor will be converted into magnetic field energy concentrated in the inductorL. The presence of an inductor leads to the fact that the current in the circuit does not increase instantly, but gradually due to the phenomenon of self-induction. As the capacitor discharges, the charge on its plates will decrease, the current in the circuit will increase. The maximum value of the loop current will reach when the charge on the plates is equal to zero. From this point on, the loop current will begin to decrease, but, due to the phenomenon of self-induction, it will be maintained by the magnetic field of the inductor, i.e. when the capacitor is fully discharged, the energy of the magnetic field stored in the inductor will begin to turn into the energy of an electric field. Because of the loop current, the capacitor will begin to recharge and a charge opposite to the original one will begin to accumulate on its plates. The capacitor will be recharged until all the energy of the magnetic field of the inductor is converted into the energy of the electric field of the capacitor. Then the process will be repeated in the opposite direction, and thus, electromagnetic oscillations will occur in the circuit.

Let us write down the 2nd Kirchhoff's law for the considered k.k.,

Differential equation k.k.

We have obtained a differential equation for charge oscillations in a c.c. This equation is similar to a differential equation describing the motion of a body under the action of a quasi-elastic force. Therefore, the solution of this equation will be written similarly

The equation of charge fluctuations in c.c.

The equation of voltage fluctuations on the capacitor plates in the c.c.

The equation of current fluctuations in k.k.

1. Damped oscillations in QC

Consider a C.C. containing capacitance, inductance, and resistance. Kirchhoff's 2nd law in this case will be written in the form

- attenuation factor,

Own cyclic frequency.

- - differential equation of damped oscillations in the c.c.

The equation of damped charge oscillations in a c.c.

The law of change of the charge amplitude during damped oscillations in the c.c.;

The period of damped oscillations.

Decrement of attenuation.

- logarithmic damping decrement.

The goodness of the circuit.

If damping is weak, then T ≈T 0

We investigate the change in voltage on the capacitor plates.

The change in current is out of phase by φ from the voltage.

at - damped oscillations are possible,

at - critical situation

at , i.e. R > RTo- fluctuations do not occur (aperiodic discharge of the capacitor).

• Electromagnetic vibrations are periodic changes over time in electrical and magnetic quantities in an electrical circuit.

• free are called such fluctuations, which arise in a closed system due to the deviation of this system from a state of stable equilibrium.

During oscillations, a continuous process of transformation of the energy of the system from one form into another takes place. In the case of oscillations of the electromagnetic field, the exchange can only take place between the electric and magnetic components of this field. The simplest system where this process can take place is oscillatory circuit.

• Ideal oscillatory circuit (LC circuit) - an electrical circuit consisting of an inductance coil L and a capacitor C.

Unlike a real oscillatory circuit, which has electrical resistance R, the electrical resistance of an ideal circuit is always zero. Therefore, an ideal oscillatory circuit is a simplified model of a real circuit.

Figure 1 shows a diagram of an ideal oscillatory circuit.

Circuit energy

Total energy of the oscillatory circuit

\(W=W_(e) + W_(m), \; \; \; W_(e) =\dfrac(C\cdot u^(2) )(2) = \dfrac(q^(2) ) (2C), \; \; \; W_(m) =\dfrac(L\cdot i^(2))(2),\)

Where We- the energy of the electric field of the oscillatory circuit at a given time, With is the capacitance of the capacitor, u- the value of the voltage on the capacitor at a given time, q- the value of the charge of the capacitor at a given time, Wm- the energy of the magnetic field of the oscillatory circuit at a given time, L- coil inductance, i- the value of the current in the coil at a given time.

Processes in the oscillatory circuit

Consider the processes that occur in the oscillatory circuit.

To remove the circuit from the equilibrium position, we charge the capacitor so that there is a charge on its plates Qm(Fig. 2, position 1 ). Taking into account the equation \(U_(m)=\dfrac(Q_(m))(C)\) we find the value of the voltage across the capacitor. There is no current in the circuit at this point in time, i.e. i = 0.

After the key is closed, under the action of the electric field of the capacitor, an electric current will appear in the circuit, the current strength i which will increase over time. The capacitor at this time will begin to discharge, because. the electrons that create the current (I remind you that the direction of the movement of positive charges is taken as the direction of the current) leave the negative plate of the capacitor and come to the positive one (see Fig. 2, position 2 ). Along with charge q tension will decrease u\(\left(u = \dfrac(q)(C) \right).\) As the current strength increases, a self-induction emf will appear through the coil, preventing a change in the current strength. As a result, the current strength in the oscillatory circuit will increase from zero to a certain maximum value not instantly, but over a certain period of time, determined by the inductance of the coil.

Capacitor charge q decreases and at some point in time becomes equal to zero ( q = 0, u= 0), the current in the coil will reach a certain value I m(see fig. 2, position 3 ).

Without the electric field of the capacitor (and resistance), the electrons that create the current continue to move by inertia. In this case, the electrons arriving at the neutral plate of the capacitor give it a negative charge, the electrons leaving the neutral plate give it a positive charge. The capacitor begins to charge q(and voltage u), but of opposite sign, i.e. the capacitor is recharged. Now the new electric field of the capacitor prevents the electrons from moving, so the current i begins to decrease (see Fig. 2, position 4 ). Again, this does not happen instantly, since now the self-induction EMF seeks to compensate for the decrease in current and “supports” it. And the value of the current I m(pregnant 3 ) turns out maximum current in contour.

And again, under the action of the electric field of the capacitor, an electric current will appear in the circuit, but directed in the opposite direction, the current strength i which will increase over time. And the capacitor will be discharged at this time (see Fig. 2, position 6 ) to zero (see Fig. 2, position 7 ). Etc.

Since the charge on the capacitor q(and voltage u) determines its electric field energy We\(\left(W_(e)=\dfrac(q^(2))(2C)=\dfrac(C \cdot u^(2))(2) \right),\) and the current in the coil i- magnetic field energy wm\(\left(W_(m)=\dfrac(L \cdot i^(2))(2) \right),\) then along with changes in charge, voltage and current, the energies will also change.

Designations in the table:

\(W_(e\, \max ) =\dfrac(Q_(m)^(2) )(2C) =\dfrac(C\cdot U_(m)^(2) )(2), \; \; \; W_(e\, 2) =\dfrac(q_(2)^(2) )(2C) =\dfrac(C\cdot u_(2)^(2) )(2), \; \; \ ; W_(e\, 4) =\dfrac(q_(4)^(2) )(2C) =\dfrac(C\cdot u_(4)^(2) )(2), \; \; \; W_(e\, 6) =\dfrac(q_(6)^(2) )(2C) =\dfrac(C\cdot u_(6)^(2) )(2),\)

\(W_(m\; \max ) =\dfrac(L\cdot I_(m)^(2) )(2), \; \; \; W_(m2) =\dfrac(L\cdot i_(2 )^(2) )(2), \; \; \; W_(m4) =\dfrac(L\cdot i_(4)^(2) )(2), \; \; \; W_(m6) =\dfrac(L\cdot i_(6)^(2) )(2).\)

The total energy of an ideal oscillatory circuit is conserved over time, since there is energy loss in it (no resistance). Then

\(W=W_(e\, \max ) = W_(m\, \max ) = W_(e2) + W_(m2) = W_(e4) + W_(m4) = ...\)

Thus, ideally LC- the circuit will experience periodic changes in current strength values i, charge q and stress u, and the total energy of the circuit will remain constant. In this case, we say that there are free electromagnetic oscillations.

• Free electromagnetic oscillations in the circuit - these are periodic changes in the charge on the capacitor plates, current strength and voltage in the circuit, occurring without consuming energy from external sources.

Thus, the occurrence of free electromagnetic oscillations in the circuit is due to the recharging of the capacitor and the occurrence of self-induction EMF in the coil, which “provides” this recharging. Note that the charge on the capacitor q and the current in the coil i reach their maximum values Qm and I m at various points in time.

Free electromagnetic oscillations in the circuit occur according to the harmonic law:

\(q=Q_(m) \cdot \cos \left(\omega \cdot t+\varphi _(1) \right), \; \; \; u=U_(m) \cdot \cos \left(\ omega \cdot t+\varphi _(1) \right), \; \; \; i=I_(m) \cdot \cos \left(\omega \cdot t+\varphi _(2) \right).\)

The smallest period of time during which LC- the circuit returns to its original state (to the initial value of the charge of this lining), is called the period of free (natural) electromagnetic oscillations in the circuit.

The period of free electromagnetic oscillations in LC-contour is determined by the Thomson formula:

\(T=2\pi \cdot \sqrt(L\cdot C), \;\;\; \omega =\dfrac(1)(\sqrt(L\cdot C)).\)

From the point of view of mechanical analogy, a spring pendulum without friction corresponds to an ideal oscillatory circuit, and to a real one - with friction. Due to the action of friction forces, the oscillations of a spring pendulum damp out over time.

*Derivation of the Thomson formula

Since the total energy of the ideal LC-circuit, equal to the sum of the energies of the electrostatic field of the capacitor and the magnetic field of the coil, is preserved, then at any time the equality

\(W=\dfrac(Q_(m)^(2) )(2C) =\dfrac(L\cdot I_(m)^(2) )(2) =\dfrac(q^(2) )(2C ) +\dfrac(L\cdot i^(2) )(2) =(\rm const).\)

We obtain the equation of oscillations in LC-circuit, using the law of conservation of energy. Differentiating the expression for its total energy with respect to time, taking into account the fact that

\(W"=0, \;\;\; q"=i, \;\;\; i"=q"",\)

we obtain an equation describing free oscillations in an ideal circuit:

\(\left(\dfrac(q^(2) )(2C) +\dfrac(L\cdot i^(2) )(2) \right)^((") ) =\dfrac(q)(C ) \cdot q"+L\cdot i\cdot i" = \dfrac(q)(C) \cdot q"+L\cdot q"\cdot q""=0,\)

\(\dfrac(q)(C) +L\cdot q""=0,\; \; \; \; q""+\dfrac(1)(L\cdot C) \cdot q=0.\ )

By rewriting it as:

\(q""+\omega ^(2) \cdot q=0,\)

note that this is the equation of harmonic oscillations with a cyclic frequency

\(\omega =\dfrac(1)(\sqrt(L\cdot C) ).\)

Accordingly, the period of the oscillations under consideration

\(T=\dfrac(2\pi )(\omega ) =2\pi \cdot \sqrt(L\cdot C).\)

Literature

1. Zhilko, V.V. Physics: textbook. allowance for grade 11 general education. school from Russian lang. training / V.V. Zhilko, L.G. Markovich. - Minsk: Nar. Asveta, 2009. - S. 39-43.

Charge the capacitor from the battery and connect it to the coil. In the circuit we have created, electromagnetic oscillations will immediately begin (Fig. 46). The discharge current of the capacitor, passing through the coil, creates a magnetic fraction around it. This means that during the discharge of a capacitor, the energy of its electric field is converted into the energy of the magnetic field of the coil, just as when a pendulum or a string vibrates, potential energy is converted into kinetic energy.

As the capacitor discharges, the voltage on its plates drops, and the current in the circuit increases, and by the time the capacitor is completely discharged, the current will be maximum (current amplitude). But even after the end of the discharge of the capacitor, the current will not stop - the decreasing magnetic field of the coil will support the movement of charges, and they will again begin to accumulate on the capacitor plates. In this case, the current in the circuit decreases, and the voltage across the capacitor increases. This process of the reverse transition of the energy of the magnetic field of the coil into the energy of the electric field of the capacitor is somewhat reminiscent of what happens when the pendulum, having skipped the middle point, rises up.

By the time the current in the circuit stops and the magnetic field of the coil disappears, the capacitor will be charged to the maximum (amplitude) voltage of reverse polarity. The latter means that on the plate where there used to be positive charges, there will now be negative ones, and vice versa. Therefore, when the discharge of the capacitor begins again (and this will happen immediately after it is fully charged), then the reverse current will flow in the circuit.

The periodically repeating exchange of energy between the capacitor and the coil is the electromagnetic oscillation in the circuit. During these oscillations, an alternating current flows in the circuit (that is, not only the magnitude, but also the direction of the current changes), and an alternating voltage acts on the capacitor (that is, not only the magnitude of the voltage changes, but also the polarity of the charges accumulating on the plates). One of the directions of the current voltage is conditionally called positive, and the opposite direction is negative.

By observing changes in voltage or current, you can plot the electromagnetic oscillations in the circuit (Fig. 46), just as we plotted the mechanical oscillations of the pendulum (). On the graph, the values ​​\u200b\u200bof positive current or voltage are plotted above the horizontal axis, and negative - below this axis. That half of the period when the current flows in the positive direction is often called the positive half-cycle of the current, and the other half is the negative half-cycle of the current. We can also talk about the positive and negative half-cycle voltage.

I would like to emphasize once again that we use the words “positive” and “negative” quite conditionally, only in order to distinguish two opposite directions of current.

Electromagnetic oscillations, which we met, are called free or natural oscillations. They arise whenever we transfer a certain amount of energy to the circuit, and then allow the capacitor and the coil to freely exchange this energy. The frequency of free oscillations (that is, the frequency of the alternating voltage and current in the circuit) depends on how quickly the capacitor and coil can store and release energy. This, in turn, depends on the inductance Lk and capacitance Ck of the circuit, just as the frequency of a string depends on its mass and elasticity. The greater the inductance L of the coil, the longer it takes to create a magnetic field in it, and the longer this magnetic field can maintain current in the circuit. The larger the capacitance C of the capacitor, the longer it will be discharged and the longer it will take to recharge this capacitor. Thus, the more Lk and C to the circuit, the slower the electromagnetic oscillations occur in it, the lower their frequency. The dependence of the frequency f about free oscillations from L to and C to the circuit is expressed by a simple formula, which is one of the basic formulas of radio engineering:

The meaning of this formula is extremely simple: in order to increase the frequency of natural oscillations f 0, it is necessary to reduce the inductance L to or the capacitance C to the circuit; to reduce f 0, the inductance and capacitance must be increased (Fig. 47).

From the formula for the frequency, one can easily derive (we already did this with the formula of Ohm's law) calculation formulas for determining one of the parameters of the circuit L k or C k at a given frequency f0 and a known second parameter. Formulas convenient for practical calculations are given on sheets 73, 74 and 75.

>> An equation describing the processes in an oscillatory circuit. Period of free electrical oscillations

§ 30 EQUATION DESCRIBING PROCESSES IN THE OSCILLATORY CIRCUIT. PERIOD OF FREE ELECTRIC OSCILLATIONS

Let us now turn to the quantitative theory of processes in an oscillatory circuit.

An equation describing the processes in an oscillatory circuit. Consider an oscillatory circuit, the resistance R of which can be neglected (Fig. 4.6).

The equation describing the free electrical oscillations in the circuit can be obtained using the law of conservation of energy. The total electromagnetic energy W of the circuit at any time is equal to the sum of its energies of the magnetic and electric fields:

This energy does not change over time if its resistance R of the circuit is zero. Hence, the time derivative of the total energy is zero. Therefore, the sum of the time derivatives of the energies of the magnetic and electric fields is equal to zero:

The physical meaning of equation (4.5) is that the rate of change in the energy of the magnetic field is equal in absolute value to the rate of change in the energy of the electric field; the "-" sign indicates that as the energy of the electric field increases, the energy of the magnetic field decreases (and vice versa).

Calculating the derivatives in equation (4.5), we get 1

But the time derivative of the charge is the current strength at a given time:

Therefore, equation (4.6) can be rewritten in the following form:

1 We calculate derivatives with respect to time. Therefore, the derivative (і 2) "is not just equal to 2 i, as it would be when calculating the derivative but i. It is necessary to multiply 2 i by the derivative i" of the current strength with respect to time, since the derivative of a complex function is calculated. The same applies to the derivative (q 2)".

The derivative of the current with respect to time is nothing but the second derivative of the charge with respect to time, just as the derivative of velocity with respect to time (acceleration) is the second derivative of the coordinate with respect to time. Substituting into equation (4.8) i "= q" and dividing the left and right parts of this equation by Li, we obtain the main equation describing free electrical oscillations in the circuit:

Now you can fully appreciate the significance of the efforts that have been expended to study the oscillations of a ball on a spring and a mathematical pendulum. After all, equation (4.9) does not differ in anything, except for the notation, from equation (3.11), which describes the vibrations of a ball on a spring. Replacing x with q, x" with q", k with 1/C, and m with L in equation (3.11), we obtain equation (4.9) exactly. But equation (3.11) has already been solved above. Therefore, knowing the formula describing the oscillations of a spring pendulum, we can immediately write down a formula for describing the electrical oscillations in the circuit.

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