Oscillatory circuit Thomson formula. SA Oscillatory circuit. Alternating electric current

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Thomson's formula looks like this:

$T\; =\; 2\backslash pi\backslash sqrt(LC)$

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Notes

An excerpt characterizing the Thomson Formula

• 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.

Thomson formula:

The period of electromagnetic oscillations in an ideal oscillatory circuit (i.e., in such a circuit where there is no energy loss) depends on the inductance of the coil and the capacitance of the capacitor and is found according to the formula first obtained in 1853 by the English scientist William Thomson:

The frequency is related to the period by an inversely proportional dependence ν = 1/Т.

For practical application, it is important to obtain undamped electromagnetic oscillations, and for this it is necessary to replenish the oscillatory circuit with electricity in order to compensate for the losses.

To obtain undamped electromagnetic oscillations, a undamped oscillation generator is used, which is an example of a self-oscillating system.

See below "Forced Electrical Vibrations"

FREE ELECTROMAGNETIC OSCILLATIONS IN THE CIRCUIT

ENERGY CONVERSION IN AN OSCILLATING CIRCUIT

See above "Oscillation circuit"

NATURAL FREQUENCY IN THE LOOP

See above "Oscillation circuit"

FORCED ELECTRICAL OSCILLATIONS

ADD DIAGRAM EXAMPLES

If in a circuit that includes inductance L and capacitance C, the capacitor is somehow charged (for example, by briefly connecting a power source), then periodic damped oscillations will occur in it:

u = Umax sin(ω0t + φ) e-αt

ω0 = (Natural oscillation frequency of the circuit)

To ensure undamped oscillations, the generator must necessarily include an element capable of connecting the circuit to the power source in time - a key or an amplifier.

In order for this switch or amplifier to open only at the right moment, feedback from the circuit to the control input of the amplifier is necessary.

An LC-type sinusoidal voltage generator must have three main components:

resonant circuit

Amplifier or key (on a vacuum tube, transistor or other element)

Feedback

Consider the operation of such a generator.

If the capacitor C is charged and it is recharged through the inductance L in such a way that the current in the circuit flows counterclockwise, then e occurs in the winding that has an inductive connection with the circuit. d.s., blocking the transistor T. The circuit is disconnected from the power source.

In the next half-cycle, when the reverse charge of the capacitor occurs, an emf is induced in the coupling winding. of another sign and the transistor opens slightly, the current from the power source passes into the circuit, recharging the capacitor.

If the amount of energy supplied to the circuit is less than the losses in it, the process will begin to decay, although more slowly than in the absence of an amplifier.

With the same replenishment and energy consumption, the oscillations are undamped, and if the replenishment of the circuit exceeds the losses in it, then the oscillations become divergent.

The following method is usually used to create a undamped character of oscillations: at small amplitudes of oscillations in the circuit, such a collector current of the transistor is provided in which the replenishment of energy exceeds its consumption. As a result, the oscillation amplitudes increase and the collector current reaches the saturation current value. A further increase in the base current does not lead to an increase in the collector current, and therefore the increase in the oscillation amplitude stops.

AC ELECTRIC CURRENT

AC GENERATOR (ac.11 class. p.131)

EMF of a frame rotating in the field

Alternator.

In a conductor moving in a constant magnetic field, an electric field is generated, an EMF of induction occurs.

The main element of the generator is a frame rotating in a magnetic field by an external mechanical motor.

Let us find the EMF induced in a frame of size a x b, rotating with an angular frequency ω in a magnetic field with induction B.

Let the angle α between the magnetic induction vector B and the frame area vector S equal zero in the initial position. In this position, no charge separation occurs.

In the right half of the frame, the velocity vector is co-directed to the induction vector, and in the left half it is opposite to it. Therefore, the Lorentz force acting on the charges in the frame is zero

When the frame is rotated through an angle of 90o, the charges are separated in the sides of the frame under the action of the Lorentz force. In the sides of the frame 1 and 3, the same induction emf arises:

εi1 = εi3 = υBb

The separation of charges in sides 2 and 4 is insignificant, and therefore the induction emf arising in them can be neglected.

Taking into account the fact that υ = ω a/2, the total EMF induced in the frame:

εi = 2 εi1 = ωB∆S

The EMF induced in the frame can be found from Faraday's law of electromagnetic induction. The magnetic flux through the area of ​​the rotating frame changes with time depending on the angle of rotation φ = wt between the lines of magnetic induction and the area vector.

When the loop rotates with a frequency n, the angle j changes according to the law j = 2πnt, and the expression for the flow takes the form:

Φ = BDS cos(wt) = BDS cos(2πnt)

According to Faraday's law, changes in the magnetic flux create an induction emf equal to minus the rate of flux change:

εi = - dΦ/dt = -Φ’ = BSω sin(ωt) = εmax sin(wt) .

where εmax = wBDS is the maximum EMF induced in the frame

Therefore, the change in the EMF of induction will occur according to a harmonic law.

If, with the help of slip rings and brushes sliding along them, we connect the ends of the coil with an electrical circuit, then under the action of the induction EMF, which changes over time according to a harmonic law, forced electrical oscillations of the current strength - alternating current - will occur in the electrical circuit.

In practice, a sinusoidal EMF is excited not by rotating a coil in a magnetic field, but by rotating a magnet or electromagnet (rotor) inside the stator - stationary windings wound on steel cores.

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If we compare Fig. 50 with fig. 17, which shows the vibrations of a body on springs, it is not difficult to establish a great similarity in all stages of the process. It is possible to compile a kind of "dictionary", with the help of which the description of electrical vibrations can be immediately translated into a description of mechanical ones, and vice versa. Here is the dictionary.

Try to reread the previous paragraph with this "dictionary". At the initial moment, the capacitor is charged (the body is deflected), i.e., a supply of electrical (potential) energy is reported to the system. The current begins to flow (the body gains speed), after a quarter of the period the current and magnetic energy are the largest, and the capacitor is discharged, the charge on it is zero (the body's speed and its kinetic energy are the largest, and the body passes through the equilibrium position), etc.

Note that the initial charge of the capacitor, and hence the voltage across it, is created by the electromotive force of the battery. On the other hand, the initial deflection of the body is created by an externally applied force. Thus, the force acting on a mechanical oscillatory system plays a role similar to the electromotive force acting on an electrical oscillatory system. Our "dictionary" can therefore be supplemented by another "translation":

7) force, 7) electromotive force.

The similarity of the regularities of both processes goes further. Mechanical oscillations are attenuated due to friction: with each oscillation, part of the energy is converted into heat due to friction, so the amplitude becomes smaller and smaller. In the same way, with each recharge of the capacitor, part of the energy of the current is converted into heat, released due to the presence of resistance at the wire of the coil. Therefore, the electrical oscillations in the circuit are also damped. Resistance plays the same role for electrical vibrations as friction plays for mechanical vibrations.

In 1853 English physicist William Thomson (Lord Kelvin, 1824-1907) showed theoretically that natural electrical oscillations in a circuit consisting of a capacitance capacitor and an inductor are harmonic, and their period is expressed by the formula

(- in henry, - in farads, - in seconds). This simple and very important formula is called the Thomson formula. The oscillatory circuits themselves with capacitance and inductance are often also called Thomson, since Thomson was the first to give a theory of electrical oscillations in such circuits. Recently, the term “-contour” is increasingly used (and similarly “-contour”, “-contour”, etc.).

Comparing Thomson's formula with the formula that determines the period of harmonic oscillations of an elastic pendulum (§ 9), we see that the mass of the body plays the same role as inductance, and the stiffness of the spring plays the same role as the reciprocal of capacitance (). In accordance with this, in our "dictionary" the second line can be written like this:

2) the stiffness of the spring 2) the reciprocal of the capacitance of the capacitor.

By choosing different and , you can get any periods of electrical oscillations. Naturally, depending on the period of electrical oscillations, it is necessary to use various methods of their observation and recording (oscillography). If we take, for example, and , then the period will be

i.e., oscillations will occur with a frequency of about . This is an example of electrical vibrations whose frequency lies in the audio range. Such fluctuations can be heard using a telephone and recorded on a loop oscilloscope. An electronic oscilloscope makes it possible to obtain a sweep of both these and higher-frequency oscillations. Radio engineering uses extremely fast oscillations - with frequencies of many millions of hertz. An electronic oscilloscope makes it possible to observe their shape just as well as we can see the shape of a pendulum with the help of the trace of a pendulum on a sooty plate (§ 3). Oscillography of free electrical oscillations with a single excitation of the oscillatory circuit is usually not used. The fact is that the state of equilibrium in the circuit is established in just a few periods, or, at best, in several tens of periods (depending on the relationship between the inductance of the circuit, its capacitance and resistance). If, say, the decay process practically ends in 20 periods, then in the above example of a circuit with periods of the entire flash of free oscillations, it will take only everything and it will be very difficult to follow the oscillogram with a simple visual observation. The problem is easily solved if the whole process - from the excitation of oscillations to their almost complete extinction - is periodically repeated. By making the scanning voltage of the electronic oscilloscope also periodic and synchronous with the process of excitation of oscillations, we will force the electron beam to “draw” the same oscillogram many times in the same place on the screen. With sufficiently frequent repetition, the picture observed on the screen will generally appear to be continuous, i.e., we will sit on a motionless and unchanging curve, an idea of ​​which is given by Fig. 49b.

In the switch circuit shown in Fig. 49, a, a multiple repetition of the process can be obtained simply by periodically tossing the switch from one position to another.

Radio engineering has for the same much more advanced and faster electrical switching methods using electronic tube circuits. But even before the invention of electronic tubes, an ingenious method was invented for periodically repeating the excitation of damped oscillations in a circuit, based on the use of a spark charge. In view of the simplicity and clarity of this method, we will dwell on it in somewhat more detail.

Rice. 51. Scheme of spark excitation of oscillations in the circuit

The oscillatory circuit is broken by a small gap (spark gap 1), the ends of which are connected to the secondary winding of step-up transformer 2 (Fig. 51). The current from the transformer charges the capacitor 3 until the voltage across the spark gap becomes equal to the breakdown voltage (see Volume II, §93). At this moment, a spark discharge occurs in the spark gap, which closes the circuit, since the column of highly ionized gas in the spark channel conducts current almost as well as metal. In such a closed circuit, electrical oscillations will occur, as described above. As long as the spark gap conducts current well, the secondary winding of the transformer is practically short-circuited by the spark, so that the entire voltage of the transformer drops on its secondary winding, the resistance of which is much greater than the resistance of the spark. Consequently, with a well-conducting spark gap, the transformer delivers practically no energy to the circuit. Due to the fact that the circuit has resistance, part of the vibrational energy is spent on Joule heat, as well as on the processes in the spark, the oscillations damp out and after a short time the amplitudes of the current and voltage drop so much that the spark goes out. Then the electrical oscillations are interrupted. From this point on, the transformer charges the capacitor again until a breakdown occurs again, and the whole process is repeated (Fig. 52). Thus, the formation of a spark and its extinction play the role of an automatic switch that ensures the repetition of the oscillatory process.

Rice. 52. Curve a) shows how the high voltage changes on the open secondary winding of the transformer. At those moments when this voltage reaches the breakdown voltage, a spark jumps in the spark gap, the circuit closes, a flash of damped oscillations is obtained - curves b)

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