The main device that determines the operating frequency of any alternator is an oscillatory circuit. The oscillatory circuit (Fig. 1) consists of an inductor L(consider the ideal case when the coil has no ohmic resistance) and the capacitor C and is called closed. The characteristic of a coil is its inductance, it is denoted L and is measured in Henry (H), the capacitor is characterized by the capacitance C, which is measured in farads (F).

Let the capacitor be charged at the initial moment of time (Fig. 1) so that one of its plates has a charge + Q 0 , and on the other - charge - Q 0 . In this case, an electric field is formed between the plates of the capacitor, which has an energy

where is the amplitude (maximum) voltage or potential difference across the capacitor plates.

After the circuit is closed, the capacitor begins to discharge and an electric current will flow through the circuit (Fig. 2), the value of which increases from zero to the maximum value. Since an alternating current flows in the circuit, an EMF of self-induction is induced in the coil, which prevents the capacitor from discharging. Therefore, the process of discharging the capacitor does not occur instantly, but gradually. At each moment of time, the potential difference across the capacitor plates

(where is the charge of the capacitor at a given time) is equal to the potential difference across the coil, i.e. equal to self-induction emf

Fig.1	Fig.2

When the capacitor is completely discharged and , the current in the coil will reach its maximum value (Fig. 3). The induction of the magnetic field of the coil at this moment is also maximum, and the energy of the magnetic field will be equal to

Then the current strength begins to decrease, and the charge will accumulate on the capacitor plates (Fig. 4). When the current decreases to zero, the charge of the capacitor reaches its maximum value. Q 0 , but the plate, previously positively charged, will now be negatively charged (Fig. 5). Then the capacitor begins to discharge again, and the current in the circuit will flow in the opposite direction.

So the process of charge flowing from one plate of the capacitor to another through the inductor is repeated again and again. They say that in the circuit occur electromagnetic oscillations. This process is associated not only with fluctuations in the magnitude of the charge and voltage on the capacitor, the current strength in the coil, but also with the transfer of energy from the electric field to the magnetic field and vice versa.

Fig.3	Fig.4

Recharging the capacitor to the maximum voltage will occur only when there is no energy loss in the oscillatory circuit. Such a circuit is called ideal.

In real circuits, the following energy losses take place:

1) heat losses, because R ¹ 0;

2) losses in the capacitor dielectric;

3) hysteresis losses in the coil core;

4) radiation losses, etc. If we neglect these energy losses, then we can write that , i.e.

Oscillations occurring in an ideal oscillatory circuit in which this condition is satisfied are called free, or own, oscillations of the contour.

In this case, the voltage U(and charge Q) on the capacitor varies according to the harmonic law:

where n is the natural frequency of the oscillatory circuit, w 0 = 2pn is the natural (circular) frequency of the oscillatory circuit. The frequency of electromagnetic oscillations in the circuit is defined as

Period T- the time during which one complete oscillation of the voltage across the capacitor and the current in the circuit takes place, is determined Thomson's formula

The current strength in the circuit also changes according to the harmonic law, but lags behind the voltage in phase by . Therefore, the dependence of the current strength in the circuit on time will have the form

Figure 6 shows graphs of voltage changes U on the capacitor and current I in a coil for an ideal oscillatory circuit.

In a real circuit, the energy will decrease with each oscillation. The amplitudes of the voltage on the capacitor and the current in the circuit will decrease, such oscillations are called damped. They cannot be used in master generators, because the device will work at best in a pulsed mode.

Fig.5	Fig.6

To obtain undamped oscillations, it is necessary to compensate for energy losses at a wide variety of operating frequencies of devices, including those used in medicine.

Lesson type: a lesson of primary acquaintance with the material and the practical application of knowledge and skills.

Lesson duration: 45 minutes.

Goals:

Didactic – generalize and systematize knowledge about the physical processes occurring in an electromagnetic oscillatory circuit

create conditions for the assimilation of new material, using active teaching methods

educational I– to show the universal nature of the theory of oscillations;

Educational - to develop the cognitive processes of students, based on the application of the scientific method of cognition: similarity and modeling; forecasting the situation; to develop among schoolchildren methods of effective processing of educational information, to continue the formation of communicative competencies.

Educational – to continue the formation of ideas about the relationship between natural phenomena and a single physical picture of the world

Lesson objectives:

1. Educational

ü formulate the dependence of the period of the oscillatory circuit on its characteristics: capacitance and inductance

ü to study the techniques for solving typical problems on the "oscillatory circuit"

2. Educational

ü continue the formation of skills to compare phenomena, draw conclusions and generalizations based on experiment

ü work on the formation of skills to analyze properties and phenomena based on knowledge.

3. Nurturers

ü to show the significance of experimental facts and experiment in human life.

ü reveal the significance of the accumulation of facts and their clarifications in the cognition of phenomena.

ü to acquaint students with the relationship and conditionality of the phenomena of the surrounding world.

TCO:computer, projector, IAD

Preliminary preparation:

- individual evaluation sheets - 24 pieces

- route sheets (colored) - 4 pieces

Technological map of the lesson:

Lesson stages	Active Methods	ICT support
1.Organizational	Epigraph of the lesson	Slide №1,2
2. Knowledge update (generalization of previously studied material - testing knowledge of formulas on the topic “Mechanical and electromagnetic vibrations”)	Get the error! Formulas are given with errors. Assignment: correct mistakes, then peer-check, scoring	Slide #3 Slide #4 slide number 5
3.Activity motivation : why this topic is studied in the 11th grade physics course (the word of the teacher-thesis) The oscillatory circuit is the main part of the radio receiver. The purpose of the receiver is to receive vibrations (waves) of various frequencies. The simplest oscillatory circuit is a coil and a capacitor with characteristics of inductance and capacitance, respectively. How does the receiving capacity of the circuit depend on the coil and the capacitor?	Keywords CMD (collective mental activity) The groups have 5 minutes to by brainstorming give a general interpretation of these terms and suggest how they will appear in the next lesson.	slide number 6
4. Goal setting Find out the dependence of the period of the electromagnetic oscillatory circuit on the capacitance of the capacitor and the inductance of the coil. Learn how to use formulas to solve problems.	(the goal is set by the students themselves, using key terms)
5. Formation of new knowledge (using the experience of students when learning new material) What period formula do you already know? T=2π/ω; ω =2πν What formula for cyclic frequency was obtained in the last lesson? Connect these two formulas and get the formula that the king of Victorian physics, William Thomson, derived: History of Lord Thomson	Virtual laboratory (video experiment) Virtual laboratory (interactive model) "Thick" questions: Explain why...? Why do you think...? What is the difference …? Guess what happens if...? "Subtle" questions: What? Where? How? Can...? Will it …? Do you agree …? Basket - method (analysis of the practical situation in groups)	Slide #9 Slide #10 Slide №11,12
6. Control of acquired knowledge Debrief one problem on the board In groups, come up with a condition for a qualitative or calculation problem, write it down on the route sheet, the next group solves this problem, the speaker shows on the board

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/T.

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 varies 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.

Go to page:

"Damped oscillations" - 26.1. Free damped mechanical oscillations; 26.2. damping factor and logarithmic damping decrement; 26.26. Self-oscillations; Today: Saturday, August 6, 2011 Lecture 26. Fig. 26.1.

"Harmonic vibrations" - The beat method is used for tuning musical instruments, hearing analysis, etc. Figure 4. View fluctuations. (2.2.4). ?1 is the phase of the 1st oscillation. - The resulting oscillation, also harmonic, with a frequency?: Projection of circular motion on the y-axis, also makes a harmonic oscillation. Figure 3

"Frequency of oscillation" - Reflection of sound. Speed ​​of sound in various media, m/s (at t = 20°C). Mechanical vibrations with a frequency of less than 20 Hz are called infrasound. Understand sound as a phenomenon. Project goals. Sound sources. The speed of sound depends on the properties of the medium in which the sound propagates. What determines the timbre of a sound?

"Mechanical vibrations and waves" - Properties of waves. Types of waves. Mathematical pendulum. The period of free oscillations of a mathematical pendulum. Energy transformation. Laws of reflection. Spring pendulum. The hearing organs are most sensitive to sounds with frequencies from 700 to 6000 Hz. Free Forced Self Oscillations.

"Mechanical vibrations" - Harmonic. Elastic waves are mechanical disturbances propagating in an elastic medium. Mathematical pendulum. Waves. Wavelength (?) is the distance between the nearest particles oscillating in the same phase. Forced. Forced vibrations. Graph of a mathematical pendulum. Waves - the propagation of vibrations in space over time.

"Mechanical resonance" - Amplitude of forced oscillations. State educational institution Gymnasium No. 363 of the Frunzensky district. The destructive role of resonance Bridges. Resonance in technology. Thomas Young. 1. Physical basis of resonance Forced vibrations. Mechanical reed frequency meter - a device for measuring the frequency of vibrations.

There are 10 presentations in total in the topic

• 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.