ELECTROMAGNETIC OSCILLATIONS.
FREE AND FORCED ELECTRIC OSCILLATIONS.

Electromagnetic oscillations - interconnected oscillations of electric and magnetic fields.

Electromagnetic oscillations appear in various electrical circuits. In this case, the magnitude of the charge, voltage, current strength, electric field strength, magnetic field induction and other electrodynamic quantities fluctuate.

Free electromagnetic oscillations occur in an electromagnetic system after it is taken out of equilibrium, for example, by charging a capacitor or changing the current in a circuit section.

These are damped oscillations, since the energy communicated to the system is spent on heating and other processes.

Forced electromagnetic oscillations - undamped oscillations in the circuit caused by an external periodically changing sinusoidal EMF.

Electromagnetic oscillations are described by the same laws as mechanical ones, although the physical nature of these oscillations is completely different.

Electrical oscillations are a special case of electromagnetic ones, when oscillations of only electrical quantities are considered. In this case, they talk about alternating current, voltage, power, etc.

OSCILLATORY CIRCUIT

An oscillatory circuit is an electrical circuit consisting of a capacitor with a capacitance C, an inductor L and a resistor with a resistance R connected in series.

The state of stable equilibrium of the oscillatory circuit is characterized by the minimum energy of the electric field (the capacitor is not charged) and the magnetic field (there is no current through the coil).

Quantities expressing the properties of the system itself (system parameters): L and m, 1/C and k

quantities characterizing the state of the system:

quantities expressing the rate of change in the state of the system: u = x"(t) and i = q"(t).

CHARACTERISTICS OF ELECTROMAGNETIC OSCILLATIONS

It can be shown that the equation of free vibrations for a charge q = q(t) capacitor in the circuit has the form

where q" is the second derivative of charge with respect to time. Value

is the cyclic frequency. The same equations describe fluctuations in current, voltage, and other electrical and magnetic quantities.

One of the solutions to equation (1) is the harmonic function

The oscillation period in the circuit is given by the formula (Thomson):

The value φ \u003d ώt + φ 0, which is under the sign of sine or cosine, is the phase of the oscillation.

The phase determines the state of the oscillating system at any time t.

The current in the circuit is equal to the derivative of the charge with respect to time, it can be expressed

To more clearly express the phase shift, let's move from cosine to sine

AC ELECTRIC CURRENT

1. Harmonic EMF occurs, for example, in a frame that rotates at a constant angular velocity in a uniform magnetic field with induction B. Magnetic flux F, penetrating the frame with the area S,

where is the angle between the normal to the frame and the magnetic induction vector.

According to Faraday's law of electromagnetic induction, the EMF of induction is equal to

where is the rate of change of the flux of magnetic induction.

A harmonically varying magnetic flux induces a sinusoidal induction EMF

where is the amplitude value of the induction emf.

2. If you connect a source of external harmonic EMF to the circuit

then forced oscillations occur in it, occurring with a cyclic frequency ώ coinciding with the frequency of the source.

In this case, the forced oscillations make a charge q, the potential difference u, current strength i and other physical quantities. These are undamped oscillations, since energy is supplied to the circuit from a source, which compensates for losses. Harmoniously changing current, voltage and other quantities in the circuit are called variables. They obviously vary in size and direction. Currents and voltages that vary only in magnitude are called pulsating.

In industrial AC circuits in Russia, a frequency of 50 Hz is adopted.

To calculate the amount of heat Q released when an alternating current passes through a conductor with active resistance R, the maximum power value cannot be used, since it is reached only at certain points in time. It is necessary to use the average power for the period - the ratio of the total energy W entering the circuit for the period to the value of the period:

Therefore, the amount of heat released during the time T:

The effective value I of the alternating current is equal to the strength of such a direct current, which, in a time equal to the period T, releases the same amount of heat as the alternating current:

Hence the effective value of the current

Similarly effective voltage value

TRANSFORMER

Transformer- a device that increases or decreases the voltage several times with virtually no energy loss.

The transformer consists of a steel core assembled from separate plates, on which two coils with wire windings are mounted. The primary winding is connected to an alternating voltage source, and devices that consume electricity are connected to the secondary.

the value

called the transformation ratio. For step-down transformer K> 1, for step-up K< 1.

Example. The charge on the plates of the capacitor of the oscillatory circuit changes over time in accordance with the equation. Find the period and frequency of oscillations in the circuit, the cyclic frequency, the amplitude of charge oscillations and the amplitude of current oscillations. Write the equation i = i(t) expressing the dependence of the current strength on time.

It follows from the equation that . The period is determined by the cyclic frequency formula

Oscillation frequency

The dependence of the current strength on time has the form:

Current amplitude.

Answer: the charge oscillates with a period of 0.02 s and a frequency of 50 Hz, which corresponds to a cyclic frequency of 100 rad / s, the amplitude of current oscillations is 510 3 A, the current changes according to the law:

i=-5000sin100t

Tasks and tests on the topic "Topic 10. "Electromagnetic oscillations and waves.""

• Transverse and longitudinal waves. Wavelength - Mechanical oscillations and waves. Sound grade 9

Free electromagnetic oscillations this is a periodic change in the charge on the capacitor, the current in the coil, as well as electric and magnetic fields in the oscillatory circuit, occurring under the influence of internal forces.

Continuous electromagnetic oscillations

Used to excite electromagnetic oscillations oscillatory circuit , consisting of an inductor L connected in series and a capacitor with a capacitance C (Fig. 17.1).

Consider an ideal circuit, i.e., a circuit whose ohmic resistance is zero (R=0). To excite oscillations in this circuit, it is necessary either to inform the capacitor plates of a certain charge, or to excite a current in the inductor. Let at the initial moment of time the capacitor be charged to a potential difference U (Fig. (Fig. 17.2, a); therefore, it has a potential energy
.At this point in time, the current in the coil I \u003d 0 . This state of the oscillatory circuit is similar to the state of a mathematical pendulum deflected by an angle α (Fig. 17.3, a). At this time, the current in the coil I=0. After connecting the charged capacitor to the coil, under the action of the electric field created by the charges on the capacitor, free electrons in the circuit will begin to move from the negatively charged capacitor plate to the positively charged one. The capacitor will begin to discharge, and an increasing current will appear in the circuit. The alternating magnetic field of this current will generate a vortex electric field. This electric field will be directed opposite to the current and therefore will not allow it to immediately reach its maximum value. The current will increase gradually. When the force in the circuit reaches its maximum, the charge on the capacitor and the voltage between the plates is zero. This will happen in a quarter of the period t = π/4. At the same time, the energy the electric field goes into the energy of the magnetic field W e =1/2C U 2 0 . At this moment, on the positively charged plate of the capacitor there will be so many electrons that have passed to it that their negative charge completely neutralizes the positive charge of the ions that was there. The current in the circuit will begin to decrease and the induction of the magnetic field created by it will begin to decrease. The changing magnetic field will again generate a vortex electric field, which this time will be directed in the same direction as the current. The current supported by this field will go in the same direction and gradually recharge the capacitor. However, as the charge accumulates on the capacitor, its own electric field will increasingly slow down the movement of electrons, and the current in the circuit will become less and less. When the current drops to zero, the capacitor will be completely recharged.

The states of the system depicted in fig. 17.2 and 17.3 correspond to successive points in time T = 0; ;;and T.

The self-induction emf that occurs in the circuit is equal to the voltage on the capacitor plates: ε = U

and

Assuming
, we get

(17.1)

Formula (17.1) is similar to the differential equation of harmonic oscillations considered in mechanics; his decision will be

q = q max sin(ω 0 t+φ 0) (17.2)

where q max is the largest (initial) charge on the capacitor plates, ω 0 is the circular frequency of natural oscillations of the circuit, φ 0 is the initial phase.

According to the accepted notation,
where

(17.3)

Expression (17.3) is called Thomson's formula and shows that at R=0, the period of electromagnetic oscillations that occur in the circuit is determined only by the values ​​of the inductance L and capacitance C.

According to the harmonic law, not only the charge on the capacitor plates changes, but also the voltage and current in the circuit:

where U m and I m are voltage and current amplitudes.

From expressions (17.2), (17.4), (17.5) it follows that the charge (voltage) and current fluctuations in the circuit are phase-shifted by π/2. Consequently, the current reaches its maximum value at those moments in time when the charge (voltage) on the capacitor plates is zero, and vice versa.

When a capacitor is charged, an electric field appears between its plates, the energy of which is

or

When a capacitor is discharged onto an inductor, a magnetic field arises in it, the energy of which is

In an ideal circuit, the maximum energy of the electric field is equal to the maximum energy of the magnetic field:

The energy of a charged capacitor periodically changes with time according to the law

or

Given that
, we get

The energy of the magnetic field of the solenoid varies with time according to the law

(17.6)

Considering that I m ​​=q m ω 0 , we obtain

(17.7)

The total energy of the electromagnetic field of the oscillatory circuit is equal to

W \u003d W e + W m \u003d (17.8)

In an ideal circuit, the total energy is conserved, electromagnetic oscillations are undamped.

Damped electromagnetic oscillations

A real oscillatory circuit has ohmic resistance, so the oscillations in it are damped. As applied to this circuit, Ohm's law for the complete circuit can be written in the form

(17.9)

Transforming this equality:

and making the substitution:

and
, where β is the attenuation coefficient, we get

(17.10) is differential equation of damped electromagnetic oscillations .

The process of free oscillations in such a circuit no longer obeys the harmonic law. For each period of oscillation, part of the electromagnetic energy stored in the circuit is converted into Joule heat, and the oscillations become fading(Fig. 17.5). At low damping ω ≈ ω 0 , the solution of the differential equation will be an equation of the form

(17.11)

Damped vibrations in an electric circuit are similar to damped mechanical vibrations of a load on a spring in the presence of viscous friction.

The logarithmic damping decrement is equal to

(17.12)

Time interval
during which the oscillation amplitude decreases by a factor of e ≈ 2.7 is called decay time .

Quality factor Q of the oscillatory system is determined by the formula:

(17.13)

For an RLC circuit, the quality factor Q is expressed by the formula

(17.14)

The quality factor of electrical circuits used in radio engineering is usually of the order of several tens or even hundreds.

Consider the following oscillatory circuit. We assume that its resistance R is so small that it can be neglected.

The total electromagnetic energy of the oscillatory circuit at any time will be equal to the sum of the energy of the capacitor and the energy of the magnetic field of the current. The following formula will be used to calculate it:

W = L*i^2/2 + q^2/(2*C).

The total electromagnetic energy will not change over time, since there is no energy loss through the resistance. Although its components will change, they will always add up to the same number. This is provided by the law of conservation of energy.

From this it is possible to obtain equations describing free oscillations in an electrical oscillatory circuit. The equation will look like this:

q"' = -(1/(L*C))*q.

The same equation, up to notation, is obtained when describing mechanical vibrations. Given the analogy between these types of oscillations, we can write down a formula describing electromagnetic oscillations.

Frequency and period of electromagnetic oscillations

But first, let's deal with the frequency and period of electromagnetic oscillations. The value of the frequency of natural vibrations can again be obtained from an analogy with mechanical vibrations. The coefficient k/m will be equal to the square of the natural frequency.

Therefore, in our case, the square frequencies free vibrations will be equal to 1/(L*C)

ω0 = 1/√(L*C).

From here period free vibrations:

T = 2*pi/ω0 = 2*pi*√(L*C).

This formula is called Thompson's formulas. It follows from it that the oscillation period increases with an increase in the capacitance of the capacitor or the inductance of the coil. These conclusions are logical, since with an increase in capacitance, the time spent on charging the capacitor increases, and with an increase in inductance, the current in the circuit will increase more slowly, due to self-induction.

Charge fluctuation equation capacitor is described by the following formula:

q = qm*cos(ω0*t), where qm is the amplitude of the capacitor charge oscillations.

The current strength in the oscillatory circuit circuit will also make harmonic oscillations:

I = q'= Im*cos(ω0*t+pi/2).

Here Im is the amplitude of current oscillations. Note that between the fluctuations of the charge and the current strength there is a difference in vases equal to pi / 2.
The figure below shows the graphs of these fluctuations.

Again, by analogy with mechanical vibrations, where fluctuations in the speed of a body are ahead by pi / 2 of the fluctuations in the coordinates of this body.
In real conditions, it is impossible to neglect the resistance of the oscillatory circuit, and therefore the oscillations will be damped.

With a very large resistance R, oscillations may not start at all. In this case, the energy of the capacitor is released in the form of heat at the resistance.

• 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 Q m(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 through the coil, self-induction emf will appear, which prevents the current strength from changing. 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 Q m 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, an ideal oscillatory circuit corresponds to a spring pendulum without friction, and 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.

An oscillatory circuit is a device designed to generate (create) electromagnetic oscillations. From its inception to the present day, it has been used in many areas of science and technology: from everyday life to huge factories producing a wide variety of products.

What does it consist of?

The oscillatory circuit consists of a coil and a capacitor. In addition, it may also contain a resistor (element with variable resistance). An inductor (or solenoid, as it is sometimes called) is a rod on which are wound several layers of winding, which, as a rule, is a copper wire. It is this element that creates oscillations in the oscillatory circuit. The rod in the middle is often called a choke or core, and the coil is sometimes called a solenoid.

An oscillatory circuit coil only oscillates when there is a stored charge. When current passes through it, it accumulates a charge, which it then gives off to the circuit if the voltage drops.

The wires of the coil usually have very little resistance, which always remains constant. In the circuit of an oscillating circuit, a change in voltage and current very often occurs. This change is subject to certain mathematical laws:

• U = U 0 *cos(w*(t-t 0) , where
  U - voltage at a given time t,
  U 0 - voltage at time t 0,
  w is the frequency of electromagnetic oscillations.

Another integral component of the circuit is the electrical capacitor. This is an element consisting of two plates, which are separated by a dielectric. In this case, the thickness of the layer between the plates is less than their sizes. This design allows you to accumulate an electric charge on the dielectric, which can then be transferred to the circuit.

The difference between a capacitor and a battery is that there is no transformation of substances under the action of an electric current, but a direct accumulation of charge in an electric field. Thus, with the help of a capacitor, it is possible to accumulate a sufficiently large charge, which can be given away all at once. In this case, the current strength in the circuit increases greatly.

Also, the oscillatory circuit consists of one more element: a resistor. This element has resistance and is designed to control the current and voltage in the circuit. If you increase at a constant voltage, then the current strength will decrease according to Ohm's law:

• I \u003d U / R, where
  I - current strength,
  U - voltage,
  R is resistance.

Inductor

Let's take a closer look at all the subtleties of the operation of an inductor and better understand its function in an oscillatory circuit. As we have already said, the resistance of this element tends to zero. Thus, when connected to a DC circuit, it would happen. However, if you connect the coil to an AC circuit, it works properly. This allows us to conclude that the element offers resistance to alternating current.

But why does this happen and how does resistance arise with alternating current? To answer this question, we need to turn to such a phenomenon as self-induction. When the current passes through the coil, it arises in it, which creates an obstacle to the change in current. The magnitude of this force depends on two factors: the inductance of the coil and the derivative of the current strength with respect to time. Mathematically, this dependence is expressed through the equation:

• E \u003d -L ​​* I "(t) , where
  E - EMF value,
  L - the value of the inductance of the coil (for each coil it is different and depends on the number of coils of the winding and their thickness),
  I "(t) - the derivative of the current strength with respect to time (the rate of change of the current strength).

The strength of direct current does not change with time, so there is no resistance when it is exposed.

But with alternating current, all its parameters are constantly changing according to a sinusoidal or cosine law, as a result of which an EMF arises that prevents these changes. Such resistance is called inductive and is calculated by the formula:

• X L \u003d w * L, where
  w is the oscillation frequency of the circuit,
  L is the inductance of the coil.

The current strength in the solenoid linearly increases and decreases according to various laws. This means that if you stop the current supply to the coil, it will continue to give charge to the circuit for some time. And if at the same time the current supply is abruptly interrupted, then a shock will occur due to the fact that the charge will try to be distributed and exit the coil. This is a serious problem in industrial production. Such an effect (although not entirely related to the oscillatory circuit) can be observed, for example, when pulling the plug out of the socket. At the same time, a spark jumps, which on such a scale is not able to harm a person. It is due to the fact that the magnetic field does not disappear immediately, but gradually dissipates, inducing currents in other conductors. On an industrial scale, the current strength is many times greater than the 220 volts we are used to, therefore, when a circuit is interrupted in production, sparks of such strength can occur that cause a lot of harm to both the plant and the person.

The coil is the basis of what the oscillatory circuit consists of. The inductances of the solenoids in series add up. Next, we will take a closer look at all the subtleties of the structure of this element.

What is inductance?

The inductance of the coil of an oscillatory circuit is an individual indicator numerically equal to the electromotive force (in volts) that occurs in the circuit when the current changes by 1 A in 1 second. If the solenoid is connected to a DC circuit, then its inductance describes the energy of the magnetic field that is created by this current according to the formula:

• W \u003d (L * I 2) / 2, where
  W is the energy of the magnetic field.

The inductance factor depends on many factors: the geometry of the solenoid, the magnetic characteristics of the core, and the number of coils of wire. Another property of this indicator is that it is always positive, because the variables on which it depends cannot be negative.

Inductance can also be defined as the property of a conductor carrying current to store energy in a magnetic field. It is measured in Henry (named after the American scientist Joseph Henry).

In addition to the solenoid, the oscillatory circuit consists of a capacitor, which will be discussed later.

Electrical Capacitor

The capacitance of the oscillatory circuit is determined by the capacitor. About his appearance was written above. Now let's analyze the physics of the processes that take place in it.

Since the capacitor plates are made of a conductor, an electric current can flow through them. However, there is an obstacle between the two plates: a dielectric (it can be air, wood or other material with high resistance. Due to the fact that the charge cannot move from one end of the wire to the other, it accumulates on the capacitor plates. This increases the power of the magnetic and electric fields around it.Thus, when the charge stops, all the electricity accumulated on the plates begins to be transferred to the circuit.

Each capacitor has an optimum for its operation. If this element is operated for a long time at a voltage higher than the rated voltage, its service life is significantly reduced. The oscillatory circuit capacitor is constantly affected by currents, and therefore, when choosing it, you should be extremely careful.

In addition to the usual capacitors that were discussed, there are also ionistors. This is a more complex element: it can be described as a cross between a battery and a capacitor. As a rule, organic substances serve as a dielectric in an ionistor, between which there is an electrolyte. Together, they create a double electrical layer, which makes it possible to store many times more energy in this design than in a traditional capacitor.

What is the capacitance of a capacitor?

The capacitance of a capacitor is the ratio of the charge on the capacitor to the voltage it is under. This value can be calculated very simply using the mathematical formula:

• C \u003d (e 0 *S) / d, where
  e 0 - dielectric material (table value),
  S is the area of ​​the capacitor plates,
  d is the distance between the plates.

The dependence of the capacitance of a capacitor on the distance between the plates is explained by the phenomenon of electrostatic induction: the smaller the distance between the plates, the more they affect each other (according to Coulomb's law), the greater the charge of the plates and the lower the voltage. And with a decrease in voltage, the value of the capacitance increases, since it can also be described by the following formula:

• C = q/U, where
  q - charge in pendants.

It is worth talking about the units of measurement of this quantity. Capacitance is measured in farads. 1 farad is a large enough value, so existing capacitors (but not ionistors) have a capacitance measured in picofarads (one trillion farads).

Resistor

The current in the oscillatory circuit also depends on the resistance of the circuit. And in addition to the two elements described that make up the oscillatory circuit (coils, capacitors), there is also a third one - a resistor. He is responsible for creating resistance. The resistor differs from other elements in that it has a large resistance, which can be changed in some models. In the oscillatory circuit, it performs the function of a magnetic field power regulator. You can connect several resistors in series or in parallel, thereby increasing the resistance of the circuit.

The resistance of this element also depends on temperature, so you should be careful about its operation in the circuit, since it heats up when current passes.

The resistance of a resistor is measured in ohms, and its value can be calculated using the formula:

• R = (p*l)/S, where
  p is the specific resistance of the resistor material (measured in (Ohm * mm 2) / m);
  l is the length of the resistor (in meters);
  S is the cross-sectional area (in square millimeters).

How to link contour parameters?

Now we have come close to the physics of the operation of an oscillatory circuit. Over time, the charge on the capacitor plates changes according to a second-order differential equation.

If this equation is solved, several interesting formulas follow from it, describing the processes occurring in the circuit. For example, the cyclic frequency can be expressed in terms of capacitance and inductance.

However, the simplest formula that allows you to calculate many unknown quantities is the Thomson formula (named after the English physicist William Thomson, who derived it in 1853):

• T = 2*n*(L*C) 1/2 .
  T - period of electromagnetic oscillations,
  L and C - respectively, the inductance of the coil of the oscillatory circuit and the capacitance of the circuit elements,
  n is the number pi.

quality factor

There is another important value that characterizes the operation of the circuit - the quality factor. In order to understand what it is, one should turn to such a process as resonance. This is a phenomenon in which the amplitude becomes maximum with a constant value of the force that supports this oscillation. The resonance can be explained with a simple example: if you start pushing the swing to the beat of its frequency, then it will accelerate, and its "amplitude" will increase. And if you push out of time, they will slow down. At resonance, a lot of energy is often dissipated. In order to be able to calculate the magnitude of the losses, they came up with such a parameter as the quality factor. It is a ratio equal to the ratio of the energy in the system to the losses occurring in the circuit in one cycle.

The quality factor of the circuit is calculated by the formula:

• Q = (w 0 *W)/P, where
  w 0 - resonant cyclic oscillation frequency;
  W is the energy stored in the oscillatory system;
  P is the dissipated power.

This parameter is a dimensionless value, since it actually shows the ratio of energies: stored to spent.

What is an ideal oscillatory circuit

To better understand the processes in this system, physicists came up with the so-called ideal oscillating circuit. This is a mathematical model that represents a circuit as a system with zero resistance. It produces undamped harmonic oscillations. Such a model makes it possible to obtain formulas for the approximate calculation of contour parameters. One of these parameters is the total energy:

• W \u003d (L * I 2) / 2.

Such simplifications significantly speed up calculations and make it possible to evaluate the characteristics of a circuit with given indicators.

How it works?

The entire cycle of the oscillatory circuit can be divided into two parts. Now we will analyze in detail the processes occurring in each part.

• First phase: A positively charged capacitor plate begins to discharge, giving current to the circuit. At this moment, the current goes from a positive charge to a negative one, passing through the coil. As a result, electromagnetic oscillations occur in the circuit. The current, having passed through the coil, passes to the second plate and charges it positively (whereas the first plate, from which the current flowed, is charged negatively).
• Second phase: the reverse process takes place. The current passes from the positive plate (which was negative at the very beginning) to the negative, passing again through the coil. And all the charges fall into place.

The cycle is repeated until the capacitor is charged. In an ideal oscillatory circuit, this process occurs endlessly, but in a real one, energy losses are inevitable due to various factors: heating, which occurs due to the existence of resistance in the circuit (Joule heat), and the like.

Loop design options

In addition to simple coil-capacitor and coil-resistor-capacitor circuits, there are other options that use an oscillating circuit as a basis. This, for example, is a parallel circuit, which differs in that it exists as an element of an electrical circuit (because, if it existed separately, it would be a series circuit, which was discussed in the article).

There are also other types of construction, including different electrical components. For example, you can connect a transistor to the network, which will open and close the circuit with a frequency equal to the oscillation frequency in the circuit. Thus, undamped oscillations will be established in the system.

Where is the oscillatory circuit used?

The most familiar application of circuit components is electromagnets. They, in turn, are used in intercoms, electric motors, sensors and in many other not so common areas. Another application is an oscillation generator. In fact, this use of the circuit is very familiar to us: in this form it is used in the microwave to create waves and in mobile and radio communications to transmit information over a distance. All this happens due to the fact that the oscillations of electromagnetic waves can be encoded in such a way that it becomes possible to transmit information over long distances.

The inductor itself can be used as an element of a transformer: two coils with a different number of windings can transfer their charge using an electromagnetic field. But since the characteristics of the solenoids are different, the current indicators in the two circuits to which these two inductors are connected will differ. Thus, it is possible to convert a current with a voltage of, say, 220 volts into a current with a voltage of 12 volts.

Conclusion

We analyzed in detail the principle of operation of the oscillatory circuit and each of its parts separately. We learned that an oscillatory circuit is a device designed to create electromagnetic waves. However, these are only the basics of the complex mechanics of these seemingly simple elements. You can learn more about the intricacies of the circuit and its components from the specialized literature.