The oscillation period of such a current is much longer than the propagation time, which means that the process will almost not change over time τ. Free oscillations in a circuit without active resistance An oscillating circuit is a circuit of inductance and capacitance. Let's find the oscillation equation.
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Lecture
electrical vibrations
Plan
- Quasi-stationary currents
- Free oscillations in a circuit without active resistance
- Alternating current
- dipole radiation
- Quasi-stationary currents
The electromagnetic field propagates at the speed of light.
l - conductor length
Quasi-stationary current condition:
The oscillation period of such a current is much longer than the propagation time, which means that the process will hardly change over time τ.
Instantaneous values of quasi-stationary currents obey Ohm's and Kirchhoff's laws.
2) Free oscillations in the circuit without active resistance
Oscillatory circuit- a circuit of inductance and capacitance.
Let's find the oscillation equation. We will consider the charging current of the capacitor as positive.
Dividing both sides of the equation by L , we get
Let be
Then the oscillation equation takes the form
The solution to such an equation is:
Thomson formula
Current is leading in phase U on π /2
- Free damped vibrations
Any real circuit has active resistance, the energy is used for heating, the oscillations are damped.
At
Decision:
Where
The frequency of damped oscillations is less than the natural frequency
At R=0
Logarithmic damping decrement:
If damping is small
Quality factor:
- Forced electrical vibrations
The voltage across the capacitance is out of phase with the current byπ /2, and the voltage across the inductance leads the current in phase byπ /2. The voltage across the resistance changes in phase with the current.
- Alternating current
Electrical impedance (impedance)
Reactive inductive reactance
Reactive capacitance
AC power
RMS values in AC circuit
with osφ - Power factor
- dipole radiation
The simplest system emitting EMW is an electric dipole.
Dipole moment
r is the charge radius vector
l - oscillation amplitude
Let be
wave zone
Wave front spherical
Sections of the wave front through the dipole - meridians , through perpendiculars to the dipole axis – parallels.
Dipole radiation power
The average radiation power of the dipole is proportional to the square of the amplitude of the electric moment of the dipole and the 4th power of the frequency.
a is the acceleration of the oscillating charge.
Most natural and artificial sources of electromagnetic radiation satisfy the condition
d- radiation area size
Or
v- average charge speed
Such a source of electromagnetic radiation is the Hertzian dipole
The range of distances to the Hertzian dipole is called the wave zone
Total average radiation intensity of the Hertzian dipole
Any charge moving with acceleration excites electromagnetic waves, and the radiation power is proportional to the square of the acceleration and the square of the charge
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Electrical oscillations and electromagnetic waves
fluctuating changes in electrical circuit quantities of charge, current or voltage are called electrical oscillations. Variable electric current is one of the types of electrical oscillations.
High-frequency electrical oscillations are obtained in most cases using an oscillatory circuit.
The oscillatory circuit is a closed circuit consisting of an inductance L and containers C.
The period of natural oscillations of the circuit:
and the current in the circuit changes according to the law of damped oscillations:
When an oscillatory circuit is exposed to a variable emf, forced oscillations are set in the circuit. Amplitude of forced current oscillations at constant values L, C, R depends on the ratio of the natural oscillation frequency of the circuit and the frequency of change of the sinusoidal EMF (Fig. 1).
According to the Biot-Savart-Laplace law, the conduction current creates a magnetic field with closed lines of force. Such a field is called eddy.
An alternating conduction current creates an alternating magnetic field. Alternating current, unlike direct current, passes through the capacitor; but this current is not a conduction current; it is called bias current. The bias current is a time-varying electric field; it creates an alternating magnetic field, like an alternating conduction current. Bias Current Density:
At each point in space, the change in time of the induction of the electric field creates an alternating vortex magnetic field (Fig. 2a). Vectors B of the emerging magnetic field lie in a plane perpendicular to the vector D. The mathematical equation that expresses this pattern is called Maxwell's first equation.
With electromagnetic induction, an electric field arises with closed lines of force (vortex field), which manifests itself as an EMF of induction. At each point in space, the change in time of the induction vector magnetic field creates an alternating vortex electric field (Fig. 2b). Vectors D of the emerging electric field lie in a plane perpendicular to the vector B. The mathematical equation that describes this pattern is called Maxwell's second equation.
The combination of variable electric and magnetic fields, which are inextricably linked with each other, is called the electromagnetic field.
It follows from Maxwell's equations that the change in time of the electric (or magnetic) field that has arisen at any point will move from one point to another, and mutual transformations of electric and magnetic fields will occur.
Electromagnetic waves are a process of simultaneous propagation in space of changing electric and magnetic fields. Vectors of strengths of electric and magnetic fields ( E and H) to the electromagnetic wave are perpendicular to each other, and the vector v propagation velocity is perpendicular to the plane in which both vectors lie E and H(Fig.3), This is true for the propagation of electromagnetic waves and unlimited space.
The speed of propagation of electromagnetic waves in vacuum does not depend on the wavelength and is equal to
The speed of electromagnetic waves in various media is less than the speed in vacuum.
Electrical oscillations are understood as periodic changes in charge, current and voltage. The simplest system in which free electrical oscillations are possible is the so-called oscillatory circuit. This is a device consisting of a capacitor and a coil connected to each other. We will assume that there is no active resistance of the coil, in this case the circuit is called ideal. When energy is communicated to this system, undamped harmonic oscillations of the charge on the capacitor, voltage and current will occur in it.
It is possible to inform the oscillatory circuit of energy different ways. For example, by charging a capacitor from a DC source or by exciting current in an inductor. In the first case, the electric field between the plates of the capacitor possesses energy. In the second, the energy is contained in the magnetic field of the current flowing through the circuit.
§1 The equation of oscillations in the circuit
Let us prove that when energy is imparted to the circuit, undamped harmonic oscillations will occur in it. To do this, it is necessary to obtain a differential equation of harmonic oscillations of the form .
Suppose the capacitor is charged and closed to the coil. The capacitor will begin to discharge, current will flow through the coil. According to Kirchhoff's second law, the sum of the voltage drops along a closed circuit is equal to the sum of the EMF in this circuit.
In our case, the voltage drop is because the circuit is ideal. The capacitor in the circuit behaves like a current source, the potential difference between the capacitor plates acts as an EMF, where is the charge on the capacitor, is the capacitance of the capacitor. In addition, when a changing current flows through the coil, an EMF of self-induction arises in it, where is the inductance of the coil, is the rate of change of current in the coil. Since the EMF of self-induction prevents the process of discharging the capacitor, the second Kirchhoff law takes the form
But the current in the circuit is the current of discharging or charging the capacitor, therefore. Then
The differential equation is transformed to the form
By introducing the notation , we obtain the well-known differential equation of harmonic oscillations.
This means that the charge on the capacitor in the oscillatory circuit will change according to the harmonic law
where is the maximum value of the charge on the capacitor, is the cyclic frequency, is the initial phase of the oscillations.
Charge oscillation period. This expression is called the Thompson formula.
Capacitor voltage
Circuit current
We see that in addition to the charge on the capacitor, according to the harmonic law, the current in the circuit and the voltage on the capacitor will also change. The voltage oscillates in phase with the charge, and the current is ahead of the charge in
phase on .
Capacitor electric field energy
The energy of the magnetic field current
Thus, the energies of the electric and magnetic fields also change according to the harmonic law, but with a doubled frequency.
Summarize
Electric oscillations should be understood as periodic changes in charge, voltage, current strength, electric field energy, magnetic field energy. These oscillations, like mechanical ones, can be both free and forced, harmonic and non-harmonic. Free harmonic electrical oscillations are possible in an ideal oscillatory circuit.
§2 Processes occurring in an oscillatory circuit
We mathematically proved the existence of free harmonic oscillations in an oscillatory circuit. However, it remains unclear why such a process is possible. What causes oscillations in a circuit?
In the case of free mechanical vibrations, such a reason was found - it is an internal force that arises when the system is taken out of equilibrium. This force at any moment is directed to the equilibrium position and is proportional to the coordinate of the body (with a minus sign). Let's try to find a similar reason for the occurrence of oscillations in the oscillatory circuit.
Let the oscillations in the circuit excite by charging the capacitor and closing it to the coil.
At the initial moment of time, the charge on the capacitor is maximum. Consequently, the voltage and energy of the electric field of the capacitor are also maximum.
There is no current in the circuit, the energy of the magnetic field of the current is zero.
First quarter of the period- capacitor discharge.
The capacitor plates, having different potentials, are connected by a conductor, so the capacitor begins to discharge through the coil. The charge, the voltage on the capacitor and the energy of the electric field decrease.
The current that appears in the circuit increases, however, its growth is prevented by the self-induction EMF that occurs in the coil. The energy of the magnetic field of the current increases.
A quarter has passed- the capacitor is discharged.
The capacitor discharged, the voltage across it became equal to zero. The energy of the electric field at this moment is also equal to zero. According to the law of conservation of energy, it could not disappear. The energy of the field of the capacitor has completely turned into the energy of the magnetic field of the coil, which at this moment reaches its maximum value. The maximum current in the circuit.
It would seem that at this moment the current in the circuit should stop, because the cause of the current, the electric field, has disappeared. However, the disappearance of the current is again prevented by the EMF of self-induction in the coil. Now it will maintain a decreasing current, and it will continue to flow in the same direction, charging the capacitor. The second quarter of the period begins.
Second quarter of the period - Capacitor recharge.
The current supported by the self-induction EMF continues to flow in the same direction, gradually decreasing. This current charges the capacitor in opposite polarity. The charge and voltage across the capacitor increase.
The energy of the magnetic field of the current, decreasing, passes into the energy of the electric field of the capacitor.
The second quarter of the period has passed - the capacitor has recharged.
The capacitor recharges as long as there is current. Therefore, at the moment when the current stops, the charge and voltage on the capacitor take on a maximum value.
The energy of the magnetic field at this moment completely turned into the energy of the electric field of the capacitor.
The situation in the circuit at this moment is equivalent to the original one. The processes in the circuit will be repeated, but in the opposite direction. One complete oscillation in the circuit, lasting for a period, will end when the system returns to its original state, that is, when the capacitor is recharged in its original polarity.
It is easy to see that the cause of oscillations in the circuit is the phenomenon of self-induction. The EMF of self-induction prevents a change in current: it does not allow it to instantly increase and instantly disappear.
By the way, it would not be superfluous to compare the expressions for calculating the quasi-elastic force in a mechanical oscillatory system and the EMF of self-induction in the circuit:
Previously, differential equations were obtained for mechanical and electrical oscillatory systems:
Despite the fundamental differences between physical processes in mechanical and electrical oscillatory systems, the mathematical identity of the equations describing the processes in these systems is clearly visible. This should be discussed in more detail.
§3 Analogy between electrical and mechanical vibrations
Careful analysis of differential equations for spring pendulum and an oscillatory circuit, as well as formulas connecting the quantities that characterize the processes in these systems, makes it possible to identify which quantities behave in the same way (Table 2).
| Spring pendulum | Oscillatory circuit |
| Body coordinate () | Charge on the capacitor () |
| body speed | Loop current |
| Potential energy of an elastically deformed spring | Capacitor electric field energy |
| Kinetic energy of the load | The energy of the magnetic field of the coil with current |
| The reciprocal of the spring stiffness | Capacitor capacity |
| Load weight | Coil inductance |
| Elastic force | EMF of self-induction, equal to the voltage on the capacitor |
table 2
It is important not just a formal similarity between the quantities that describe the processes of pendulum oscillation and the processes in the circuit. The processes themselves are identical!
The extreme positions of the pendulum are equivalent to the state of the circuit when the charge on the capacitor is maximum.
The equilibrium position of the pendulum is equivalent to the state of the circuit when the capacitor is discharged. At this moment, the elastic force vanishes, and there is no voltage on the capacitor in the circuit. The speed of the pendulum and the current in the circuit are maximum. The potential energy of elastic deformation of the spring and the energy of the electric field of the capacitor are equal to zero. The energy of the system consists of the kinetic energy of the load or the energy of the magnetic field of the current.
The discharge of the capacitor proceeds similarly to the movement of the pendulum from the extreme position to the equilibrium position. The process of recharging the capacitor is identical to the process of removing the load from the equilibrium position to the extreme position.
The total energy of an oscillatory system or remains unchanged over time.
A similar analogy can be traced not only between a spring pendulum and an oscillatory circuit. General patterns of free oscillations of any nature! These patterns, illustrated by the example of two oscillatory systems (a spring pendulum and an oscillatory circuit), are not only possible, but must see in the vibrations of any system.
In principle, it is possible to solve the problem of any oscillatory process by replacing it with pendulum oscillations. To do this, it is enough to competently build an equivalent mechanical system, solve a mechanical problem and change the values in the final result. For example, you need to find the period of oscillation in a circuit containing a capacitor and two coils connected in parallel.
The oscillatory circuit contains one capacitor and two coils. Since the coil behaves like the weight of a spring pendulum and the capacitor behaves like a spring, the equivalent mechanical system must contain one spring and two weights. The whole problem is how the weights are attached to the spring. Two cases are possible: one end of the spring is fixed, and one weight is attached to the free end, the second is on the first one, or the weights are attached to different ends of the spring.
When coils of different inductances are connected in parallel, the currents flow through them are different. Consequently, the speeds of the loads in an identical mechanical system must also be different. Obviously, this is possible only in the second case.
We have already found the period of this oscillatory system. He is equal. Replacing the masses of the weights by the inductance of the coils, and the reciprocal of the spring stiffness by the capacitance of the capacitor, we obtain .
§4 Oscillatory circuit with a direct current source
Consider an oscillatory circuit containing a direct current source. Let the capacitor be initially uncharged. What will happen in the system after the key K is closed? Will oscillations be observed in this case and what is their frequency and amplitude?
Obviously, after the key is closed, the capacitor will begin to charge. We write Kirchhoff's second law:
The current in the circuit is the charging current of the capacitor, therefore. Then . The differential equation is transformed to the form
*Solve the equation by change of variables.
Let's denote . Differentiate twice and, taking into account that , we obtain . The differential equation takes the form
This is a differential equation of harmonic oscillations, its solution is the function
where is the cyclic frequency, the integration constants and are found from the initial conditions.
The charge on a capacitor changes according to the law
Immediately after the switch is closed, the charge on the capacitor is zero and there is no current in the circuit. Taking into account the initial conditions, we obtain a system of equations:
Solving the system, we get and . After the key is closed, the charge on the capacitor changes according to the law.
It is easy to see that harmonic oscillations occur in the circuit. The presence of a direct current source in the circuit did not affect the oscillation frequency, it remained equal. The “equilibrium position” has changed - at the moment when the current in the circuit is maximum, the capacitor is charged. The amplitude of the charge oscillations on the capacitor is equal to Cε.
The same result can be obtained more simply by using the analogy between oscillations in a circuit and oscillations of a spring pendulum. A constant current source is equivalent to a constant force field in which a spring pendulum is placed, for example, a gravitational field. The absence of charge on the capacitor at the moment of closing the circuit is identical to the absence of deformation of the spring at the moment of bringing the pendulum into oscillatory motion.
In a constant force field, the period of oscillation of a spring pendulum does not change. The oscillation period in the circuit behaves in the same way - it remains unchanged when a direct current source is introduced into the circuit.
In the equilibrium position, when the load speed is maximum, the spring is deformed:
When the current in the oscillatory circuit is maximum. Kirchhoff's second law is written as follows
At this moment, the charge on the capacitor is equal to The same result could be obtained based on the expression (*) by replacing
§5 Examples of problem solving
Task 1 Law of energy conservation
L\u003d 0.5 μH and a capacitor with a capacitance With= 20 pF electrical oscillations occur. What is the maximum voltage across the capacitor if the amplitude of the current in the circuit is 1 mA? The active resistance of the coil is negligible.
Decision:
2 At the moment when the voltage on the capacitor is maximum (maximum charge on the capacitor), there is no current in the circuit. The total energy of the system consists only of the energy of the electric field of the capacitor
3 At the moment when the current in the circuit is maximum, the capacitor is completely discharged. The total energy of the system consists only of the energy of the magnetic field of the coil
4 Based on expressions (1), (2), (3) we obtain the equality . The maximum voltage across the capacitor is
Task 2 Law of energy conservation
In an oscillatory circuit consisting of an inductance coil L and a capacitor WITH, electrical oscillations occur with a period T = 1 μs. The maximum charge value. What is the current in the circuit at the moment when the charge on the capacitor is equal to? The active resistance of the coil is negligible.
Decision:
1 Since the active resistance of the coil can be neglected, the total energy of the system, consisting of the energy of the electric field of the capacitor and the energy of the magnetic field of the coil, remains unchanged over time:
2 At the moment when the charge on the capacitor is maximum, there is no current in the circuit. The total energy of the system consists only of the energy of the electric field of the capacitor
3 Based on (1) and (2) we obtain the equality . The current in the circuit is .
4 The oscillation period in the circuit is determined by the Thomson formula. From here. Then for the current in the circuit we obtain
Task 3 Oscillatory circuit with two capacitors connected in parallel
In an oscillatory circuit consisting of an inductance coil L and a capacitor WITH, electrical oscillations occur with an amplitude of charge. At the moment when the charge on the capacitor is maximum, the key K is closed. What will be the period of oscillations in the circuit after the key is closed? What is the amplitude of the current in the circuit after closing the switch? Ignore the ohmic resistance of the circuit.
Decision:
1 Closing the key leads to the appearance in the circuit of another capacitor connected in parallel to the first one. The total capacitance of two capacitors connected in parallel is .
The period of oscillations in the circuit depends only on its parameters and does not depend on how oscillations were excited in the system and what energy was imparted to the system for this. According to the Thomson formula.
2 To find the amplitude of the current, let's find out what processes occur in the circuit after the key is closed.
The second capacitor was connected at the moment when the charge on the first capacitor was maximum, therefore, there was no current in the circuit.
The loop capacitor should begin to discharge. The discharge current, having reached the node, should be divided into two parts. However, in the branch with the coil, an EMF of self-induction occurs, which prevents the increase in the discharge current. For this reason, the entire discharge current will flow into the branch with the capacitor, the ohmic resistance of which is zero. The current will stop as soon as the voltages on the capacitors are equal, while the initial charge of the capacitor is redistributed between the two capacitors. The charge redistribution time between two capacitors is negligible due to the absence of ohmic resistance in the capacitor branches. During this time, the current in the branch with the coil will not have time to appear. Oscillations in the new system will continue after the charge is redistributed between the capacitors.
It is important to understand that in the process of redistributing the charge between two capacitors, the energy of the system is not conserved! Before the key was closed, one capacitor, a loop capacitor, had energy:
After the charge is redistributed, a battery of capacitors possesses energy:
It is easy to see that the energy of the system has decreased!
3 We find the new amplitude of the current using the law of conservation of energy. In the process of oscillations, the energy of the capacitor bank is converted into the energy of the magnetic field of the current:
Please note that the law of conservation of energy begins to "work" only after the completion of the redistribution of charge between the capacitors.
Task 4 Oscillatory circuit with two capacitors connected in series
The oscillatory circuit consists of a coil with an inductance L and two capacitors C and 4C connected in series. A capacitor with a capacity of C is charged to a voltage, a capacitor with a capacity of 4C is not charged. After the key is closed, oscillations begin in the circuit. What is the period of these oscillations? Determine the amplitude of the current, the maximum and minimum voltage values on each capacitor.
Decision:
1 At the moment when the current in the circuit is maximum, there is no self-induction EMF in the coil. We write down for this moment the second law of Kirchhoff
We see that at the moment when the current in the circuit is maximum, the capacitors are charged to the same voltage, but in the opposite polarity:
2 Before closing the key, the total energy of the system consisted only of the energy of the electric field of the capacitor C:
At the moment when the current in the circuit is maximum, the energy of the system is the sum of the energy of the magnetic field of the current and the energy of two capacitors charged to the same voltage:
According to the law of conservation of energy
To find the voltage on the capacitors, we use the law of conservation of charge - the charge of the lower plate of the capacitor C has partially transferred to the upper plate of the capacitor 4C:
We substitute the found voltage value into the law of conservation of energy and find the amplitude of the current in the circuit:
3 Let's find the limits within which the voltage on the capacitors changes during the oscillation process.
It is clear that at the moment the circuit was closed, there was a maximum voltage on the capacitor C. Capacitor 4C was not charged, therefore, .
After the switch is closed, capacitor C begins to discharge, and a capacitor with a capacity of 4C begins to charge. The process of discharging the first and charging the second capacitors ends as soon as the current in the circuit stops. This will happen in half a period. According to the laws of conservation of energy and electric charge:
Solving the system, we find:
The minus sign means that after half a period, the capacitance C is charged in the reverse polarity of the original.
Task 5 Oscillatory circuit with two coils connected in series
The oscillating circuit consists of a capacitor with a capacitance C and two coils with an inductance L1 and L2. At the moment when the current in the circuit has reached its maximum value, an iron core is quickly introduced into the first coil (compared to the oscillation period), which leads to an increase in its inductance by μ times. What is the voltage amplitude in the process of further oscillations in the circuit?
Decision:
1 With the rapid introduction of the core into the coil, the magnetic flux(the phenomenon of electromagnetic induction). Therefore, a rapid change in the inductance of one of the coils will result in a rapid change in the current in the circuit.
2 During the introduction of the core into the coil, the charge on the capacitor did not have time to change, it remained uncharged (the core was introduced at the moment when the current in the circuit was maximum). After a quarter of the period, the energy of the magnetic field of the current will turn into the energy of a charged capacitor:
Substitute in the resulting expression the value of the current I and find the amplitude of the voltage across the capacitor:
Task 6 Oscillatory circuit with two coils connected in parallel
The inductors L 1 and L 2 are connected through the keys K1 and K2 to a capacitor with a capacitance C. At the initial moment, both keys are open, and the capacitor is charged to a potential difference. First, the key K1 is closed and, when the voltage across the capacitor becomes equal to zero, K2 is closed. Determine the maximum voltage across the capacitor after closing K2. Ignore coil resistances.
Decision:
1 When the key K2 is open, oscillations occur in the circuit consisting of the capacitor and the first coil. By the time K2 is closed, the energy of the capacitor has transferred into the energy of the magnetic field of the current in the first coil:
2 After closing K2, two coils connected in parallel appear in the oscillatory circuit.
The current in the first coil cannot stop due to the phenomenon of self-induction. At the node, it divides: one part of the current goes to the second coil, and the other part charges the capacitor.
3 The voltage on the capacitor will become maximum when the current stops I charging capacitor. It is obvious that at this moment the currents in the coils will be equal.
:The pendulum moves forward with the speed of the center of mass and oscillates about the center of mass.
The elastic force is maximum at the moment of maximum deformation of the spring. Obviously, at this moment, the relative speed of the weights becomes equal to zero, and relative to the table, the weights move at the speed of the center of mass. We write down the law of conservation of energy:
Solving the system, we find
We make a replacement
and we get the previously found value for the maximum voltage
§6 Tasks for independent solution
Exercise 1 Calculation of the period and frequency of natural oscillations
1 The oscillatory circuit includes a coil of variable inductance, varying within L1= 0.5 µH to L2\u003d 10 μH, and a capacitor, the capacitance of which can vary from From 1= 10 pF to
From 2\u003d 500 pF. What frequency range can be covered by tuning this circuit?
2 How many times will the frequency of natural oscillations in the circuit change if its inductance is increased by 10 times, and the capacitance is reduced by 2.5 times?
3 An oscillatory circuit with a 1 uF capacitor is tuned to a frequency of 400 Hz. If you connect a second capacitor in parallel to it, then the oscillation frequency in the circuit becomes equal to 200 Hz. Determine the capacitance of the second capacitor.
4 The oscillatory circuit consists of a coil and a capacitor. How many times will the frequency of natural oscillations in the circuit change if a second capacitor is connected in series in the circuit, the capacitance of which is 3 times less than the capacitance of the first?
5 Determine the oscillation period of the circuit, which includes a coil (without core) of length in= 50 cm m cross-sectional area
S\u003d 3 cm 2, having N\u003d 1000 turns, and a capacitance capacitor With= 0.5 uF.
6 The oscillatory circuit includes an inductor L\u003d 1.0 μH and an air capacitor, the areas of the plates of which S\u003d 100 cm 2. The circuit is tuned to a frequency of 30 MHz. Determine the distance between the plates. The active resistance of the circuit is negligible.
The most important parts of radio transmitters and radio receivers are oscillatory circuits in which electrical oscillations are excited, that is, high-frequency alternating currents.
For a clearer idea of the operation of oscillatory circuits, let us first consider the mechanical oscillations of the pendulum (Fig. 1).
Fig.1 - Oscillations of the pendulum
If he is given a certain amount of energy, for example, if you push him or take him aside and let him go, then he will oscillate. Such oscillations occur without the participation of external forces only due to the initial energy reserve, and therefore are called free oscillations.
The movement of the pendulum from position 1 to position 2 and back is one oscillation. The first oscillation is followed by the second, then the third, the fourth, and so on.
The greatest deviation of the pendulum from position 0 is called the amplitude of the oscillation. The time of one complete oscillation is called a period and is denoted by the letter T. The number of oscillations in one second is the frequency f. The period is measured in seconds and the frequency is in hertz (Hz). Free oscillations of a pendulum have the following properties:
one). They are always damped, i.e. their amplitude gradually decreases (fades) due to energy losses to overcome air resistance and friction at the suspension point;
3). The frequency of free oscillations of the pendulum depends on its length and does not depend on the amplitude. When the oscillations are damped, the amplitude decreases, but the period and frequency remain unchanged;
4). The amplitude of free oscillations depends on the initial energy reserve. The more you push the pendulum or the farther you move it from the equilibrium position, the greater the amplitude.
In the process of pendulum oscillations, the potential mechanical energy goes into kinetic and vice versa. In position 1 or 2, the pendulum stops and has the highest potential energy, and its kinetic energy is zero. As the pendulum moves to position 0, the speed of movement increases and the kinetic energy - the energy of movement - increases. When the pendulum passes through position 0, its velocity and kinetic energy have a maximum value, and the potential energy is zero. Further, the speed decreases and the kinetic energy is converted into potential energy. If there were no energy losses, then such a transition of energy from one state to another would continue indefinitely and the oscillations would be undamped. However, there are almost always energy losses. Therefore, to create undamped oscillations, it is necessary to push the pendulum, i.e. add to it periodically energy that compensates for losses, as is done, for example, in a clockwork.
Let us now turn to the study of electrical oscillations. The oscillatory circuit is a closed circuit consisting of a coil L and a capacitor C. In the diagram (Fig. 2), such a circuit is formed at position 2 of switch P. Each circuit also has an active resistance, the influence of which we will not consider yet.
Fig. 2 - Scheme for excitation of free oscillations in the circuit
The purpose of the oscillatory circuit is the creation of electrical oscillations.
If a charged capacitor is connected to the coil, then its discharge will have an oscillatory character. To charge the capacitor, it is necessary in the circuit (Fig. 2) to put switch P in position 1. If then it is transferred to contact 2, the capacitor will begin to discharge to the coil.
It is convenient to follow the oscillation process using a graph showing changes in voltage and current i (Fig. 3).
Fig.3 - The process of free electrical oscillations in the circuit
At the beginning, the capacitor is charged to the largest potential difference Um, and the current I is zero. As soon as the capacitor starts to discharge, a current arises, which gradually increases. On (Fig. 3) the direction of movement of the ejectrons of this current is shown by arrows. Prevents rapid current change emf self-induction coils. As the current increases, the voltage across the capacitor decreases, at some point (moment 1 in Fig. 3) the capacitor is completely discharged. The current will return to the initial state of the circuit (moment 4 in Fig. 3).
The electrons in the oscillatory circuit made one complete oscillation, the period of which is shown in (Fig. 3) by the letter T. This oscillation is followed by the second, third, etc.
Free electrical oscillations occur in the circuit. They are made independently without the influence of any external emf, only due to the initial charge of the capacitor.
These oscillations are harmonic, that is, they represent a sinusoidal alternating current.
In the process of oscillation, electrons do not move from one plate of the capacitor to another. Although the speed of current propagation is very high (close to 300,000 km / s), electrons move in conductors at a very low speed - fractions of a centimeter per second. During one half-cycle, electrons can only pass a small section of the wire. They leave the plate with a negative charge to the nearest section of the connecting wire, and the same number of electrons come to the other plate from the section of the wire closest to this plate. Thus, in the wires of the circuit, only a small displacement of electrons takes place.
A charged capacitor has a store of potential electrical energy concentrated in an electric field between the plates. The movement of electrons is accompanied by the appearance of a magnetic field. Therefore, the kinetic energy of moving electrons is the energy of the magnetic field.
The electrical oscillation in the circuit is a periodic transition potential energy electric field into the kinetic energy of the magnetic field and vice versa.
At the initial moment, all the energy is concentrated in the electric field of a charged capacitor. When the capacitor is discharged, its energy decreases and the energy of the coil's magnetic field increases. At maximum current, all the energy of the circuit is concentrated in the magnetic field.
Then the process goes in reverse order: the magnetic energy decreases and the energy of the electric field arises. Half a period after the start of oscillations, all the energy will again be concentrated in the capacitor, and then the transition of the energy of the electric field into the energy of the magnetic field will begin again, etc.
The maximum current (or magnetic energy) corresponds to zero voltage (or zero electrical energy) and vice versa, i.e., the phase shift between voltage and current is equal to a quarter of the period, or 90 °. In the first and third quarters of the period, the capacitor plays the role of a generator, and the coil is an energy receiver. In the second and fourth quarters, on the contrary, the coil works as a generator, giving energy back to the capacitor.
A feature of the circuit is the equality of the inductive resistance of the coil and the capacitance of the capacitor for the current of free oscillations. This follows from the following.
Lecture plan
1. Oscillatory contours. Quasi-stationary currents.
2. Own electrical oscillations.
2.1. Own undamped oscillations.
2.2. Natural damped oscillations.
3. Forced electrical oscillations.
3.1. Resistance in an alternating current circuit.
3.2. Capacitance in the AC circuit.
3.3. Inductance in an alternating current circuit.
3.4. Forced vibrations. Resonance.
3.5. Cosine phi problem.
oscillatory contours. Quasi-stationary currents.
Fluctuations in electrical quantities - charge, voltage, current - can be observed in a circuit consisting of series-connected resistances ( R), capacities ( C) and inductors ( L) (Fig. 11.1).
Rice. 11.1.
At switch position 1 To, the capacitor is charged from the source.
If we now switch it to position 2, then in the circuit RLC there will be fluctuations with a period T similar to the vibrations of a load on a spring.
Oscillations that occur only due to the internal energy resources of the system are called own. Initially, energy was imparted to the capacitor and localized in an electrostatic field. When the capacitor closes to the coil, a discharge current appears in the circuit, and a magnetic field appears in the coil. emf The self-induction of the coil will prevent the instantaneous discharge of the capacitor. After a quarter period, the capacitor will be completely discharged, but the current will continue to flow, supported by the electromotive force of self-induction. To the moment 
this emf recharge the capacitor. The current in the circuit and the magnetic field will decrease to zero, the charge on the capacitor plates will reach its maximum value.
These fluctuations in electrical quantities in the circuit will occur indefinitely if the resistance of the circuit R= 0. Such a process is called own undamped oscillations. We observed similar oscillations in a mechanical oscillatory system when there is no resistance force in it. If the resistance of the resistor R(resistance force in a mechanical oscillator) cannot be neglected, then in such systems there will be own damped oscillations.
On the graphs of Fig. 11.2. the dependences of the capacitor charge on time are presented in the case of undamped ( a) and decaying ( b,in,G) fluctuations. The nature of the damped oscillations changes with an increase in the resistance of the resistor R. When the resistance exceeds a certain critical meaning R k, there are no oscillations in the system. There is a monotonous periodic capacitor discharge (Fig. 11.2. G.).
Rice. 11.2.
Before proceeding to the mathematical analysis of oscillatory processes, we will make one important remark. When compiling the oscillation equations, we will use Kirchhoff's rules (Ohm's laws), which are valid, strictly speaking, for direct current. But in oscillatory systems, the current changes with time. However, in this case, you can use these laws for the instantaneous value of the current, if the rate of current change is not too high. Such currents are called quasi-stationary (“quasi” (lat.) - as if). But what does the speed "too" or "not too" mean? If the current changes in some section of the circuit, then the impulse of this change will reach the farthest point of the circuit after a while:
.
Here l is the characteristic size of the contour, and with is the speed of light at which the signal propagates in the circuit.
The rate of change of the current is considered not too high, and the current is quasi-stationary, if:
,
where T- the period of change, that is, the characteristic time of the oscillatory process.
For example, for a chain 3 m long, the signal delay will be == 
= 10 -8 s. That is, the alternating current in this circuit can be considered quasi-stationary if its period is more than10 -6 s, which corresponds to the frequency= 
10 6 Hz. Thus, for frequencies 010 6 Hz in the circuit under consideration, Kirchhoff's rules for instantaneous values of current and voltage can be used.