An oscillating circuit is a device designed to generate (create) electromagnetic waves. 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 oscillating circuit consists of a coil and a capacitor. In addition, it may also contain a resistor (variable resistance element). An inductor (or a solenoid, as it is sometimes called) is a rod on which several layers of winding are wound, which, as a rule, is a copper wire. It is this element that creates vibrations in the oscillatory circuit. The rod in the middle is often called the choke, or core, and the coil is sometimes called the solenoid.
The coil of the oscillating circuit creates oscillations only in the presence of a stored charge. When a current passes through it, it accumulates a charge, which then gives it back to the circuit if the voltage drops.
Coil wires usually have very little resistance, which always remains constant. In the circuit of the oscillatory circuit, voltage and current changes very often. This change obeys certain mathematical laws:
- U \u003d U 0 * cos (w * (t-t 0), where
U - voltage at a given time t,
U 0 - voltage during t 0,
w is the frequency of electromagnetic oscillations.
Another integral component of the circuit is an 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 dimensions. This design allows an electric charge to accumulate on the dielectric, which can then be given to the circuit.
The difference between a capacitor and a battery is that there is no transformation of substances under the influence of an electric current, but a direct accumulation of charge in an electric field occurs. Thus, with the help of a capacitor, a sufficiently large charge can be accumulated, which can be given all at once. In this case, the current in the circuit increases greatly.
Also, the oscillating circuit consists of one more element: a resistor. This element has a resistance and is designed to control the current and voltage in the circuit. If you increase at constant voltage, then the current strength will decrease according to Ohm's law:
- I \u003d U / R, where
I - current strength,
U - voltage,
R - 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 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 resists 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 current passes through the coil, it arises in it, which creates an obstacle to changing the current. The magnitude of this force depends on two factors: the inductance of the coil and the time derivative of the current. Mathematically, this dependence is expressed through the equation:
- E \u003d -L * I "(t), where
E - EMF value,
L is the value of the inductance of the coil (it is different for each coil and depends on the number of windings and their thickness),
I "(t) - time derivative of current (rate of change of current).
The strength of the direct current does not change over time, so no resistance arises when exposed to it.
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. This 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 increases and decreases linearly according to various laws. This means that if you stop supplying current 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 distribute and exit the coil. This is a serious problem in industrial production. This 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 out, 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 higher than the 220 volts we are used to, therefore, when the circuit is interrupted in production, sparks of such strength can arise that they cause a lot of harm to both the plant and the person.
The coil is the basis of what the oscillating circuit consists of. The inductances of the solenoids connected in series are added. 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 the oscillating circuit is an individual indicator, numerically equal to the electromotive force (in volts) that occurs in the circuit when the current strength 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: on the geometry of the solenoid, on the magnetic characteristics of the core, and on the number of wire coils. 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 current-carrying conductor 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 oscillating circuit consists of a capacitor, which will be discussed below.
Electric capacitor
The capacity of the oscillating circuit is determined by the capacitor. Its appearance was described above. Now let's analyze the physics of the processes that take place in it.
Since the plates of a capacitor 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 pass from one end of the wire to the other, it accumulates on the plates of the capacitor. This increases the power of the magnetic and electrical Thus, when the charge ceases to flow, all the electricity accumulated on the plates begins to be transmitted into the circuit.
Each capacitor is optimized for its performance. If you operate this element for a long time at a voltage higher than the rated voltage, its service life will be significantly reduced. The capacitor of the oscillating circuit is constantly affected by currents, and therefore, when choosing it, you should be extremely careful.
In addition to the usual capacitors, which were discussed, there are also supercapacitors. This is a more complex element: it can be described as a cross between a battery and a capacitor. As a rule, organic substances, between which there is an electrolyte, serve as a dielectric in a supercapacitor. Together they create a double electrical layer, which allows this structure to store many times more energy than a traditional capacitor.
What is the capacitance of a capacitor?
The capacity of a capacitor is the ratio of the capacitor's charge to the voltage under which it is located. You can calculate this value very simply using the mathematical formula:
- C \u003d (e 0 * S) / d, where
e 0 - dielectric material (tabular value),
S - area of \u200b\u200bcapacitor plates,
d is the distance between the plates.
The dependence of the capacitance of the 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 decreasing voltage, the capacitance value increases, since it can also be described by the following formula:
- C \u003d q / U, where
q is the charge in coulombs.
It is worth talking about the units of this quantity. Capacitance is measured in farads. 1 farad is a fairly large value, so existing capacitors (but not supercapacitors) have a capacitance measured in picofarads (one trillion farad).
Resistor
The current in the oscillating circuit also depends on the resistance of the circuit. And besides the described two elements, of which the oscillatory circuit (coil, capacitor) consists, there is also a third - a resistor. He is responsible for creating resistance. The resistor differs from other elements in that it has a high resistance, which in some models can be changed. In the oscillatory circuit, it performs the function of a magnetic field power regulator. Several resistors can be connected in series or in parallel, thereby increasing the resistance of the circuit.
The resistance of this element also depends on temperature, therefore, 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 \u003d (p * l) / S, where
p is the resistivity of the resistor material (measured in (Ohm * mm 2) / m);
l is the length of the resistor (in meters);
S - sectional area (in square millimeters).
How to link path parameters?
Now we come close to the physics of the oscillatory circuit. Over time, the charge on the capacitor plates changes according to a second-order differential equation.
If you solve this equation, 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 Thomson's formula (named after the English physicist William Thomson, who derived it in 1853):
- T \u003d 2 * n * (L * C) 1/2.
T is the period of electromagnetic oscillations,
L and C - respectively, the inductance of the coil of the oscillating circuit and the capacitance of the circuit elements,
n is 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 at a constant magnitude of the force that supports this vibration. The resonance can be explained using a simple example: if you start pushing the swing in time with their frequency, then they will accelerate, and their "amplitude" will increase. And if you push out of time, they will slow down. Resonance often dissipates a lot of energy. In order to be able to calculate the values \u200b\u200bof losses, they invented such a parameter as figure of merit. It is a coefficient equal to the ratio of the energy in the system to the losses occurring in the circuit in one cycle.
The contour quality factor is calculated by the formula:
- Q \u003d (w 0 * W) / P, where
w 0 - resonant cyclic oscillation frequency;
W is the energy stored in the oscillatory system;
P is the power dissipation.
This parameter is a dimensionless quantity, since it actually shows the ratio of energies: stored to spent.
What is an ideal oscillating circuit
For a better understanding of the processes in this system, physicists invented the so-called ideal oscillating circuit... It is a mathematical model that represents a circuit as a zero resistance system. Continuous harmonic oscillations appear in it. Such a model allows one to obtain formulas for the approximate calculation of the contour parameters. One of these parameters is the total energy:
- W \u003d (L * I 2) / 2.
Such simplifications significantly speed up the calculations and allow you to evaluate the characteristics of the chain with the given indicators.
How it works?
The entire operating cycle of the oscillating circuit can be divided into two parts. Now we will analyze in detail the processes taking place in each part.
- First phase:the capacitor plate, charged positively, begins to discharge, giving current to the circuit. At this moment, the current goes from positive to negative charge, while passing through the coil. As a result, electromagnetic oscillations arise in the circuit. The current, passing through the coil, goes to the second plate and charges it positively (while the first plate, from which the current passed, is charged negatively).
- Second phase:the opposite process takes place. The current passes from the positive plate (which at the very beginning was negative) to the negative one, again passing 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.
Contour design options
In addition to simple "coil-capacitor" and "coil-resistor-capacitor" circuits, there are other options that use an oscillatory circuit as a basis. This is, for example, 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 sequential circuit, which was discussed in the article).
There are also other types of construction that include different electrical components. For example, you can connect a transistor to the network that will open and close the circuit with a frequency equal to the oscillation frequency in the circuit. Thus, continuous oscillations will be established in the system.
Where is the oscillating circuit used?
The most familiar applications for circuit components are electromagnets. They, in turn, are used in intercoms, electric motors, sensors, and in many other less mundane areas. Another application is an oscillator. 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 will be possible to transmit information over long distances.
An inductor itself can be used as a transformer element: two coils with different numbers of windings can transmit their charge using an electromagnetic field. But since the characteristics of the solenoids are different, then 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 to a current with a voltage of 12 volts.
Conclusion
We have analyzed in detail the principle of operation of the oscillating circuit and each of its parts separately. We learned that an oscillating circuit is a device designed to create electromagnetic waves. However, these are just the basics of the complex mechanics of these seemingly simple elements. You can learn more about the intricacies of the contour and its components from specialized literature.
Electrical fluctuations 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 together. 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, continuous harmonic oscillations of the charge on the capacitor, voltage and current will occur in it.
Energy can be imparted to the oscillatory circuit in different ways. For example, by charging a capacitor from a direct current source or by inducing a current in an inductor. In the first case, the energy is possessed by the electric field between the capacitor plates. In the second, the energy is contained in the magnetic field of the current flowing through the circuit.
§1 Equation of oscillations in the circuit
Let us prove that when the energy is communicated to the contour, undamped harmonic oscillations will occur in it. For this, it is necessary to obtain a differential equation of harmonic oscillations of the form.
Let's say the capacitor is charged and shorted to the coil. The capacitor will begin to discharge, current will flow through the coil. According to the second Kirchhoff's law, the sum of voltage drops along a closed circuit is equal to the sum of the EMF in this circuit.
In our case, the voltage drop since 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 varying 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 the current in the coil. Since the EMF of self-induction prevents the capacitor discharge process, the second Kirchhoff's law takes the form
But the loop current is the capacitor discharge or charge current, therefore. Then
The differential equation is converted to the form
Introducing the notation, we obtain the known to us 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.
Period of charge fluctuations. 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 across the capacitor will also change. The voltage fluctuates in one phase with the charge, and the amperage is ahead of the charge in
phase on.
Capacitor electric field energy
Current magnetic field energy
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 vibrations, as well as mechanical ones, can be both free and forced, harmonic and inharmonic. Free harmonic electrical oscillations are possible in an ideal oscillating circuit.
§2 Processes occurring in the oscillatory circuit
We have mathematically proven the existence of free harmonic oscillations in an oscillatory circuit. However, it remains unclear why such a process is possible. What is the cause of oscillation in the circuit?
In the case of free mechanical vibrations, such a reason was found - this is the internal force arising when the system is removed from the equilibrium position. 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 be excited 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, which have different potentials, are connected with a conductor, so the capacitor begins to discharge through the coil. The charge, the voltage across 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 EMF of self-induction that occurs in the coil. The energy of the magnetic field of the current increases.
A quarter of the period has passed - the capacitor is discharged.
The capacitor is discharged, the voltage across it has become zero. The energy of the electric field at this moment is also zero. According to the law of conservation of energy, it could not disappear. The field energy of the capacitor is completely converted into the magnetic field energy 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 is recharged as long as there is current. Therefore, at the moment when the current stops, the charge and voltage across the capacitor take on a maximum value.
The energy of the magnetic field at this moment completely transformed into the energy of the electric field of the capacitor.
The situation in the loop at this moment is equivalent to the original one. The processes in the loop will be repeated, but in the opposite direction. One complete oscillation in the circuit, lasting over a period, will end when the system returns to its original state, that is, when the capacitor recharges 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 the current: it does not allow it to instantly grow and disappear instantly.
By the way, it will 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:
Earlier, differential equations were obtained for mechanical and electrical oscillatory systems:
Despite the fundamental differences in physical processes to 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
A careful analysis of the differential equations for the spring pendulum and the oscillatory circuit, as well as the formulas connecting the quantities characterizing 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 () | Capacitor charge () |
| Body speed | Loop current |
| Potential energy of an elastically deformed spring | Capacitor electric field energy |
| Kinetic energy of the load | Magnetic field energy of a coil with current |
| The inverse of the spring stiffness | Capacitor capacity |
| Weight of cargo | Coil inductance |
| Elastic force | EMF of self-induction, equal to the voltage across the capacitor |
table 2
It is not just the formal similarity between the quantities describing the processes of oscillation of the pendulum and processes in the circuit that is important. 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 turns to zero, and there is no voltage in the capacitor in the circuit. The pendulum speed and 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 weight from the equilibrium position to the extreme position.
The total energy of the oscillatory system either remains unchanged over time.
A similar analogy can be traced not only between a spring pendulum and an oscillating circuit. The laws of free vibrations of any nature are universal! These patterns, illustrated by the example of two oscillatory systems (a spring pendulum and an oscillatory circuit) are not just possible, but need to see in the vibrations of any system.
In principle, it is possible to solve the problem of any oscillatory process by replacing it with the oscillations of the mint. To do this, it is enough to competently construct an equivalent mechanical system, solve a mechanical problem, and replace the values \u200b\u200bin 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 oscillating 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, or the weights are attached to different ends of the spring.
When coils of different inductances are connected in parallel, different currents flow through them. Consequently, the speeds of loads in an identical mechanical system should also be different. Obviously, this is only possible in the second case.
We have already found the period of this oscillatory system. It is equal. Replacing the masses of the weights with the inductances of the coils, and the reciprocal of the spring stiffness, with the capacitance of the capacitor, we obtain.
§4 Oscillating circuit with a constant current source
Consider an oscillating circuit containing a constant current source. Let the capacitor be initially uncharged. What will happen in the system after the K key 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 start charging. We write down the second Kirchhoff's law:
The loop current is the capacitor charging current, hence. Then. The differential equation is converted to the form
* Solve the equation by changing variables.
Let us denote. Differentiate twice and, taking into account that, we get. 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 the capacitor changes according to the law
Immediately after the key 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 the 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 the same. 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 direct 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 the circuit closure is identical to the absence of deformation of the spring at the moment the pendulum is set in oscillatory motion.
In a constant force field, the oscillation period of the spring pendulum does not change. The period of oscillation in the circuit behaves 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 at its maximum, the spring is deformed:
When the current in the oscillating 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
Problem 1Law of energy conservation
L \u003d 0.5 μH and a capacitor with a capacity FROM\u003d 20 pF, electrical vibrations occur. What is the maximum voltage across the capacitor if the amplitude of the current in the loop is 1 mA? The coil resistance is negligible.
Decision:
2 At the moment when the voltage across the capacitor is maximum (the 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 equality. The maximum voltage across the capacitor is
Problem 2Law of energy conservation
In an oscillatory circuit consisting of an inductor L and a capacitor with a capacity FROM,electrical oscillations occur with a period of T \u003d 1 μs. Maximum charge value. What is the current in the circuit at the moment when the charge on the capacitor is equal? The coil resistance 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 equality. The loop current is.
4 The period of oscillation in the circuit is determined by the Thomson formula. From here. Then for the current in the circuit we obtain
Problem 3Oscillating circuit with two parallel-connected capacitors
In an oscillatory circuit consisting of an inductor L and a capacitor with a capacity FROM,electrical oscillations occur with the amplitude of the charge. At the moment when the charge on the capacitor is maximum, the key K is closed. What will be the period of oscillation in the circuit after the key is closed? What is the amplitude of the current in the circuit after the key is closed? Neglect the ohmic resistance of the circuit.
Decision:
1 Closing the key leads to the appearance of another capacitor in the circuit, connected in parallel to the first. The total capacitance of two parallel-connected capacitors is equal.
The period of oscillations in the circuit depends only on its parameters and does not depend on how the oscillations were excited in the system and what energy was communicated to the system for this. According to Thomson's formula.
2 To find the current amplitude, we 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 start to discharge. The discharge current, reaching the node, should be divided into two parts. However, in the branch with the coil, an EMF of self-induction arises, 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 across the capacitors become equal, and the initial charge of the capacitor is redistributed between the two capacitors. The time for redistribution of charge between two capacitors is negligible due to the absence of ohmic resistance in the branches with capacitors. During this time, the current in the branch with the coil will not have time to arise. Oscillations in the new system will continue after the charge is redistributed between the capacitors.
It is important to understand that during the redistribution of charge between two capacitors, the energy of the system is not conserved! Before the key was closed, one capacitor possessed energy, a circuit one:
After the redistribution of the charge, the capacitor bank possesses energy:
It is easy to see that the energy of the system has decreased!
3 The new amplitude of the current is found 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:
Pay attention, the law of conservation of energy begins to "work" only after the completion of the redistribution of charge between the capacitors.
Task 4 Oscillating circuit with two series-connected capacitors
The oscillating circuit consists of a coil with inductance L and two capacitors C and 4C connected in series. The C capacitor is charged to voltage, the 4C capacitor 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 \u200b\u200bon 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 Kirchhoff's law
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 the key was closed, 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 consists 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 across the capacitors, we use the law of conservation of charge - the charge of the lower plate of capacitor C has partially transferred to the upper plate of capacitor 4C:
We substitute the found voltage value into the energy conservation law and find the current amplitude in the circuit:
3 Let us find the limits within which the voltage across 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. The 4C capacitor was not charged, therefore.
After the key is closed, the capacitor C begins to discharge, and the capacitor with a capacity of 4C starts 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 the 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 capacitor of capacitance C is charged in the reverse polarity of the initial one.
Problem 5Oscillatory circuit with two series-connected coils
The oscillating circuit consists of a capacitor with a capacity of C and two inductors L 1 and L 2 ... At the moment when the current in the circuit has taken the maximum value, an iron core is quickly introduced into the first coil (in comparison with the oscillation period), which leads to an increase in its inductance by a factor of μ. What is the voltage amplitude in the process of further oscillations in the circuit?
Decision:
1 When the core is quickly inserted into the coil, the magnetic flux must be maintained (the phenomenon of electromagnetic induction). Therefore, a rapid change in the inductance of one of the coils will lead to a rapid change in the current in the loop.
2 During the insertion 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 a period, the energy of the magnetic field of the current will be converted into the energy of a charged capacitor:
Substitute the current value into the resulting expression I and find the amplitude of the voltage across the capacitor:
Problem 6Oscillatory circuit with two parallel-connected coils
Inductors L 1 and L 2 are connected through keys K1 and K2 to a capacitor with a capacity C. At the initial moment, both keys are open, and the capacitor is charged to a potential difference. First, switch K1 is closed and when the voltage across the capacitor becomes zero, K2 is closed. Determine the maximum voltage across the capacitor after K2 is closed. Disregard the coil resistance.
Decision:
1 When the K2 switch is open, oscillations occur in the circuit consisting of a capacitor and the first coil. By the time K2 closes, the energy of the capacitor has passed into the energy of the magnetic field of the current in the first coil:
2 After K2 closes, 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 into the second coil, and the other charges the capacitor.
3 The voltage across the capacitor will become maximum when the current stops Icharging the 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 spring force is at its maximum at the moment of maximum spring deformation. Obviously, at this moment the relative speed of the weights becomes equal to zero, and relative to the table, the weights move with 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 obtain the previously found value for the maximum voltage
§6 Tasks for independent solution
Exercise 1 Calculating the period and natural frequency
1 The oscillating circuit includes a variable inductance coil, varying within L 1 \u003d 0.5 μH to L 2 \u003d 10 μH, and a capacitor, the capacitance of which can vary from C 1 \u003d 10 pF to
C 2 \u003d 500 pF. What frequency range can be covered by tuning this loop?
2 How many times will the natural frequency in the circuit change if its inductance is increased by 10 times, and the capacitance is reduced by 2.5 times?
3 An oscillating circuit with a capacitor with a capacitance of 1 μF is tuned to a frequency of 400 Hz. If you connect a second capacitor to it in parallel, then the oscillation frequency in the circuit becomes equal to 200 Hz. Determine the capacity of the second capacitor.
4 The oscillating circuit consists of a coil and a capacitor. How many times will the natural frequency in the circuit change if a second capacitor is sequentially connected to the circuit, the capacity of which is 3 times less than the capacity of the first?
5 Determine the period of oscillation of the circuit, which includes a coil (without core) length in\u003d 50 cm m cross-sectional area
S \u003d 3 cm 2 having N \u003d 1000 turns, and capacitor capacitance FROM \u003d 0.5 μF.
6 The oscillating circuit includes an inductor L \u003d 1.0 μH and an air condenser, the area of \u200b\u200bthe plates of which S \u003d 100 cm 2. The loop is tuned to 30 MHz. Determine the distance between the plates. The loop resistance is negligible.
The main device that determines the operating frequency of any alternator is the oscillating circuit. The oscillating circuit (Fig. 1) consists of an inductor L (consider the ideal case when the coil has no ohmic resistance) and a capacitor C and is called closed. The characteristic of the coil is inductance, it is denoted L and is measured in Henry (H), the capacitor is characterized by the capacity C, which is measured in farads (F).
Let at the initial moment of time the capacitor is charged in such a way (Fig. 1) that on one of its plates there is 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, having the 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 flows 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 in 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. is equal to the EMF of self-induction
| Fig. 1 | Fig. 2 |
When the capacitor is completely discharged and the current in the coil reaches 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 capacitor charge will reach its maximum value Q 0, but the plate, previously charged positively, will now be charged negatively (Fig. 5). Then the capacitor starts to discharge again, and the current in the circuit will flow in the opposite direction.
So the process of charge flow from one capacitor plate to another through the inductor is repeated over and over again. They say that in the circuit occur electromagnetic vibrations ... This process is associated not only with fluctuations in the magnitude of the charge and voltage on the capacitor, the current 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 |
The capacitor will be recharged to the maximum voltage only if there is no energy loss in the oscillatory circuit. Such a contour 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, ie
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 changes according to the harmonic law:
where n is the natural frequency of the oscillating circuit, w 0 \u003d 2pn is the natural (circular) frequency of the oscillating 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 occurs, is determined by the Thomson formula
The current in the circuit also changes harmonically, but lags behind the voltage in phase by. Therefore, the time dependence of the current in the circuit will have the form
Figure 6 shows graphs of voltage changes U on the capacitor and current I in the coil for a perfect oscillatory circuit.
In a real circuit, the energy will decrease with each oscillation. The amplitudes of the voltage across the capacitor and the current in the circuit will decrease, such oscillations are called damped. They cannot be used in master oscillators, because the device will operate in a pulsed mode at best.
| Fig. 5 | Fig. 6 |
To obtain sustained oscillations, it is necessary to compensate for energy losses at a wide variety of operating frequencies of devices, including those used in medicine.
– an electrical circuit consisting of a series-connected capacitor with a capacitor, a coil with an inductance and an electrical resistance.
Ideal oscillating circuit - a circuit consisting only of an inductor (not having its own resistance) and a capacitor (-circuit). Then, in such a system, continuous electromagnetic oscillations of the current in the circuit, the voltage across the capacitor and the charge of the capacitor are maintained. Let's take a look at the contour and think about where the vibrations come from. Let the initially charged capacitor be placed in the circuit we are describing.
Figure: 1. Oscillatory circuit
At the initial moment of time, all the charge is concentrated on the capacitor, there is no current on the coil (Fig. 1.1). Because there is no external field on the capacitor plates either, then the electrons from the plates begin to "leave" into the circuit (the charge on the capacitor begins to decrease). In this case (due to the freed electrons) the current in the circuit increases. The direction of the current, in this case, is from plus to minus (however, as always), and the capacitor is the source of alternating current for this system. However, with an increase in the current on the coil, as a result, a reverse induction current () appears. The direction of the induction current, according to Lenz's rule, should neutralize (decrease) the growth of the main current. When the charge of the capacitor becomes zero (all the charge will drain), the strength of the induction current in the coil will become maximum (Fig. 1.2).
However, the current charge in the circuit cannot disappear (the law of conservation of charge), then this charge, which left one plate through the circuit, ended up on the other plate. Thus, the capacitor is recharged in the opposite direction (Fig. 1.3). The induction current on the coil is reduced to zero because the change in magnetic flux also tends to zero.
When the capacitor is fully charged, the electrons begin to move in the opposite direction, i.e. the capacitor is discharged in the opposite direction and a current arises, reaching its maximum when the capacitor is completely discharged (Fig. 1.4).
Further reverse charging of the capacitor brings the system to the position in Figure 1.1. This behavior of the system is repeated as long as you want. Thus, we get the fluctuation of various parameters of the system: the current in the coil, the charge on the capacitor, the voltage on the capacitor. In the case of the ideality of the circuit and wires (no intrinsic resistance), these vibrations are.
For the mathematical description of these parameters of this system (first of all, the period of electromagnetic oscillations), the calculated before us is introduced thomson's formula:
Imperfect outline is the same ideal circuit that we have considered, with one small inclusion: with the presence of resistance (-contour). This resistance can be either the resistance of the coil (it is not ideal), or the resistance of the conducting wires. The general logic of the occurrence of oscillations in a non-ideal circuit is similar to that in an ideal one. The only difference is in the vibrations themselves. In the case of the presence of resistance, part of the energy will be dissipated into the environment - the resistance will heat up, then the energy of the oscillatory circuit will decrease and the oscillations themselves will become decaying.
To work with circuits in the school, only general energy logic is used. In this case, we assume that the total energy of the system is initially concentrated on and / or, and is described:
For an ideal circuit, the total energy of the system remains constant.
Fluctuations called movements or processes that are characterized by a certain repetition in time. Oscillations can be different in physical nature (mechanical, electromagnetic, gravitational), but they are described by equations that are identical in structure.
The simplest type of vibration is harmonic vibrations, at which the fluctuating quantity changes according to the harmonic law, that is, according to the sine or cosine law.
Oscillations are free and forced... Free vibrations are divided into undamped (own) and fading.
Free undamped, or natural, oscillations are those oscillations that occur due to the energy imparted to the oscillatory system at the initial moment of time, in the absence of further external influence on the system.
Differential equation of natural electrical harmonic oscillations contour (fig. 4.1)
where is the electric charge of the capacitor; Is the cyclic (circular) frequency of free undamped oscillations, (here is the inductance of the circuit; is the electric capacitance of the circuit).
Electric harmonic vibration equation:
where is the amplitude of the capacitor charge; - the initial phase.
Current in the oscillating circuit
where is the amplitude of the current,.
Figure: 4.1. Ideal oscillating circuit
Oscillation period - the time of one complete oscillation. During this time, the oscillation phase is incremented.
Oscillation frequency - the number of vibrations per unit of time,
Formulas connecting period, frequency and cyclic frequency:
Period of free undamped oscillations in the electromagnetic oscillatory circuit is determined by the Thomson formula
Amplitude of the resulting oscillation of the charge arising in two different circuits and added on one load (added oscillations of one direction and the same frequency)
where and are the amplitudes of two oscillations; and - the initial phases of two oscillations.
The initial phase of the resulting oscillation of the charge participating in two oscillations of the same direction and the same frequency,
The equation of beats, i.e., nonharmonic vibrations arising when harmonic vibrations are superimposed, the frequencies of which are sufficiently close:
where is the beat amplitude; - beat frequency,.
Charge trajectory equationparticipating in two mutually perpendicular oscillations of the same frequency:
Free damped oscillations - these are such oscillations, the amplitude of which decreases over time due to energy losses by the oscillatory system. In an electric oscillatory circuit, energy is spent on Joule heat and on electromagnetic radiation.
Differential equation of damped electrical oscillations in a circuit with electrical resistance:
where is the attenuation coefficient, (here is the loop inductance).
Damped oscillation equation in the case of weak attenuation () (Fig.4.2):
where is the amplitude of the damped oscillations of the capacitor charge; - initial amplitude of oscillations; - cyclic frequency of damped oscillations,.
Figure: 4.2. Change in charge over time with weak damped oscillations
Relaxation time - this is the period of time during which the amplitude of the oscillations decreases by a factor of:
Relaxation time is associated with attenuation coefficient ratio
Logarithmic decrement of vibration damping
where is the period of damped oscillations.
The formula connecting the logarithmic decrement of oscillations with the damping coefficient and the period of damped oscillations:
Forced vibrations - these are such vibrations that occur in the presence of an external periodically changing influence.
Differential equation of forced electrical oscillations in a circuit with electrical resistance, in the presence of a forcing EMF, changing according to a harmonic law, where is the amplitude value of the EMF, and is the cyclic frequency of EMF change (Fig.4.3):
where is the attenuation coefficient,; - loop inductance.
Figure: 4.3. Forced electrical vibration observation circuit
Equation of steady-state forced electrical oscillations:
where is the phase difference between the oscillations of the capacitor charge and the forcing EMF of the current source.
Amplitude of steady-state forced oscillations capacitor charge
The phase difference between the oscillations of the capacitor charge and the driving EMF of the current source
The amplitude of the forced vibrations depends on the relationship between the cyclic frequencies of the forcing and natural vibrations. Resonant frequency and resonant amplitude.
