Statement of the problem: We already know a lot about mechanical vibrations: free and forced vibrations, self-oscillations, resonance, etc. Let's start studying electrical vibrations. The topic of today's lesson: obtaining free electromagnetic oscillations.
Let's remember first: What conditions should an oscillatory system meet, a system in which free oscillations can occur. Answer: a restoring force must arise in the oscillatory system and the transformation of energy from one type to another must occur.
(Analysis of new material based on the presentation with a detailed explanation of all processes and recording in a notebook the first two quarters of the period, describe the 3rd and 4th quarters at home, according to the model).
An oscillatory circuit is an electrical circuit in which free electromagnetic oscillations can be obtained. K.K. consists of only two devices: a coil with inductance L and a capacitor with electrical capacity C. An ideal oscillatory circuit has no resistance.
To impart energy to K.K., i.e. To remove it from the equilibrium position, you need to temporarily open its circuit and install a key with two positions. When the switch is closed to the current source, the capacitor is charged to its maximum charge. This is what they serve at K.K. energy in the form of electric field energy. When the key is closed to the right position, the current source is turned off, K.K. left to his own devices.
This is the state of K.K. corresponds to the position of the mathematical pendulum in the extreme right position when it was brought out of rest. The oscillatory circuit is removed from the equilibrium position. The charge of the capacitor is maximum and the energy of the charged capacitor is the energy of the electric field is maximum. We will consider the entire process that occurs in it in quarters of the period.
At the first moment, the capacitor is charged to its maximum charge (the lower plate is positively charged), the energy in it is concentrated in the form of electric field energy. The capacitor closes on itself and it begins to discharge. Positive charges, according to Coulomb's law, are attracted to negative ones, and a discharge current appears, directed counterclockwise. If there were no inductor in the path of the current, then everything would happen instantly: the capacitor would simply discharge. The accumulated charges would compensate each other, and electrical energy would turn into thermal energy. But a magnetic field arises in the coil, the direction of which can be determined by the gimlet rule - “up”. The magnetic field is growing and the phenomenon of self-induction occurs, which prevents the growth of current in it. The current does not grow instantly, but gradually, throughout the entire 1st quarter of the period. During this time, the current will increase as long as the capacitor supports it. As soon as the capacitor is discharged, the current no longer increases; by this time it will reach its maximum value. The capacitor is discharged, the charge is 0, which means the energy of the electric field is 0. But the maximum current flows in the coil, there is a magnetic field around the coil, which means that the energy of the electric field has been converted into the energy of the magnetic field. By the end of the 1st quarter of the period in the K.K. the current is maximum, the energy is concentrated in the coil in the form of magnetic field energy. This corresponds to the position of the pendulum when it passes the equilibrium position.
At the beginning of the 2nd quarter of the period, the capacitor is discharged, and the current has reached its maximum value and it should instantly disappear, because the capacitor does not support it. And the current really begins to decrease sharply, but it flows through the coil, and the phenomenon of self-induction arises in it, which prevents any change in the magnetic field that causes this phenomenon. The self-induction emf maintains the vanishing magnetic field, the induced current has the same direction as the existing one. In K.K. current flows counterclockwise into an empty capacitor. An electric charge accumulates in the capacitor - a positive charge on the top plate. The current flows as long as it is supported by the magnetic field, until the end of the 2nd quarter of the period. The capacitor will charge to its maximum charge (if no energy leakage occurs), but in the opposite direction. They say the capacitor has overcharged. By the end of the 2nd quarter of the period, the current disappears, which means that the magnetic field energy is equal to 0. The capacitor is recharged, its charge is equal to (– maximum). The energy is concentrated in the form of electric field energy. During this quarter, the energy of the magnetic field was converted into energy of the electric field. The state of the oscillatory circuit corresponds to the position of the pendulum in which it deviates to the extreme left position.
In the 3rd quarter of the period, everything happens the same as in the 1st quarter, only in the opposite direction. The capacitor begins to discharge. The discharge current increases gradually throughout the entire quarter, because its rapid growth is hampered by the phenomenon of self-induction. The current increases to a maximum value until the capacitor is discharged. By the end of the 3rd quarter, the energy of the electric field will turn into energy of the magnetic field, completely, if there is no leakage. This corresponds to the position of the pendulum when it again passes the equilibrium position, but in the opposite direction.
In the 4th quarter of the period everything happens the same as in the 2nd quarter, only in the opposite direction. The current maintained by the magnetic field gradually decreases, supported by the self-inductive emf and recharges the capacitor, i.e. returns it to its original position. The energy of the magnetic field is converted into energy of the electric field. Which corresponds to the return of the mathematical pendulum to its original position.
Analysis of the material considered:
1. Can an oscillatory circuit be considered as an oscillatory system? Answer: 1. In an oscillatory circuit, the energy of the electric field is converted into the energy of the magnetic field and vice versa. 2. The phenomenon of self-induction plays the role of a restoring force. Therefore, the oscillatory circuit should be considered as an oscillatory system. 3. Oscillations in K.K. can be considered free.
2. Is it possible to oscillate in K.K. considered as harmonic? We analyze the change in the magnitude and sign of the charge on the capacitor plates and the instantaneous value of the current and its direction in the circuit.
The graph shows:
3. What oscillates in the oscillatory circuit? What physical bodies perform oscillatory movements? Answer: electrons vibrate, they perform free vibrations.
4. What physical quantities change during operation of the oscillatory circuit? Answer: the current strength in the circuit, the charge in the capacitor, the voltage on the capacitor plates, the energy of the electric field and the energy of the magnetic field change.
5. The period of oscillation in the oscillatory circuit depends only on the inductance of the coil L and the capacitance of the capacitor C. Thomson's formula: T = 2π can also be compared with the formulas for mechanical oscillations.
In the article we will tell you what an oscillatory circuit is. Series and parallel oscillatory circuit.
Oscillatory circuit - a device or electrical circuit containing the necessary radio-electronic elements to create electromagnetic oscillations. Divided into two types depending on the connection of elements: consistent And parallel.
The main radio element base of the oscillatory circuit: Capacitor, power supply and inductor.
A series oscillatory circuit is the simplest resonant (oscillatory) circuit. The series oscillatory circuit consists of an inductor and a capacitor connected in series. When such a circuit is exposed to an alternating (harmonic) voltage, an alternating current will flow through the coil and capacitor, the value of which is calculated according to Ohm’s law:I = U / X Σ, Where X Σ— the sum of the reactances of a series-connected coil and capacitor (the sum module is used).
To refresh your memory, let's remember how the reactance of a capacitor and inductor depends on the frequency of the applied alternating voltage. For an inductor, this dependence will look like:
The formula shows that as the frequency increases, the reactance of the inductor increases. For a capacitor, the dependence of its reactance on frequency will look like this:
Unlike inductance, with a capacitor everything happens the other way around - as the frequency increases, the reactance decreases. The following figure graphically shows the dependences of the coil reactances X L and capacitor X C from cyclic (circular) frequency ω , as well as a graph of frequency dependence ω their algebraic sum X Σ. The graph essentially shows the frequency dependence of the total reactance of a series oscillating circuit.
The graph shows that at a certain frequency ω=ω р, at which the reactances of the coil and capacitor are equal in magnitude (equal in value, but opposite in sign), the total resistance of the circuit becomes zero. At this frequency, a maximum current is observed in the circuit, which is limited only by ohmic losses in the inductor (i.e., the active resistance of the coil winding wire) and the internal resistance of the current source (generator). The frequency at which the phenomenon under consideration, called resonance in physics, is observed is called the resonant frequency or the natural frequency of the circuit. It is also clear from the graph that at frequencies below the resonance frequency the reactance of the series oscillatory circuit is capacitive in nature, and at higher frequencies it is inductive. As for the resonant frequency itself, it can be calculated using Thomson’s formula, which we can derive from the formulas for the reactances of the inductor and capacitor, equating their reactances to each other:
The figure on the right shows the equivalent circuit of a series resonant circuit taking into account ohmic losses R, connected to an ideal harmonic voltage generator with amplitude U. The total resistance (impedance) of such a circuit is determined by: Z = √(R 2 +X Σ 2), Where X Σ = ω L-1/ωC. At the resonant frequency, when the coil reactance values XL = ωL and capacitor X C = 1/ωС equal in modulus, value X Σ goes to zero (hence, the circuit resistance is purely active), and the current in the circuit is determined by the ratio of the generator voltage amplitude to the resistance of ohmic losses: I=U/R. At the same time, the same voltage drops on the coil and on the capacitor, in which reactive electrical energy is stored U L = U C = IX L = IX C.
At any other frequency other than the resonant one, the voltages on the coil and capacitor are not the same - they are determined by the amplitude of the current in the circuit and the values of the reactance modules X L And X C Therefore, resonance in a series oscillatory circuit is usually called voltage resonance. The resonant frequency of the circuit is the frequency at which the resistance of the circuit is purely active (resistive) in nature. The resonance condition is the equality of the reactance values of the inductor and capacitance.
One of the most important parameters of an oscillatory circuit (except, of course, the resonant frequency) is its characteristic (or wave) impedance ρ and circuit quality factor Q. Characteristic (wave) impedance of the circuit ρ is the value of the reactance of the capacitance and inductance of the circuit at the resonant frequency: ρ = X L = X C at ω =ω р. The characteristic impedance can be calculated as follows: ρ = √(L/C). Characteristic impedance ρ is a quantitative measure of the energy stored by the reactive elements of the circuit - the coil (magnetic field energy) W L = (LI 2)/2 and a capacitor (electric field energy) W C =(CU 2)/2. The ratio of the energy stored by the reactive elements of the circuit to the energy of ohmic (resistive) losses over a period is usually called the quality factor Q contour, which literally means “quality” in English.
Quality factor of the oscillatory circuit- a characteristic that determines the amplitude and width of the frequency response of the resonance and shows how many times the energy reserves in the circuit are greater than the energy losses during one oscillation period. The quality factor takes into account the presence of active load resistance R.
For a series oscillatory circuit in RLC circuits, in which all three elements are connected in series, the quality factor is calculated:
Where R, L And C
The reciprocal of the quality factor d = 1/Q called circuit attenuation. To determine the quality factor, the formula is usually used Q = ρ/R, Where R- resistance of the ohmic losses of the circuit, characterizing the power of the resistive (active losses) of the circuit P = I 2 R. The quality factor of real oscillatory circuits made on discrete inductors and capacitors ranges from several units to hundreds or more. The quality factor of various oscillatory systems built on the principle of piezoelectric and other effects (for example, quartz resonators) can reach several thousand or more.
It is customary to evaluate the frequency properties of various circuits in technology using amplitude-frequency characteristics (AFC), while the circuits themselves are considered as four-terminal networks. The figures below show two simple two-port networks containing a series oscillatory circuit and the frequency response of these circuits, which are shown (shown by solid lines). The vertical axis of the frequency response graphs shows the value of the circuit's voltage transfer coefficient K, showing the ratio of the circuit's output voltage to the input.
For passive circuits (i.e., those not containing amplifying elements and energy sources), the value TO never exceeds one. The alternating current resistance of the circuit shown in the figure will be minimal at an exposure frequency equal to the resonant frequency of the circuit. In this case, the circuit transmission coefficient is close to unity (determined by ohmic losses in the circuit). At frequencies very different from the resonant one, the resistance of the circuit to alternating current is quite high, and therefore the transmission coefficient of the circuit will drop to almost zero.
When there is resonance in this circuit, the input signal source is actually short-circuited by a small circuit resistance, due to which the transmission coefficient of such a circuit at the resonant frequency drops to almost zero (again due to the presence of finite loss resistance). On the contrary, at input frequencies significantly distant from the resonant one, the circuit transmission coefficient turns out to be close to unity. The property of an oscillatory circuit to significantly change the transmission coefficient at frequencies close to the resonant one is widely used in practice when it is necessary to isolate a signal with a specific frequency from many unnecessary signals located at other frequencies. Thus, in any radio receiver, tuning to the frequency of the desired radio station is ensured using oscillatory circuits. The property of an oscillatory circuit to select one from many frequencies is usually called selectivity or selectivity. In this case, the intensity of the change in the transmission coefficient of the circuit when the frequency of influence is detuned from resonance is usually assessed using a parameter called the passband. The passband is taken to be the frequency range within which the decrease (or increase, depending on the type of circuit) of the transmission coefficient relative to its value at the resonant frequency does not exceed 0.7 (3 dB).
The dotted lines in the graphs show the frequency response of exactly the same circuits, the oscillatory circuits of which have the same resonant frequencies as for the case discussed above, but have a lower quality factor (for example, the inductor is wound with a wire that has a high resistance to direct current). As can be seen from the figures, this expands the bandwidth of the circuit and deteriorates its selective properties. Based on this, when calculating and designing oscillatory circuits, one must strive to increase their quality factor. However, in some cases, the quality factor of the circuit, on the contrary, has to be underestimated (for example, by including a small resistor in series with the inductor), which avoids distortion of broadband signals. Although, if in practice it is necessary to isolate a sufficiently broadband signal, selective circuits, as a rule, are built not on single oscillatory circuits, but on more complex coupled (multi-circuit) oscillatory systems, incl. multi-section filters.
Parallel oscillatory circuit
In various radio engineering devices, along with serial oscillatory circuits, parallel oscillatory circuits are often used (even more often than serial ones). The figure shows a schematic diagram of a parallel oscillatory circuit. Here, two reactive elements with different reactivity patterns are connected in parallel. As is known, when elements are connected in parallel, you cannot add their resistances - you can only add their conductivities. The figure shows graphical dependences of the reactive conductivities of the inductor B L = 1/ωL, capacitor B C = -ωC, as well as total conductivity In Σ, these two elements, which is the reactive conductivity of a parallel oscillatory circuit. Similarly, as for a series oscillating circuit, there is a certain frequency, called resonant, at which the reactance (and therefore conductivity) of the coil and capacitor are the same. At this frequency, the total conductivity of the parallel oscillatory circuit without loss becomes zero. This means that at this frequency the oscillatory circuit has an infinitely large resistance to alternating current.
If we plot the dependence of the circuit reactance on frequency X Σ = 1/B Σ, this curve, shown in the following figure, at the point ω = ω р will have a discontinuity of the second kind. The resistance of a real parallel oscillatory circuit (i.e. with losses), of course, is not equal to infinity - it is lower, the greater the ohmic resistance of losses in the circuit, that is, it decreases in direct proportion to the decrease in the quality factor of the circuit. In general, the physical meaning of the concepts of quality factor, characteristic impedance and resonant frequency of an oscillatory circuit, as well as their calculation formulas, are valid for both series and parallel oscillatory circuits.
For a parallel oscillating circuit in which inductance, capacitance and resistance are connected in parallel, the quality factor is calculated:
Where R, L And C- resistance, inductance and capacitance of the resonant circuit, respectively.
Consider a circuit consisting of a harmonic oscillation generator and a parallel oscillatory circuit. In the case when the oscillation frequency of the generator coincides with the resonant frequency of the circuit, its inductive and capacitive branches have equal resistance to alternating current, as a result of which the currents in the branches of the circuit will be the same. In this case, they say that there is a current resonance in the circuit. As in the case of a series oscillating circuit, the reactance of the coil and capacitor cancel each other, and the resistance of the circuit to the current flowing through it becomes purely active (resistive). The value of this resistance, often called equivalent in technology, is determined by the product of the quality factor of the circuit and its characteristic resistance R eq = Q ρ. At frequencies other than resonant, the resistance of the circuit decreases and becomes reactive at lower frequencies - inductive (since the reactance of inductance decreases as the frequency decreases), and at higher frequencies - on the contrary, capacitive (since the reactance of the capacitance decreases with increasing frequency) .
Let us consider how the transmission coefficients of quadripole networks depend on frequency when they include not serial oscillatory circuits, but parallel ones.
The four-terminal network shown in the figure at the resonant frequency of the circuit represents a huge current resistance, therefore, when ω=ω р its transmission coefficient will be close to zero (taking into account ohmic losses). At frequencies other than the resonant one, the circuit resistance will decrease, and the transmission coefficient of the four-terminal network will increase.
For the four-terminal network shown in the figure above, the situation will be the opposite - at the resonant frequency the circuit will have a very high resistance and almost all of the input voltage will go to the output terminals (that is, the transmission coefficient will be maximum and close to unity). If the frequency of the input action differs significantly from the resonant frequency of the circuit, the signal source connected to the input terminals of the quadripole will be practically short-circuited, and the transmission coefficient will be close to zero.
Oscillatory circuit- an electrical circuit in which oscillations can occur with a frequency determined by the parameters of the circuit.
The simplest oscillatory circuit consists of a capacitor and an inductor connected in parallel or in series.
Capacitor C– reactive element. Has the ability to accumulate and release electrical energy.
- Inductor L– reactive element. Has the ability to accumulate and release magnetic energy.
Free electrical oscillations in a parallel circuit.
Basic properties of inductance:
The current flowing in the inductor creates a magnetic field with energy.
- A change in current in a coil causes a change in the magnetic flux in its turns, creating an EMF in them that prevents a change in current and magnetic flux.
Period of free oscillations of the circuit L.C. can be described as follows:
If the capacitor has a capacity C charged to voltage U, the potential energy of its charge will be 
.
If you connect an inductor in parallel to a charged capacitor L, its discharge current will flow through the circuit, creating a magnetic field in the coil.
The magnetic flux, increasing from zero, will create an EMF in the direction opposite to the current in the coil, which will prevent the current from increasing in the circuit, so the capacitor will not discharge instantly, but after a while t 1, which is determined by the inductance of the coil and the capacitance of the capacitor from the calculation t 1 = .
After time has passed t 1, when the capacitor is discharged to zero, the current in the coil and the magnetic energy will be maximum.
The magnetic energy accumulated by the coil at this moment will be.
In an ideal consideration, with complete absence of losses in the circuit, E C will be equal E L. Thus, the electrical energy of the capacitor will be converted into magnetic energy of the coil.
A change (decrease) in the magnetic flux of the accumulated energy of the coil will create an EMF in it, which will continue the current in the same direction and the process of charging the capacitor with induced current will begin. Decreasing from maximum to zero over time t 2 = t 1, it will recharge the capacitor from zero to the maximum negative value ( -U).
So the magnetic energy of the coil will be converted into electrical energy of the capacitor.
Described intervals t 1 and t 2 will be half the period of complete oscillation in the circuit.
In the second half, the processes are similar, only the capacitor will discharge from a negative value, and the current and magnetic flux will change direction. Magnetic energy will again accumulate in the coil over time t 3, changing the polarity of the poles.
During the final stage of oscillation ( t 4), the accumulated magnetic energy of the coil will charge the capacitor to its original value U(in the absence of losses) and the oscillation process will repeat.
In reality, in the presence of energy losses on the active resistance of the conductors, phase and magnetic losses, the oscillations will be damped in amplitude.
Time t 1 + t 2 + t 3 + t 4 will be the oscillation period 
.
Frequency of free oscillations of the circuit ƒ = 1 / T
The free oscillation frequency is the resonance frequency of the circuit at which the inductance reactance X L =2πfL equal to the reactance of the capacitance X C =1/(2πfC).
Resonance Frequency Calculation L.C.-contour:
A simple online calculator is provided to calculate the resonant frequency of an oscillating circuit.
The main device that determines the operating frequency of any alternating current generator is the oscillating circuit. The oscillatory 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 a coil is inductance, it is designated L and measured in Henry (H), the capacitor is characterized by capacitance C, which is measured in farads (F).
Let at the initial moment of time the capacitor be 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 with energy is formed between the plates of the capacitor
where is the amplitude (maximum) voltage or potential difference across the capacitor plates.
After closing the circuit, 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 a current of variable magnitude flows in the circuit, a self-inductive emf is induced in the coil, which prevents the capacitor from discharging. Therefore, the process of discharging the capacitor does not occur instantly, but gradually. At each moment of time, the potential difference across the capacitor plates
(where is the charge of the capacitor at a given time) is equal to the potential difference across the coil, i.e. equal to the self-induction emf
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| Fig.1 | Fig.2 |
When the capacitor is completely discharged and , the current in the coil will reach its maximum value (Fig. 3). The induction of the magnetic field of the coil at this moment is also maximum, and the energy of the magnetic field will be equal to
Then the current begins to decrease, and the charge will accumulate on the capacitor plates (Fig. 4). When the current decreases to zero, the capacitor charge reaches its maximum value Q 0, but the plate, previously positively charged, will now be negatively charged (Fig. 5). Then the capacitor begins to discharge again, and the current in the circuit flows in the opposite direction.
So the process of charge flowing from one capacitor plate to another through the inductor is repeated again and again. They say that in the circuit there are electromagnetic vibrations. This process is associated not only with fluctuations in the amount of charge and voltage on the capacitor, the current strength in the coil, but also with the transfer of energy from the electric field to the magnetic field and vice versa.
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| Fig.3 | Fig.4 |
Recharging the capacitor to the maximum voltage will occur only if there is no energy loss in the oscillatory circuit. Such a contour is called ideal.
In real circuits the following energy losses occur:
1) heat losses, because R ¹ 0;
2) losses in the dielectric of the capacitor;
3) hysteresis losses in the coil core;
4) radiation losses, etc. If we neglect these energy losses, then we can write that, i.e.
Oscillations occurring in an ideal oscillatory circuit in which this condition is met are called free, or own, circuit vibrations.
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 oscillatory circuit, w 0 = 2pn is the natural (circular) frequency of the oscillatory circuit. The frequency of electromagnetic oscillations in the circuit is defined as
Period T- the time during which one complete oscillation of the voltage on the capacitor and the current in the circuit occurs is determined Thomson's formula
The current strength in the circuit also changes according to the harmonic law, but lags behind the voltage in phase by . Therefore, the dependence of the current strength in the circuit on time will have the form
. (9)
Figure 6 shows graphs of voltage changes U on the capacitor and current I in the coil for an ideal oscillating circuit.
In a real circuit, the energy will decrease with each oscillation. The amplitudes of the voltage on the capacitor and the current in the circuit will decrease; such oscillations are called damped. They cannot be used in master oscillators, because The device will work at best in pulse mode.
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| Fig.5 | Fig.6 |
To obtain undamped oscillations, it is necessary to compensate for energy losses at a wide variety of operating frequencies of devices, including those used in medicine.
Topics of the Unified State Examination codifier: free electromagnetic oscillations, oscillatory circuit, forced electromagnetic oscillations, resonance, harmonic electromagnetic oscillations.
Electromagnetic vibrations- These are periodic changes in charge, current and voltage that occur in an electrical circuit. The simplest system for observing electromagnetic oscillations is an oscillatory circuit.
Oscillatory circuit
Oscillatory circuit is a closed circuit formed by a capacitor and a coil connected in series.
Let's charge the capacitor, connect the coil to it and close the circuit. Will start to happen free electromagnetic oscillations- periodic changes in the charge on the capacitor and the current in the coil. Let us remember that these oscillations are called free because they occur without any external influence - only due to the energy stored in the circuit.
The period of oscillations in the circuit will be denoted, as always, by . We will assume the coil resistance to be zero.
Let us consider in detail all the important stages of the oscillation process. For greater clarity, we will draw an analogy with the oscillations of a horizontal spring pendulum.
Starting moment: . The capacitor charge is equal to , there is no current through the coil (Fig. 1). The capacitor will now begin to discharge.
Rice. 1.
Even though the coil resistance is zero, the current will not increase instantly. As soon as the current begins to increase, a self-induction emf will arise in the coil, preventing the current from increasing.
Analogy. The pendulum is pulled to the right by an amount and released at the initial moment. The initial speed of the pendulum is zero.
First quarter of the period: . The capacitor is discharging, its charge is currently equal to . The current through the coil increases (Fig. 2).
Rice. 2.
The current increases gradually: the vortex electric field of the coil prevents the current from increasing and is directed against the current.
Analogy. The pendulum moves to the left towards the equilibrium position; the speed of the pendulum gradually increases. The deformation of the spring (aka the coordinate of the pendulum) decreases.
End of first quarter: . The capacitor is completely discharged. The current strength has reached its maximum value (Fig. 3). The capacitor will now begin recharging.
Rice. 3.
The voltage across the coil is zero, but the current will not disappear instantly. As soon as the current begins to decrease, a self-induction emf will arise in the coil, preventing the current from decreasing.
Analogy. The pendulum passes through its equilibrium position. Its speed reaches its maximum value. The spring deformation is zero.
Second quarter: . The capacitor is recharged - a charge of the opposite sign appears on its plates compared to what it was at the beginning (Fig. 4).
Rice. 4.
The current strength decreases gradually: the eddy electric field of the coil, supporting the decreasing current, is co-directed with the current.
Analogy. The pendulum continues to move to the left - from the equilibrium position to the right extreme point. Its speed gradually decreases, the deformation of the spring increases.
End of second quarter. The capacitor is completely recharged, its charge is again equal (but the polarity is different). The current strength is zero (Fig. 5). Now the reverse recharging of the capacitor will begin.
Rice. 5.
Analogy. The pendulum has reached the far right point. The speed of the pendulum is zero. The spring deformation is maximum and equal to .
Third quarter: . The second half of the oscillation period began; processes went in the opposite direction. The capacitor is discharged (Fig. 6).
Rice. 6.
Analogy. The pendulum moves back: from the right extreme point to the equilibrium position.
End of the third quarter: . The capacitor is completely discharged. The current is maximum and again equal to , but this time it has a different direction (Fig. 7).
Rice. 7.
Analogy. The pendulum again passes through the equilibrium position at maximum speed, but this time in the opposite direction.
Fourth quarter: . The current decreases, the capacitor charges (Fig. 8).
Rice. 8.
Analogy. The pendulum continues to move to the right - from the equilibrium position to the extreme left point.
End of the fourth quarter and the entire period: . Reverse charging of the capacitor is completed, the current is zero (Fig. 9).
Rice. 9.
This moment is identical to the moment, and this figure is identical to Figure 1. One complete oscillation took place. Now the next oscillation will begin, during which the processes will occur exactly as described above.
Analogy. The pendulum returned to its original position.
The considered electromagnetic oscillations are undamped- they will continue indefinitely. After all, we assumed that the coil resistance is zero!
In the same way, the oscillations of a spring pendulum will be undamped in the absence of friction.
In reality, the coil has some resistance. Therefore, the oscillations in a real oscillatory circuit will be damped. So, after one complete oscillation, the charge on the capacitor will be less than the original value. Over time, the oscillations will completely disappear: all the energy initially stored in the circuit will be released in the form of heat at the resistance of the coil and connecting wires.
In the same way, the oscillations of a real spring pendulum will be damped: all the energy of the pendulum will gradually turn into heat due to the inevitable presence of friction.
Energy transformations in an oscillatory circuit
We continue to consider undamped oscillations in the circuit, considering the coil resistance to be zero. The capacitor has a capacitance and the inductance of the coil is equal to .
Since there are no heat losses, energy does not leave the circuit: it is constantly redistributed between the capacitor and the coil.
Let's take a moment in time when the charge of the capacitor is maximum and equal to , and there is no current. The energy of the magnetic field of the coil at this moment is zero. All the energy of the circuit is concentrated in the capacitor:
Now, on the contrary, let’s consider the moment when the current is maximum and equal to , and the capacitor is discharged. The energy of the capacitor is zero. All the circuit energy is stored in the coil:
At an arbitrary moment in time, when the charge of the capacitor is equal and current flows through the coil, the energy of the circuit is equal to:
Thus,
(1)
Relationship (1) is used to solve many problems.
Electromechanical analogies
In the previous leaflet about self-induction, we noted the analogy between inductance and mass. Now we can establish several more correspondences between electrodynamic and mechanical quantities.
For a spring pendulum we have a relationship similar to (1):
(2)
Here, as you already understood, is the spring stiffness, is the mass of the pendulum, and is the current values of the coordinates and speed of the pendulum, and is their greatest values.
Comparing equalities (1) and (2) with each other, we see the following correspondences:
(3)
(4)
(5)
(6)
Based on these electromechanical analogies, we can foresee a formula for the period of electromagnetic oscillations in an oscillatory circuit.
In fact, the period of oscillation of a spring pendulum, as we know, is equal to:
In accordance with analogies (5) and (6), here we replace mass with inductance, and stiffness with inverse capacitance. We get:
(7)
Electromechanical analogies do not fail: formula (7) gives the correct expression for the period of oscillations in the oscillatory circuit. It is called Thomson's formula. We will present its more rigorous conclusion shortly.
Harmonic law of oscillations in a circuit
Recall that oscillations are called harmonic, if the oscillating quantity changes over time according to the law of sine or cosine. If you have forgotten these things, be sure to repeat the “Mechanical Vibrations” sheet.
The oscillations of the charge on the capacitor and the current in the circuit turn out to be harmonic. We will prove this now. But first we need to establish rules for choosing the sign for the capacitor charge and for the current strength - after all, when oscillating, these quantities will take on both positive and negative values.
First we choose positive bypass direction contour. The choice does not matter; let this be the direction counterclock-wise(Fig. 10).
Rice. 10. Positive bypass direction
The current strength is considered positive class="tex" alt="(I > 0)"> , если ток течёт в положительном направлении. В противном случае сила тока будет отрицательной .!}
The charge on a capacitor is the charge on its plate to which positive current flows (i.e., the plate to which the bypass direction arrow points). In this case - charge left capacitor plates.
With such a choice of signs of current and charge, the following relation is valid: (with a different choice of signs it could happen). Indeed, the signs of both parts coincide: if class="tex" alt="I > 0"> , то заряд левой пластины возрастает, и потому !} class="tex" alt="\dot(q) > 0"> !}.
The quantities and change over time, but the energy of the circuit remains unchanged:
(8)
Therefore, the derivative of energy with respect to time becomes zero: . We take the time derivative of both sides of relation (8); do not forget that complex functions are differentiated on the left (If is a function of , then according to the rule of differentiation of a complex function, the derivative of the square of our function will be equal to: ):
Substituting and here, we get:
But the current strength is not a function that is identically equal to zero; That's why
Let's rewrite this as:
(9)
We have obtained a differential equation of harmonic oscillations of the form , where . This proves that the charge on the capacitor oscillates according to a harmonic law (i.e., according to the law of sine or cosine). The cyclic frequency of these oscillations is equal to:
(10)
This quantity is also called natural frequency contour; It is with this frequency that free (or, as they also say, own fluctuations). The oscillation period is equal to:
We again come to Thomson's formula.
The harmonic dependence of charge on time in the general case has the form:
(11)
The cyclic frequency is found by formula (10); the amplitude and initial phase are determined from the initial conditions.
We will look at the situation discussed in detail at the beginning of this leaflet. Let the charge of the capacitor be maximum and equal (as in Fig. 1); there is no current in the circuit. Then the initial phase is , so that the charge varies according to the cosine law with amplitude:
(12)
Let's find the law of change in current strength. To do this, we differentiate relation (12) with respect to time, again not forgetting about the rule for finding the derivative of a complex function:
We see that the current strength also changes according to a harmonic law, this time according to the sine law:
(13)
The amplitude of the current is:
The presence of a “minus” in the law of current change (13) is not difficult to understand. Let's take, for example, a time interval (Fig. 2).
The current flows in the negative direction: . Since , the oscillation phase is in the first quarter: . The sine in the first quarter is positive; therefore, the sine in (13) will be positive on the time interval under consideration. Therefore, to ensure that the current is negative, the minus sign in formula (13) is really necessary.
Now look at fig. 8 . The current flows in the positive direction. How does our “minus” work in this case? Figure out what's going on here!
Let us depict graphs of charge and current fluctuations, i.e. graphs of functions (12) and (13). For clarity, let us present these graphs in the same coordinate axes (Fig. 11).
Rice. 11. Graphs of charge and current fluctuations
Please note: charge zeros occur at current maxima or minima; conversely, current zeros correspond to charge maxima or minima.
Using the reduction formula
Let us write the law of current change (13) in the form:
Comparing this expression with the law of charge change, we see that the current phase, equal to, is greater than the charge phase by an amount. In this case they say that the current ahead in phase charge on ; or phase shift between current and charge is equal to ; or phase difference between current and charge is equal to .
The advance of the charge current in phase is graphically manifested in the fact that the current graph is shifted left on relative to the charge graph. The current strength reaches, for example, its maximum a quarter of a period earlier than the charge reaches its maximum (and a quarter of a period exactly corresponds to the phase difference).
Forced electromagnetic oscillations
As you remember, forced oscillations arise in the system under the influence of a periodic forcing force. The frequency of forced oscillations coincides with the frequency of the driving force.
Forced electromagnetic oscillations will occur in a circuit connected to a sinusoidal voltage source (Fig. 12).
Rice. 12. Forced vibrations
If the source voltage changes according to the law:
then oscillations of charge and current occur in the circuit with a cyclic frequency (and with a period, respectively). The AC voltage source seems to “impose” its oscillation frequency on the circuit, making you forget about its own frequency.
The amplitude of forced oscillations of charge and current depends on frequency: the amplitude is greater, the closer to the natural frequency of the circuit. When resonance- a sharp increase in the amplitude of oscillations. We'll talk about resonance in more detail in the next worksheet on alternating current.
