How the oscillatory process occurs in the circuit. Voltage resonance in a series oscillating circuit

In radio engineering, electrical circuits consisting of an inductor and a capacitor are widely used. In radio engineering such circuits are called oscillatory circuits. An alternating current source can be connected to an oscillating circuit in two ways: in series (Figure 1a) and in parallel (Figure 1b).

Picture 1. Schematic designation of an oscillatory circuit.a) series oscillatory circuit; b) parallel oscillatory circuit.

Let us consider the behavior of an oscillatory circuit in an alternating current circuit at serial connection of the circuit and the current source(Figure 1a).

We know that such a circuit provides alternating current with reactance equal to:

Where R L is the active resistance of the inductor in ohms;

ωL, is the inductive reactance of the inductor in ohms;

1/ωC- capacitance of the capacitor in ohms.

Coil resistance R L practically changes very little as the frequency changes (if we neglect the surface effect). Inductive and capacitive reactance are very dependent on frequency, namely: inductive reactance ωL increases in direct proportion to the frequency of the current, and capacitance 1/ωC decreases with increasing current frequency, i.e. it is inversely proportional to the current frequency.

It immediately follows that the reactance of a series oscillatory circuit also depends on frequency, and the oscillatory circuit will provide unequal resistance to currents of different frequencies.

If we measure the reactance of the oscillating circuit at different frequencies, we will find that in the low frequency region the resistance of the series circuit is very high; as the frequency increases, it decreases to a certain limit, and then begins to increase again.

This is explained by the fact that in the region of low frequencies the current experiences high resistance from the capacitor, but as the frequency increases, inductive reactance begins to act, compensating for the effect of capacitive reactance.

At a certain frequency, the inductive reactance becomes equal to the capacitive reactance, i.e.

They will cancel each other out and the total reactance of the circuit will become zero:

In this case, the reactance of the series oscillatory circuit will be equal only to its active resistance, since

With a further increase in frequency, the current will experience more and more resistance from the inductance of the coil, while the compensating effect of capacitance decreases. Therefore, the reactance of the circuit will begin to increase again.

Figure 2a shows a curve showing the change in the reactance of a series oscillatory circuit when the current frequency changes.

Figure 2. Voltage resonance.a) dependence of the change in impedance on frequency; b) dependence of reactance on the active resistance of the circuit; c) resonance curves.

The current frequency at which the resistance of the oscillating circuit is minimal is called resonance frequency or resonant frequency oscillatory circuit.

At the resonant frequency the equality holds:

using which it is easy to determine the resonance frequency:

(1)

The units in these formulas are hertz, henry and farad.

From formula (1) it is clear that the smaller the capacitance and self-inductance of the oscillatory circuit, the greater will be its resonant frequency.

Active resistance value R L does not affect the resonant frequency, but the nature of the change depends on it Z. Figure 2b shows a number of graphs of changes in the reactance of the oscillatory circuit at the same values L And WITH, but at different R L. From this figure it can be seen that the greater the active resistance of the series oscillatory circuit, the dumber the reactance change curve becomes.

Now let's look at how the current strength in the oscillatory circuit will change if we change the frequency of the current. In this case, we will assume that the voltage developed by the alternating current source remains the same all the time.

Since the current source is connected in series with L And WITH circuit, then the strength of the current flowing through the coil and capacitor will be greater, the lower the reactance of the oscillatory circuit as a whole, since

It immediately follows that at resonance the current strength in the oscillatory circuit will be greatest. The magnitude of the current at resonance will depend on the voltage of the AC source and on the active resistance of the circuit:

Figure 2d shows a series of graphs of changes in current strength in a series oscillatory circuit when changing the frequency of the current, the so-called resonance curves. From this figure it can be seen that the greater the active resistance of the circuit, the dumber the resonance curve.

At resonance, the current strength can reach enormous values with a relatively small external EMF. Therefore, voltage drops across the inductive and capacitive resistances of the circuit, i.e., across the coil and across the capacitor, can reach very large values and far exceed the magnitude of the external voltage.

The last statement may seem somewhat strange at first glance, but you need to remember that the phases of the voltages on the capacitive and inductive reactances are shifted relative to each other by 180°, i.e., the instantaneous voltage values on the coil and capacitor are always directed in opposite directions. As a result, large voltages that exist during resonance inside the circuit on its coil and capacitor do not reveal themselves outside the circuit, mutually compensating each other.

The case of series resonance we analyzed is called voltage resonance, since in this case at the moment of resonance there is a sharp increase in voltage on L and C of the oscillatory circuit.

In any broadcast receiver, regardless of its complexity, there are absolutely three elements that ensure its performance. These elements are an oscillating circuit, a detector and telephones, or, if the receiver has an AF amplifier, a direct radiation dynamic head. Your first receiver, assembled and tested during the previous conversation, consisted of only these three elements. An oscillatory circuit, which included a grounded antenna, provided the receiver with tuning to the wave of the radio station; the detector converted modulated radio frequency oscillations into audio frequency oscillations, which the telephones converted into sound. Without them, or without any of them, radio reception is impossible.

What is the essence of the action of these mandatory elements of a radio receiver?

OSCILLATION CIRCUIT

The structure of the simplest oscillatory circuit and its diagram are shown in Fig. 38. As you can see, it consists of a coil L and a capacitor C, forming a closed electrical circuit. Under certain conditions, electrical oscillations may occur and exist in the circuit. That is why it is called an oscillatory circuit.

Have you ever observed such a phenomenon: when the power to an electric lighting lamp is turned off, a spark appears between the opening contacts of the switch. If you accidentally connect the terminals of the battery poles of an electric flashlight (which should be avoided), at the moment they are disconnected, a small spark also jumps between them. And in factories, in factory workshops, where switches break electrical circuits through which high currents flow, sparks can be so significant that measures have to be taken to prevent them from causing harm to the person turning on the current. Why do these sparks occur?

From the first conversation, you already know that there is a magnetic field around a current-carrying conductor, which can be depicted in the form of closed magnetic lines of force that penetrate the space surrounding it. This field, if it is constant, can be detected using a magnetic compass needle. If you disconnect a conductor from a current source, then its disappearing magnetic field, dissipating in space, will induce currents in other conductors closest to it. The current is also induced in the conductor that created this magnetic field. And since it is located in the very midst of its own magnetic lines of force, a stronger current will be induced in it than in any other conductor. The direction of this current will be the same as it was at the moment the conductor broke. In other words, a disappearing magnetic field will support the current creating it until it disappears itself, i.e. the energy contained in it is not completely consumed.

Rice. 38. The simplest electrical oscillatory circuit

Consequently, the current flows in the conductor even after the current source is turned off, but, of course, not for long - an insignificant fraction of a second.

But in an open circuit, the movement of electrons is impossible, you will object. Yes it is. But after opening the circuit, electric current can flow for some time through the air gap between the disconnected ends of the conductor, between the contacts of a switch or switch. It is this current through the air that forms an electric spark.

This phenomenon is called self-induction, and the electric force (do not confuse it with the phenomenon of induction, familiar to you from the first conversation), which, under the influence of a vanishing magnetic field, maintains a current in it - the electromotive force of self-induction or, in short, the EMF of self-induction. The greater the self-induction EMF, the more significant the spark can be at the point where the electrical circuit breaks.

The phenomenon of self-induction is observed not only when turning off the current, but also when turning on the current. In the space surrounding the conductor, a magnetic field appears immediately when the current is turned on. At first it is weak, but then it intensifies very quickly. The increasing magnetic field of the current also excites a self-induction current, but this current is directed towards the main current. The self-induction current prevents the instantaneous increase in the main current and the growth of the magnetic field. However, after a short period of time, the main current in the conductor overcomes the countercurrent of self-induction and reaches its greatest value, the magnetic field becomes constant and the effect of self-induction ceases.

The phenomenon of self-induction can be compared with the phenomenon of inertia. A sled, for example, is difficult to move. But when they pick up speed and store up kinetic energy - the energy of motion, they cannot be stopped instantly. When braking, the sled continues to slide until the energy stored in it is used up to overcome friction with the snow.

Do all conductors have the same self-inductance? No! The longer the conductor, the greater the self-induction. In a conductor coiled into a coil, the phenomenon of self-induction is more pronounced than in a straight conductor, since the magnetic field of each turn of the coil induces a current not only in this turn, but also in the neighboring turns of this coil. The longer the wire in the coil is, the longer the self-induction current will exist in it after the main current is turned off. Conversely, it will take more time after turning on the main current for the current in the circuit to increase to a certain value and to establish a constant magnetic field strength.

Remember: the property of conductors to influence the current in a circuit when its value changes is called inductance, and the coils in which this property is most strongly manifested are called self-inductance or inductance coils. The greater the number of turns and the size of the coil, the greater its inductance, the more significant its effect on the current in the electrical circuit.

So, the inductor prevents both the increase and decrease of current in the electrical circuit. If it is in a direct current circuit, its influence is felt only when the current is turned on and off. In an alternating current circuit, where the current and its magnetic field are constantly changing, the self-inductive emf of the coil acts all the time the current flows. This electrical phenomenon is used in the first element of the oscillating circuit of the receiver - the inductor.

The second element of the oscillatory circuit of the receiver is a “storage” of electrical charges - a capacitor. The simplest capacitor consists of two conductors of electric current, for example, two metal plates, called capacitor plates, separated by a dielectric, such as air or paper. You have already used such a capacitor during experiments with the simplest receiver. The larger the area of the capacitor plates and the closer they are located to each other, the greater the electrical capacitance of this device.

If a direct current source is connected to the plates of the capacitor (Fig. 39, a), then a short-term current will arise in the resulting circuit and the capacitor will be charged to a voltage equal to the voltage of the current source.

You may ask: why does current occur in a circuit where there is a dielectric?

Rice. 39. Charging and discharging a capacitor

When we connect a direct current source to a capacitor, free electrons in the conductors of the resulting circuit begin to move towards the positive pole of the current source, forming a short-term flow of electrons throughout the circuit. As a result, the plate of the capacitor, which is connected to the positive pole of the current source, is depleted of free electrons and is charged positively, and the other plate is enriched in free electrons and, therefore, is charged negatively. Once the capacitor is charged, the short-term current in the circuit, called the capacitor charging current, will stop.

If the current source is disconnected from the capacitor, the capacitor will be charged (Fig. 39, b). The dielectric prevents the transfer of excess electrons from one plate to another. There will be no current between the plates of the capacitor, and the electrical energy accumulated by it will be concentrated in the electric field of the dielectric. But as soon as the plates of a charged capacitor are connected with some kind of conductor (Fig. 39, c), the “extra” electrons of the negatively charged plate will pass through this conductor to another plate, where they are missing, and the capacitor will be discharged. In this case, a short-term current also arises in the resulting circuit, called the capacitor discharge current. If the capacity of the capacitor is large and it is charged to a significant voltage, the moment it is discharged is accompanied by the appearance of a significant spark and crackling sound.

The property of a capacitor to accumulate electrical charges and discharge through conductors connected to it is used in the oscillating circuit of a radio receiver.

And now, young friend, remember an ordinary swing. You can swing on them in such a way that it will take your breath away. What needs to be done for this? First push to bring the swing out of its resting position, and then apply some force, but only in time with its vibrations. Without much difficulty, you can achieve strong swings of the swing and obtain large amplitudes of vibration. Even a little boy can push an adult on a swing if he applies his strength skillfully. Having rocked the swing harder to achieve larger vibration amplitudes, let’s stop pushing it. What happens next? Due to the stored energy, they swing freely for some time, the amplitude of their oscillations gradually decreases, as they say, the oscillations die out, and finally the swing stops.

With free oscillations of a swing, as well as a freely suspended pendulum, the stored potential energy turns into kinetic energy, which at the highest point again turns into potential, and after a split second - again into kinetic. And so on until the entire energy reserve is used up to overcome the friction of the ropes in the places where the swing is suspended and air resistance. With an arbitrarily large supply of energy, free oscillations are always damped: with each oscillation their amplitude decreases and the oscillations gradually die out completely - the swing stops. But the period, i.e. the time during which one oscillation occurs, and therefore the frequency of oscillations remains constant.

However, if the swing is constantly pushed in time with its oscillations and thereby replenishes the loss of energy spent on overcoming various braking forces, the oscillations will become undamped. These are no longer free, but forced vibrations. They will last until the external pushing force ceases to act.

I remembered the swing here because the physical phenomena occurring in such a mechanical oscillatory system are very similar to the phenomena in an electrical oscillatory circuit. In order for electrical oscillations to occur in the circuit, it must be given energy that would “push” the electrons in it. This can be done by charging, for example, its capacitor.

Let us break the oscillatory circuit with switch S and connect a direct current source to the plates of its capacitor, as shown in Fig. 40 on the left.

Rice. 40. Electrical vibrations in the circuit

The capacitor will charge to the battery voltage GB. Then we disconnect the battery from the capacitor, and close the circuit with switch S. The phenomena that will now occur in the circuit are shown graphically in Fig. 40 on the right.

At the moment the circuit is closed by the switch, the upper plate of the capacitor has a positive charge, and the bottom one has a negative charge (Fig. 40, a). At this time (point 0 on the graph) there is no current in the circuit, and all the energy accumulated by the capacitor is concentrated in the electric field of its dielectric. When a capacitor is shorted to the coil, the capacitor will begin to discharge. A current appears in the coil, and a magnetic field appears around its turns. By the time the capacitor is completely discharged (Fig. 40, b), marked on the graph by number 1, when the voltage on its plates decreases to zero, the current in the coil and the energy of the magnetic field will reach their highest values. It would seem that at this moment the current in the circuit should have stopped. This, however, will not happen, since due to the action of the self-induction EMF, which tends to maintain the current, the movement of electrons in the circuit will continue. But only until all the energy of the magnetic field is used up. At this time, an induced current will flow in the coil, decreasing in value but in the original direction.

By the moment of time, marked on the graph by number 2, when the energy of the magnetic field is used up, the capacitor will again be charged, only now there will be a positive charge on its lower plate, and a negative one on the top (Fig. 40, c). Now the electrons will begin to move in the opposite direction - in the direction from the upper plate through the coil to the bottom plate of the capacitor. By time 3 (Fig. 40, d) the capacitor will be discharged, and the magnetic field of the coil will reach its greatest value. And again, the self-induction EMF will “drive” electrons along the coil wire, thereby recharging the capacitor.

At time 4 (Fig. 40, e), the state of the electrons in the circuit will be the same as at the initial time 0. One complete oscillation has ended. Naturally, the charged capacitor will again be discharged into the coil, recharged, and a second, followed by a third, fourth, etc. oscillation will occur. In other words, an alternating electric current, electrical oscillations, will appear in the circuit. But this oscillatory process in the circuit is not endless. It continues until all the energy received by the capacitor from the battery is spent on overcoming the resistance of the circuit coil wire. Oscillations in the circuit are free and, therefore, damped.

What is the frequency of such oscillations of electrons in the circuit? To understand this issue in more detail, I advise you to carry out such an experiment with a simple pendulum.

Rice. 41. Graphs of oscillations of the simplest pendulum

Suspend on a thread 100 cm long a ball molded from plasticine, or another load weighing 20-40 g (in Fig. 41, the length of the pendulum is indicated by the Latin letter 1). Take the pendulum out of its equilibrium position and, using a clock with a second hand, count how many complete oscillations it makes in 1 minute. Approximately 30. Therefore, the frequency of oscillation of this pendulum is 0.5 Hz, and the period is 2 s. During the period, the potential energy of the pendulum transforms twice into kinetic energy, and kinetic energy into potential energy. Shorten the thread by half. The frequency of the pendulum will increase by about one and a half times and the period of oscillation will decrease by the same amount.

This experience allows us to conclude: as the length of the pendulum decreases, the frequency of its own oscillations increases, and the period proportionally decreases.

By changing the length of the pendulum suspension, ensure that its oscillation frequency is 1 Hz. This should be with a thread length of about 25 cm. In this case, the period of oscillation of the pendulum will be equal to 1 s. No matter how you try to create the initial swing of the pendulum, the frequency of its oscillations will remain unchanged. But as soon as you shorten or lengthen the thread, the oscillation frequency immediately changes. With the same length of thread there will always be the same oscillation frequency. This is the natural frequency of the pendulum. You can obtain a given oscillation frequency by selecting the length of the thread.

The oscillations of a thread pendulum are damped. They can become undamped only if the pendulum is slightly pushed in time with its oscillations, thus compensating for the energy it expends on overcoming the resistance provided to it by the air, the energy of friction, and gravity.

The natural frequency is also characteristic of an electric oscillatory circuit. It depends, firstly, on the inductance of the coil. The greater the number of turns and the diameter of the coil, the greater its inductance, the longer will be the duration of the period of each oscillation. The natural frequency of oscillations in the circuit will be correspondingly lower. And, conversely, with a decrease in the inductance of the coil, the period of oscillation will decrease - the natural frequency of oscillations in the circuit will increase. Secondly, the natural frequency of oscillations in the circuit depends on the capacitance of its capacitor. The larger the capacitance, the more charge the capacitor can accumulate, the longer it will take to recharge it, and the lower the oscillation frequency in the circuit. As the capacitor capacity decreases, the oscillation frequency in the circuit increases. Thus, the natural frequency of damped oscillations in the circuit can be adjusted by changing the inductance of the coil or the capacitance of the capacitor.

But in an electrical circuit, as in a mechanical oscillatory system, it is possible to obtain undamped, i.e. forced oscillations, if at each oscillation the circuit is replenished with additional portions of electrical energy from some alternating current source.

How are undamped electrical oscillations excited and maintained in the receiver circuit? Radio frequency oscillations excited in the receiver antenna. These vibrations impart an initial charge to the circuit, and they also maintain the rhythmic oscillations of electrons in the circuit. But the strongest undamped oscillations in the receiver circuit occur only at the moment of resonance of the circuit’s natural frequency with the frequency of the current in the antenna. What does it mean?

People of the older generation say that in St. Petersburg the Egyptian Bridge collapsed from soldiers marching in step. And this could happen, apparently, under such circumstances. All the soldiers walked rhythmically along the bridge. As a result, the bridge began to sway and oscillate. By chance, the bridge's natural vibration frequency coincided with the soldiers' step frequency, and the bridge is said to have entered into resonance.

The rhythm of the formation imparted more and more energy to the bridge. As a result, the bridge swayed so much that it collapsed: the coherence of the military formation caused damage to the bridge. If there had been no resonance of the bridge’s natural vibration frequency with the soldiers’ step frequency, nothing would have happened to the bridge. Therefore, by the way, when soldiers pass over weak bridges, it is customary to give the command “knock down your leg.”

Here's the experience. Go to some stringed musical instrument and shout “a” loudly: one of the strings will respond and sound. The one that is in resonance with the frequency of this sound will vibrate more strongly than the other strings - it will respond to the sound.

Another experiment - with a pendulum. Stretch a thin rope horizontally. Tie the same pendulum made of thread and plasticine to it (Fig. 42). Throw another similar pendulum over the rope, but with a longer thread. The length of the suspension of this pendulum can be changed by pulling the free end of the thread with your hand. Set the pendulum into oscillatory motion. In this case, the first pendulum will also begin to oscillate, but with a smaller amplitude. Without stopping the oscillations of the second pendulum, gradually reduce the length of its suspension - the amplitude of oscillations of the first pendulum will increase. In this experiment, illustrating the resonance of mechanical vibrations, the first pendulum is a receiver of vibrations excited by the second pendulum. The reason that forces the first pendulum to oscillate is the periodic oscillations of the tension rod with a frequency equal to the oscillation frequency of the second pendulum. The forced oscillations of the first pendulum will have maximum amplitude only when its natural frequency coincides with the oscillation frequency of the second.

Rice. 42. Experience illustrating the phenomenon of resonance

Such or similar phenomena, only, of course, of electrical origin, are also observed in the oscillatory circuit of the receiver. From the action of waves from many radio stations, currents of various frequencies are excited in the receiving antenna. From all the oscillations of radio frequencies, we need to select only the carrier frequency of the radio station whose broadcasts we want to listen to. To do this, we must select the number of coil turns and the capacitance of the oscillating circuit capacitor so that its natural frequency coincides with the frequency of the current created in the antenna by the radio waves of the station of interest to us. In this case, the strongest oscillations will occur in the circuit with the carrier frequency of the radio station to which it is tuned. This is the setting of the receiver circuit in resonance with the frequency of the transmitting station. In this case, the signals of other stations are not heard at all or are heard very quietly, since the oscillations they excite in the circuit will be many times weaker.

Thus, by tuning the circuit of your first receiver into resonance with the carrier frequency of the radio station, you, with its help, sort of selected and isolated the frequency oscillations of only this station. The better the circuit isolates the necessary vibrations from the antenna, the higher the selectivity of the receiver, the weaker the interference from other radio stations will be.

Until now, I have told you about a closed oscillatory circuit, that is, a circuit whose natural frequency is determined only by the inductance of the coil and the capacitance of the capacitor that forms it. However, the receiver input circuit also includes an antenna and ground. This is no longer a closed, but an open oscillatory circuit. The fact is that the antenna wire and ground are the “plates” of a capacitor (Fig. 43), which has a certain electrical capacitance. Depending on the length of the wire and the height of the antenna above the ground, this capacitance can be several hundred picofarads. Such a capacitor in Fig. ZCH, but was shown with dashed lines. But the antenna and the ground can also be considered as an incomplete turn of a large coil.

Rice. 43. Antenna and grounding - open oscillatory circuit

Therefore, the antenna and grounding, taken together, also have inductance. And capacitance together with inductance form an oscillatory circuit.

Such a circuit, which is an open oscillatory circuit, also has its own oscillation frequency. By connecting inductors and capacitors between the antenna and the ground, we can change its natural frequency, tune it into resonance with the frequencies of different radio stations. You already know how this is done in practice.

I will not be mistaken if I say that the oscillatory circuit is the “heart” of the radio receiver. And not just a radio. You will be convinced of this later. That's why I paid a lot of attention to him.

I turn to the second element of the receiver - the detector.

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 XL 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 X L = 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 XL 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 oscillatory 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.

A series oscillatory circuit is a circuit consisting of an inductor and a capacitor, which are connected in series. On the diagrams ideal A series oscillating circuit is designated like this:

A real oscillating circuit has a loss resistance of a coil and a capacitor. This total loss resistance is denoted by the letter R. As a result, real the series oscillatory circuit will look like this:

R is the total loss resistance of the coil and capacitor

L – the actual inductance of the coil

C is the capacitance of the capacitor itself

Oscillatory circuit and frequency generator

Let's do a classic experiment that is in every electronics textbook. To do this, let's put together the following diagram:

Our generator will produce sine.

In order to record an oscillogram through a series oscillating circuit, we will connect a shunt resistor with a low resistance of 0.5 Ohms to the circuit and remove the voltage from it. That is, in this case we use a shunt to monitor the current strength in the circuit.

And here is the diagram itself in reality:

From left to right: shunt resistor, inductor and capacitor. As you already understand, resistance R is the total loss resistance of the coil and capacitor, since there are no ideal radio elements. It is “hidden” inside the coil and capacitor, so in a real circuit we will not see it as a separate radio element.

Now all we have to do is connect this circuit to a frequency generator and an oscilloscope, and run it through some frequencies, taking an oscillogram from the shunt U w, as well as taking an oscillogram from the generator itself U GENE.

From the shunt we will remove the voltage, which reflects the behavior of the current in the circuit, and from the generator the generator signal itself. Let's run our circuit through some frequencies and see what is what.

The influence of frequency on the resistance of the oscillatory circuit

So, let's go. In the circuit, I took a 1 µF capacitor and a 1 mH inductor. On the generator I set up a sine wave with a swing of 4 Volts. Let us remember the rule: if in a circuit the connection of radio elements occurs in series one after another, it means that the same current flows through them.

The red waveform is the voltage from the frequency generator, and the yellow waveform is a display of the current through the voltage across the shunt resistor.

Frequency 200 Hertz with kopecks:

As we see, at this frequency there is a current in this circuit, but it is very weak

Adding frequency. 600 Hertz with kopecks

Here we can clearly see that the current strength has increased, and we also see that the current oscillogram is ahead of the voltage. Smells like a capacitor.

Adding frequency. 2 Kilohertz

The current strength became even greater.

3 Kilohertz

The current strength has increased. Notice also that the phase shift has begun to decrease.

4.25 Kilohertz

The oscillograms are almost merging into one. The phase shift between voltage and current becomes almost imperceptible.

And at some frequency, the current strength became maximum, and the phase shift became zero. Remember this moment. It will be very important for us.

Just recently, the current was ahead of the voltage, but now it has already begun to lag after it has aligned with it in phase. Since the current already lags behind the voltage, it already smells like the reactance of the inductor.

We increase the frequency even more

The current strength begins to drop, and the phase shift increases.

22 Kilohertz

74 Kilohertz

As you can see, as the frequency increases, the shift approaches 90 degrees, and the current becomes less and less.

Resonance

Let's take a closer look at the very moment when the phase shift was zero and the current passing through the series oscillatory circuit was maximum:

This phenomenon is called resonance.

As you remember, if our resistance becomes small, and in this case the loss resistances of the coil and capacitor are very small, then a large current begins to flow in the circuit according to Ohm's law: I=U/R. If the generator is powerful, then the voltage on it does not change, and the resistance becomes negligible and voila! The current grows like mushrooms after rain, which is what we saw by looking at the yellow oscillogram at resonance.

Thomson's formula

If, at resonance, the reactance of the coil is equal to the reactance of the capacitor X L =X C, then you can equalize their reactances and from there calculate the frequency at which the resonance occurred. So, the reactance of the coil is expressed by the formula:

The reactance of a capacitor is calculated using the formula:

We equate both sides and calculate from here F:

In this case we got the formula resonant frequency. This formula is called differently Thomson's formula, as you understand, in honor of the scientist who brought it out.

Let's use Thomson's formula to calculate the resonant frequency of our series oscillatory circuit. For this I will use my RLC transistormeter.

We measure the inductance of the coil:

And we measure our capacity:

We calculate our resonant frequency using the formula:

I got 5.09 Kilohertz.

Using frequency adjustment and an oscilloscope, I caught a resonance at a frequency of 4.78 Kilohertz (written in the lower left corner)

Let's write off an error of 200 kopecks Hertz to the measurement error of the instruments. As you can see, Thompson's formula works.

Voltage resonance

Let's take other parameters of the coil and capacitor and see what is happening on the radio elements themselves. We need to find out everything thoroughly ;-). I take an inductor with an inductance of 22 microhenry:

and a 1000 pF capacitor

So, in order to catch the resonance, I will not add . I'll do something more cunning.

Since my frequency generator is Chinese and low-power, during resonance we only have active loss resistance R in the circuit. The total resistance is still a small value, so the current at resonance reaches its maximum values. As a result of this, a decent voltage drops across the internal resistance of the frequency generator and the output frequency amplitude of the generator drops. I will catch the minimum value of this amplitude. Therefore, this will be the resonance of the oscillatory circuit. Overloading a generator is not good, but what can’t you do for the sake of science!

Well, let's get started ;-). Let's first calculate the resonant frequency using Thomson's formula. To do this, I open an online calculator on the Internet and quickly calculate this frequency. I got 1.073 Megahertz.

I catch resonance on the frequency generator by its minimum amplitude values. It turned out something like this:

Peak-to-peak amplitude 4 Volts

Although the frequency generator has a swing of more than 17 Volts! This is how the tension dropped a lot. And as you can see, the resonant frequency turned out to be a little different than the calculated one: 1.109 Megahertz.

Now a little fun ;-)

This is the signal we apply to our serial oscillatory circuit:

As you can see, my generator is not able to deliver a large current to the oscillating circuit at the resonant frequency, so the signal turned out to be even slightly distorted at the peaks.

Well, now the most interesting part. Let's measure the voltage drop across the capacitor and coil at the resonant frequency. That is, it will look like this:

We look at the voltage on the capacitor:

The amplitude swing is 20 Volts (5x4)! Where? After all, we supplied a sine wave to the oscillatory circuit with a frequency of 2 Volts!

Okay, maybe something happened to the oscilloscope? Let's measure the voltage on the coil:

People! Freebie!!! We supplied 2 Volts from the generator, but received 20 Volts both on the coil and on the capacitor! Energy gain 10 times! Just have time to remove energy from either the capacitor or the coil!

Well, okay, since this is the case... I take a 12-volt moped light bulb and connect it to a capacitor or coil. The light bulb seems to know what frequency to operate at and what current to consume. I set the amplitude so that there is somewhere around 20 Volts on the coil or capacitor since the root mean square voltage will be somewhere around 14 Volts, and I attach a light bulb to them one by one:

As you can see - complete zero. The light is not going to light up, so shave, fans of free energy). You haven't forgotten that power is determined by the product of current and voltage, right? There seems to be enough voltage, but alas, the current strength! Therefore, the series oscillatory circuit is also called narrowband (resonant) voltage amplifier, not power!

Let's summarize what we found in these experiments.

At resonance, the voltage on the coil and on the capacitor turned out to be much greater than what we applied to the oscillatory circuit. In this case, we got 10 times more. Why is the voltage on the coil at resonance equal to the voltage on the capacitor? This is easy to explain. Since in a series oscillating circuit the coil and the conductor follow each other, therefore, the same current flows in the circuit.

At resonance, the reactance of the coil is equal to the reactance of the capacitor. According to the shunt rule, we find that the voltage drops across the coil U L = IX L, and on the capacitor U C = IX C. And since at resonance we have X L = X C, then we get that U L = U C, the current in the circuit is the same ;-). Therefore, resonance in a series oscillatory circuit is also called voltage resonance, because the voltage across the coil at the resonant frequency is equal to the voltage across the capacitor.

Quality factor

Well, since we started to push the topic of oscillatory circuits, we cannot ignore such a parameter as quality factor oscillatory circuit. Since we have already carried out some experiments, it will be easier for us to determine the quality factor based on the voltage amplitude. Quality factor is indicated by the letter Q and is calculated using the first simple formula:

Let's calculate the quality factor in our case.

Since the cost of dividing one square vertically is 2 Volts, therefore, the amplitude of the frequency generator signal is 2 Volts.

And this is what we have at the terminals of the capacitor or coil. Here the price of dividing one square vertically is 5 Volts. We count squares and multiply. 5x4=20 Volts.

We calculate using the quality factor formula:

Q=20/2=10. In principle, a little and not a little. It'll do. This is how quality factor can be found in practice.

There is also a second formula for calculating the quality factor.

Where

R – loss resistance in the circuit, Ohm

L – inductance, Henry

C – capacitance, Farad

Knowing the quality factor, you can easily find the loss resistance R series oscillatory circuit.

I also want to add a few words about quality factor. The quality factor of the circuit is a qualitative indicator of the oscillatory circuit. Basically, they always try to increase it in various possible ways. If you look at the formula above, you can understand that in order to increase the quality factor, we need to somehow reduce the loss resistance of the oscillating circuit. The lion's share of losses relates to the inductor, since it already has large losses structurally. It is wound from wire and in most cases has a core. At high frequencies, a skin effect begins to appear in the wire, which further introduces losses into the circuit.

Summary

A series oscillating circuit consists of an inductor and a capacitor connected in series.

At a certain frequency, the reactance of the coil becomes equal to the reactance of the capacitor and a phenomenon such as resonance.

At resonance, the reactances of the coil and capacitor, although equal in magnitude, are opposite in sign, so they are subtracted and add up to zero. Only the active loss resistance R remains in the circuit.

At resonance, the current strength in the circuit becomes maximum, since the loss resistance of the coil and capacitor R add up to a small value.

At resonance, the voltage across the coil is equal to the voltage across the capacitor and exceeds the voltage across the generator.

The coefficient showing how many times the voltage on the coil or capacitor exceeds the voltage on the generator is called the quality factor Q of the series oscillatory circuit and shows a qualitative assessment of the oscillatory circuit. Basically they try to make Q as big as possible.

At low frequencies, the oscillatory circuit has a capacitive current component before resonance, and after resonance, an inductive current component.

ELECTROMAGNETIC OSCILLATIONS.
FREE AND FORCED ELECTRICAL VIBRATIONS.

Electromagnetic oscillations are interconnected oscillations of electric and magnetic fields.

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

Free electromagnetic oscillations arise in an electromagnetic system after removing it from a state of equilibrium, for example, by imparting a charge to a capacitor or changing the current in a section of the circuit.

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

Forced electromagnetic oscillations are undamped oscillations in a circuit caused by an external periodically changing sinusoidal EMF.

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

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

OSCILLATION CIRCUIT

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

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

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

quantities characterizing the state of the system:

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

CHARACTERISTICS OF ELECTROMAGNETIC VIBRATIONS

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

Where q" is the second derivative of the charge with respect to time. Magnitude

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

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

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

The quantity φ = ώt + φ 0, standing under the sine or cosine sign, is the oscillation phase.

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

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

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

ALTERNATING ELECTRIC CURRENT

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

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

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

where is the rate of change of magnetic induction flux.

A harmonically changing magnetic flux causes a sinusoidal induced emf

where is the amplitude value of the induced emf.

2. If a source of external harmonic EMF is connected to the circuit

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

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

In industrial AC circuits in Russia, the accepted frequency is 50 Hz.

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

Therefore, the amount of heat released during time T:

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

Hence the effective current value

Similarly, the effective voltage value

TRANSFORMER

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

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

Size

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

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

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

Oscillation frequency

The dependence of current strength on time has the form:

Current amplitude.

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

i=-5000 sin100t

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

Transverse and longitudinal waves. Wavelength - Mechanical vibrations and waves. Sound 9th grade

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