Oscillatory circuit with current source. Oscillatory circuit and its operation. Real oscillatory circuit

Encyclopedic YouTube

• 1 / 5

  For example, under initial conditions φ = 0 (\displaystyle \varphi =0) and the amplitude of the initial current, the solution will be reduced to:

  i (t) = I a sin ⁡ (ω t) (\displaystyle i(t)=I_(a)\sin((\omega )t))

  The solution can also be written in the form

  i (t) = I a 1 sin ⁡ (ω t) + I a 2 cos ⁡ (ω t) (\displaystyle i(t)=I_(a1)\sin((\omega )t)+I_(a2) \cos((\omega )t))

  Where I a 1 (\displaystyle I_(a1)) And I a 2 (\displaystyle I_(a2))- some constants that are related to amplitude I a (\displaystyle I_(a)) and phase φ (\displaystyle \varphi ) the following trigonometric relations:

  I a 1 = I a cos ⁡ (φ) (\displaystyle I_(a1)=I_(a)\cos ((\varphi))), I a 2 = I a sin ⁡ (φ) (\displaystyle I_(a2)=I_(a)\sin ((\varphi))).

  Complex resistance (impedance) of the oscillating circuit

  The oscillatory circuit can be considered as a two-terminal network, which is a parallel connection of a capacitor and an inductor. The complex resistance of such a two-terminal network can be written as

  z ^ (i ω) = i ω L 1 − ω 2 L C (\displaystyle (\hat (z))(i\omega)\;=(\frac (i\omega L)(1-\omega ^(2 )LC)))

  For such a two-terminal network, the so-called characteristic frequency (or resonant frequency), when the impedance of the oscillatory circuit tends to infinity (the denominator of the fraction tends to zero).

  This frequency is

  ω h = 1 L C (\displaystyle \omega _(h)=(\frac (1)(\sqrt (LC))))

  and coincides in value with the natural frequency of the oscillatory circuit.

  From this equation it follows that many circuits with different values ​​of L and C, but with the same product LC, can operate at the same frequency. However, the choice of the relationship between L and C is often not completely arbitrary, since it is determined by the required value of the quality factor of the circuit.

  For a series circuit, the quality factor increases with increasing L:

  Q = 1 R L C (\displaystyle Q=(\frac (1)(R))(\sqrt (\frac (L)(C)))), where R is the active resistance of the circuit.

  For a parallel path:

  Q = R e C L (\displaystyle Q=R_(e)(\sqrt (\frac (C)(L)))),

  Where R e = L C R L + C (\displaystyle R_(e)=(\frac (L)(CR_(L+C)))), which in a series circuit is connected in series with L and C, and in a parallel circuit - in parallel with them. Low losses (that is, high quality factor) mean that there is little loss in a series circuit, and a large loss in a parallel circuit. In a low frequency series circuit R e (\displaystyle R_(e)) easily takes on a physical meaning - it is mainly the active resistance of the coil wire and circuit conductors.

  Sub-exciter generator(the generator itself produces 400 Hz). When the frequency deviates from the nominal value, the reactance of one of the circuits becomes greater than the other, and the BRF issues a control signal to the constant-speed generator drive to correct the generator speed. If the frequency rises above the nominal one, the resistance of the second circuit will become less than that of the first, and the BRF will issue a signal to reduce the generator speed; if the frequency drops, then vice versa. This maintains a constant generator voltage frequency when engine speed changes.

  In the last article, we looked at a series oscillatory circuit, since all the radioelements participating in it were connected in series. In the same article we will look at a parallel oscillating circuit in which a coil and a capacitor are connected in parallel.

  Parallel oscillatory circuit in the diagram

  On the diagram ideal oscillating circuit looks like this:

  In reality, our coil has a decent loss resistance, since it is wound from wire, and the capacitor also has some loss resistance. Capacitance losses are very small and are usually neglected. Therefore, we leave only one coil loss resistance R. Then the circuit real oscillatory circuit will look like this:

  
  

  Where

  R is the circuit loss resistance, Ohm

  L is the inductance itself, Henry

  C – the capacitance itself, Farad

  Operation of a parallel oscillatory circuit

  Let's connect a real parallel oscillatory circuit to the frequency generator
  

  
  

  What will happen if we apply a current to the circuit with a frequency of zero Hertz, that is, direct current? It will calmly run through the coil and will be limited only by the losses R of the coil itself. No current will flow through the capacitor, because the capacitor does not allow direct current to pass through. I wrote about this in the article: capacitor in direct and alternating current circuits.

  Let's add frequency then. So, as the frequency increases, our capacitor and coil will begin to provide reactance to the electric current.

  The reactance of the coil is expressed by the formula

  

  and the capacitor according to the formula

  

  If you gradually increase the frequency, you can understand from the formulas that at the very beginning, with a smooth increase in frequency, the capacitor will have greater resistance than the inductor. At some frequency, the reactances of the coil X L and the capacitor X C will be equal. If you further increase the frequency, then the coil will already have greater resistance than the capacitor.
  

  Resonance of a parallel oscillatory circuit

  A very interesting property of a parallel oscillatory circuit is that when X L = X C our oscillatory circuit will enter resonance. At resonance, the oscillatory circuit will begin to provide greater resistance to alternating electric current. This resistance is also often called resonant resistance contour and it is expressed by the formula:

  

  Where

  Rres is the circuit resistance at the resonant frequency

  L – the actual inductance of the coil

  C is the actual capacitance of the capacitor

  R – coil loss resistance

  Resonance formula

  For a parallel oscillatory circuit, Thomson’s formula for the resonant frequency also works as for a series oscillatory circuit:

  

  Where

  F is the resonant frequency of the circuit, Hertz

  L – coil inductance, Henry

  C – capacitance of the capacitor, Farads

  How to find resonance in practice

  Okay, let's get to the point. We take the soldering iron in our hands and solder the coil and capacitor in parallel. The coil is 22 µH, and the capacitor is 1000pF.

  
  

  So, the real diagram of this circuit will be like this:

  In order to show everything clearly and clearly, let’s add a 1 KOhm resistor in series to the circuit and assemble the following circuit:

  

  
  

  We will change the frequency on the generator, and we will remove the voltage from terminals X1 and X2 and watch it on an oscilloscope.

  It is not difficult to guess that the resistance of the parallel oscillatory circuit will depend on the frequency of the generator, since in this oscillatory circuit we see two radio elements whose reactance directly depends on the frequency, so we will replace the oscillatory circuit with the equivalent resistance of the circuit R con.

  A simplified diagram would look like this:

  

  I wonder what this circuit looks like? Is it a voltage divider? Exactly! So, remember the rule of the voltage divider: at a lower resistance, a smaller voltage drops, at a higher resistance, a larger voltage drops. What conclusion can be drawn in relation to our oscillatory circuit? Yes, everything is simple: at the resonant frequency, the resistance Rcon will be maximum, as a result of which a greater voltage will “drop” at this resistance.

  Let's begin our experience. We increase the frequency on the generator, starting with the lowest frequencies.

  200 Hertz.
  

  
  

  As you can see, a small voltage “drops” on the oscillatory circuit, which means, according to the voltage divider rule, we can say that now the circuit has a low resistance R con

  Adding frequency. 11.4 Kilohertz

  
  

  As you can see, the voltage on the circuit has increased. This means that the resistance of the oscillatory circuit has increased.

  Let's add another frequency. 50 Kilohertz

  
  

  Notice that the voltage on the circuit has increased even more. This means his resistance has increased even more.

  723 Kilohertz

  
  

  Pay attention to the cost of dividing one square vertically, compared to past experience. There was 20 mV per square, and now it’s 500 mV per square. The voltage increased as the resistance of the oscillatory circuit became even greater.

  And so I caught the frequency at which the maximum voltage on the oscillating circuit was obtained. Pay attention to the vertical division price. It is equal to two Volts.

  
  

  A further increase in frequency causes the voltage to begin to drop:

  
  

  We add the frequency again and see that the voltage has become even less:

  
  

  Let's analyze the resonance frequency

  Let's take a closer look at this waveform when we had the maximum voltage from the circuit.

  What happened here?

  Since there was a voltage surge at this frequency, therefore, at this frequency the parallel oscillatory circuit had the highest resistance R con. At this frequency X L = X C. Then, with increasing frequency, the circuit resistance dropped again. This is the same resonant resistance of the circuit, which is expressed by the formula:

  Current resonance

  So, let's say we have driven our oscillatory circuit into resonance:

  

  What will the resonant current be equal to? I cut? We calculate according to Ohm's law:

  I res = U gen /R res, where R res = L/CR.

  But the funny thing is that when we resonate in the circuit, our own circuit current appears I con, which does not go beyond the contour and remains only in the contour itself! Since I have a hard time with mathematics, I will not give various mathematical calculations with derivatives and complex numbers and explain where the loop current comes from during resonance. That is why the resonance of a parallel oscillating circuit is called current resonance.

  Quality factor

  By the way, this loop current will be much greater than the current that passes through circuit. And do you know how many times? That's right, Q times. Q – this is quality factor! In a parallel oscillatory circuit, it shows how many times the current strength in the circuit I con is greater than the current strength in the common circuit I res

  Or the formula:

  

  If we also add loss resistance here, the formula will take the following form:

  

  Where

  Q – quality factor

  R – loss resistance on the coil, Ohm
  

  C – capacity, F
  

  L – inductance, H
  

  Conclusion

  Well, in conclusion, I would like to add that a parallel oscillatory circuit is used in radio receiving equipment, where it is necessary to select the frequency of a station. Also, with the help of an oscillatory circuit, we can build different ones that would highlight the frequency we need, and pass other frequencies through themselves, which is basically what we did in our experiment.

  An electrical oscillating circuit is a mandatory element of any radio receiver, regardless of its complexity. Without an oscillating circuit, receiving radio signals is generally impossible.

  The simplest electrical oscillatory circuit (Fig. 20) is a closed circuit consisting of an inductor L and capacitor C. Under certain conditions, electrical oscillations can arise and be maintained in it.

  To understand the essence of this phenomenon, first conduct several experiments with a thread pendulum (Fig. 21). Hang a ball made of plasticine or another weight weighing 20...40 g on a thread 100 cm long. Bring the pendulum out of its equilibrium position and, using a clock with a second hand, count how many complete oscillations it makes per minute. Approximately 30. Therefore, the natural frequency of oscillation of this pendulum is 0.5 Hz, and the period (time of one complete oscillation) 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 pendulum thread by half. The natural frequency of oscillation of the pendulum will increase by one and a half times and the period of oscillation will decrease by the same amount. Conclusion: as the length of the pendulum decreases, the frequency of its natural oscillations increases, and the period decreases proportionally.

  By changing the length of the pendulum suspension, ensure that its natural frequency of oscillation is 1 Hz (one complete oscillation per second). 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.

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

  The frequency of natural oscillations of the pendulum depends on its mass and the length of the suspension.

  Now stretch a thin rope or twine horizontally. Tie the same pendulum to the stretcher (Fig. 22). 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. Bring it into an oscillating motion. In this case, the first pendulum will also begin to oscillate, but with a smaller swing (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 oscillations, the first pendulum is a receiver of mechanical oscillations excited by the second pendulum, the transmitter of these oscillations. 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 pendulum.

  The natural frequency, forced oscillations and resonance that you observed in these experiments are phenomena that are also characteristic of an electric oscillatory circuit.

  Electrical vibrations in the circuit. To excite oscillations in the circuit, it is necessary to charge its capacitor from a constant voltage source, and then turn off the source and close the circuit circuit (Fig. 23). From this moment, the capacitor will begin to discharge through the inductor, creating an increasing current in the circuit circuit; and around the inductor there is a magnetic field of current. When the capacitor is completely discharged and the current in the circuit becomes zero, the magnetic field around the coil will be at its strongest - the electrical charge of the capacitor has been converted into the magnetic field of the coil. The current in the circuit will continue to flow in the same direction for some time, but due to the decreasing energy of the magnetic field accumulated by the coil, and the capacitor will begin to charge. As soon as the magnetic field of the coil disappears, the current in the circuit will stop for a moment. But by this moment the condenser-fop will be overcharged, so current will flow in the circuit circuit again, but in the opposite direction. As a result, oscillations of electric current occur in the circuit, which continue until the energy stored by the capacitor is used up to overcome the resistance of the circuit conductors.

  Electrical oscillations excited in the circuit by the capacitor charge are free and therefore damped. By charging the capacitor again, a new series of damped oscillations can be excited in the circuit.

  Connect electromagnetic headphones to the 3336L battery. At the moment the circuit is closed, a sound resembling a click will appear in the phones. The same click is heard when the phones are disconnected from the battery. Charge a paper capacitor with the largest possible capacity from this battery, and then, disconnecting the battery, connect the same telephones to it. On phones you will hear a short low-pitched sound. But when the phones are disconnected from the capacitor, there will be no such sound.

  In the first of these experiments, clicks in telephones are a consequence of single oscillations of their membranes when the strength of the magnetic fields of the coils of electromagnetic systems of telephones changes at the moments of the appearance and disappearance of current in them. In the second experiment, the sound in phones is vibrations of their membranes under the influence of alternating magnetic fields of the phone coils. They are created by a short burst of damped oscillations of very low frequency excited in. this circuit after connecting a charged capacitor.

  The natural frequency of electrical oscillations in the circuit depends on the inductance of its coil and the capacitance of the capacitor. The larger they are, the lower the frequency of oscillations in the circuit and, conversely, the smaller they are, the higher the frequency of oscillations in the circuit. By changing the inductance (number of turns) of the coil and the capacitance of the capacitor, you can vary the frequency of natural electrical oscillations in the circuit within a wide range.

  In order for the forced oscillations in the circuit to be undamped, the circuit must be replenished with additional energy in time with the oscillations in it. For the receiving circuit, the source of this energy can be high-frequency electrical oscillations induced by radio waves in the radio receiver antenna.

  

  Circuit in a radio receiver. If you connect an antenna, grounding and a circuit made up of a diode that acts as a detector and telephones to the oscillatory circuit, you will get the simplest radio receiver - a detector (Fig. 24).

  For the oscillatory circuit of such a receiver, use the inductor coil that you wound during the third workshop. Variable capacitor (G2) for smooth and. To fine-tune the circuit to the frequency of the radio station, make it from two tin plates, soldering conductors to them. Between the plates, so that they do not short, place a sheet of dry writing paper or newsprint. The larger the area of ​​mutual overlap of the plates and the smaller the distance between them, the greater the capacitance of such a capacitor. With plate sizes of 150X250 mm and a distance between them equal to the thickness of the paper, the largest capacitance of that capacitor can be 400...450 pF, which will suit you perfectly, and the smallest is several picofarads. Temporary antenna (W1) can serve as a piece of wire 10...15 m long, well insulated from the ground and from the walls of the building, suspended at a height of 10...12 m. For grounding, you can use a metal pin driven into the ground, water supply or central heating pipes, having, as generally good contact with the ground.

  Role of the detector (VI) can perform a point diode, for example, the D9 or D2 series with any letter index. IN 1— electromagnetic, high-ohm headphones (with electromagnet coils with a direct current resistance of 1500...2200 Ohms), for example, type TON-1. Connect a capacitor in parallel to the phones (NW) capacity 3300...6200 pF.

  All connections must be electrically reliable. It's better if they are soldered. Due to poor contact in any of the connections, the receiver will not work. The receiver will not work if there are short circuits or incorrect connections in its circuits.

  Tuning the receiver circuit to the frequency of the radio station is carried out: coarse - by abruptly changing the number of coil turns included in the circuit (shown in Fig. 24 by a dashed line with an arrow); smooth and accurate - by changing the capacitance of the capacitor by displacing one of its plates relative to the other. If in the city, region or region where you live there is a long-wave radio station (735.3...2000 m, which corresponds to frequencies 408...150 kHz), then include all turns of the coil in the circuit, and if the station is medium-wave (186.9...571.4 m, which corresponds to frequencies of 1.608 MHz. "525 kHz), then only part of its turns.

  If the transmissions of two radio stations are audible at the same time, connect a capacitor with a capacity of 62...82 pF between the antenna and the circuit (in Fig. 24 - capacitor C1, shown in dashed lines). This will reduce the sound volume of telephones somewhat, but the selectivity of the receiver, that is, its ability to tune out interfering stations, will improve.

  How does such a receiver work in general? Modulated high-frequency oscillations, induced in the antenna wire by radio waves from many stations, excite oscillations of different frequencies and amplitudes in the receiver circuit, which includes the antenna itself. In the circuit, the strongest oscillations will occur only at the frequency to which it is tuned into resonance. The circuit weakens oscillations of all other frequencies. The better (higher quality) the circuit is, the more clearly it identifies vibrations corresponding to vibrations of its own frequency, and the greater their amplitude.

  The detector is also an important element of the receiver. Possessing one-way conductivity of current, it rectifies high-frequency modulated oscillations coming to it from the oscillatory circuit, converting them into oscillations of low, that is, sound, frequencies, which telephones convert into sound vibrations.

  Capacitor NW, connected in parallel to the telephones, it is an auxiliary element of the receiver: by smoothing out the ripples of the current rectified by the detector, it improves the operating conditions of the telephones.

  Do some experiments.

  1. Having tuned the receiver to a radio station, insert a thick nail into the coil, and then use a variable capacitor to adjust the circuit to restore the previous volume of telephone sounds.

  2. Do the same, but instead of a nail, take a copper or brass rod.

  3. Connect to the loop coil instead of a variable capacitor a constant capacitor (select experimentally) so that the receiver is tuned to the frequency of the local station.

  Remember the final results of these experiments. By introducing a metal core inside the coil, you, of course, noticed that the natural frequency of the circuit changes: a steel core reduces the natural frequency of oscillations in the circuit, and a copper or brass core, on the contrary, increases it. This can be judged by the fact that in the first case, in order to adjust the circuit to signals from the same station, the capacity of the circuit capacitor had to be reduced, and in the second case, it had to be increased.

  Contour coil with high frequency core. The vast majority of loop coils in modern receivers have high-frequency, usually ferrite, cores in the form of rods, cups or rings. Ferrite rods, in addition, are mandatory elements of the input circuits of all transistor portable and so-called “pocket” receivers.

  

  The high-frequency core, as it were, “thickens” the magnetic field lines of the coil, increasing its inductance and quality factor. The movable core, in addition, allows you to adjust the inductance of the coil, which is used to tune circuits to a given frequency, and sometimes even tune circuits to the frequencies of radio stations. As an experiment, make a receiver with an oscillating circuit, an adjustable ferrite rod of grade 400NN or 600NN, 120...150 mm long (Fig. 25). Such rods are used for magnetic antennas of transistor receivers. From a strip of paper, wrapping it around the rod 3...4 times, glue and dry well a sleeve 80...90 mm long. The rod should fit freely inside the sleeve. Cut out 9...10 rings from cardboard and glue them to the sleeve at a distance of 6...7 mm from each other. On the resulting sectioned frame, wind 300...350 turns of PEV, PEL or PELSHO 0.2...0.25 wire, laying it 35...40 turns in each section. From the 35...40th and from the 75...80th turns, make two taps in the form of loops in order to be able to change the number of coil turns included in the circuit.

  Connect the antenna, grounding and the detector-phone circuit to the coil. The more turns of the coil are involved in the operation of the circuit and the deeper the ferrite rod is inserted inside the coil, the longer the wavelength the receiver can be tuned to.

  The detector receiver operates solely on the electromagnetic energy emitted by the radio station's transmitter antenna. That's why phones don't sound loud. To increase the volume of the detector receiver, you need to add an amplifier, for example a transistor, to it.

  Literature: Borisov V.G. Workshop for a beginner radio amateur. 2nd ed., revised. and additional - M.: DOSAAF, 1984. 144 p., ill. 55k.

  An electrical oscillatory circuit is a system for exciting and maintaining electromagnetic oscillations. In its simplest form, this is a circuit consisting of a coil with inductance L, a capacitor with capacitance C, and a resistor with resistance R connected in series (Fig. 129). When switch P is set to position 1, capacitor C is charged to voltage U T. In this case, an electric field is formed between the plates of the capacitor, the maximum energy of which is equal to

  When the switch is moved to position 2, the circuit closes and the following processes take place in it. The capacitor begins to discharge and current flows through the circuit i, the value of which increases from zero to the maximum value , and then decreases to zero again. Since an alternating current flows in the circuit, an 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. As a result of the appearance of current in the coil, a magnetic field arises, the energy of which
  reaches its maximum value at a current equal to . The maximum magnetic field energy will be equal to

  

  After reaching the maximum value, the current in the circuit will begin to decrease. In this case, the capacitor will be recharged, the energy of the magnetic field in the coil will decrease, and the energy of the electric field in the capacitor will increase. Upon reaching the maximum value. The process will begin to repeat itself and oscillations of the electric and magnetic fields will occur in the circuit. If we assume that resistance
  (i.e. energy is not spent on heating), then according to the law of conservation of energy, the total energy W remains constant

  

  And
  ;
  .

  A circuit in which there is no energy loss is called ideal. The voltage and current in the circuit vary according to the harmonic law

  ;
  

  Where - circular (cyclic) oscillation frequency
  .

  Circular frequency is related to the oscillation frequency and periods of oscillations T ratio.

  N and fig. 130 shows graphs of changes in voltage U and current I in the coil of an ideal oscillating circuit. It can be seen that the current is out of phase with the voltage by .

  ;
  ;
  - Thomson's formula.

  In the case where the resistance
  , Thomson's formula takes the form

  .

  Basics of Maxwell's theory

  Maxwell's theory is the theory of a single electromagnetic field created by an arbitrary system of charges and currents. The theory solves the main problem of electrodynamics - using a given distribution of charges and currents, the characteristics of the electric and magnetic fields they create are found. Maxwell's theory is a generalization of the most important laws describing electrical and electromagnetic phenomena - the Ostrogradsky-Gauss theorem for electric and magnetic fields, the law of total current, the law of electromagnetic induction and the theorem on the circulation of the electric field strength vector. Maxwell's theory is phenomenological in nature, i.e. it does not consider the internal mechanism of phenomena occurring in the environment and causing the appearance of electric and magnetic fields. In Maxwell's theory, the medium is described using three characteristics - dielectric ε and magnetic permeability μ of the medium and specific electrical conductivity γ.

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

  What does it consist of?

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

  The coil of the oscillating circuit creates oscillations only in the presence of stored charge. When current passes through it, it accumulates a charge, which it then releases into the circuit if the voltage drops.

  Coil wires usually have very little resistance, which always remains constant. In the oscillatory circuit circuit, changes in voltage and current very often occur. This change obeys certain mathematical laws:

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

  

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

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

  

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

  • I = U/R, where
    I - current strength,
    U - voltage,
    R - resistance.

  

  Inductor

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

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

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

  The strength of direct current does not change over time, so resistance does not arise when exposed to it.

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

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

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

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

  What is inductance?

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

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

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

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

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

  Electric capacitor

  The capacitance of the oscillating circuit is determined by the capacitor. His appearance was described above. Now let's look at the physics of the processes that take place in it.

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

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

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

  What is the capacitance of a capacitor?

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

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

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

  • C = q/U, where
    q is the charge in coulombs.

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

  Resistor

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

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

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

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

  

  How to link contour parameters?

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

  If you solve this equation, several interesting formulas follow that describe the processes occurring in the circuit. For example, cyclic frequency can be expressed in terms of capacitance and inductance.

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

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

  

  Quality factor

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

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

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

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

  What is an ideal oscillating circuit

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

  • W = (L*I 2)/2.

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

  

  How it works?

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

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

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

  Circuit design options

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

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

  Where is the oscillating circuit used?

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

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

  Conclusion

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