Basic elements of an oscillatory circuit. Resonance frequency: formula

Oscillatory circuit

an electrical circuit containing an inductor and a capacitor in which electrical oscillations can be excited. If at some point in time the capacitor is charged to voltage V 0, then the energy concentrated in the electric field of the capacitor is equal to E s = , where C is the capacitance of the capacitor. When the capacitor discharges, current will flow in the coil. I, which will increase until the capacitor is completely discharged. At this moment, the electrical energy of the coil is E c = 0, and the magnetic energy concentrated in the coil, E L = L, is the inductance of the coil, I 0 is the maximum current value. Then the current in the coil begins to fall, and the voltage across the capacitor increases in absolute value, but with the opposite sign. After some time, the current through the inductance will stop, and the capacitor will charge to voltage - V 0. The energy of the QC will again concentrate in the charged capacitor. Then the process is repeated, but with the opposite direction of the current. The voltage on the capacitor plates changes according to the law V= V 0 cos ω 0 t, a inductor current I = I 0 sin ω 0 t, i.e., natural harmonic oscillations of voltage and current are excited in the CC with a frequency ω 0 = 2 π/T 0, where T0- period of natural oscillations equal to T0= 2π

In real cosmic rays, however, some of the energy is lost. It is spent on heating the coil wires, which have active resistance, on the radiation of electromagnetic waves into the surrounding space and losses in dielectrics (see Dielectric losses) , which leads to damping of oscillations. The amplitude of oscillations gradually decreases, so that the voltage on the capacitor plates changes according to the law: V = V 0 e -δt cosω t, where coefficient δ = R/2L - attenuation indicator (coefficient), and ω = - frequency of damped oscillations. Thus, losses lead to a change not only in the amplitude of oscillations, but also in their period T = 2π/ω. The quality of a capacitor is usually characterized by its quality factor Q, which determines the number of oscillations that a capacitor will perform after charging its capacitor once, before the amplitude of the oscillations decreases by e once ( e- the base of natural logarithms).

If you include a generator with a variable emf in the KK: U = U 0 cosΩ t(), then a complex oscillation will arise in the QC, which is the sum of its own oscillations with a frequency ω 0 and forced oscillations with a frequency Ω. Some time after turning on the generator, the natural oscillations in the circuit will die out and only forced ones will remain. The amplitude of these stationary forced oscillations is determined by the relation

That is, it depends not only on the amplitude of the external emf U0, but also on its frequency Ω. Dependence of the amplitude of oscillations in K. k.

on the frequency of the external emf is called the resonant characteristic of the circuit. A sharp increase in amplitude occurs at values of Ω close to the natural frequency ω 0 K.c. At Ω = ω 0 the amplitude of oscillations V makc is Q times greater than the amplitude of the external emf U. Since usually 10 Q 100, the QC makes it possible to select from the set of oscillations those whose frequencies are close to ω 0. It is this property (selectivity) of CC that is used in practice. The region (band) of frequencies ΔΩ near ω 0, within which the amplitude of oscillations in a QC changes little, depends on its quality factor Q. Numerically, Q is equal to the ratio of the frequency ω 0 of natural oscillations to the frequency bandwidth ΔΩ.

To increase the selectivity of the Q-factor, it is necessary to increase Q. However, an increase in the quality factor is accompanied by an increase in the time it takes to establish oscillations in the Q-box. Changes in the amplitude of oscillations in a circuit with a high quality factor do not have time to follow rapid changes in the amplitude of the external emf. The requirement for high selectivity of CC contradicts the requirement for the transmission of rapidly changing signals. Therefore, for example, in television signal amplifiers, the quality factor of the QCs is artificially reduced. Circuits with two or more interconnected QCs are often used. Such systems, with properly selected connections, have an almost rectangular resonance curve (dotted line).

In addition to the described linear coefficients with constants L and C, nonlinear QKs are used, the parameters of which L or C depend on the amplitude of oscillations. For example, if an iron core is inserted into the inductance coil of a coil, then the magnetization of the iron, and with it the inductance L coil changes with the change in current flowing through it. The period of oscillation in such a cosmic ring depends on the amplitude, so the resonance curve acquires a slope, and at large amplitudes it becomes ambiguous (). In the latter case, amplitude jumps occur with a smooth change in the frequency Ω of the external emf. Nonlinear effects are more pronounced, the lower the losses in a resonant circuit. In a resonant circuit with a low quality factor, nonlinearity does not affect the character of the resonance curve at all.

Lit.: Strelkov S.P.. Introduction to the theory of oscillations, M. - L., 1951.

V. N. Parygin.

Rice. 2. Oscillatory circuit with a source of variable emf U=U 0 cos Ωt.

Rice. 3. Resonance curve of the oscillatory circuit: ω 0 - frequency of natural oscillations; Ω - frequency of forced oscillations; ΔΩ - frequency band near ω 0, at the boundaries of which the amplitude of oscillations V = 0,7 V makc. The dotted line is the resonance curve of two connected circuits.

Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .

An oscillatory circuit is a simple electrical circuit consisting of an inductor and a capacitor. In such a circuit, fluctuations in current or voltage may occur. The resonant frequency of such oscillations is determined by Thomson's formula.

This type of LC oscillatory circuit (OC) is the simplest example of a resonant oscillatory circuit. Consists of a series-connected inductor and capacitor. When alternating current flows through such a circuit, its value is determined by: I = U / X Σ, Where X Σ- the sum of the reactances of the inductor and capacitance.

Let me remind you that the reactance of capacitance and inductance depends on the voltage frequency; their formulas look like this:

It is clearly seen from the formulas that as the frequency increases, the inductance reactance increases. Unlike a coil, a capacitor's reactance decreases as the frequency increases. The figure below shows the graphical dependences of the reactance of the inductor X L and containers X C from cyclic frequency omega ω , and the dependence graph ω from their algebraic sum X Σ. The graph shows the frequency dependence of the total reactance of a series oscillating circuit consisting of a capacitor and inductance.

The graph clearly shows that at a certain frequency ω=ω р, the reactances of inductance and capacitance are the same in value, but opposite in sign, and the total resistance of the circuit is zero. At this frequency, the maximum possible current will flow in the circuit, limited only by ohmic losses in the inductance (i.e., the active resistance of the coil) and the internal active resistance of the current source. This frequency at which this phenomenon occurs is called the resonance frequency. In addition, the following conclusion can be drawn from the graph: at frequencies below the resonant frequency, the reactance of a series CC has a capacitive factor, and at higher frequencies it is inductive in nature. The resonant frequency can be found using Thomson's formula, which is easily derived from the formulas of the reactances of both components of the CC, equating their reactances:

In the figure below, we display the equivalent circuit of a series resonant circuit taking into account active ohmic losses R, with an ideal current source of harmonic voltage with a certain amplitude U. The impedance, or also called the impedance of the circuit, is calculated: Z = √(R 2 +X Σ 2), Where X Σ = ω L-1/ωC. At the resonance frequency, when both reactances XL = ωL And X C = 1/ωС equal in modulus, X Σ tends to zero and is only active in nature, and the current in the circuit is calculated by the ratio of the voltage amplitude of the current source to the loss resistance according to Ohm’s law: I=U/R. In this case, the same voltage value drops on the coil and the container, in which there is a supply of reactive components of energy, i.e. U L = U C = IX L = IX C.

At any frequency except the resonant one, the voltages on the inductance and capacitance are different - they depend on the amplitude of the current in the circuit and the ratings of the reactance modules X L And X C Therefore, resonance in a series oscillatory circuit is called voltage resonance.

Very important characteristics of the CC are also its wave impedance ρ and quality factor QC Q. Wave impedance ρ calculate the reactance value of both components (L,C) at the resonant frequency: ρ = X L = X C at ω =ω р. Characteristic impedance can be calculated using the following formula: ρ = √(L/C). Characteristic impedance ρ considered a quantitative measure of the energy stored by the reactive components of a circuit - W L = (LI 2)/2 And W C =(CU 2)/2. The ratio of the energy stored by the reactive elements of the CC to the energy of resistive losses over a period is called the quality factor Q KK. Quality factor of the oscillatory circuit- a quantity that determines the amplitude and width of the amplitude-frequency characteristic of the resonance and indicates how many times the stored energy in the spacecraft is greater than the energy loss over a single period of oscillation. The quality factor also takes into account active resistance R. For a series QC in RLC circuits, in which all three passive components are connected in series, the quality factor is calculated by the expression:

Where R, L And C- resistance, inductance and capacitance of the resonant circuit KK.

The reciprocal of the quality factor d = 1/Q physicists called it KK damping. To determine the quality factor, the expression is usually used Q = ρ/R, Where R- resistance of ohmic losses of the CC, characterizing the power of active losses of the CC P = I 2 R. The quality factor of most oscillatory circuits varies from several units to hundreds and higher. The quality factor of such oscillatory systems as piezoelectric or can be several thousand or even more.

The frequency properties of CC are usually assessed using the frequency response, while the circuits themselves are considered as four-terminal networks. The figures below show elementary quadripole networks containing sequential CC and frequency response of these circuits. The X-axis of the graphs shows the circuit's voltage transfer coefficient K, or the ratio of the output voltage to the input.

For passive circuits (without amplifying elements and energy sources), the value TO never higher than one. AC resistance will be minimal at the resonant frequency. Then the transmission coefficient tends to unity. At frequencies other than resonant, the AC resistance to alternating current is high and the transmission coefficient will be close to zero.

At resonance, the input signal source is practically short-circuited by the low resistance KK, so the transmission coefficient drops to almost zero. On the contrary, at input frequencies farther from the resonant one, the coefficient tends to unity. The property of CC to change the transmission coefficient at frequencies near resonant ones is widely used in amateur radio practice, when it is necessary to select a signal with the required frequency from many similar ones, but at different frequencies. So, in any radio receiver, using CC, tuning is performed to the frequency of the required radio station. The property of selecting only one from many frequencies is called selectivity. In this case, the intensity of the change in the transmission coefficient when adjusting the frequency of the influence from the resonance is described by the passband. It is taken to be the frequency range in which the decrease (increase) in the transmission coefficient relative to its value at the resonant frequency is not higher than 0.7 (dB).

The dotted lines in the figures indicate the frequency response of similar circuits, the CCs of which have the same resonances, but have a lower quality factor. As we can see from the graphs, the bandwidth increases and its selectivity decreases.

In this circuit, two reactive elements with different levels of reactivity are connected in parallel. The figure below shows the graphical dependences of the reactive conductivities of inductance B L = 1/ωL and capacitance of the capacitor B C = -ωC, as well as general conductivity In Σ. And in this oscillatory circuit, there is a resonant frequency at which the reactances of both components are the same. This suggests that at this frequency the parallel CC has enormous resistance to alternating current.

The resistance of a real parallel CC (with losses), of course, does not tend to infinity - it is lower, the higher the ohmic resistance of losses in the circuit, i.e. it decreases in direct proportion to the decrease in the quality factor.

Let's consider the simplest circuit consisting of a source of harmonic oscillations and a parallel CC. If the natural frequency of the generator (voltage source) coincides with the resonant frequency of the circuit, then the inductive and capacitive branches have the same resistance to alternating current, and the currents in the branches will be exactly the same. Therefore, we can confidently say that in this scheme there is current resonance. The reactivity of both components quite successfully compensate each other, and the CC's resistance to the flowing current becomes fully active (has only a resistive component). The value of this resistance is calculated by multiplying the quality factor of the QC and the characteristic resistance R eq = Q ρ. At other frequencies, the resistance of the parallel CC falls and becomes reactive at lower frequencies, inductive, and at higher frequencies, capacitive.

Let's consider the dependence of the transmission coefficients of four-terminal networks on frequency in this case.

A four-terminal network at the resonance frequency represents a fairly large resistance to the flowing alternating current, therefore, when ω=ω р its transmission coefficient tends to zero (and this even taking into account real ohmic losses). At other frequencies other than the resonant one, the resistance of the CC will fall, and the transmission coefficient of the quadripole will increase. For the two-terminal network of the second option, the situation will be diametrically opposite - at the resonant frequency the CC will have a very large resistance, i.e. the transmission coefficient will be maximum and tends to unity). If the frequency differs significantly from the resonant one, the signal source will be practically shunted, and the transmission coefficient will tend to zero.

Suppose we need to manufacture a parallel CC with a resonance frequency of 1 MHz. Let us carry out a preliminary simplified calculation of such a QC. That is, we calculate the required values of capacitance and inductance. Let's use a simplified formula:

L=(159.1/F) 2 / C where:

L coil inductance in µH; WITH capacitor capacity in pF; F resonant frequency in MHz

Let's set a frequency of 1 MHz and a capacity of 1000 pF. We get:

L=(159.1/1) 2 /1000 = 25 µH

Thus, if our homemade amateur radio uses CC at a frequency of 1 MHz, then we need to take a capacitance of 1000 pF and an inductance of 25 μH. The capacitor is quite easy to select, but IMHO it’s easier to make the inductor yourself.

To do this, calculate the number of turns for a coil without a core

N=32 *v(L/D) Where:

N required number of turns; L specified inductance in µH; D is the diameter of the coil frame.

Suppose the frame diameter is 5 mm, then:

N=32*v(25/5) = 72 turns

This formula is considered approximate; it does not take into account the own interturn inductance capacitance. The formula serves to preliminary calculate the coil parameters, which are then adjusted when adjusting the circuit in the device.

In amateur radio practice, coils with a tuning core made of ferrite, having a length of 12-14 mm and a diameter of 2.5 - 3 mm, are very often used. Such cores are actively used in oscillatory circuits of receivers.

Practical calculation of series or parallel LC circuit.

Good afternoon, dear radio amateurs!
Today we will look at procedure for calculating the LC circuit.

Some of you may ask, why the hell do we need this? Well, firstly, extra knowledge never hurts, and secondly, there are times in life when you may need knowledge of these calculations. For example, many beginning radio amateurs (naturally, mostly young) are keen on assembling so-called “bugs” - devices that allow you to listen to something from a distance. Of course, I am sure that this is done without any bad (even dirty) thoughts of eavesdropping on someone, but for good purposes. For example, they install a “bug” in the room with the baby, and listen to the broadcast receiver to see if he has woken up. All “radio bug” schemes operate at a certain frequency, but what to do when this frequency does not suit you. This is where knowledge of the article below will come to your aid.

LC oscillating circuits are used in almost any equipment operating at radio frequencies. As you know from a physics course, an oscillatory circuit consists of an inductor and a capacitor (capacitance), which can be connected in parallel ( parallel circuit) or sequentially ( series circuit), as in Fig. 1:

The reactances of inductance and capacitance are known to depend on the frequency of alternating current. As frequency increases, inductance reactance increases and capacitance reactance decreases. As the frequency decreases, on the contrary, the inductive reactance decreases and the capacitive reactance increases. Thus, for each circuit there is a certain resonance frequency at which the inductive and capacitive reactances are equal. At the moment of resonance, the amplitude of the alternating voltage on a parallel circuit sharply increases or the amplitude of the current on a series circuit sharply increases. Figure 2 shows a graph of voltage on a parallel circuit or current on a series circuit versus frequency:

At the resonance frequency these quantities have their maximum value. And the bandwidth of the circuit is determined at the level of 0.7 of the maximum amplitude that exists at the resonance frequency.

Now let's move on to practice. Suppose we need to make a parallel circuit that has a resonance at a frequency of 1 MHz. First of all, you need to make a preliminary calculation of such a circuit. That is, determine the required capacitance of the capacitor and the inductance of the coil. For preliminary calculation there is a simplified formula:

L=(159.1/F) 2/C Where:
L– coil inductance in µH;
WITH– capacitance of the capacitor in pF;
F– frequency in MHz

Let's set a frequency of 1 MHz and a capacitance of, for example, 1000 pF. We get:

L=(159.1/1) 2 /1000 = 25 µH

Thus, if we want a circuit with a frequency of 1 MHz, then we need a 1000 pF capacitor and a 25 μH inductor. You can select a capacitor, but you need to make the inductance yourself.

N=32 *√(L/D) Where:
N– required number of turns;
L– specified inductance in µH;
D– diameter of the frame in mm on which the coil is supposed to be wound.

Suppose the frame diameter is 5 mm, then:

N=32*√(25/5) = 72 turns.

This formula is approximate; it does not take into account the coil’s own interturn capacitance. The formula serves to pre-calculate the coil parameters, which are then adjusted when tuning the circuit.

In amateur radio practice, coils with tuning ferrite cores having a length of 12-14 mm and a diameter of 2.5 - 3 mm are more often used. Such cores, for example, are used in the circuits of televisions and receivers. To preliminary calculate the number of turns for such a core, there is another approximate formula:

N=8.5*√L, substitute the values for our contour N=8.5*√25 = 43 turns. That is, in this case, you will not need to wind 43 turns of wire onto the coil.

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 is the actual inductance of the coil

C is the actual capacitance of the capacitor

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 the capacitor is calculated by 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 either from the capacitor or from 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 oscillatory 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.

Today we are interested in the simplest oscillatory circuit, its working principle and application.

For useful information on other topics, go to our telegram channel.

Oscillations– a process that repeats over time and is characterized by a change in the parameters of the system around the equilibrium point.

The first thing that comes to mind is the mechanical vibrations of a mathematical or spring pendulum. But vibrations can also be electromagnetic.

A-priory oscillatory circuit(or is an electrical circuit in which free electromagnetic oscillations occur.

Such a circuit is an electrical circuit consisting of an inductance coil L and a capacitor with a capacity C . These two elements can be connected in only two ways - in series and in parallel. Let us show in the figure below an image and a diagram of a simple oscillatory circuit.

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Operating principle of the oscillatory circuit

Let's look at an example where we first charge the capacitor and complete the circuit. After this, a sinusoidal electric current begins to flow in the circuit. The capacitor is discharged through the coil. In a coil, when current flows through it, a Self-induced emf, directed in the direction opposite to the capacitor current.

Having completely discharged, the capacitor thanks to the energy EMF the coil, which at this moment will be maximum, will begin to charge again, but only in reverse polarity.

Oscillations that occur in the circuit - free damped oscillations. That is Without additional energy supply, oscillations in any real oscillatory circuit will sooner or later stop, like any oscillations in nature.

This is due to the fact that the circuit consists of real materials (capacitor, coil, wires) that have such a property as electrical resistance, and energy losses in a real oscillating circuit are inevitable. Otherwise, this simple device could become a perpetual motion machine, the existence of which, as we know, is impossible.

Another important characteristic is quality factor Q . The quality factor determines the amplitude of the resonance and shows how many times the energy reserves in the circuit exceed the energy losses during one oscillation period. The higher the quality factor of the system, the slower the oscillations will decay.

LC circuit resonance

Electromagnetic oscillations occur at a certain frequency, which is called resonant. Read more about it in our separate article. The oscillation frequency can be changed by varying circuit parameters such as capacitor capacity C , coil inductance L , resistor resistance R (For LCR circuit).

Application of an oscillating circuit

The oscillatory circuit is widely used in practice. Frequency filters are built on its basis; not a single radio receiver or signal generator of a certain frequency can do without it.

If you don’t know how to approach calculating an LC circuit or you have absolutely no time for it, contact a professional student service. High-quality and fast help in solving any problems will not keep you waiting!