Resonance in series and parallel LC circuit. Parallel oscillatory circuit
In this 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 alternating (harmonic) voltage, an alternating current will flow through the coil and capacitor, the value of which is calculated according to Ohm’s law:I = U / X Σ, Where X Σ— the sum of the reactances of a series-connected coil and capacitor (the sum module is used).
To refresh your memory, let's remember how the reactance of a capacitor and inductor depends on the frequency of the applied alternating voltage. For an inductor, this dependence will look like:
The formula shows that as the frequency increases, the reactance of the inductor increases. For a capacitor, the dependence of its reactance on frequency will look like this:
Unlike inductance, with a capacitor everything happens the other way around - as the frequency increases, the reactance decreases. The following figure graphically shows the dependences of the coil reactances X L and capacitor X C from cyclic (circular) frequency ω , as well as a graph of frequency dependence ω their algebraic sum X Σ. The graph essentially shows the frequency dependence of the total reactance of a series oscillating circuit.
The graph shows that at a certain frequency ω=ω р, at which the reactances of the coil and capacitor are equal in magnitude (equal in value, but opposite in sign), the total resistance of the circuit becomes zero. At this frequency, a maximum current is observed in the circuit, which is limited only by ohmic losses in the inductor (i.e., the active resistance of the coil winding wire) and the internal resistance of the current source (generator). The frequency at which the considered phenomenon, 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 X L And X C Therefore, resonance in a series oscillatory circuit is usually called voltage resonance. The resonant frequency of the circuit is the frequency at which the resistance of the circuit is purely active (resistive) in nature. The resonance condition is the equality of the reactance values of the inductor and capacitance.
One of the most important parameters of an oscillatory circuit (except, of course, the resonant frequency) is its characteristic (or wave) impedance ρ and circuit quality factor Q. Characteristic (wave) impedance of the circuit ρ is the value of the reactance of the capacitance and inductance of the circuit at the resonant frequency: ρ = X L = X C at ω =ω р. The characteristic impedance can be calculated as follows: ρ = √(L/C). Characteristic impedance ρ is a quantitative measure of the energy stored by the reactive elements of the circuit - the coil (magnetic field energy) W L = (LI 2)/2 and a capacitor (electric field energy) W C =(CU 2)/2. The ratio of the energy stored by the reactive elements of the circuit to the energy of ohmic (resistive) losses over a period is usually called the quality factor Q contour, which literally means “quality” in English.
Quality factor of the oscillatory circuit- a characteristic that determines the amplitude and width of the frequency response of the resonance and shows how many times the energy reserves in the circuit are greater than the energy losses during one oscillation period. The quality factor takes into account the presence of active load resistance R.
For a series oscillatory circuit in RLC circuits, in which all three elements are connected in series, the quality factor is calculated:
Where R, L And C
The reciprocal of the quality factor d = 1/Q called circuit attenuation. To determine the quality factor, the formula is usually used Q = ρ/R, Where R- resistance of the ohmic losses of the circuit, characterizing the power of the resistive (active losses) of the circuit P = I 2 R. The quality factor of real oscillatory circuits made on discrete inductors and capacitors ranges from several units to hundreds or more. The quality factor of various oscillatory systems built on the principle of piezoelectric and other effects (for example, quartz resonators) can reach several thousand or more.
It is customary to evaluate the frequency properties of various circuits in technology using amplitude-frequency characteristics (AFC), while the circuits themselves are considered as four-terminal networks. The figures below show two simple two-port networks containing a series oscillatory circuit and the frequency response of these circuits, which are shown (shown by solid lines). The vertical axis of the frequency response graphs shows the value of the circuit's voltage transfer coefficient K, showing the ratio of the circuit's output voltage to the input.
For passive circuits (i.e., those not containing amplifying elements and energy sources), the value TO never exceeds one. The alternating current resistance of the circuit shown in the figure will be minimal at an exposure frequency equal to the resonant frequency of the circuit. In this case, the circuit transmission coefficient is close to unity (determined by ohmic losses in the circuit). At frequencies very different from the resonant one, the resistance of the circuit to alternating current is quite high, and therefore the transmission coefficient of the circuit will drop to almost zero.
When there is resonance in this circuit, the input signal source is actually short-circuited by a small circuit resistance, due to which the transmission coefficient of such a circuit at the resonant frequency drops to almost zero (again due to the presence of finite loss resistance). On the contrary, at input frequencies significantly distant from the resonant one, the circuit transmission coefficient turns out to be close to unity. The property of an oscillatory circuit to significantly change the transmission coefficient at frequencies close to the resonant one is widely used in practice when it is necessary to isolate a signal with a specific frequency from many unnecessary signals located at other frequencies. Thus, in any radio receiver, tuning to the frequency of the desired radio station is ensured using oscillatory circuits. The property of an oscillatory circuit to select one from many frequencies is usually called selectivity or selectivity. In this case, the intensity of the change in the transmission coefficient of the circuit when the frequency of influence is detuned from resonance is usually assessed using a parameter called the passband. The passband is taken to be the frequency range within which the decrease (or increase, depending on the type of circuit) of the transmission coefficient relative to its value at the resonant frequency does not exceed 0.7 (3 dB).
The dotted lines in the graphs show the frequency response of exactly the same circuits, the oscillatory circuits of which have the same resonant frequencies as for the case discussed above, but have a lower quality factor (for example, the inductor is wound with a wire that has a high resistance to direct current). As can be seen from the figures, this expands the bandwidth of the circuit and deteriorates its selective properties. Based on this, when calculating and designing oscillatory circuits, one must strive to increase their quality factor. However, in some cases, the quality factor of the circuit, on the contrary, has to be underestimated (for example, by including a small resistor in series with the inductor), which avoids distortion of broadband signals. Although, if in practice it is necessary to isolate a sufficiently broadband signal, selective circuits, as a rule, are built not on single oscillatory circuits, but on more complex coupled (multi-circuit) oscillatory systems, incl. multi-section filters.
Parallel oscillatory circuit
In various radio engineering devices, along with serial oscillatory circuits, parallel oscillatory circuits are often used (even more often than serial ones). The figure shows a schematic diagram of a parallel oscillatory circuit. Here, two reactive elements with different reactivity patterns are connected in parallel. As is known, when elements are connected in parallel, you cannot add their resistances - you can only add their conductivities. The figure shows graphical dependences of the reactive conductivities of the inductor B L = 1/ωL, capacitor B C = -ωC, as well as total conductivity In Σ, these two elements, which is the reactive conductivity of a parallel oscillatory circuit. Similarly, as for a series 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.
Oscillatory circuit is called ideal if it consists of a coil and a capacitance and there is no loss resistance.
Consider the physical processes in the following chain:
1 The key is in position 1. The capacitor begins to charge from the voltage source and the energy of the electric field accumulates in it,
i.e. the capacitor becomes a source of electrical energy.
2. Key in position 2. The capacitor will begin to discharge. The electrical energy stored in the capacitor is converted into the energy of the magnetic field of the coil.
The current in the circuit reaches its maximum value (point 1). The voltage across the capacitor plates decreases to zero.
In the period from point 1 to point 2, the current in the circuit decreases to zero, but as soon as it begins to decrease, the magnetic field of the coil decreases and a self-inductive emf is induced in the coil, which counteracts the decrease in current, so it decreases to zero not abruptly, but smoothly. Since self-induction emf occurs, the coil becomes a source of energy. From this EMF, the capacitor begins to charge, but with reverse polarity (the capacitor voltage is negative) (at point 2 the capacitor is charged again).
Conclusion: in an LC circuit there is a continuous oscillation of energy between the electric and magnetic fields, therefore such a circuit is called an oscillatory circuit.
The resulting oscillations are called free or own, since they occur without the help of an external source of electrical energy previously introduced into the circuit (into the electric field of the capacitor). Since the capacitance and inductance are ideal (there is no loss resistance) and energy does not leave the circuit, the amplitude of the oscillations does not change over time and the oscillations will undamped.
Let us determine the angular frequency of free oscillations:
We use the equality of energies of electric and magnetic fields
Where ώ is the angular frequency of free oscillations.
[ ώ ]=1/s
f0= ώ /2π [Hz].
Free oscillation period Т0=1/f.
The frequency of free vibrations is called the frequency of natural vibrations of the circuit.
From the expression: ώ²LC=1 we get ώL=1/Cώ, therefore, with a current in a circuit with a frequency of free oscillations, the inductive reactance is equal to the capacitive reactance.
Characteristic resistances.
Inductive or capacitive reactance in an oscillatory circuit at a free oscillation frequency is called characteristic resistance.
Characteristic resistance is calculated using the formulas:
5.2 Real oscillatory circuit
A real oscillatory circuit has active resistance, therefore, when exposed to free oscillations in the circuit, the energy of a pre-charged capacitor is gradually spent, converted into heat.
Free oscillations in the circuit are damped, since in each period the energy decreases and the amplitude of oscillations in each period will decrease.
The picture is a real oscillatory circuit.
Angular frequency of free oscillations in a real oscillatory circuit:
If R=2..., then the angular frequency is zero, therefore free oscillations will not occur in the circuit.
Thus oscillatory circuit is an electrical circuit consisting of inductance and capacitance and having a low active resistance, less than twice the characteristic resistance, which ensures the exchange of energy between the inductance and capacitance.
In a real oscillatory circuit, free oscillations decay faster, the greater the active resistance.
To characterize the intensity of attenuation of free oscillations, the concept of “circuit attenuation” is used - the ratio of active resistance to characteristic resistance.
In practice, the reciprocal of attenuation is used - the quality factor of the circuit.
To obtain undamped oscillations in a real oscillatory circuit, it is necessary to replenish electrical energy at the active resistance of the circuit during each period of oscillation in time with the frequency of natural oscillations. This is done using a generator.
If you connect an oscillating circuit to an alternating current generator, the frequency of which differs from the frequency of free oscillations of the circuit, then a current flows in the circuit with a frequency equal to the frequency of the generator voltage. These oscillations are called forced.
If the frequency of the generator differs from the natural frequency of the circuit, then such an oscillatory circuit is untuned relative to the frequency of the external influence, but if the frequencies coincide, then it is tuned.
Task: Determine the inductance, angular frequency of the circuit, characteristic resistance, if the capacitance of the oscillatory circuit is 100 pF, the frequency of free oscillations is 1.59 MHz.
Solution:
Test tasks:
Lesson 8: VOLTAGE RESONANCE
Voltage resonance is the phenomenon of increasing voltages on reactive elements exceeding the voltage at the circuit terminals at a maximum current in the circuit, which is in phase with the input voltage.
Conditions for the occurrence of resonance:
Serial connection of L&C with alternator;
The frequency of the generator must be equal to the frequency of natural oscillations of the circuit, while the characteristic resistances are equal;
The resistance should be less than 2ρ, since only in this case free oscillations will occur in the circuit, supported by an external source.
Circuit impedance:
since the characteristic resistances are equal. Consequently, during resonance, the circuit is purely active in nature, which means that the input voltage and current are in phase at the moment of resonance. The current reaches its maximum value.
At the maximum current value, the voltage in sections L and C will be large and equal to each other.
Circuit terminal voltage:
Consider the following relationships:
, hence
Q – circuit quality factor - at voltage resonance, it shows how many times the voltage on the reactive elements is greater than the input voltage of the generator feeding the circuit. At resonance, the transmission coefficient of the series oscillating circuit
resonance.
Example:
Uc=Ul=QU=100V,
that is, the voltage at the terminals is less than the voltage at the capacitance and inductance. This phenomenon is called voltage resonance
At resonance, the transmission coefficient is equal to the quality factor.
Let's build a vector voltage diagram
The voltage across the capacitance is equal to the voltage across the inductance, therefore the voltage across the resistance is equal to the voltage at the terminals and is in phase with the current.
Let's consider the energy process in an oscillatory circuit:
In the circuit there is an exchange of energy between the electric field of the capacitor and the magnetic field of the coil. The coil energy does not return to the generator. The amount of energy supplied to the circuit from the generator is the amount of energy that is spent on the resistor. This is necessary so that undamped oscillations are observed in the circuit. The power in the circuit is only active.
Let's prove this mathematically:
, the total power of the circuit, which is equal to the active power.
Reactive power.
8.1 Resonant frequency. Upset.
Lώ=l/ώC, hence
, angular resonant frequency.
From the formula it is clear that resonance occurs if the frequency of the supply generator is equal to the natural oscillations of the circuit.
When working with an oscillatory circuit, it is necessary to know whether the frequency of the generator and the frequency of natural oscillations of the circuit coincide. If the frequencies match, then the circuit remains tuned to resonance; if they do not match, then there is a detuning in the circuit.
There are three ways to tune the oscillating circuit to resonance:
1 Change the frequency of the generator, with values of capacitance and inductance const, that is, by changing the frequency of the generator we adjust this frequency to the frequency of the oscillatory circuit
2 Change the inductance of the coil, at supply frequency and capacitance const;
3 Change the capacitance of the capacitor, at supply frequency and inductance const.
In the second and third methods, by changing the natural frequency of the circuit, we adjust it to the frequency of the generator.
When the circuit is not tuned, the frequency of the generator and the circuit are not equal, that is, there is detuning.
Detuning is a frequency deviation from the resonant frequency.
There are three types of derangement:
Absolute – the difference between a given frequency and the resonant one
Generalized - the ratio of reactance to active resistance:
Relative – the ratio of absolute detuning to resonant frequency:
At resonance, all detunings are zero , if the generator frequency is less than the circuit frequency, then the detuning is considered negative,
If more - positive.
Thus, the quality factor characterizes the quality of the circuit, and the generalized detuning characterizes the distance from the resonant frequency.
8.2 Building dependencies X, X L , X C from f.
Tasks:
Circuit resistance 15 Ohms, inductance 636 μH, Capacitance 600 pF, supply voltage 1.8 V. Find the natural frequency of the circuit, circuit attenuation, characteristic resistance, current, active power, quality factor, voltage at the circuit terminals.
Solution:
The voltage at the generator terminals is 1 V, the supply frequency is 1 MHz, the quality factor is 100, the capacitance is 100 pF. Find: attenuation, characteristic resistance, active resistance, inductance, circuit frequency, current, power, voltage across capacitance and inductance.
Solution:
Test tasks:
Lesson topic 9 : Input and transfer frequency response and phase response of a series oscillatory circuit.
9.1 Input frequency response and phase response.
In a series oscillating circuit:
R – active resistance;
X – reactance.
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 QKs 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
.
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 is 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 is 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 1000 pF.
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 oscillating 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 cool 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 is the 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, using an oscillatory circuit, it is possible to construct different ones that would highlight the frequency we need, and pass other frequencies through themselves, which is basically what we did in our experiment.
The main device that determines the operating frequency of any alternating current generator is the oscillating circuit. The oscillatory circuit (Fig. 1) consists of an inductor L(consider the ideal case when the coil has no ohmic resistance) and a capacitor C and is called closed. The characteristic of a coil is inductance, it is designated L and measured in Henry (H), the capacitor is characterized by capacitance C, which is measured in farads (F).
Let at the initial moment of time the capacitor be charged in such a way (Fig. 1) that on one of its plates there is a charge + Q 0, and on the other - charge - Q 0 . In this case, an electric field with energy is formed between the plates of the capacitor
where is the amplitude (maximum) voltage or potential difference across the capacitor plates.
After closing the circuit, the capacitor begins to discharge and an electric current flows through the circuit (Fig. 2), the value of which increases from zero to the maximum value. Since a current of variable magnitude flows in the circuit, a self-inductive emf is induced in the coil, which prevents the capacitor from discharging. Therefore, the process of discharging the capacitor does not occur instantly, but gradually. At each moment of time, the potential difference across the capacitor plates
(where is the charge of the capacitor at a given time) is equal to the potential difference across the coil, i.e. equal to the self-induction emf
 |
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| Fig.1 | Fig.2 |
When the capacitor is completely discharged and , the current in the coil will reach its maximum value (Fig. 3). The induction of the magnetic field of the coil at this moment is also maximum, and the energy of the magnetic field will be equal to
Then the current begins to decrease, and the charge will accumulate on the capacitor plates (Fig. 4). When the current decreases to zero, the capacitor charge reaches its maximum value Q 0, but the plate, previously positively charged, will now be negatively charged (Fig. 5). Then the capacitor begins to discharge again, and the current in the circuit flows in the opposite direction.
So the process of charge flowing from one capacitor plate to another through the inductor is repeated again and again. They say that in the circuit there are electromagnetic vibrations. This process is associated not only with fluctuations in the amount of charge and voltage on the capacitor, the current strength in the coil, but also with the transfer of energy from the electric field to the magnetic field and vice versa.
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| Fig.3 | Fig.4 |
Recharging the capacitor to the maximum voltage will occur only if there is no energy loss in the oscillating circuit. Such a contour is called ideal.
In real circuits the following energy losses occur:
1) heat losses, because R ¹ 0;
2) losses in the dielectric of the capacitor;
3) hysteresis losses in the coil core;
4) radiation losses, etc. If we neglect these energy losses, then we can write that, i.e.
Oscillations occurring in an ideal oscillatory circuit in which this condition is met are called free, or own, circuit vibrations.
In this case the voltage U(and charge Q) on the capacitor changes according to the harmonic law:
where n is the natural frequency of the oscillatory circuit, w 0 = 2pn is the natural (circular) frequency of the oscillatory circuit. The frequency of electromagnetic oscillations in the circuit is defined as
Period T- the time during which one complete oscillation of the voltage on the capacitor and the current in the circuit occurs is determined Thomson's formula
The current strength in the circuit also changes according to the harmonic law, but lags behind the voltage in phase by . Therefore, the dependence of the current in the circuit on time will have the form
. (9)
Figure 6 shows graphs of voltage changes U on the capacitor and current I in the coil for an ideal oscillating circuit.
In a real circuit, the energy will decrease with each oscillation. The amplitudes of the voltage on the capacitor and the current in the circuit will decrease; such oscillations are called damped. They cannot be used in master oscillators, because The device will work at best in pulse mode.
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| Fig.5 | Fig.6 |
To obtain undamped oscillations, it is necessary to compensate for energy losses at a wide variety of operating frequencies of devices, including those used in medicine.
