The phenomenon of resonance in an electrical circuit. VI. Resonance phenomena in electrical circuits Voltage resonance. Thus, a comparative conclusion can be drawn

The phenomenon of resonance. An electrical circuit containing inductance and capacitance can serve as an oscillatory circuit, where the process of oscillations of electrical energy occurs, passing from inductance to capacitance and back. In an ideal oscillatory circuit, these oscillations will be undamped. When connecting an oscillating circuit to a source alternating current angular frequency of the source? may be equal to the angular frequency? 0, from which electrical energy oscillates in the circuit. In this case, does the phenomenon of resonance occur, i.e. does the frequency of free vibrations coincide? 0 arising in any physical system, with a frequency forced oscillations?, communicated to this system by external forces.

Resonance in an electrical circuit can be achieved in three ways: by changing the angular frequency? alternating current source, inductance L or capacitance C. A distinction is made between resonance when connected in series L and C - voltage resonance and when they are connected in parallel - current resonance. Angular frequency? 0 at which resonance occurs is called resonant, or natural frequency of oscillations of the resonant circuit.

Voltage resonance. At voltage resonance (Fig. 196, a) the inductive reactance X L is equal to the capacitive reactance X c and the total resistance Z becomes equal to the active resistance R:

Z = ?(R 2 + [? 0 L - 1/(? 0 C)] 2) = R

In this case, the voltages across the inductance U L and capacitance U c are equal and are in antiphase (Fig. 196, b), therefore, when added, they compensate each other. If the active resistance of the circuit R is small, the current in the circuit increases sharply, since the reactance of the circuit X = X L -X s becomes equal to zero. In this case, the current I is in phase with the voltage U and I=U/R. A sharp increase in current in the circuit during voltage resonance causes the same increase in voltages U L and U c , and their values ​​can be many times higher than the voltage U of the source feeding the circuit.

The angular frequency?0, at which resonance conditions occur, is determined from the equality ? o L = 1/(? 0 C).

From here we have

? o = 1/?(LC) (74)

If you smoothly change the angular frequency? source, then the total resistance Z first begins to decrease, reaches its lowest value at voltage resonance (at? o), and then increases (Fig. 197, a). In accordance with this, the current I in the circuit first increases, reaches its highest value at resonance, and then decreases.

Resonance of currents. Current resonance can occur when inductance and capacitance are connected in parallel (Fig. 198, a). In the ideal case, when there is no active resistance in parallel branches (R 1 = R 2 = 0), the condition for current resonance is the equality of the reactances of the branches containing inductance and capacitance, i.e. ? o L = 1/(? o C). Since in the case under consideration the active conductivity G = 0, the current in the unbranched part
circuits at resonance I=U?(G 2 +(B L -B C) 2)= 0. The values ​​of the currents in branches I 1 and I 2 will be equal (Fig. 198, b), but the currents will be shifted in phase by 180° (the current IL in the inductance lags in phase from the voltage U by 90°, and the current in the capacitance I c is ahead voltage U at 90°). Consequently, such a resonant circuit represents an infinitely large resistance for current I and electrical energy does not enter the circuit from the source. At the same time, currents I L and I c flow inside the circuit, i.e. there is a process of continuous exchange of energy within the circuit. This energy moves from inductance to capacitance and back.

As follows from formula (74), by changing the values ​​of capacitance C or inductance L, can the oscillation frequency be changed? 0 electrical energy and current in the circuit, i.e., adjust the circuit to the required frequency. If there were no active resistance in the branches in which the inductance and capacitance are included, this process of energy oscillation would continue indefinitely, i.e., undamped oscillations of energy and currents I L and I s would arise in the circuit. However, real inductors and capacitors always absorb electrical energy (due to the presence of active wire resistance in the coils and the occurrence of

in bias current capacitors that heat the dielectric), therefore, when the currents resonate, some electrical energy enters the real circuit from the source and some current I flows through the unbranched part of the circuit.

The condition for resonance in a real resonant circuit containing active resistances R 1 and R 2 will be the equality of the reactive conductivities B L = B C of the branches, which include inductance and capacitance.

From Fig. 198, c it follows that the current I in the unbranched part of the circuit is in phase with the voltage U, since the reactive currents 1 L and I c are equal, but opposite in phase, as a result of which their vector sum is zero.

What if the frequency in the parallel circuit under consideration is changed? about the alternating current source, then the total resistance of the circuit begins to increase, reaches its highest value at resonance, and then decreases (see Fig. 197, b). In accordance with this, the current I begins to decrease, reaches its lowest value I min = I a at resonance, and then increases.

In real oscillatory circuits containing active resistance, each current fluctuation is accompanied by energy losses. As a result, the energy imparted to the circuit is consumed quite quickly and the current fluctuations gradually die out. To obtain undamped oscillations, it is necessary to constantly replenish the energy losses in the active resistance, i.e., such a circuit must be connected to an alternating current source of the appropriate frequency? 0 .

The phenomena of voltage and current resonance and the oscillatory circuit have become very widely used in radio engineering and high-frequency installations. Using oscillatory circuits we obtain currents high frequency in various radio devices and high-frequency generators. The oscillatory circuit is the most important element of any radio receiver. It ensures its selectivity, i.e. the ability to distinguish from radio signals with different wavelengths (i.e. with different frequencies) sent various radio stations, signals from a specific radio station.

Let's start with basic definitions.

Definition 1

Resonance is a phenomenon in which the oscillation frequency of any system is increased by fluctuations in an external force.

Forced vibrations, the source of which is an external force, increase even those vibrations whose amplitude is quite small. The maximum resonance with the greatest amplitude is possible precisely when the frequencies of the external influence and the system under consideration coincide.

An example of resonance is the rocking of a bridge by a company of soldiers. The step frequency of the soldiers, which is an example of forced vibrations in relation to the bridge, is synchronized and can coincide with the natural frequency of vibrations of the bridge. As a result, the bridge may collapse.

Electrical resonance in physics is considered one of the most common physical phenomena in the world, without which it would be impossible, for example, television and diagnostics using medical devices.

Some of the most useful types of resonance in an electrical circuit are:

• current resonance;
• voltage resonance.

The occurrence of resonance in an electrical circuit

Note 1

The occurrence of resonance in an electrical circuit is facilitated by a sharp increase in the amplitude of the stationary natural oscillations of the system, provided that the frequency of the external side of the influence coincides with the corresponding oscillatory resonant frequency of the system.

The $RLC$ circuit represents an electrical circuit with elements (resistor, inductor, capacitor) connected in series or parallel. The name $RLC$ consists of simple symbols for electrical elements: resistance, capacitance, inductance.

The vector diagram of a sequential $RLC$ circuit is presented in one of three variations:

• capacitive;
• active;
• inductive.

In the last variation, voltage resonance occurs under the condition of zero phase shift, and the values ​​of inductive and capacitive reactance coincide.

Voltage resonance

When an active element $r$, a capacitive element $C$, and an inductive element $L$ are connected in series in alternating current circuits, a physical phenomenon such as voltage resonance may occur. The oscillations of the voltage source in this case will be equal in frequency to the oscillations of the circuit. At the same time, both the usefulness (for example, in radio engineering) of this phenomenon and the negative consequences (for high-power electrical installations) are known, for example, with a sharp surge in voltage in the systems, a malfunction or even a fire may occur.

Voltage resonance is usually achieved in three ways:

• selection of coil inductance;
• selection of capacitor capacity;
• selection of angular frequency $w_0$.

In this case, all values ​​of capacitance, frequency and inductance are determined using the formulas:

$L_0 = \frac(1)(w^2C)$

$C_0 = \frac(1)(w^2L)$

The frequency $w_0$ is considered resonant. Provided that both the voltage and active resistance $r$ in the circuit remain constant, the current strength at voltage resonance in it will be maximum and equal to:

This assumes that the current is completely independent of the reactance of the circuit. In a situation where the reactance $XC = XL$ exceeds the active resistance $r$ in value, a voltage will appear at the coil and capacitor terminals that significantly exceeds the voltage at the circuit terminals.

The excess voltage ratio at the terminals of the capacitive and inductive element relative to the network is determined by the expression:

$Q = \frac(U_c0)(U)$

The value $Q$ characterizes the resonant properties of the circuit, and is called the quality factor of the circuit. Also, resonant properties are characterized by the value $\frac(1)(Q)$, that is, the damping of the circuit.

Resonance of currents through reactive elements

Resonance of currents appears in electrical circuits of alternating current circuits under the condition of parallel connection of branches with different reactances. In the resonant mode of currents, the reactive inductive conductivity of the circuit will be equivalent to its own reactive capacitive conductivity, i.e. $BL = BC$.

The oscillations of the circuit, the frequency of which has a certain value, in this case coincide in frequency with the voltage source.

The simplest electrical circuit in which we observe current resonance is considered to be a circuit with a parallel connection of a capacitor to an inductor.

Since the reactivity resistances are equal in magnitude, the amplitudes of the currents $I_c$ and $I_u$ will be the same and can reach their maximum amplitude. Based on Kirchhoff's first law, $IR$ is equal to the source current. The source current, in other words, flows only through the resistor. When considering a separate parallel circuit $LC$, at the resonant frequency its resistance turns out to be infinitely large: $ZL = ZC$. When a harmonic mode with a resonant frequency is established, the circuit is observed to provide a certain steady-state oscillation amplitude with a source, and the power of the current source is spent exclusively on replenishing losses in the active resistance.

Thus, the impedance of a series $RLC$ circuit turns out to be minimal at the resonant frequency and equal to the active resistance of the circuit. At the same time, the impedance of a parallel $RLC$ circuit is maximum at the resonant frequency and is considered equal to the leakage resistance, which is actually also the active resistance of the circuit. In order to ensure conditions for resonance of current or voltage, it is necessary to check the electrical circuit to predetermine its complex resistance or conductivity. In addition, its imaginary part must be equal to zero.

Application of resonance phenomenon

A good example of the use of the resonant phenomenon is the electrical resonant transformer developed by Nikola Tesla back in 1891. The scientist conducted experiments on different configurations, consisting of a combination of two, and often three, resonant electrical circuits.

Note 2

The term "Tesla coils" is applied to high-voltage resonant transformers. The devices are used to produce high voltage, alternating current frequency. Regular transformer necessary for efficient transfer of energy from the primary to the secondary winding, resonant is used for temporary storage of electricity.

The device is responsible for controlling the air core of a resonantly tuned transformer in order to obtain high voltage at low current strengths. Each winding has a capacitance and functions as a resonant circuit. To produce the highest output voltage, the primary and secondary circuits are tuned into resonance with each other.

Fundamentals > Theoretical foundations of electrical engineering

Resonance phenomena in electrical circuits

The ideal active resistance does not depend on frequency, inductive reactance depends linearly on frequency, capacitive reactance depends on frequency according to the hyperbolic law:

Voltage resonance

Resonance in electrical circuits, the mode of a section of an electrical circuit containing inductive and capacitive elements is called, in which the phase difference between voltage and current is zero. Resonance mode can be obtained by changing frequencysupply voltage or changing the parameters of elements L and C.
When connected in series, voltage resonance occurs.

Serial connection R, L, C.

The denominator of this expression is the module of the complex resistance, which depends on the frequency. When a certain frequency is reached, the reactive component of the resistance disappears, the resistance modulus becomes minimal, the current in this circuit increases to the maximum value, and the current vector coincides with the voltage vector in phase:

The maximum amplitude of the current is achieved under the condition of a minimum impedance, i.e. when

Where
- resonant voltage frequency, determined from the condition

When a capacitor and solenoid are connected in series in a circuit, the current strengths in each section of the circuit are known to be equal. Therefore, multiplying the left and right sides of the last relation by the current strength I'm getting it

In this expression on the left is the voltage amplitudeat the ends of the solenoid, and on the right is the voltage amplitudeon the capacitor plates.
We see that . From here we get

The minus sign indicates that voltage fluctuations in areas with inductance and capacitance occur in antiphase.
The mode of an electrical circuit with a series connection of inductance and capacitance, characterized by equality of voltages across the inductance and capacitance, is calledvoltage resonance.

Wave or characteristic impedance of a series circuit

The ratio of the voltage across the inductance or capacitance to the voltage at the input in resonance mode is calledcircuit quality factor:

The quality factor of the circuit is the voltage gain and in inductors can reach hundreds of units:

At the voltage across the inductance (or capacitance) can be much more voltage at the input, which is widely used in radio engineering. In industrial networks, voltage resonance is an emergency mode, since an increase in voltage on a capacitor can lead to its breakdown, and an increase in current can lead to heating of wires and insulation.

Current resonance

When connecting a capacitor and solenoid in parallel (see figure), as well as in series, the current strength in the circuit depends on the values ​​of capacitance and inductance. When capacitance and inductance change at a certain ratio, the current strength in the unbranched section of the circuit turns out to be minimal (almost close to zero).
In this case:

Parallel connection of reactive elements

Then

At a certain frequency, called resonant, the reactive components of conductivity can be equal in magnitude and the total conductivity will be minimal. In this case, the total resistance becomes maximum, the total current is minimum, the current vector coincides with the voltage vector.This phenomenon is calledcurrent resonance.
Wave conduction

At the current in the branch with inductance is much greater than the total current, therefore this phenomenon is called current resonance and is widely used in power networks of industrial enterprises to compensate for reactive power.
Resonant frequency of currentwe find from the condition of equality of the reactive conductivities of the branches.

After a series of transformations we get:

From the formula it follows that:

1) the resonant frequency depends on the parameters of not only reactances, but also active ones;
2) resonance is possible if And more or less r , otherwise the frequency will be an imaginary quantity and resonance is impossible;
3) if , then the frequency will have an indefinite value, which means that resonance can exist at any frequency when the phases of the supply voltage and total current coincide;
4) when the resonant frequency of the voltage is equal to the resonant frequency of the current.

Energy processes in a circuit during current resonance are similar to the processes occurring during voltage resonance.
Reactive energy circulates within the circuit: in one part of the period the energy magnetic field inductance is converted into the energy of the electric field of the capacitance, in the next part of the period the reverse process occurs.
When the currents resonate, the reactive power is zero.
Most industrial AC consumers are active-inductive in nature and therefore consume reactive power. Such consumers include asynchronous motors, electric welding installations, etc.
To reduce reactive power and increase the power factor, a capacitor bank is connected in parallel to the consumer, which leads to a decrease in the current in the wires connecting the consumer to the energy source.

Resonance is a mode of a passive circuit containing inductors and capacitors in which its input reactance or its input reactance is zero. At resonance, the current at the input of the circuit, if it is different from zero, is in phase with the voltage.

Consider a series connection of resistance, inductance and capacitance (Fig. 3-8). Such a circuit is often called a series circuit. For it, resonance occurs when or, i.e.

When the values ​​of voltages opposite in phase on the inductance and capacitance are equal (Fig. 3-11, b), therefore the resonance in the circuit under consideration is called voltage resonance.

The voltages across the inductance and capacitance at resonance can significantly exceed the voltage at the circuit terminals, which is equal to the voltage across the active resistance. The total resistance of the circuit at a minimum: , and the current at a given

voltage U reaches its highest value. In the theoretical case, at the total resistance of the circuit in the resonance mode is also zero, and the current at any finite voltage value U is infinitely large. In the same way, the voltages on inductance and capacitance are infinitely large.

It follows from the condition that resonance can be achieved by changing either the source voltage frequency or the circuit parameters - inductance or capacitance. The angular frequency at which resonance occurs is called the resonant angular frequency

Inductive and capacitive reactance at resonance

The value is called the characteristic impedance of the circuit or circuit.

The ratio of the voltage across an inductance or capacitance to the voltage applied to the circuit at resonance

called the circuit quality factor or resonance coefficient. The resonance coefficient indicates how many times the voltage across the inductance or capacitance at resonance is greater than the voltage applied to the circuit: if . The name “quality factor” of the circuit will be explained in the next paragraph.

To understand the energy processes during resonance, let us determine the sum of the energies of the magnetic and electric fields of the circuit. Let the current in the circuit be . Then the voltage across the capacitance

Total energy

and therefore

that is, the sum of the energies of the magnetic and electric fields does not change over time. A decrease in the electric field energy is accompanied by an increase in the magnetic field energy and vice versa. Thus, there is a continuous transition of energy from the electric field to the magnetic field and back.

The energy supplied to the circuit from the power source is completely converted into heat at any given time. Therefore, for a power supply, the entire circuit is equivalent to one active resistance.

The name “resonance” for the considered circuit mode is borrowed from the theory of oscillations. As is known, resonance is a process of forced oscillations with a frequency at which the intensity of oscillations, other things being equal, is maximum. But the intensity of the oscillatory process can be characterized by various manifestations, the maxima of which are observed at different frequencies. Therefore, it is necessary to agree on the criterion of resonance.

Charges oscillate in an electrical circuit. One could take as a resonance criterion the maximum amplitude value of the charge on the capacitor, which corresponds to the maximum amplitude of the voltage on the capacitor. This criterion determines the amplitude resonance. For the resonance criterion adopted at the beginning of the paragraph, the current at resonance is in phase with the applied voltage, this is the so-called phase resonance. In the circuit under consideration (Fig. 3-8), phase resonance occurs when maximum speed movement of oscillating charges or maximum current.

If a charged capacitor is connected to an inductance coil, then in such a circuit, with a sufficiently low resistance of the coil, a process of damped oscillations of voltage and current is observed. The frequency of these vibrations is called the frequency of natural or free vibrations. Note that the frequencies at which phase and amplitude resonances are observed do not coincide with the frequency of natural oscillations (they coincide only in the theoretical case when the circuit resistance is zero). The resonance criterion adopted here is also applicable in the case when natural oscillations are impossible in the circuit due to high resistance.

Knowledge of physics and the theory of this science is directly related to housekeeping, repairs, construction and mechanical engineering. We propose to consider what resonance of currents and voltages in a series RLC circuit is, what the main condition for its formation is, as well as the calculation.

What is resonance?

Definition of the phenomenon by TOE: electrical resonance occurs in an electrical circuit at a certain resonant frequency, when some parts of the resistance or conductivity of the circuit elements cancel each other. In some circuits, this occurs when the impedance between the input and output of the circuit is almost zero and the signal transfer function is close to unity. In this case, the quality factor of this circuit is very important.

Signs of resonance:

1. The components of the reactive branches of the current are equal to each other IPC = IPL, antiphase is formed only when the net active energy at the input is equal;
2. The current in individual branches exceeds the entire current of a particular circuit, while the branches are in phase.

In other words, resonance in an AC circuit implies a special frequency, and is determined by the values ​​of resistance, capacitance and inductance. There are two types of current resonance:

1. Consistent;
2. Parallel.

For series resonance, the condition is simple and is characterized by minimal resistance and zero phase, it is used in reactive circuits, and it is also used in branched circuits. Parallel resonance or the concept of an RLC circuit occurs when the inductive and capacitive inputs are equal in magnitude but cancel each other out since they are at an angle of 180 degrees from each other. This connection must be constantly equal to the specified value. It has received wider practical application. The sharp minimum impedance that it exhibits is beneficial for many electrical household appliances. The sharpness of the minimum depends on the resistance value.

An RLC circuit (or circuit) is electrical diagram, which consists of a resistor, inductor, and capacitor connected in series or parallel. The RLC parallel oscillating circuit gets its name from the abbreviation physical quantities, representing resistance, inductance and capacitance, respectively. The circuit forms a harmonic oscillator for the current. Any oscillation of the current induced in the circuit fades over time if the movement of the directed particles is stopped by the source. This resistor effect is called attenuation. The presence of resistance also reduces the peak resonant frequency. Some resistance is inevitable in real circuits, even if the resistor is not included in the circuit.

Application

Almost all power electrical engineering uses just such an oscillatory circuit, say, a power transformer. The circuit is also necessary for setting up the operation of a TV, capacitive generator, welding machine, radio receiver; it is used by the “matching” technology of television broadcast antennas, where you need to select a narrow frequency range of some of the waves used. The RLC circuit can be used as a band-pass filter, notch filter, for low-pass or high-pass distribution sensors.

Resonance is even used in aesthetic medicine (microcurrent therapy) and bioresonance diagnostics.

Principle of current resonance

We can make a resonant or oscillating circuit at its natural frequency, say, to power a capacitor, as the following diagram demonstrates:

Circuit for powering a capacitor

The switch will be responsible for the direction of vibration.

Circuit: resonant circuit switch

The capacitor stores all the current at the moment when time = 0. Oscillations in the circuit are measured using ammeters.

Scheme: the current in the resonant circuit is zero

Directed particles move in right side. The inductor receives current from the capacitor.

When the polarity of the circuit returns to its original form, the current returns to the heat exchanger.

Now the directed energy goes back into the capacitor, and the circle repeats again.

In real mixed circuit circuits there is always some resistance which causes the amplitude of the directed particles to grow smaller with each circle. After several changes in the polarity of the plates, the current decreases to 0. This process is called a damped sine wave signal. How quickly this process occurs depends on the resistance in the circuit. But the resistance does not change the frequency of the sine wave. If the resistance is high enough, the current will not fluctuate at all.

The AC designation means that the energy leaving the power supply oscillates at a certain frequency. An increase in resistance helps to reduce the maximum size of the current amplitude, but this does not lead to a change in the resonance frequency. But an eddy current process can form. After its occurrence, network interruptions are possible.

Resonant circuit calculation

It should be noted that this phenomenon requires very careful calculation, especially if a parallel connection is used. In order to avoid interference in technology, you need to use various formulas. They will be useful to you for solving any problem in physics from the corresponding section.

It is very important to know the power value in the circuit. The average power dissipated in a resonant circuit can be expressed in terms of rms voltage and current as follows:

R av = I 2 contact * R = (V 2 contact / Z 2) * R.

At the same time, remember that the power factor at resonance is cos φ = 1

The resonance formula itself has the following form:

ω 0 = 1 / √L*C

Zero impedance at resonance is determined using the following formula:

F res = 1 / 2π √L*C

The resonant frequency of oscillation can be approximated as follows:

F = 1/2 r (LC) 0.5

Where: F = frequency

L = inductance

C = capacity

Generally, a circuit will not oscillate unless the resistance (R) is low enough to satisfy the following requirements:

R = 2 (L/C) 0.5

To obtain accurate data, you should try not to round the obtained values ​​due to calculations. Many physicists recommend using a method called vector diagram of active currents. With proper calculation and configuration of devices, you will get good savings on alternating current.