Voltage resonance in an electrical circuit and its consequences. Resonance in a series circuit (voltage resonance) How to detect voltage resonance in a circuit
Oscillatory circuit- an electrical circuit in which oscillations can occur with a frequency determined by the parameters of the circuit.
The simplest oscillatory circuit consists of a capacitor and an inductor connected in parallel or in series.
Capacitor C– reactive element. Has the ability to accumulate and release electrical energy.
- Inductor L– reactive element. Has the ability to accumulate and release magnetic energy.
Free electrical oscillations in a parallel circuit.
Basic properties of inductance:
The current flowing in the inductor creates a magnetic field with energy.
- A change in current in a coil causes a change in the magnetic flux in its turns, creating an EMF in them that prevents a change in current and magnetic flux.
Period of free oscillations of the circuit L.C. can be described as follows:
If the capacitor has a capacity C charged to voltage U, the potential energy of its charge will be 
.
If you connect an inductor in parallel to a charged capacitor L, its discharge current will flow through the circuit, creating a magnetic field in the coil.
The magnetic flux, increasing from zero, will create an EMF in the direction opposite to the current in the coil, which will prevent the current from increasing in the circuit, so the capacitor will not discharge instantly, but after a while t 1, which is determined by the inductance of the coil and the capacitance of the capacitor from the calculation t 1 = .
After time has passed t 1, when the capacitor is discharged to zero, the current in the coil and the magnetic energy will be maximum.
The magnetic energy accumulated by the coil at this moment will be.
In an ideal consideration, with complete absence of losses in the circuit, E C will be equal E L. Thus, the electrical energy of the capacitor will be converted into magnetic energy of the coil.
A change (decrease) in the magnetic flux of the accumulated energy of the coil will create an EMF in it, which will continue the current in the same direction and the process of charging the capacitor with induced current will begin. Decreasing from maximum to zero over time t 2 = t 1, it will recharge the capacitor from zero to the maximum negative value ( -U).
So the magnetic energy of the coil will be converted into electrical energy of the capacitor.
Described intervals t 1 and t 2 will be half the period of complete oscillation in the circuit.
In the second half, the processes are similar, only the capacitor will discharge from a negative value, and the current and magnetic flux will change direction. Magnetic energy will again accumulate in the coil over time t 3, changing the polarity of the poles.
During the final stage of oscillation ( t 4), the accumulated magnetic energy of the coil will charge the capacitor to its original value U(in the absence of losses) and the oscillation process will repeat.
In reality, in the presence of energy losses on the active resistance of the conductors, phase and magnetic losses, the oscillations will be damped in amplitude.
Time t 1 + t 2 + t 3 + t 4 will be the oscillation period 
.
Frequency of free oscillations of the circuit ƒ = 1 / T
The free oscillation frequency is the resonance frequency of the circuit at which the inductance reactance X L =2πfL equal to the reactance of the capacitance X C =1/(2πfC).
Resonance Frequency Calculation L.C.-contour:
A simple online calculator is provided to calculate the resonant frequency of an oscillating circuit.
The power factor cosφ at voltage resonance is equal to unity.
2. Condition, sign and application of stress resonance. When is voltage resonance harmful? Why?
A mode in which, in a circuit with a series connection of an inductive and capacitive element, the input voltage is in phase with the current, voltage resonance.
The sudden occurrence of a resonant mode in high-power circuits can cause emergency situations, lead to breakdown of the insulation of wires and cables and create a danger for personnel.
3. In what ways can voltage resonance be achieved?
When connecting an oscillating circuit consisting of an inductor and a capacitor to an energy source, a resonant phenomenon may occur. Two main types of resonance are possible: when the coil and capacitor are connected in series, there is voltage resonance, and when they are connected in parallel, there is current resonance.
4. Why during voltage resonanceU 2 >U 1 ?
Where R is active resistance
I – current strength
XL – coil inductance
XC – capacitance of the capacitor
Z – AC impedance
At resonance: UL = UC,
Where UC is the coil voltage,
UL – capacitor voltage
The voltage can be found:
U=UR+UL+UC =>U=UR,
Where UR is the voltage of the coil to which the voltmeter V2 is connected, which means voltage V2=V1
5. What is the feature of voltage resonance? Explain it.
Consequently, the resonance mode can be achieved by changing the inductance of the coil L, the capacitance of the condensate C or the frequency of the input voltage ω.
6. Write down the expression for Ohm’s law in terms of conductivity for a circuit with a parallel connection of a capacitor and an inductive coil. What is the total conductivity?
Ohm's law through conductivity for an alternating current circuit with parallel connections of branches.
7. Condition, sign and application of current resonance.
i.e. equality of inductive and capacitive conductivity.
8 . In what ways can current resonance be achieved?
A mode in which in a circuit containing parallel branches with inductive and capacitive elements, the current of the unbranched section of the circuit is in phase with the voltage, the resonance of the currents.
9. Why during resonance of currentsI 2 > I 1 ?
Because, based on the vector diagram of currents at resonance, the graph will be a right triangle, where currents I and I 1 will be legs, and current I 2 will be the hypotenuse. Consequently, I 2 will be greater than I 1.
10. What is the feature of current resonance? Explain it.
With current resonance, the currents in the branches are significantly greater than the current in the unbranched part of the circuit. This property—current strength—is the most important feature of current resonance.
11. Explain the construction of vector diagrams.
The purpose of its construction is to determine the active and reactive components of the voltage on the coil and the phase shift angle between the voltage at the circuit input and the current
Calculations
LIST OF SOURCES USED
Electrical and Electronics. Book 1. Electric and magnetic circuits. - B 3 books: book 1 /B. G. Gerasimov and others; Ed. V. G. Gerasimova. M.: Energoatomizdat, 1996. – 288 p.
Kasatkin A. S., Nemtsov M. V. Electrical engineering. M.: Higher. school, 1999. – 542 p.
Electrical engineering /Ed. Yu. L. Khotuntseva. M.: AGAR, 1998. – 332 p.
Borisov Yu. M., Lipatov D. N., Zorin Yu. N. Electrical engineering. Energoatomizdat, 1985. – 550 p.
GOST 19880-74. Electrical engineering. Basic concepts. Terms and Definitions. M.: Standards Publishing House, 1974.
Variable EMF. It changes according to the law:
Picture 1.
A current of the form flows in the circuit:
The amplitude of the current $(\ (I)_m)$ is related to the amplitude $((\mathcal E))_m$ by “Ohm’s law” for alternating current:
Expression:
total electrical resistance. The angle ($\varphi $) by which current fluctuations lag behind voltage fluctuations is determined by the expression:
If you change the oscillation frequency ($\omega $). As follows from formulas (3), (5), there will be a change in the amplitude of the current ($I_m$) and the phase shift ($\varphi $).
If $\omega =0$, then the expression is $\frac(1)(\omega C)\to \infty $. The impedance ($Z$) becomes infinite, therefore $I_m=0.$ At $\omega =0$ we are dealing with a direct current that does not pass through the capacitor. If you start to increase the frequency, then the value of the reactance ($(\left(\omega L-\frac(1)(\omega C)\right))^2$) first decreases, therefore, the impedance decreases, $I_m.$ increases. When the frequency ($\omega $) becomes equal to the resonant frequency of the circuit ($(\omega )_0$):
the total resistance of the circuit ($Z$) becomes minimal and equal to the active resistance of the circuit ($R$). The current strength reaches its maximum. For $\omega >(\omega )_0$ the expression $(\left(\omega L-\frac(1)(\omega C)\right))^2\ne 0$ and increases with increasing frequency. The impedance increases again, the amplitude of the current decreases, approaching zero asymptotically.
The process described above is graphically depicted in Fig. 2.
Figure 2.
The amplitude of the current at the resonant frequency ($\omega =(\omega )_0$) is equal to:
in this case, the phase difference is zero ($\varphi =0$). There is no capacitance or inductance in the circuit. At this frequency, the voltages on the capacitance and inductance are completely mutually compensated, becoming equal in magnitude, since they are always opposite in phase. This resonance is called voltage resonance. The vector diagram of voltage resonance is shown in Fig. 3. At resonance, the circuit behaves like an active resistance.
Figure 3.
Comment
So, the case of forced oscillations, when the frequency of the EMF generator (or applied external voltage) is equal to the resonant frequency, is of particular interest. In this case, the current amplitude reaches a maximum, and the phase shift between current and voltage is zero. The circuit acts as an active resistance.
Application of Voltage Resonance
The phenomenon of voltage resonance is used in radio engineering if it is necessary to amplify voltage fluctuations of any frequency, for example, in devices of the input part of a radio receiver. There is an oscillating circuit ($LC$) in this part. The quality factor of this circuit is high, the voltage from the circuit capacitor is supplied to the input of the amplifier. The input signals cause an alternating current of fairly high frequency in the antenna, which causes a mutual induction emf in the $L$ coil, the amplitude of which is $((\mathcal E))_m\ \ $. Due to resonance, a voltage appears at the capacitor (and therefore at the input) with an amplitude $((\mathcal E))_mO>((\mathcal E))_m.$ This amplification works only in a narrow frequency range, around the resonant frequency, which will allow you to select only vibrations of the desired frequency from a large number of signals from different radio stations.
Example 1
Exercise: What is the voltage amplitude across the capacitor ($U_(mC)$) at voltage resonance if the oscillations are weakly damped? The quality factor of the circuit is $\O$. External EMF changes in accordance with the law: $(\mathcal E)=((\mathcal E))_m(sin \left(\omega t\right)\ ).$
Solution:
The current amplitude at resonance reaches a maximum, it is equal to:
where $(\omega )_0$ is the resonant frequency.
Therefore, the voltage amplitude across the capacitor will be equal to:
where the capacitance is equal to:
Substituting $X_C$ from (1.3) and $I_(m\ )$ from (1.1) into formula (1.2) we obtain the voltage amplitude on the capacitor at resonance:
Let's take into account that:
\[(\omega )_0=\frac(1)(\sqrt(LC))(1.5)\]
Substituting the expression for the resonant frequency into formula (1.4), we obtain:
where $O=\frac(1)(R)\sqrt(\frac(L)(C))$ is the quality factor of the circuit.
Answer:$U_(mC)=((\mathcal E))_mO.$
Example 2
Exercise: What is the voltage amplitude across the inductance ($U_(mL)$) at voltage resonance if the oscillations are weakly damped? The quality factor of the circuit is $\O$. External EMF changes in accordance with the law: $(\mathcal E)=((\mathcal E))_m(sin \left(\omega t\right)\ ).$
Solution:
The expression for the voltage across the inductance can be written as:
where the expression for the current amplitude ($I_m(\omega_0)$) at voltage resonance:
Let's replace:
\[(\omega )_0=\frac(1)(\sqrt(LC))\left(2.4\right).\]
We find that the voltage amplitude across the inductance is equal to:
Answer:$U_(mL)(=(\mathcal E))_mO.$
The voltage fluctuations across the capacitor and inductance have equal amplitudes, but their phase difference is equal to $\pi$.
Then they have their own effect on the generator powering the circuit and on the phase relationships between current and voltage.
An inductor introduces a phase shift in which the current lags behind the voltage by a quarter of a period, while a capacitor, on the contrary, causes the voltage in the circuit to lag in phase with the current by a quarter of a period. Thus, the effect of inductive reactance on the phase shift between current and voltage in a circuit is opposite to the effect of capacitive reactance.
This leads to the fact that the overall phase shift between current and voltage in the circuit depends on the ratio of the inductive and capacitive reactance values.
If the value of the capacitive resistance of the circuit is greater than the inductive one, then the circuit is capacitive in nature, that is, the voltage lags in phase with the current. If, on the contrary, the inductive reactance of the circuit is greater than the capacitive reactance, then the voltage leads the current, and, therefore, the circuit is inductive in nature.
The total reactance Xtot of the circuit we are considering is determined by adding the inductive reactance of the coil X L and the capacitive reactance of the capacitor X C.
But since the action of these resistances in the circuit is opposite, then one of them, namely Xc, is assigned a minus sign, and the total reactance is determined by the formula:
Applying to this circuit, we get:
This formula can be transformed as follows:
In the resulting equality, I X L is the effective value of the component of the total voltage of the circuit going to overcome the inductive reactance of the circuit, and I X C is the effective value of the component of the total voltage of the circuit going to overcome the capacitive reactance.
Thus, the total voltage of a circuit consisting of a series connection of a coil and a capacitor can be considered as consisting of two terms, the values of which depend on the values of the inductive and capacitive reactances of the circuit.
We believed that such a circuit does not have active resistance. However, in cases where the active resistance of the circuit is not so small that it can be neglected, the total resistance of the circuit is determined by the following formula:
where R is the total active resistance of the circuit, X L -X C is its total reactance. Moving on to the formula of Ohm's law, we have the right to write:
Voltage resonance in the AC circuit
Inductive and capacitive reactances connected in series cause less phase shift between current and voltage in an alternating current circuit than if they were connected separately in the circuit.
In other words, from the simultaneous action of these two reactive resistances of different nature in the circuit, compensation (mutual destruction) of the phase shift occurs.
Full compensation, i.e. complete elimination of the phase shift between current and voltage in such a circuit, will occur when the inductive reactance is equal to the capacitive reactance of the circuit, i.e. when X L = X C or, which is the same, whenω L = 1 / ωС.
The circuit in this case will behave as a purely active resistance, that is, as if it had neither a coil nor a capacitor. The value of this resistance is determined by the sum of the active resistances of the coil and connecting wires. In this case, it will be the largest in the chain and is determined by the formula of Ohm’s law I = U / R, where R is now placed instead of Z.
At the same time, the effective voltages both on the coil U L = I X L and on the capacitor Uc = I X C will be equal and will be as large as possible. With low active resistance of the circuit, these voltages can be many times higher than the total voltage U at the circuit terminals. This interesting phenomenon is called in electrical engineering voltage resonance.
In Fig. Figure 1 shows the voltage, current and power curves at voltage resonance in the circuit.
It should be firmly remembered that the resistances X L and X C are variable, depending on the frequency of the current, and it is worth at least slightly changing its frequency, for example, increasing it, as X L =ω Lwill increase, and X C == 1 / ωС will decrease, and thus the voltage resonance in the circuit will immediately be disrupted, and along with active resistance, reactive resistance will also appear in the circuit. The same thing will happen if you change the value of the inductance or capacitance of the circuit.
With voltage resonance, the power of the current source will be spent only on overcoming the active resistance of the circuit, i.e., on heating the conductors.
Indeed, in a circuit with one inductor, energy oscillates, i.e., periodically transfers energy from the generator to the coils. In a circuit with a capacitor, the same thing happens, but due to the energy of the electric field of the capacitor. In a circuit with a capacitor and an inductor at stress resonance(X L = X C) the energy, once stored by the circuit, periodically passes from the coil to the capacitor and back, and the current source receives only the energy consumption necessary to overcome the active resistance of the circuit. Thus, energy exchange occurs between the capacitor and the coil almost without the participation of the generator.
All you have to do is break voltage resonance appreciate how the energy of the magnetic field of the coil will not be equal to the energy of the electric field of the capacitor, and in the process of energy exchange between these fields, an excess of energy will appear, which will periodically either flow from the source into the circuit, or be returned to it back by the circuit.
This phenomenon is very similar to what happens in a clock mechanism. The pendulum of a clock could continuously oscillate without the help of a spring (or a load in a walking clock), if not for the frictional forces that slow down its movement.
The spring, imparting part of its energy to the pendulum at the right moment, helps it overcome the forces of friction, which ensures continuity of oscillations.
Similarly, in an electrical circuit, when resonance occurs in it, the current source spends its energy only to overcome the active resistance of the circuit, thereby supporting the oscillatory process in it.
So we come to the conclusion that an alternating current circuit consisting of a generator and a series-connected inductor and capacitor, under certain conditions X L = X C turns into an oscillatory system. This chain is called oscillatory circuit.
From the equality X L = X C we can determine generator frequency at which voltage resonance occurs:
Along with the beneficial use of the phenomenon of voltage resonance in electrical engineering, there are often cases when voltage resonance is harmful. A large increase in voltage in individual sections of the circuit (on a coil or on a capacitor) compared to the generator voltage can lead to damage to individual parts and measuring instruments.
In electrical engineering, when analyzing the operating modes of electrical circuits, the concept of a two-terminal network is widely used. Two-terminal network it is customary to call a part of an electrical circuit of arbitrary configuration, considered relative to two selected terminals (poles). Two-terminal circuits that do not contain energy sources are called passive. Any passive two-terminal network is characterized by one quantity - input resistance, i.e. resistance measured (or calculated) relative to the two terminals of a two-terminal network. Input resistance and input conductance are mutually inverse quantities.
Let a passive two-terminal network contain one or more inductors and one or more capacitors. Under resonant mode The operation of such a two-terminal network is understood as the mode(s) of a two-terminal network in which the input resistance is purely active. In relation to the external circuit, the two-terminal network behaves like an active resistance, as a result of which the input voltage and current are in phase. There are two types of resonant modes: voltage resonance and current resonance.
Voltage resonance
In the simplest case, voltage resonance can be obtained in an AC electrical circuit by connecting an inductor and capacitors in series. At the same time, by changing the capacitance of the capacitors at constant coil parameters, a voltage resonance is obtained at constant values of voltage and inductance, frequency and active resistance of the circuit. When changing the capacitance of capacitors WITH there is a change in reactance capacitance. At the same time, the total resistance of the circuit also changes, therefore, the current, power factor, voltage on the inductor, capacitors, as well as the active, reactive and apparent power of the electrical circuit change. Current dependencies I, power factor cosand impedance Z AC circuits as a function of capacitance (resonance curves) for the circuit under consideration are shown in Fig. 9, A. The vector diagram of the current and voltage of this circuit at resonance is shown in Fig. 9, b.
As can be seen from this diagram, the reactive voltage component U L on the coil at resonance is equal to the voltage U C on the capacitor. In this case, the voltage across the inductor U to at resonance due to the fact that the coil, in addition to the reactance X L also has active resistance R, slightly greater than the voltage across the capacitor.
Analysis of the presented expressions (2), as well as Fig. 9, A And b show that voltage resonance has a number of distinctive features.
1. With voltage resonance, the total resistance of the AC electrical circuit takes on a minimum value and turns out to be equal to its active resistance, i.e.
2. From this it follows that at a constant supply voltage ( U= const) at voltage resonance, the current in the circuit reaches its greatest value I=U/Z=U/R. Theoretically, the current can reach large values determined by the network voltage and the active resistance of the coil.
A)b)
3. Power factor at resonance cos= R/Z=R/R= 1, i.e. takes the largest value, which corresponds to the angle = 0. This means that the current vector and the network voltage vector coincide in direction, since they have equal initial phases i = u.
4. Active power at resonance P=R.I. 2 has the largest value equal to the full power S, at the same time the reactive power of the circuit Q=XI 2 = (X L - X C) I 2 turns out to be zero: Q=Q L - Q C = 0.
5. When voltage resonance occurs, the voltages on the capacitance and inductance are equal U C = U L= X C I=X L I and depending on the current and reactance, they can take on large values, many times higher than the supply voltage. In this case, the voltage across the active resistance turns out to be equal to the voltage of the supply network, i.e. U R= U.
Voltage resonance in industrial electrical installations is an undesirable and dangerous phenomenon, since it can lead to an accident due to unacceptable overheating of individual elements of the electrical circuit or breakdown of the insulation of the windings of electrical machines and devices, insulation of cables and capacitors with possible overvoltage in certain sections of the circuit. At the same time, voltage resonance is widely used in various types of instruments and electronic devices.
