The role of resonance in an oscillatory circuit. Resonance of currents and its useful applications in electrical engineering
In an oscillating circuit with inductance L, capacitance C and resistance R, free electrical oscillations tend to attenuate. To prevent the oscillations from fading, it is necessary to periodically replenish the circuit with energy, then forced oscillations will arise that will not decay, because the external EMF variable will now support oscillations in the circuit.
If the oscillations are supported by a source of external harmonic EMF, the frequency of which f is very close to the resonant frequency of the oscillatory circuit F, then the amplitude of electrical oscillations U in the circuit will begin to increase sharply, that is, electrical resonance phenomenon.
Let us first consider the behavior of capacitor C in an alternating current circuit. If a capacitor C is connected to a generator, the voltage U at the terminals of which varies according to a harmonic law, then the charge q on the plates of the capacitor will also change according to a harmonic law, as does the current I in the circuit. The greater the capacitance of the capacitor, and the higher the frequency f of the harmonic EMF applied to it, the greater the current I will be.
Associated with this fact is the idea of the so-called capacitive reactance of the capacitor XC, which it introduces into the alternating current circuit, limiting the current like active resistance R, but in comparison with active resistance, the capacitor does not dissipate energy in the form of heat.
If active resistance dissipates energy and thus limits the current, then the capacitor limits the current simply because it does not have time to accommodate more charge than the generator can provide in a quarter of the period, and in the next quarter of the period the capacitor releases energy, which has accumulated in the electric field of its dielectric, back to the generator, that is, although the current is limited, the energy is not dissipated (we will neglect losses in the wires and in the dielectric).
Now consider the behavior of inductance L in an alternating current circuit. If, instead of a capacitor, a coil with inductance L is connected to the generator, then when a sinusoidal (harmonic) EMF is supplied from the generator to the terminals of the coil, it will begin to generate Self-induced emf, since when the current changes through the inductance, the increasing magnetic field of the coil tends to prevent the current from growing (Lenz’s law), that is, it turns out that the coil introduces inductive reactance XL into the alternating current circuit - additional to the resistance of the wire R.
The greater the inductance of a given coil, and the higher the frequency F of the generator current, the higher the inductive reactance XL and the lower the current I, because the current simply does not have time to establish itself, because the self-inductive emf of the coil interferes with it. And every quarter of a period, the energy accumulated in the magnetic field of the coil returns to the generator (we will neglect losses in the wires for now).
In any real oscillatory circuit, inductance L, capacitance C and active resistance R are connected in series.
Inductance and capacitance act oppositely on the current in each quarter of the period of the harmonic EMF of the source: on the plates of the capacitor, although the current decreases, and when the current increases through the inductance, the current, although it experiences inductive resistance, increases and is maintained.
And during the discharge: the discharge current of the capacitor is initially large, the voltage on its plates tends to establish a large current, and the inductance prevents the current from increasing, and the greater the inductance, the lower the discharge current will occur. In this case, active resistance R introduces purely active losses. That is, the total resistance of Z, series-connected L, C and R, at source frequency f, will be equal to:
From Ohm's law for alternating current it is obvious that the amplitude of forced oscillations is proportional to the amplitude of the emf and depends on frequency. The total resistance of the circuit will be the smallest, and the amplitude of the current will be the largest, provided that the inductive and capacitive reactance at a given frequency are equal to each other, in which case resonance will occur. From here it follows formula for the resonant frequency of an oscillatory circuit:
When an EMF source, capacitance, inductance and resistance are connected in series with each other, the resonance in such a circuit is called series resonance or voltage resonance. A characteristic feature of voltage resonance is significant voltages on the capacitance and inductance, compared to the source emf.
The reason for this picture is obvious. According to Ohm's law, there will be a voltage Ur across the active resistance, Uc across the capacitance, and Ul across the inductance, and by making the ratio of Uc to Ur, you can find the value of the quality factor Q. The voltage across the capacitance will be Q times greater than the emf of the source, the same voltage will be applied to the inductance.
That is, voltage resonance leads to an increase in the voltage on the reactive elements by Q times, and the resonant current will be limited by the emf of the source, its internal resistance and the active resistance of the circuit R. Thus, the resistance of the series circuit at the resonant frequency is minimal.
The phenomenon of voltage resonance is used in, for example, if it is necessary to eliminate a current component of a certain frequency from the transmitted signal, then a chain of a capacitor and an inductor connected in series is placed parallel to the receiver so that the current of the resonant frequency of this LC chain is closed through it and does not reach the receiver .
Then currents of a frequency far from the resonant frequency of the LC circuit will pass into the load unhindered, and only currents close to the resonance frequency will find the shortest path through the LC circuit.
Or vice versa. If it is necessary to pass only a current of a certain frequency, then the LC circuit is connected in series with the receiver, then the signal components at the resonant frequency of the circuit will pass to the load almost without losses, and frequencies far from resonance will be greatly attenuated and we can say that they will not reach the load at all. This principle is applicable to radio receivers, where a tunable oscillating circuit is tuned to receive a strictly defined frequency of the desired radio station.
In general, voltage resonance in electrical engineering is an undesirable phenomenon, since it causes overvoltages and equipment failure.
A simple example would be a long cable line that for some reason is not connected to the load, but is still powered by an intermediate transformer. Such a line with distributed capacitance and inductance, if its resonant frequency coincides with the frequency of the supply network, will simply be broken and fail. To prevent cable destruction from accidental voltage resonance, an auxiliary load is used.
But sometimes voltage resonance plays into our hands, and not only in radios. For example, it happens that in rural areas the voltage in the network has dropped unpredictably, and the machine needs a voltage of at least 220 volts. In this case, the phenomenon of voltage resonance saves.
It is enough to connect several capacitors per phase in series with the machine (if the drive is an asynchronous motor), and thus the voltage on the stator windings will rise.
Here it is important to choose the right number of capacitors so that they accurately compensate with their capacitance, together with the inductive reactance of the windings, for the voltage drop in the network, that is, by slightly bringing the circuit closer to resonance, you can increase the dropped voltage even under load.
When an EMF source, capacitance, inductance and resistance are connected in parallel, the resonance in such a circuit is called parallel resonance or current resonance. A characteristic feature of current resonance is significant currents through capacitance and inductance, compared to the source current.
The reason for this picture is obvious. The current through the active resistance according to Ohm's law will be equal to U/R, through the capacitance U/XC, through the inductance U/XL, and by making the ratio of IL to I, you can find the value of the quality factor Q. The current through the inductance will be Q times greater than the source current, the same current will flow every half cycle into and out of the capacitor.
That is, the resonance of the currents leads to an increase in the current through the reactive elements by Q times, and the resonant EMF will be limited by the EMF of the source, its internal resistance and the active resistance of the circuit R. Thus, at the resonant frequency, the resistance of the parallel oscillatory circuit is maximum.
Similar to voltage resonance, current resonance is used in various filters. But when included in a circuit, a parallel circuit acts in the opposite way than in the case of a series one: installed parallel to the load, a parallel oscillating circuit will allow the current of the resonant frequency of the circuit to pass into the load, since the resistance of the circuit itself at its own resonant frequency is maximum.
Installed in series with the load, a parallel oscillating circuit will not pass the resonant frequency signal, since all the voltage will drop across the circuit, and the load will receive a tiny fraction of the resonant frequency signal.
Thus, the main application of current resonance in radio engineering is the creation of high resistance for a current of a certain frequency in tube oscillators and high-frequency amplifiers.
In electrical engineering, current resonance is used to achieve a high power factor for loads that have significant inductive and capacitive components.
For example, they are capacitors connected in parallel to the windings of asynchronous motors and transformers operating under a load below the rated load.
Such solutions are resorted to precisely in order to achieve current resonance (parallel resonance), when the inductive reactance of the equipment is made equal to the capacitive reactance of the connected capacitors at the network frequency, so that reactive energy circulates between the capacitors and the equipment, and not between the equipment and the network; so that the network supplies energy only when the equipment is loaded and consumes active power.
When the equipment is idle, the network is connected in parallel to a resonant circuit (external capacitors and inductance of the equipment), which represents a very large complex resistance for the network and allows it to decrease.
It is clear that when w = w 0, when φ v = φ f, force and speed have the same directions at any moment of time, the work of force is positive all the time. This means that the energy of the oscillatory system is replenished all the time. Under these conditions, equilibrium between the replenishment of energy into the oscillatory system and its conversion into internal energy occurs when the oscillations build up to the greatest amplitude. If w > w 0 or w 0< 0 то, между f и v имеется разность фаз. В этом случае сила и скорость имеют одинаковые направления лишь в течение части периода. В течение же другой части периода эти величины имеют противоположные направления. В первом случае работа положительна, и энергия колебательной системы пополняется, а во втором случае работа отрицательна, и энергия от колебательной системы отнимается. В результате общее поступление энергии в колебательную систему при малых и очень больших частотах невелико, и при данном трении устанавливаются вынужденные колебания малой амплитуды.
7 Accounting and use of the phenomenon of resonance during mechanical forced vibrations
The residential buildings and industrial buildings around us, railways and bridges, airplanes and ships, spaceships and rockets, hydraulic turbines and internal combustion engines are oscillatory systems in which forced oscillations can occur under certain conditions. With large amplitudes of these vibrations, the structure may collapse. Therefore, it is necessary to take into account the possibility of resonance. In some cases, the phenomenon of resonance in mechanical oscillatory systems can be used to achieve a certain positive effect
a) Examples of positive effects. The phenomenon of resonance is widely used in technology. Thus, special vibrator-compactors are used to compact loose base under foundations and roads, as well as to compact concrete. There are a large number of designs of such vibrators, but the main part of each of them is a solid base on which a motor with an unbalanced flywheel or a system of unbalanced weights is mounted. When the engine is running, loads mounted on its axle (or the flywheel) cause vibrations of the entire installation. To obtain large amplitudes, the natural frequency of vibration of the seal is made equal to the frequency of vibration of the motor shaft. The vibrations of the vibration compactor are transmitted through the platform to the soil or concrete.
Vibrators similar to those described above are used for vibratory driving of piles, shunts, pipes, etc. To vibrate piles, a powerful vibrator is installed on its upper base. When the engine is turned on, the pile begins to vibrate, the soil under the pile “liquefies” and it sinks under its own weight. This method of driving piles and pipes has found particularly wide application in the construction of marine and lake structures.
b) Examples of dangerous resonant vibrations in mechanical systems.Electric engines, steam and gas turbines, internal combustion engines, due to the imbalance of the rotating masses, are a source of vibrations transmitted to the bases on which they are installed.
If the engine is rigidly mounted on the foundation, then vibrations from it are transmitted to the building in which the machine is installed, as well as to nearby structures through the ground.
If the oscillatory system has low friction, then only a small part of the energy supplied to it is converted into the internal energy of the system. Under these conditions, when the frequency of the forcing vibrations coincides with the natural frequency of the oscillatory system, resonance occurs, and the amplitude of the forced vibrations can reach large values and cause destruction of the building or foundation.
If the frequency of natural oscillations of the circuit coincides with the frequency of changes in the external force, then the phenomenon of resonance occurs. In an electric oscillatory circuit, the role of an external periodic force is played by a generator, which ensures a change in the electromotive force according to the harmonic law:
whereas natural electromagnetic oscillations occur in the circuit with a frequency ω o. if the active resistance of the circuit is small, then the natural frequency of oscillations is determined by the formula:
The current strength during forced oscillations (or the voltage on the capacitor) should reach its maximum value when the frequency of the external emf (1) is equal to the natural frequency of the oscillatory circuit:
Resonance in an electrical oscillatory circuit is the phenomenon of a sharp increase in the amplitude of forced oscillations of current (voltage on a capacitor, inductor) when the natural frequency of oscillations of the circuit and the external emf coincide. Such changes during resonance can reach multiples of hundreds of times.
In a real oscillatory circuit, the establishment of amplitude oscillations in the circuit does not occur immediately. The maximum at resonance is higher and sharper, the lower the active resistance and the greater the inductance of the circuit: . Active resistance R plays a major role in the circuit. After all, it is the presence of this resistance that leads to the conversion of the electric field energy into the internal energy of the conductor (the conductor heats up). This suggests that resonance in the electrical oscillating circuit should be clearly expressed at low active resistance. In this case, the establishment of amplitude oscillations occurs gradually. Thus, the amplitude of current fluctuations increases until the energy released during the period on the resistor is equal to the energy entering the circuit during this time. Thus, at R → 0, the resonant value of the current increases sharply. Whereas, with increasing active resistance, the maximum value of the current decreases, and it makes no sense to talk about resonance at large values of R.
Rice. 2. Dependence of the voltage amplitude on the capacitor on the emf frequency:
1 – resonance curve with circuit resistance R1;
2 – resonance curve with circuit resistance R2;
3 – resonance curve with circuit resistance R3
The phenomenon of electrical resonance is widely used in radio communications. Radio waves from various transmitting stations excite alternating currents of different frequencies in the radio receiver antenna, since each transmitting radio station operates at its own frequency.
An oscillating circuit is inductively coupled to the antenna. Due to electromagnetic induction, alternating emfs of the corresponding frequencies and forced oscillations of the current strength of the same frequencies arise in the loop coil. But only at resonance will the fluctuations in current in the circuit and voltage in the circuit be significant. Therefore, of all the frequencies excited in the antenna, the circuit selects only oscillations whose frequency is equal to the natural frequency of the circuit. Tuning the circuit to the desired frequency ω0 is usually done by changing the capacitance of the capacitor.
In some cases, resonance in an electrical circuit can be harmful. So, if the circuit is not designed to operate under resonance conditions, then the occurrence of resonance will lead to an accident: high voltages will lead to insulation breakdown. This kind of accident often happened in the 19th century, when people had a poor understanding of the laws of electrical vibrations and did not know how to calculate electrical circuits.
Resonance. Its application
Resonance in an electrical oscillatory circuit is the phenomenon of a sharp increase in the amplitude of forced oscillations of current strength when the frequency of the external alternating voltage coincides with the natural frequency of the oscillatory circuit.
Use of Resonance in Medicine
Magnetic resonance imaging, or its abbreviated name MRI, is considered one of the most reliable methods of radiation diagnostics. The obvious advantage of using this method to check the condition of the body is that it is not ionizing radiation and gives fairly accurate results when studying the muscular and joint systems of the body, and helps with high probability to diagnose various diseases of the spine and central nervous system.
The examination process itself is quite simple and absolutely painless - all you will hear is just loud noise, but the headphones that the doctor will give you before the procedure will protect you well from it. There are only two types of inconvenience that cannot be avoided. First of all, this applies to those people who are afraid of closed spaces - the diagnosed patient lies down on a horizontal bed and automatic relays move him inside a narrow pipe with a strong magnetic field, where he remains for about 20 minutes. During the diagnosis, you should not move so that the results are as accurate as possible. The second inconvenience that resonance imaging causes when examining the pelvis is the need to fill the bladder.
If your loved ones wish to be present during the diagnosis, they are required to sign an information document according to which they are familiar with the rules of behavior in the diagnostic room and have no contraindications for being near a strong magnetic field. One of the reasons for the impossibility of being in the MRI control room is the presence of foreign metal components in the body.
Use of resonance in radio communications
The phenomenon of electrical resonance is widely used in radio communications. Radio waves from various transmitting stations excite alternating currents of different frequencies in the radio antenna, since each transmitting radio station operates at its own frequency. An oscillatory circuit is inductively coupled to the antenna (Fig. 4.20). Due to electromagnetic induction in the loop coil, alternating emfs of the corresponding frequencies and forced oscillations of the current strength of the same frequencies arise. But only at resonance the fluctuations in the current in the circuit and the voltage in it will be significant, i.e., from the oscillations of various frequencies excited in the antenna, the circuit selects only those whose frequency is equal to its own frequency. Tuning the circuit to the desired frequency is usually done by changing the capacitance of the capacitor. This usually involves tuning the radio to a specific radio station. The need to take into account the possibility of resonance in an electrical circuit. In some cases, resonance in an electrical circuit can cause great harm. If the circuit is not designed to operate under resonance conditions, its occurrence can lead to an accident.
Excessively high currents can overheat the wires. High voltages lead to insulation breakdown.
Accidents of this kind often happened relatively recently, when people had a poor understanding of the laws of electrical oscillations and did not know how to correctly calculate electrical circuits.
With forced electromagnetic oscillations, resonance is possible - a sharp increase in the amplitude of current and voltage oscillations when the frequency of the external alternating voltage coincides with the natural frequency of oscillations. All radio communications are based on the phenomenon of resonance.
