Parallel resonance in an alternating current circuit: physical meaning and application, formulas and calculation methods

The physical phenomenon of parallel resonance is widely used in radio electronics. To build oscillatory circuits consisting of active and reactive resistances, a circuit should be assembled from resistance, capacitance, and inductance. To do this, you need to understand the purpose of resonance, finding the resistance of radio components, its main application in radio engineering, as well as the conditions for its occurrence.

General information

The electrical resistance of a conductor is the property of carrying an electric current. To build and calculate an oscillatory circuit, you need to know how to find the active and reactive resistances. Resistance for circuits powered by alternating current (CCT) is of the following types: active, reactive and total.

Active resistance ® is an ordinary resistor. Reactive consists of the following types of load: inductive and capacitive. Inductive (Xl) is the resistance of the inductor in the AC circuit, and the capacitive (Xc) is determined by the presence of capacitance in the circuit (capacitor).

When the active and reactance is added, the total resistance of the section of the electrical circuit is obtained, which is denoted by the letter Z.

Active resistance

The active resistance in the CLT is the presence of any non-reactive load. It can be calculated in the following ways: by measuring the value of resistance and by calculation method. To measure R, a device called an ohmmeter is used. The ohmmeter is part of the combined instruments for measuring electrical quantities, which are called multimeters. It is connected in parallel with the load, and the electrical circuit must be turned off to carry out measurements, since the presence of current will cause the device to fail.

There is one more method, which is calculated, but it requires knowledge in the field of physics. When calculating the value R measurements of current and voltage should be made, or rather, their amplitude values ​​(Um and Im, respectively). This can be done using appropriate devices.

A voltmeter is used to measure the voltage, and the current can be measured using an ammeter. In addition, these devices measure only the effective values ​​of voltage (Ud) and current (Id). To calculate the amplitude values, use the following formulas:

1. Um = Ud * sqrt (2).
2. Im = Id * sqrt (2).

To calculate R, which can be found using Ohm's law for the circuit section (Im = Um / R): R = Um / Im. Using the ratios of the dependences of the amplitude values ​​on the effective ones, it is possible to calculate R: R = Ud * sqrt (2) / Id * sqrt (2) = Ud / Id. In practice, a method of measuring resistance with an ohmmeter is used.

Other types of loads

If there is an inductor in the CPT, Xl appears, which only needs to be calculated. The inductive reactance is calculated using the formula, which requires the cyclic frequency (w) and the inductance of the coil (L): Xl = w * L.

Cyclic frequency calculated by the following formula, for which you only need to know the frequency of the alternating current (f) and the number of PI (3.1416): w = 2 * 3.1416 * f. The coil inductance is calculated based on the values ​​of the coil diameter (D in mm), the number of turns (n) and the winding length (l): L = (sqr (D / 10) * sqr (n)) / (4.5 * D + 10 * l). If we substitute all the ratios in the formula for calculating the inductive resistance, it turns out: Xl = 2 * 3.1416 * f * (sqr (D / 10) * sqr (n)) / (4.5 * D + 10 * l).

If a capacitor with a capacitance C is present in the CPT, then a capacitive resistance - Xl is also added, which is calculated using the following formula: Xc = 1 / (w * C) = 1 / (2 * 3.1416 * f * C). The total resistance in the CCT is indicated by the letter Z and is calculated by the formula: Z = sqrt [sqr® + sqr (Xc - Xl)]. If we substitute into the formula for the total resistance the ratios by which R, Xl and Xc are found, then we get the following formula: Z = sqrt [sqr (Ud / Id) + sqr ((1 / (2 * 3.1416 * f * C)) - (2 * 3.1416 * f * (sqr (D / 10) * sqr ( n)) / (4.5 * D + 10 * l))]. To simplify the calculation, the R, Xc and Xl values ​​can be calculated separately.

Understanding resonance

Resonance in an alternating current circuit occurs when a resonant frequency is formed, at which some resistances cancel each other out. The main signs of resonance are:

1. Phase match of U and I in the circuit.
2. The values ​​of active and impedance are the same: Z = R.
3. The amperage is maximum.
4. The drop in the value of U by R is equal to U, which is applied to the LC circuit.
5. Equality of drops U on inductance and capacitance is fulfilled, as well as the opposite in phase and more than the applied voltage: Ul> U, Ul = I * Xl = I * Xc and U = I * R.

In the latter case, the voltage gain is calculated as follows: Ku = Ul / U = sqrt (L / C) / R = p / R. This factor is called the quality factor of the circuit and is denoted by the letter Q. The characteristic impedance of the circuit is denoted by p, which is calculated by the formula: p = sqrt (L / C).

There are two types of resonance in the CLT: serial and parallel. For series resonance, minimum resistance and zero phase are the prerequisites. It is mainly used in circuits with reactive components L and C. With the parallel type of resonance, the capacitance and inductive resistances are equal, which cancel each other out. This type of connection must always be equal to the calculated value. It is widely used due to its sharp minimum impedance. Impedance is the impedance in an alternating current circuit, which is denoted by Z.

A circuit is a circuit in which the following elements are connected in parallel or in series: a resistor, an inductor and a capacitor.

This circuit forms an oscillator for the harmonic current. The presence of resistance in the circuit leads to attenuation and reduces the resonant peak frequency.

Oscillatory circuits are used in all power electronics. An example of this is a power transformer. In addition, the circuit is used to tune TVs, match antennas. It can be used as bandpass and notch filters, which are used in sensors to distribute low and high frequencies. The resonance effect is also used in medicine for microcurrent therapy and for bioresonance diagnostics.

Cases for current and voltage

In electronics, the resonance of voltages and currents is used. They differ from each other and apply in certain cases. Voltage resonance occurs when connected in series in an RLC circuit (diagram 1):

Scheme 1 - Series connection of elements.

The main condition for the occurrence of resonance is the equality of the frequencies of the power source and the oscillating circuit. In addition, Xc = Xl, they are opposite values ​​(in sign) and equal to 0. Voltages Uc and Ul are opposite in phase and cancel each other, therefore, Z = R. As a result of this, an increase in current occurs, since with a decrease in resistance according to Ohm's law, an increase in I occurs. Not only I grows, but also the U values ​​on the circuit elements. At resonance, the voltages across the capacitor and the inductor can be higher than the power supply voltage.

As the frequency increases, the value of Xl increases, and Xc decreases. When the resonant and power supply frequencies are equal, the value of Z will decrease. The resonant frequency is found by the formula: w = sqrt (1 / (L * C)). The resonance in the CLT depends on the following quantities: the frequency of the power supply - f, parameters L and C. Electrical energy is exchanged between the coil and the capacitor through a power source.

Resonance of currents in an alternating current circuit occurs when active and reactive loads are connected in parallel. Diagram 2 shows a parallel circuit:

Diagram 2 - Parallel connection in the RLC circuit.

In this case, resonance occurs when the frequencies of the power source and the resonant frequency are equal, as well as the conductivity of the capacitor (Bc) and the coil (Bl) are equal. Conductivity is the reciprocal of resistance. As the frequency of the power supply increases, the impedance rises, at which the current decreases. As a result, the current decreases and is equal to the active component. To determine the resonant frequency, you should use the algorithm for finding this value:

1. Specific conductivity for resistor, inductor and capacitor: G = 1 / R, Bl = 1 / (w * L) and Bc = w * C, respectively.
2. 1 / (w * L) = w * C.
3. The resonant frequency is calculated using the formula: w = sqrt (1 / (L * C)).

The phenomenon of resonance can lead to failure of circuit elements, devices or devices. In order to avoid this, it is necessary to make accurate calculations of the oscillatory circuits.

Parallel contour calculation

It is necessary to make a parallel circuit, the resonance frequency of which is 1.5 MHz. For its manufacture, it is necessary to carry out a calculation, based on which it will be possible to make it. The contour should be calculated accurately, since any inaccuracy can lead to negative consequences. The main task is to calculate the required coil inductance and capacitance. The calculation is carried out according tothe following algorithm:

1. Calculate the required inductance in μH at a given capacitance and frequency: L = sqr (159.12 / f) / C.
2. Calculate the number of turns (n) and the skeleton diameter (d in mm) of the coil: n = 32 * sqrt (L / d).

Let C = 2000 pF, then L = sqr (159.12 / 2) / 2000 = 5.6 μH. The number of turns for a coil with d = 3 mm: n = 32 * sqr (5.6 / 3) = 112.

This method is approximate, since the turn-to-turn space of the coil is not taken into account. Radio amateurs often use ready-made coils with a length of 15 mm with a diameter of d = 3 mm. You can calculate using another formula: n = 8.5 * sqrt (L) = 8.5 * 2.3664 = 21.

Thus, the phenomenon of resonance is used in the construction of various radio equipment and requires performing correct calculations, since even with minor errors expensive details.