Experimental Q-meter. Oscillatory circuit Quality factor depends on what

When working with equalizers, we most often use only two parameters - Freq, which defines the center frequency of the filter and Gain, which specifies the gain at the center frequency of the filter. To this list, you can add another choice of the type of equalizer filters, but in almost all modern software equalizers this choice is automatic and depends on the initial placement of the node in the frequency range. If you click in the 20-30 Hz region, a high-pass filter will most likely be created; if you create a node in the region of 60-70 Hz, a low-frequency shelf will most likely be created; if you create a node above 100 Hz, a bell will be created, and so on. Of course, for each equalizer, the frequency values ​​​​for determining the type of filters will be different, but the market trend is as follows - a modern equalizer should determine the types of equalizer filter curves automatically. Thus, we have only two parameters left (Freq, Gain), with which we carry out manipulations. Something is missing from this list, isn't it?

Along with the parameters of the center frequency and filter gain, there is another extremely important parameter - the quality factor of the filters ( Q), which defines the width of the boosted or attenuated bandwidth and is defined as the ratio of the center frequency to the width of this band, which lies within 3 dB of the gain at the center frequency. Simply put, the higher the Q factor, the narrower the bandwidth, and the lower the Q factor, the wider the bandwidth. All this, first of all, concerns bell-shaped filters. For shelving and low-cut filters, the Q factor determines how steeply the filters roll off at the center frequency. Thus, a tool appears in your hands that can form frequency landscapes - from gentle hills to sheer cliffs.

How to use the quality factor (Q) in practice?

There are a few important things to consider when setting the quality factor:

1. By amplifying the frequency band, we reduce the value of the quality factor

The main task of equalization is, first of all, to obtain the optimal balance of frequencies within individual instruments, which ultimately contributes to balancing the entire mix. Based on this, any frequency amplification should be soft and accurate. The human ear is very sensitive to too loud frequency ranges, so in order to maintain sound balance when boosting frequencies, it is important to use wide bands corresponding to low Q values.

2. By weakening the frequency band, we increase the value of the quality factor

Any cut or attenuation of frequencies entails a fairly significant change in the internal balance of the instrument and, accordingly, its sound. With the help of attenuation of frequency bands, many issues can be solved, including the suppression of dirt, noise, humming, rumble, wadding, whistling and other unwanted overtones, but at the same time incorrect setting Adjusting the quality factors of the filters can significantly damage the instrument, making it sound dull, thin, and sluggish. To avoid these unpleasant things, it is enough to increase the quality factor of the filters and attenuate rather narrow frequency ranges. Thus, you will remove the excess, while leaving all the useful frequencies. By using extremely high Q values ​​of the bell filter, you can create a notch filter that is great for suppressing a particular frequency or narrow band of frequencies. This is useful when you want to suppress very strong resonances or remove static noise such as 50 or 60 Hz power hum, depending on the region in which the recording was made.

3. Don't use high cut-off slopes for low-cut filters

At one time, I dreamed of finding an equalizer that would have a low-cut filter that could cut frequencies at an angle of 90 degrees, that is, a kind of brickwall filter. But when I found such a filter in IZotope Ozone and turned it on, I realized that it sounds very unmusical. Indeed, the rejection of frequencies below the center frequency of the filter was impressive - the filter cut everything, but was that what I really needed? I wanted to get a clean, neat, accurate and pleasant to hear cut, but in the end I got a beautiful picture for the eyes and a terrible phase shift for the ears. Thus, I realized that when adjusting the quality factor (slope) of low-cut filters, it is rather not the degree of frequency suppression that should be taken into account, but rather the suppression / musicality tandem. The most musical sound cut filters with suppression of 6 and 12 dB per octave. If you need to use filters with a rejection of 24 dB per octave or higher, it is better to use linear phase filters that do not create phase distortion. When using high-slope low-cut filters on individual tracks, there may not be any particular problems, but if you use such filters on subgroups, or even more so on the master channel, be prepared for the fact that the instruments may lose their localization, and the stereo image will “float”.

4. Study the documentation for your equalizers

Many classic analog equalizers (for example, API 550), and their emulations, respectively, do not use constant value quality factor relative to the gain, and proportional, that is, the lower the gain, the lower the quality factor, and vice versa, the higher the gain, the higher the quality factor. Consider such features in the behavior of individual devices so that the mixing process is meaningful, and not blind work. The dependence of the Q parameter on Gain can also be found in many software equalizers - Type 3 and Type 4 in Sonnox Oxford EQ work in an "analog" way: the difference between these modes is that at the same gain level, the bandwidth at low Gain values ​​for Type 3 will be narrower than Type 4, but at the maximum Gain value, the bandwidth for Type 3 will be the same as for Type 4.

5. The low Q bandwidth affects more than the narrow region around the center frequency of the filter

Have you ever wondered why when you use a 10 kHz high-frequency shelf, instruments start to sound very juicy, and not just airy? The thing is, the more you boost the 10kHz high-frequency shelf, the more it will capture the lower frequencies, thereby amplifying not only the high frequencies, but also the high mids. The amplification of these lower frequencies, and not the top from 10 kHz, gives this effect of brightness and juiciness. The flatter the slopes of the shelving filters, the more frequencies will be captured away from the center frequency of the filter. Keep this in mind and always ask yourself what do you really want to increase or decrease in reality? Do you want to manipulate all this huge frequency range inside the shelf, or are you really interested in a particular frequency near it?

Experimental Q-meter

Lloyd Butler, VK5BR
The article describes the quality factor Q, a method for measuring the quality factor, inductance, capacitance using a Q-meter and the development of an experimental Q meter.

Introduction

For many years, the Q-meter (Q-meter) has been the instrument of choice in RF circuit research labs. In modern labs, the Q-meter is being replaced, in most cases, by more exotic (and more expensive) impedance meters, and today it is no longer possible to find a manufacturer that still produces Q-meters. For a radio amateur, a Q-meter is a very important piece of equipment needed, and the author gives some of his thoughts on how a simple Q-meter can be made for your laboratory. For those unfamiliar with this device, some introductory concepts about Q and its measurement are included.

What is quality factor (Q) and how is it measured?

The Q factor or quality factor of an inductor is usually expressed as the ratio of its series reactance to its active resistance. We can also express the quality factor of a capacitor as the ratio of its series reactance to its resistance, although capacitors are usually characterized by a D factor or dissipation, which is the reciprocal of Q.

A tuned circuit at resonance is characterized by a quality factor (which is denoted) Q. In this case, Q is equal to the ratio of the reactance of an inductive or capacitive nature to the total series resistance of losses in the resonant circuit. The greater the loss resistance and the lower the quality factor Q, the greater the power loss in each generation cycle in the resonant circuit and, hence, the greater the power required for generation to occur.

In another way, the quality factor Q can be derived as follows:

Q = fo/Δf, where fo is the resonant frequency, Δf is the level band - 3 dB
(See Note)

Sometimes we use the expression: “loaded quality”, for example, in the case of transmitter circuits, and, in this case, the active resistance for calculating the value of the quality factor (Q) is the value of the series resistance of the unloaded resonant circuit plus the additional active loss resistance, reflected, in its own turn back into the circuit from its associated load.

There are other ways to express Q. The quality factor can be expressed as the ratio of the equivalent parallel (loop) active resistance to the reactance of an inductive or capacitive nature. Series loss resistance can be converted to equivalent parallel resistance using the following formula:

R(shunt) = R(series). (Q² + 1)

Finally, the Q or quality factor of the resonant circuit is equal to the voltage increase factor and Q can also be expressed as the ratio of the voltage developed across the reactive elements to the voltage applied in series with the circuit to obtain the effective voltage. To measure the quality factor, Q-meters use precisely this principle.

The basic circuit of the Q-meter is shown in Fig. 1. The output terminals are used to connect the tested inductances (Lx), which in the circuit diagram are tuned to the resonant frequency using the KPI (C). Clamps are also provided for additional connection capacitance (Cx) if needed. The resonant circuit is excited from a tunable signal source, which develops a voltage across a resistor connected in series with the circuit. The resistor should have a small resistance compared to the loss resistance of the measured components, such that it can be neglected. The required resistance value is a small fraction of an ohm. Measurements are made to establish the value of the input AC voltage across the resistor connected in series and the value of the output AC voltage at the terminals of the KPI setting. For output measurements, a high input impedance circuit must be used so as not to load the resonant circuit with the measurement circuit.

Rice. 1. Block diagram of a Q-meter.
At resonance Lx and Cx, Q = V2/V1
*V2 meter is calibrated to read the voltage across capacitor C.

The Q-factor is measured by tuning the signal generator and/or setting the device tuning KPI to the loop resonance position corresponding to the maximum output voltage. The quality factor Q is calculated as the ratio of the output voltage on the resonant circuit to the voltage applied to it. In practice, the level of the signal source (signal generator) is adjusted to the calibration point on the scale of the meter that measures the applied voltage, and Q is directly read from the calibrated scale of the device that measures the output voltage of the circuit.

Some applications of the Q-meter

The Q-meter can be used for many purposes. As its name suggests, it can be used to measure the quality factor Q and is commonly used to measure the quality factor of inductors. Since the internal capacitor has an air dielectric, its loss resistance is negligible compared to that of inductors and therefore the quality factor is measured from them.

The Q value can be measured over a wide range different types coils and in different frequency ranges. Industrially manufactured miniature coils such as the Siemens B78108 or the Lenox-Fugal Nanored types, made with ferrite cores and operating at frequencies up to 1 MHz, have a typical quality factor Q in the region of 50 to 100. transmitter output and operating at frequencies above 10 MHz have an expected Q value in the region of 200…500. For some coils, the quality factor is quite low and amounts to 5 ... 10 at some frequencies; such coils are usually not used in selective systems or narrow-band filters. The Q-meter will be of invaluable help here.

(At one time, I was approached by a shortwaver who did not have a band-pass filter tuned in a newly built transceiver. The quality factor of his coils turned out to be so low that it was impossible to catch any resonances. Upon closer examination, it turned out that the PF coils were wound with wire not PELSHO, but PELSKO, i.e., constantan! The quality factor of the coils strongly depends on the active resistance of the wire, the smaller it is, the higher the quality factor of the coil, all other things being equal. If then a Q-meter was at hand, you would not have to rack your brains for a long time and analyze the cause - UA9LAQ).

The tuning capacitor (C) of the Q-meter has a graduated scale in picofarads (pF), so in conjunction with a calibrated signal generator, from which the measurement voltage is applied to the Q-meter, the value of inductance (Lx) can also be determined. The oscillatory circuit is simply tuned to resonance at the frequency of the signal generator or by changing the frequency of the latter or / and using the KPI in the Q-meter (or external in the circuit) according to the maximum voltage that is recorded on the instrument's meter, the desired inductance (Lx) is then calculated from the known formula:

Lx = 1/4π²f²C

If we take L, μH, C, pF and f, MHz, then the formula will turn into:

25330/f²C

Another use of the Q-meter would be to measure the capacitance values ​​of small (in terms of capacitance) capacitors. Provided that the capacitance of the measured capacitor is less than the maximum capacitance of the internal KPI, it is very easy to measure. Firstly, the connected capacitor resonates with the selected inductance at a certain frequency, which is determined when setting the voltage from the signal generator, with the KPI of the device settings set to the minimum mark of its capacitance on a calibrated scale. Then, the capacitor under test is turned off, at the same frequency from the signal generator, the KPI of the setting is set to the resonance position again (by increasing its capacitance). The difference in capacitance between the two values ​​on the KPI scale will be equal to the capacitance of the capacitor connected to determine the capacitance (i.e., the capacitance is measured by the substitution method in the resonant circuit - UA9LAQ). Large capacitance values ​​can be measured by changing the frequency of the signal generator to achieve resonance and using the appropriate formula for resonance.

Not only does the choice of an “insignificant” inductor lead to a low quality factor of the circuit, some types of capacitors (and instances) used in circuits have a high loss resistance, which also leads to a decrease in the quality factor of the circuit. Small ceramic capacitors are often used in resonant circuits, but many have high loss resistances that vary widely within the same type. If it is necessary that ceramic capacitors be used in a high-Q resonant circuit, it is prudent to select them for the lowest loss resistance and a Q-meter can be of invaluable service here. To do this, you need to take a high-quality coil (with Q at least 200) and connect it to the device, bring it into resonance with the KPI (C) included in the Q-meter, and then, with separate capacitors taken for testing, connected in parallel. A large loss in the quality factor of the circuit, when connecting capacitors, will quickly identify instances unsuitable for use.

Distributed coil capacity

The direct measurement of the quality factor of inductors mentioned above is based on a circuit consisting of two components: inductance and capacitance. The coils also have a distributed (interturn) capacitance (C d), and if this capacitance is a significant part of the tuning (concentrated), then we will get a lower quality factor of the circuit than expected. A large distributed capacitance value is a common occurrence when we deal with multi-turn, turn-by-turn and multi-layer coils.

The actual quality factor can be calculated from Q e , as follows:

Q = Q e (1 + C d / C)
where C d = allocated capacity
C = setting capacity

The error in the Q value is reduced when resonating with a large value of the tuning capacitor, or the distributed capacitance can be measured and substituted into the formula above. Two methods for measuring distributed capacitance are described in the "Boonton Q Meter Handbook". The simplest of these is considered to be fairly accurate for distributed capacitances greater than 10 pF and is described as follows:

1. Using the device tuning capacitor (C), set the value of C1 (say 50 pF), bring the oscillatory circuit formed together with the reference inductance into resonance by adjusting the frequency of the signal generator.

2. Set the signal oscillator frequency to half the resonance frequency and tune the circuit back to resonance by rotating rotor C to obtain a new capacitance value C2.

3. Calculate the distributed capacity using the formula: C d \u003d (C2 -4C1) / 3

Another manifestation of distributed capacitance in an inductor is that the inductance (calculated from the settings of the tuning capacitor and the signal generator) is higher than it actually is. And, again, the error value can be reduced by using a larger value of the tuning capacitor C and / or adding to C the separately calculated capacitance C d in the calculation.

Experimental copy

From a small scheme and experiments, let's move on to practical scheme Q-meter shown in Fig. 2. The source of the signal is not given here, since the laboratory of an experimenter in the field of radio is unthinkable without such devices as a signal generator, GSS, and they can be used with a Q-meter as an attachment. Adding a signal source inside the package (as would be the case with a commercially manufactured Q-meter) will lead to a complication of the circuit and dimensions of the device, which is undesirable, especially at the initial stage of design activity.

Rice. 2. Diagram of a Q-meter.
Tested inductances Lx and capacitances Cx are connected to terminals 1-4.
R13 (0.2 ohm) consists of five 1 ohm resistors connected in parallel. For calibration, set the GSS signal level to the middle of the M1 scale.

In designing the circuit, the biggest problem was how to measure the voltage of the signal source across a resistance in small fractions of an ohm. My first thought was to use a multi-wire toroidal step-down transformer connected to a high impedance source. (In such a transformer, the coupling coefficient is high and the leakage inductance is low.) But, in this case, the leakage inductance reflected in series with secondary winding It turns out, after all, that it was big and the idea had to be abandoned.

Another idea was to use the low resistance of a powerful voltage follower source to directly inject the signal into the measuring circuit. For these purposes, a repeater circuit was used, which is designated as V2-V3 in Fig. 2. This type of circuit has a wide bandwidth with a very low source impedance and was previously used as a buffer for transmitting video signals into a low-impedance transmission line. To achieve a low source resistance, the follower is set to a high collector current of 100mA. Hence, transistors V2 and V3 in TO5 cases get quite hot. The circuit works well for low frequencies, and at high (10 ... 30 MHz) the source resistance begins to grow, which affects the Q values, which become lower than expected.

In the scheme of Fig. 2, a voltage follower stage is used, but the stage is used to obtain a voltage across the resistor R13, the resistance of which is only a fraction of an ohm, as already mentioned earlier. The resistance value is indeed 0.2 ohm. Of course, the follower cannot directly drive such a low-resistance load, which is connected through resistors R11 and R12 (the sum of the resistance of which is 25 ohms), so the output voltage is 125 times less than injected into the resonant circuit.

The final stage of the power amplifier is driven by an emitter follower (V1). It has a high input impedance and hence the load resistance applied to the external signal source is mainly determined by resistors Rl and R3 connected in parallel (approximately 2300 ohms).

The inductance to be tested (Lx) is connected to terminals 1 and 2, and an external capacitance (Cx) if needed is connected to terminals 3 and 4. Tuning is done by KPI Ca, a conventional sectional capacitor from a broadcast receiver, with the sections connected in parallel to obtain the maximum total capacitance about 800 pF.

The high-impedance input to the voltmeter is provided by a stage on a V4 field-effect transistor, turned on by a source follower, a peak detector (C6, D1, R17, C8, R20) and an operational amplifier N1-A ensure the operation of the device with a maximum needle deflection current of 100 μA. A second NI-B op amp in the uA747 package provides voltage offset for the N1-A.

The switch (S1) has three positions. The first position, labeled CAL, is used to set the signal level, which is set by the deviation of the arrow of the M1 instrument to the middle position. (At input V1, the signal level should be about 1 Vpp). If the signal level is set correctly, switch position 2 provides a direct reading of Q from 0 to 100 on the instrument scale, and switch position 3 provides a direct reading of Q from 0 to 500. For low Q values, the calibration level in switch position 1 is set to the full scale of the instrument, so that in position 2 of the switch, you can measure the quality factor Q in the range of 0 ... 50.

The levels of the signals applied to the AC voltmeter circuit are proportional to be above the non-linear portion of the diode characteristics, but within the signal voltage swing due to the supply voltage. In position 1 of the switch (CAL) - “Calibration”, the voltage gain of N1-A is 2, in position 2 - 5, in position 3 - 1.

The supply voltage is chosen to be 12 V, but its exact value is not critical. The current consumed by the power supply is quite large (about 100 mA) due to the high consumption of the repeater on V2-V3.

Job

Comparing the Q values ​​with the values ​​obtained on other instruments, we find that the Q-meter is quite accurate and quite suitable for amateur radio measurements. For very high Q values ​​(approximately 400), with Ca set to minimum, the Q factor is slightly lower. This is due to losses in the resistor R14 connected in series with the input capacitance V4. (The result obtained can be increased by the exclusion of R14, but, without it, V4 is prone to instability when Ca is connected directly to its input). For a larger value of Ca, the input capacitance V4 is masked, since the error, in this case, is a smaller percentage and is less noticeable.

The accuracy of measuring inductance and capacitance is determined by the accuracy of the signal source and the accuracy of the graduation of the scale of the instrument's capacitor. For those interested in instrument manufacture, scale calibration can be done by direct capacitance measurement using a capacitive bridge or other Q-meter. Another method is to use signal source calibration in conjunction with a calibrated inductor. For various positions of the KPI rotor, the frequency of the signal source is set so as to obtain resonance in the circuit with a calibrated inductor, then the capacitance is calculated by the formula. Taking the value of the inductance of the reference coil and the frequency of the signal generator as precision values, we will thus obtain, probably, the most The best way, since this takes into account both the additional capacitance of the wires and the active input capacitance V4.

The device worked perfectly in the frequency range of 100 kHz ... 40 MHz. An attempt to use the device at frequencies above 40 MHz led to false results, but the operation of the device in the VHF band can probably be carried out by applying the appropriate installation, details, possibly correction calibration tables.

Assembly notes

Transistors V2-V3 (type 2N2218) have a maximum operating frequency of 250 MHz and a power dissipation of 680 mW at 50 degrees Celsius. They can be replaced by other transistors with identical characteristics. In the same way, transistors: V1 (2N3563) and V4 (FET (PT) - 2N3819) can be replaced by other small signal transistors having a high cutoff frequency.

Results

This article provides ideas on how to build a simple Q-meter and how to get it up and running. Other applications of this versatile instrument can be found in manual pages such as those prepared by Boonton Radio Corporation.

Literature:

1. Manual of Radio Frequency Measurements for the Q Meter. Boonton Radio Corp.

Application. Source Preamplifier

The experimental Q-meter shown above requires an oscillator input level of about 1 Vpp. Not all signal generators provide such a level at their output; to work with such generators, a preamplifier must be turned on at the signal input of the device.

Rice. 3. Q-meter preamplifier (100 kHz…40 MHz).

The broadband amplifier shown in Fig. 3 provides a gain of about 10 over the Q-meter operating range of 100 kHz to 40 MHz. Installed at the input of the Q-meter, it increases the sensitivity of its input to about 0.1 Vpp, which expands the fleet of connected signal sources, generators. There are no gain controls in the device, as generators usually have them: adjustable attenuators for setting the output signal level.

For those who will repeat the Q-meter: a preamplifier is a useful addition when working with signal generators that have a low output voltage level.

RF divider circuit modifications

The original divider circuit in Fig. 1 consists of R11, R12 and R13. This divider divides the RF voltage by 125 so that the voltage across resistor R13 (0.2 ohms) is 1/125 of the voltage coming from the power amplifier. All this works great at low frequencies, but as the frequency increases, the shift (offset) factor decreases (frequency dependence of the voltage divider in conjunction with the connecting wires - UA9LAQ), which gives an overestimation of the Q value, relative to the real ones.

The explanation for this is as follows: the circuit from pin 1 through R13 to pin 3 is a short conductor that has a finite amount of inductance. If we take the length of the conductor equal to 5 cm, then its inductance will be approximately 0.02 ... 0.03 μH, depending on the diameter of the conductor. If this inductance has a small value, then its reactance at frequencies of 6 ... 8 MHz will be approximately 1 Ohm. It is quite clear that such a high reactance, connected in series with the 0.2 ohm resistor R13, increases the proportion of voltage at pins 1 and 3 with increasing frequency.

To neutralize this effect, a modification of the circuit was carried out, shown in Fig. 4. The idea is to create an opposite field around R13 by the current flowing through it, while the existing inductance is destroyed (inductance compensation, like compensation for the resistance of connecting wires with acoustic systems in UZCH, a special case - UA9LAQ). To obtain a field of sufficient magnitude, three conductors connected in series, carrying the input current, are attached to five resistors connected in parallel, forming R13 with a resistance of 0.2 ohms.

Another addition is the 43 ohm resistor R25. The wires wrapped around R13 form a coil, and resistor R43 is added to lower the Q of that coil and prevent instability in the amplifier circuits that would result if resistor R25 were not added.

It was checked that in the Q-meter the bias ratio remained practically unchanged up to 40 MHz, with slight fluctuations in the frequency range of 20…30 MHz. The modification significantly increases the accuracy of the direct measurement of Q.

Rice. 4. Modifications of the RF divider circuit

Q - the meter is still working for me, but to improve the accuracy of setting the frequency, I connect a frequency meter to the signal generator (GSS). The circuit is brought into resonance and the M1 device is set to the last scale mark (full scale) by adjusting the voltage coming from the GSS. The frequencies are set, then, according to the indications of the M1 device at the level of 0.7 from the maximum on one and the other side of the resonant one, their values ​​are read from the frequency meter scale and recorded. The ratio of the center frequency (resonant) to the difference between the two side frequencies (recorded at a level of 0.7) is calculated as Q.

(Letters come to me asking me to issue a universal formula for calculating the inductance of coils, since, more and more often, in the descriptions of designs, not winding data are given, but the inductance of these circuit elements. I answer that there is no universal formula, since the coil inductance depends on many factors and, taking advantage of the moment, I would like to suggest using the above-described device for preliminary adjustment of coils made from your materials with loop capacitors to the frequencies you need - UA9LAQ).

(From the Australian magazine "Amateur Radio" November 1988)

Free translation from English with the permission of the author: Viktor Besedin (UA9LAQ) [email protected]
Tyumen April, 2005

Any radio receiver is based on the principle of selective reproduction of a signal modulated by a certain carrier frequency, which, in turn, is determined by the resonance of the oscillatory circuit, which is the main element of the receiver circuit. The quality of the received signal depends on how correctly this frequency is chosen.

The selectivity, or selectivity of the receiver, is determined by how much the signals that interfere with stable reception will be attenuated, and the useful ones will be amplified. The quality factor of the circuit is a value that objectively demonstrates in numerical terms the success of solving this problem.

The resonant frequency of the circuit is determined by the Thompson formula:

f=1/(2π√LC), where

L is the value of the inductance;

In order to understand how oscillations occur in a circuit, you need to understand how it works.

Both capacitive and inductive loads prevent electric current, but they do it in antiphase. Thus, they create the conditions for the occurrence of an oscillatory process, much like what happens on a swing, when two riders push them in different directions alternately. Theoretically, by changing the value of the capacitance of a capacitor or coil, it is possible to ensure that the resonant frequency of the circuit coincides with the carrier frequency of the transmitting radio station. The more they differ, the less quality the signal will be. In practice, the receiver is tuned by changing

The whole question is how sharp the peak will be on the chart. frequency response receiving device. This is how you can visually understand how the useful signal will be amplified, how much interference is suppressed. The quality factor of the circuit is the parameter that determines the selectivity of reception.

It is determined by the formula:

Q=2πFW/P, where

F is the resonant frequency of the circuit;

W - energy in the oscillatory circuit;

P - power dissipation.

The quality factor of the circuit when the capacitor and inductance are connected in parallel is determined by the following formula:

With the values ​​​​of the inductance and capacitance of the capacitor, everything is clear, and as for R, it reminds that, in addition to the coil, it also has an active component. Therefore, the circuit diagram is often depicted, including three elements in it: capacitance C, inductance L and R.

The quality factor of the circuit is a value inversely proportional to the damping rate of oscillations in it. The larger it is, the slower the relaxation of the system occurs.

In practice, the most significant factor affecting the quality factor of the circuit is the quality of the coil, depending on its core, on the number of turns, the degree of insulation of the wire, and on its resistance, as well as on losses during the passage of high frequency currents. Therefore, to adjust the receive frequency, variable capacitors are usually used, which are two sets of plates that enter and leave each other as they rotate. Such a system is typical for almost all non-digital radio receivers.

However, in receivers with digital tuning also have their own oscillatory circuits, just their resonant frequency changes differently.

In the article we will tell you what an oscillatory circuit is. Serial and parallel oscillatory circuit.

Oscillatory circuit - device or electrical circuit containing the necessary radio-electronic elements to create electromagnetic oscillations. It is divided into two types depending on the connection of elements: consistent And parallel.

The main radioelement base of the oscillatory circuit: Capacitor, power supply and inductor.

A series oscillatory circuit is the simplest resonant (oscillatory) circuit. It consists of a series oscillatory circuit, of series-connected inductors and capacitors. When an alternating (harmonic) voltage is applied to such a circuit, an alternating current will flow through the coil and the capacitor, the value of which is calculated according to Ohm's law:I \u003d U / X Σ, Where X Σ- the sum of the reactances of the coil and capacitor connected in series (the sum module is used).

To refresh our memory, let's recall how the reactances of a capacitor and an inductor depend on the frequency of the applied alternating voltage. For an inductor, this dependence will look like:

It can be seen from the formula 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 an inductor, a capacitor does the opposite - as the frequency increases, the reactance decreases. The following figure graphically represents the dependencies of the reactances of the coil X L and capacitor X C from cyclic (circular) frequency ω , as well as a graph of the dependence on the frequency ω their algebraic sum X Σ. The graph, in fact, shows the dependence on frequency of the total reactance of a series oscillatory circuit.

It can be seen from the graph that at some frequency ω=ω p, on which the reactances of the coil and capacitor are equal in absolute value (equal in value, but opposite in sign), the total resistance of the circuit vanishes. 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 internal resistance current source (generator). Such a frequency at which the considered phenomenon is observed, called resonance in physics, is called the resonant frequency or the natural frequency of the circuit oscillations. It can also be seen 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 the Thomson 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 reactance values ​​of the coil XL = ωL and capacitor X С = 1/ωС are equal in absolute value, the value X Σ vanishes (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 \u003d U C \u003d IX L \u003d 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 has a purely active (resistive) character. The resonance condition is the equality of the reactances of the inductor and capacitance.

One of the most important parameters of the oscillatory circuit (except, of course, the resonant frequency) is its characteristic (or wave) resistance ρ and quality factor of the circuit Q. Characteristic (wave) resistance of the circuit ρ the value of the reactance of the capacitance and inductance of the circuit at the resonant frequency is called: ρ = X L = X C at ω =ω p. The characteristic impedance can be calculated as follows: ρ = √(L/C). Characteristic resistance ρ is a quantitative measure for estimating the energy stored by the reactive elements of the circuit - the coil (energy of the magnetic field) 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 the period is commonly called the quality factor Q contour, which is literally translated from in English stands for "quality".

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 loss in one period of oscillation. 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 is called loop damping. To determine the quality factor, the formula is usually used Q = ρ / R, Where R-resistance of ohmic losses of the circuit, characterizing the power of resistive (active losses) of the circuit P \u003d 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.

The frequency properties of various circuits in technology are usually evaluated using amplitude-frequency characteristics (AFC), while the circuits themselves are considered as four-terminal networks. The figures below show two simple quadripoles containing a series oscillatory circuit and the frequency response of these circuits, which are shown (shown by solid lines). On the vertical axis of the graphs of the frequency response, the magnitude of the voltage transfer coefficient of the circuit, K, is plotted, showing the ratio of the output voltage of the circuit to the input.

For passive circuits (i.e. not containing amplifying elements and energy sources), the value TO never exceeds one. The resistance to alternating current of the circuit shown in the figure will be minimal at an impact frequency equal to the resonant frequency of the circuit. In this case, the transfer coefficient of the circuit is close to unity (determined by ohmic losses in the circuit). At frequencies that are very different from the resonant one, the resistance of the circuit to alternating current is quite large, and, consequently, the transfer coefficient of the circuit will drop to almost zero.

At resonance in this circuit, the input signal source is actually short-circuited by a low loop resistance, due to which the gain of such a circuit at the resonant frequency drops to almost zero (again, due to the presence of a finite loss resistance). On the contrary, at frequencies of the input action that are significantly different from the resonant one, the transfer coefficient of the circuit 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 required to isolate a signal with a specific frequency from a multitude of unnecessary signals located at other frequencies. So, in any radio receiver, with the help of oscillatory circuits, tuning to the frequency of the desired radio station is provided. The property of an oscillatory circuit to single out one frequency from a set is commonly called selectivity or selectivity. In this case, the intensity of the change in the transmission coefficient of the circuit when the frequency of the impact is detuned from the resonance is usually estimated using a parameter called the bandwidth. The bandwidth is 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 on the graphs show the frequency response of exactly the same circuits, the oscillatory circuits of which have the same resonant frequencies as in the case discussed above, but with a lower quality factor (for example, an inductor is wound with a wire with a large resistance direct current). As can be seen from the figures, in this case, the bandwidth of the circuit expands and its selective (selective) properties deteriorate. Based on this, when calculating and designing oscillatory circuits, it is necessary to 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 resistance resistor in series with the inductor), which makes it possible to avoid distortions of broadband signals. Although, if in practice it is required 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. multilayer filters.

Parallel oscillating circuit

In various radio engineering devices, along with serial oscillatory circuits, parallel oscillatory circuits are often (even more often than serial) used. The figure shows circuit diagram parallel oscillatory circuit. Here, two reactive elements with a different nature of reactivity are connected in parallel. As is known, when elements are connected in parallel, it is impossible to add their resistances - you can only add conductivities. The figure shows the graphical dependences of the reactive conductances of the inductor B L = 1/ωL, condenser In C=-ωC, as well as the total conductivity In Σ, these two elements, which is the reactive conduction of a parallel oscillatory circuit. Similarly, as for a series oscillatory circuit, there is a certain frequency, called resonant, at which the reactances (and hence conductances) of the coil and capacitor are the same. At this frequency, the total conductivity of the parallel oscillatory circuit vanishes without losses. This means that at this frequency the oscillatory circuit has an infinitely large resistance to alternating current.

If we build the dependence of the reactance of the circuit on the frequency X Σ = 1/B Σ, this curve, depicted in the following figure, at the point ω = ω p will have a discontinuity of the second kind. The resistance of a real parallel oscillatory circuit (that is, with losses), of course, is not equal to infinity - it is the smaller, 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. Generally, physical meaning 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 resonant circuit, in which the 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 generator of harmonic oscillations 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 circuit branches will be the same. In this case, the currents are said to be in resonance in the circuit. As in the case of a series oscillating circuit, the reactances of the coil and capacitor cancel each other out, 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 the inductance decreases with decreasing frequency), and at higher frequencies, on the contrary, capacitive (that is, the reactance of the capacitance decreases with increasing frequency) .

Let us consider how the transmission coefficients of quadripoles depend on frequency, when they include not serial oscillatory circuits, but parallel ones.

The quadripole shown in the figure, at the resonant frequency of the circuit, is a huge current resistance, therefore, when ω=ω p its transfer coefficient will be close to zero (including ohmic losses). At frequencies other than resonant, the resistance of the circuit will decrease, and the transfer coefficient of the quadripole will increase.

For the four-pole shown in the figure above, the situation will be the opposite - at the resonant frequency, the circuit will be a very large resistance and almost all of the input voltage will go to the output terminals (i.e., the transfer coefficient will be maximum and close to unity). With a significant difference in the frequency of the input action 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 transfer coefficient will be close to zero.

quality factor- a property of an oscillatory system that determines the resonance band and shows how many times the energy reserves in the system are greater than the energy losses in one period of oscillation.

The quality factor is inversely proportional to the damping rate of natural oscillations in the system. That is, the higher the quality factor of the oscillatory system, the less energy loss for each period and the slower the oscillations decay.

The general formula for the quality factor of any oscillatory system:

- resonant oscillation frequency

- energy stored in the oscillatory system

· - dissipated power.

For example, in an electric resonant circuit, energy is dissipated due to the finite resistance of the circuit; in a quartz crystal, oscillation damping is due to internal friction in the crystal; in volume electromagnetic resonators, it is lost in the walls of the resonator, in its material and in the coupling elements;

For Oscillating circuit in RLC circuits:

where , and are the resistance, inductance and capacitance of the resonant circuit, respectively.

6) Addition of harmonic oscillations of the same direction and the same frequency. beats

Let two harmonic oscillations of the same direction and the same frequency take place

(4.1)

The equation of the resulting oscillation will have the form

We verify this by adding the equations of system (4.1)

Applying the sum cosine theorem and making algebraic transformations:

One can find such quantities A and φ0 that satisfy the equations

(4.3)

Considering (4.3) as two equations with two unknowns A and φ0, we find by squaring and adding them, and then dividing the second by the first:

Substituting (4.3) into (4.2), we get:

Or finally, using the sum cosine theorem, we have:

The body, participating in two harmonic oscillations of the same direction and the same frequency, also performs a harmonic oscillation in the same direction and with the same frequency as the summed oscillations. The amplitude of the resulting oscillation depends on the phase difference (φ2-φ1) of the smoothed oscillations.

Depending on the phase difference (φ2-φ1):

1) (φ2-φ1) = ±2mπ (m=0, 1, 2, ...), then A= A1+A2, i.e. the amplitude of the resulting oscillation A is equal to the sum of the amplitudes of the added oscillations;

2) (φ2-φ1) = ±(2m+1)π (m=0, 1, 2, ...), then A= |A1-A2|, i.e. the amplitude of the resulting oscillation is equal to the difference in the amplitudes of the added oscillations

beat

Periodic changes in the amplitude of oscillations that occur when two harmonic oscillations with close frequencies are added are called beats.

Let two oscillations differ little in frequency. Then the amplitudes of the added oscillations are equal to A, and the frequencies are equal to ω and ω + Δω, and Δω is much less than ω. Let us choose the reference point so that the initial phases of both oscillations are equal to zero.