Active filters are implemented using amplifiers (usually op-amps) and passive RC filters. Among the advantages of active filters compared to passive ones, the following should be highlighted:

· lack of inductors;

· better selectivity;

· compensation for the attenuation of useful signals or even their amplification;

· suitability for implementation in the form of an IC.

Active filters also have disadvantages:

¨ energy consumption from the power source;

¨ limited dynamic range;

¨ additional nonlinear signal distortions.

We also note that the use of active filters with op-amps at frequencies above tens of megahertz is difficult due to the low unity gain frequency of most widely used op-amps. The advantage of active filters on op-amps is especially evident at the most low frequencies ah, down to fractions of hertz.

In the general case, we can assume that the op-amp in the active filter corrects the frequency response of the passive filter by providing different conditions for the passage of different frequencies of the signal spectrum, compensates for losses at given frequencies, which leads to steep drops in the output voltage on the slopes of the frequency response. For these purposes, various frequency-selective feedback loops are used in op-amps. Active filters ensure that the frequency response of all types of filters is obtained: low pass (LPF), high pass (HPF) and band pass (PF).

The first stage of the synthesis of any filter is to specify a transfer function (in operator or complex form), which meets the conditions of practical feasibility and at the same time ensures the required frequency response or phase response (but not both) of the filter. This stage is called filter characteristic approximation.

The operator function is a ratio of polynomials:

K( p)=A( p)/B( p),

and is uniquely determined by zeros and poles. The simplest numerator polynomial is a constant. The number of poles of the function (and in active filters on an op-amp, the number of poles is usually equal to the number of capacitors in the circuits that form the frequency response) determines the order of the filter. The order of the filter indicates the decay rate of its frequency response, which for the first order is 20 dB/dec, for the second - 40 dB/dec, for the third - 60 dB/dec, etc.

The approximation problem is solved for a low-pass filter, then using the frequency inversion method, the resulting dependence is used for other types of filters. In most cases, the frequency response is set, taking the normalized transmission coefficient:

,

where f(x) is the filtering function; - normalized frequency; - filter cutoff frequency; e is the permissible deviation in the passband.

Depending on which function is taken as f(x), filters (starting from the second order) of Butterworth, Chebyshev, Bessel, etc. are distinguished. Figure 7.15 shows their comparative characteristics.

The Butterworth filter (Butterworth function) describes the frequency response with the most flat part in the passband and a relatively low decay rate. The frequency response of such a low-pass filter can be presented in the following form:

where n is the filter order.

The Chebyshev filter (Chebyshev function) describes the frequency response with a certain unevenness in the passband, but not a higher decay rate.

The Bessel filter is characterized by a linear phase response, as a result of which signals whose frequencies lie in the passband pass through the filter without distortion. In particular, Bessel filters do not produce emissions when processing square-wave oscillations.

In addition to the listed approximations of the frequency response of active filters, others are known, for example, the inverse Chebyshev filter, Zolotarev filter, etc. Note that the active filter circuits do not change depending on the type of frequency response approximation, but the relationships between the values ​​of their elements change.

The simplest (first order) HPF, LPF, PF and their LFC are shown in Figure 7.16.

In these filters, the capacitor that determines the frequency response is included in the OOS circuit.

For a high-pass filter (Figure 7.16a), the transmission coefficient is equal to:

,

The frequency of conjugation of asymptotes is found from the condition, from where

.

For the low-pass filter (Figure 7.16b) we have:

,

.

The PF (Figure 7.16c) contains elements of a high-pass filter and a low-pass filter.

You can increase the slope of the LFC rolloff by increasing the order of the filters. Active low-pass filters, high-pass filters and second-order filter filters are shown in Figure 7.17.

The slope of their asymptotes can reach 40 dB/dec, and the transition from low-pass filter to high-pass filter, as can be seen from Figures 7.17a, b, is carried out by replacing resistors with capacitors, and vice versa. The PF (Figure 7.17c) contains high-pass filter and low-pass filter elements. The transfer functions are equal:

¨ for low-pass filter:

;

¨ for high-pass filter:

.

For PF, the resonant frequency is equal to:

.

For low-pass filter and high-pass filter, the cutoff frequencies are respectively equal to:

;

.

Quite often, second-order PFs are implemented using bridge circuits. The most common are double T-shaped bridges, which “do not pass” the signal at the resonance frequency (Figure 7.18a) and Wien bridges, which have a maximum transmission coefficient at the resonant frequency (Figure 7.18b).

Bridge circuits are included in the PIC and OOS circuits. In the case of a double T-bridge, the feedback depth is minimal at the resonance frequency, and the gain at this frequency is maximum. When using a Wien bridge, the gain at the resonance frequency is maximum, because maximum depth of POS. At the same time, to maintain stability, the depth of the OOS introduced using resistors and must be greater than the depth of the POS. If the depths of the POS and OOS are close, then such a filter can have an equivalent quality factor Q»2000.

Resonant frequency of a double T-bridge at and , and the Wien Bridge And , is equal , and it is chosen based on the stability condition , because The transmission coefficient of the Wien bridge at frequency is 1/3.

To obtain a notch filter, a double T-shaped bridge can be connected as shown in Figure 7.18c, or a Wien bridge can be included in the OOS circuit.

To build an active tunable filter, a Wien bridge is usually used, whose resistors are made in the form of a dual variable resistor.

It is possible to construct an active universal filter (LPF, HPF and PF), a circuit version of which is shown in Figure 7.19.

It consists of an op-amp adder and two first-order low-pass filters on the op-amp and , which are connected in series. If , then the coupling frequency . The LFC has a slope of asymptotes of the order of 40 dB/dec. The universal active filter has good stability of parameters and high quality factor (up to 100). Quite often used in serial ICs similar principle building filters.

Gyrators

It's called a gyrator electronic device, which converts the total resistance of reactive elements. Typically this is a capacitance-to-inductance converter, i.e. equivalent to inductance. Sometimes gyrators are called inductance synthesizers. The widespread use of gyrators in ICs is explained by the great difficulties in manufacturing inductors using solid-state technology. The use of gyrators makes it possible to obtain a relatively large inductance with good weight and size characteristics.

Figure 7.20 shows an electrical diagram of one of the options for a gyrator, which is an op-amp repeater covered by a frequency-selective PIC ( and ).

Since the capacitance of the capacitor decreases with increasing signal frequency, the voltage at the point A will increase. Along with it, the voltage at the output of the op-amp will increase. The increased voltage from the output through the PIC circuit is supplied to the non-inverting input, which leads to a further increase in voltage at the point A, and the more intense, the higher the frequency. Thus, the voltage at the point A behaves like the voltage across an inductor. The synthesized inductance is determined by the formula:

.

The quality factor of a gyrator is defined as:

.

One of the main problems when creating gyrators is the difficulty in obtaining the equivalent of an inductance in which both terminals are not connected to a common bus. Such a gyrator is performed on at least four op-amps. Another problem is the relatively narrow range of operating frequencies of the gyrator (up to several kilohertz for widely used op amps).

"—meaning an active low-pass filter. It is especially useful when expanding a stereo sound system with an additional speaker that reproduces only the lowest frequencies. This project consists of a second-order active filter with an adjustable cutoff frequency of 50 - 250 Hz, an input amplifier with gain control (0.5 - 1.5) and output stages.

The design allows direct connection to a bridge amplifier, since the signals are 180 degrees out of phase with each other. Thanks to the built-in power supply and stabilizer on the board, it is possible to supply the filter with symmetrical voltage from a power amplifier - usually a bipolar 20 - 70 V. The low-pass filter is ideal for working with industrial and homemade amplifiers and preamplifiers.

Low-pass filter circuit diagram

The filter circuit for the subwoofer is shown in the figure. It works based on two operational amplifiers U1-U2 (NE5532). The first of them is responsible for summing and filtering the signal, while the second ensures its caching.

Schematic diagram of a low-pass filter to a subwoofer

The stereo input signal is supplied to connector GP1, and then through capacitors C1 (470nF) and C2 (470nF), resistors R3 (100k) and R4 (100k) it goes to the inverting input of amplifier U1A. This element implements a signal adder with adjustable gain, assembled according to a classical circuit. Resistor R6 (27k) together with P1 (50k) allows you to adjust the gain in the range from 0.5 to 1.5, which will allow you to select the gain of the subwoofer as a whole.

Resistor R9 (100k) improves the stability of amplifier U1A and ensures its good polarization in the event of no input signal.

The signal from the amplifier output goes to a second-order active low-pass filter built by U1B. This is a typical Sallen-Key architecture, which allows you to get filters with different slopes and amplitudes. The shape of this characteristic is directly affected by capacitors C8 (22nF), C9 (22nF) and resistors R10 (22k), R13 (22k) and potentiometer P2 (100k). The logarithmic scale of the potentiometer allows you to achieve a linear change in the cutoff frequency while rotating the knob. A wide frequency range (up to 260 Hz) is achieved with the extreme left position of potentiometer P2, turning to the right causes a narrowing of the frequency band to 50 Hz. The figure below shows the measured amplitude response of the entire circuit for the two extreme and middle positions of potentiometer P2. In each case, potentiometer P1 was set to the middle position, providing a gain of 1 (0 dB).

The signal from the filter output is processed using amplifier U2. Elements C16 (10pF) and R17 (56k) ensure stable operation of the U2A m/s. Resistors R15-R16 (56k) determine the gain of U2B, and C15 (10pF) increases its stability. Both outputs of the circuit use filters consisting of elements R18-R19 (100 Ohm), C17-C18 (10uF/50V) and R20-R21 (100k), through which the signals are sent to the GP3 output connector. Thanks to this design, at the output we receive two signals shifted in phase by 180 degrees, which allows direct connection of two amplifiers and a bridge amplifier.

The filter uses a simple bipolar voltage power supply based on zener diodes D1 (BZX55-C16V), D2 (BZX55-C16V) and two transistors T1 (BD140) and T2 (BD139). Resistors R2 (4.7k) and R8 (4.7k) are current limiters for the zener diodes, and were selected in such a way that at the minimum supply voltage the current is about 1 mA, and at the maximum it is safe for D1 and D2.

Elements R5 (510 Ohm), C4 (47uF/25V), R7 (510 Ohm), C6 (47uF/25V) are simple voltage smoothing filters based on T1 and T2. Resistors R1 (10 Ohm), R11 (10 Ohm) and capacitors C3 (100uF/25V), C7 (100uF/25V) are also a supply voltage filter. Power connector - GP2.

Connecting a subwoofer filter

It is worth noting that the subwoofer filter module should be connected to the preamplifier output after the volume control, which will improve the volume control of the entire system. Using the gain potentiometer, you can adjust the ratio of the subwoofer volume to the volume of the entire signal path. Any power amplifier operating in a classic configuration must be connected to the module output. If necessary, use only one of the output signals, 180 degrees out of phase with each other. Both output signals can be used if you need to build an amplifier in a bridge configuration.

In your life, you have heard the word “filter” more than once. Water filter, air filter, oil filter, “filter the market” after all). Air, water, oil and other types of filters remove foreign particles and impurities. But what does an electric filter filter? The answer is simple: frequency.

What is an electric filter

Electric filter is a device for highlighting desired spectrum components (frequencies) and/or suppressing unwanted ones. For other frequencies that are not included in , the filter creates a large attenuation, up to their complete disappearance.

The characteristics of an ideal filter should cut out a strictly defined frequency band and “squeeze” other frequencies until they are completely attenuated. Below is an example of an ideal filter that passes frequencies up to a certain cutoff frequency value.

In practice, such a filter is impossible to implement. When designing filters, they try to get as close as possible to the ideal characteristic. The closer to the ideal filter, the better it will perform its signal filtering function.

Filters that are assembled only on passive radio elements, such as, are called passive filters. Filters that contain one or more active radioelements, type or, are called active filters.

In our article we will look at passive filters and start with the simplest filters, consisting of a single radio element.

Single element filters

As you understand from the name, single-element filters consist of one radio element. This can be either a capacitor or an inductor. The coil and capacitor themselves are not filters - they are essentially just radio elements. But together with and with the load, they can already be considered as filters. Everything is simple here. The reactance of the capacitor and coil depends on frequency. You can read more about reactance in the article.

Single-element filters are mainly used in audio technology. For filtering, either a coil or a capacitor is used, depending on which frequencies need to be isolated. For a high-frequency speaker (tweeter), we connect a capacitor in series with the speaker, which will pass the high-frequency signal through it almost without loss, and will dampen low frequencies.

For the subwoofer speaker, we need to highlight low frequencies (LF), so we connect an inductor in series with the subwoofer.

The ratings of single radioelements can, of course, be calculated, but they are mainly selected by ear.

For those who don’t want to bother, hardworking Chinese create ready-made filters for tweeters and subwoofers. Here is one example:

On the board we see 3 terminal blocks: input terminal block (INPUT), output terminal block for bass (BASS) and terminal block for tweeter (TREBLE).

L-shaped filters

L-shaped filters consist of two radio elements, one or two of which have a nonlinear frequency response.

RC filters

I think we'll start with the filter we know best, consisting of a resistor and a capacitor. It has two modifications:

At first glance, you might think that these are two identical filters, but this is not the case. This is easy to verify if you build the frequency response for each filter.

Proteus will help us in this matter. So, the frequency response for this circuit

will look like this:

As we can see, the frequency response of such a filter allows low frequencies to pass through unhindered, and with increasing frequency it attenuates high frequencies. Therefore, such a filter is called a low-pass filter (LPF).

But for this chain

The frequency response will look like this

Here it's just the opposite. Such a filter attenuates low frequencies and passes high frequencies, which is why such a filter is called a high-pass filter (HPF).

Frequency response slope

The slope of the frequency response in both cases is 6 dB/octave after the point corresponding to the gain value of -3 dB, that is, the cutoff frequency. What does 6 dB/octave notation mean? Before or after the cutoff frequency, the slope of the frequency response takes the form of an almost straight line, provided that the transmission coefficient is measured in . An octave is a two-to-one ratio of frequencies. In our example, the slope of the frequency response is 6 dB/octave, which means that when the frequency is doubled, our direct frequency response increases (or falls) by 6 dB.

Let's look at this example

Let's take a frequency of 1 KHz. At frequencies from 1 KHz to 2 KHz, the drop in frequency response will be 6 dB. In the interval from 2 KHz to 4 KHz the frequency response again drops by 6 dB, in the interval from 4 KHz to 8 KHz it again drops by 6 dB, at a frequency from 8 KHz to 16 KHz the attenuation of the frequency response will again be 6 dB, and so on. Therefore, the frequency response slope is 6 dB/octave. There is also such a thing as dB/decade. It is used less frequently and denotes a difference between frequencies of 10 times. How to find dB/decade can be found in the article.

The steeper the slope of the direct frequency response, the better the selective properties of the filter:

A filter with a slope characteristic of 24 dB/octave will clearly be better than one with a slope of 6 dB/octave, since it becomes closer to the ideal.

RL filters

Why not replace the capacitor with an inductor? We again get two types of filters:

For this filter

The frequency response takes the following form:

We got the same low-pass filter

and for such a chain

The frequency response will take this form

The same high-pass filter

RC and RL filters are called first order filters and they provide a frequency response slope of 6 dB/octave after the cutoff frequency.

LC filters

What if you replace the resistor with a capacitor? In total, we have two radio elements in the circuit, the reactance of which depends on frequency. There are also two options here:

Let's look at the frequency response of this filter

As you may have noticed, its frequency response in the low frequency region is the flattest and ends with a spike. Where did he even come from? Not only is the circuit assembled from passive radio elements, but it also amplifies the voltage signal in the area of ​​the spike!? But don't rejoice. It amplifies by voltage, not power. The fact is that we got , which, as you remember, has a voltage resonance at the resonance frequency. With voltage resonance, the voltage across the coil is equal to the voltage across the capacitor.

But that is not all. This voltage is Q times greater than the voltage applied to the series tank. What is Q? This . This spike should not confuse you, since the height of the peak depends on the quality factor, which in real circuits is a small value. This circuit is also notable for the fact that its characteristic slope is 12 dB/octave, which is two times better than that of RC and RL filters. By the way, even if the maximum amplitude exceeds the value of 0 dB, then we still determine the passband at a level of -3 dB. This too should not be forgotten.

The same applies to the high-pass filter.

As I already said, LC filters are already called second order filters and they provide a frequency response slope of 12 dB/octave.

Complex filters

What happens if you connect two first-order filters one after the other? Oddly enough, this will result in a second order filter.

Its frequency response will be steeper, namely 12 dB/octave, which is typical for second-order filters. Guess what slope the third order filter will have ;-) ? That's right, add 6 dB/octave and get 18 dB/octave. Accordingly, for a 4th order filter the frequency response slope will already be 24 dB/octave, etc. That is, the more links we connect, the steeper the slope of the frequency response will be and the better the filter characteristics will be. This is all true, but you forgot that each subsequent stage contributes to the weakening of the signal.

In the above diagrams we built the frequency response of the filter without internal resistance generator and also without load. That is, in this case, the resistance at the filter output is infinity. This means that it is advisable to make sure that each subsequent stage has a significantly higher input impedance than the previous one. Currently, cascading links have already sunk into oblivion and now they use active filters that are built on op-amps.

Analysis of the filter from Aliexpress

In order for you to grasp the previous idea, we will analyze a simple example from our narrow-eyed brothers. Aliexpress sells various subwoofer filters. Let's consider one of them.

As you noticed, the filter characteristics are written on it: this type The filter is designed for a 300 Watt subwoofer, its characteristic slope is 12 dB/octave. If you connect a subwoofer with a coil resistance of 4 ohms to the filter output, the cutoff frequency will be 150 Hz. If the resistance of the subwoofer coil is 8 ohms, then the cutoff frequency will be 300 Hz.

For full teapots, the seller even provided a diagram in the product description. She looks like this:

Most often you can see directly on the speakers the value of the coil resistance at DC: 2 Ω, 4 Ω, 8 Ω. Less often 16 Ω. The Ω symbol after the numbers indicates Ohms. Also remember that the coil in the speaker is inductive.

How does an inductor behave at different frequencies?

As you can see, at direct current the speaker coil has active resistance, since it is wound from copper wire. At low frequencies, it comes into play, which is calculated by the formula:

Where

X L - coil resistance, Ohm

P is constant and equal to approximately 3.14

F - frequency, Hz

L - inductance, H

Since the subwoofer is designed specifically for low frequencies, this means that the reactance of the same coil is added in series with the active resistance of the coil itself. But in our experiment we will not take this into account, since we do not know the inductance of our imaginary speaker. Therefore, we take all experimental calculations with a decent error.

According to the Chinese, when the speaker filter is loaded with 4 Ohms, its bandwidth will reach up to 150 Hertz. Let's check if this is true:

Its frequency response

As you can see, the cutoff frequency at -3 dB was almost 150 Hz.

We load our filter with an 8 ohm speaker

The cutoff frequency was 213 Hz.

The product description stated that the cutoff frequency for an 8-ohm sub would be 300 Hz. I think you can trust the Chinese, since, firstly, all the data is approximate, and secondly, the simulation in the programs is far from reality. But that was not the essence of the experience. As we see in the frequency response, loading the filter with a resistance of a higher value, the cutoff frequency shifts upward. This must also be taken into account when designing filters.

Bandpass filters

In the last article we looked at one example of a bandpass filter

This is what the frequency response of this filter looks like.

The peculiarity of such filters is that they have two cutoff frequencies. They are also determined at a level of -3 dB or at a level of 0.707 from the maximum value of the transmission coefficient, or more precisely K u max /√2.

Bandpass Resonant Filters

If we need to select some narrow frequency band, LC resonant filters are used for this. They are also often called selective. Let's look at one of their representatives.

The LC circuit in combination with resistor R forms. A coil and a capacitor in a pair create a voltage that at the resonance frequency will have a very high impedance, popularly known as an open circuit. As a result, at the output of the circuit at resonance there will be the value of the input voltage, provided that we do not connect any load to the output of such a filter.

The frequency response of this filter will look something like this:

If we take the transmission coefficient value along the Y axis, the frequency response graph will look like this:

Construct a straight line at a level of 0.707 and estimate the bandwidth of such a filter. As you can see, it will be very narrow. The quality factor Q allows you to evaluate the characteristics of the circuit. The higher the quality factor, the sharper the characteristic.

How to determine the quality factor from the graph? To do this, you need to find the resonant frequency using the formula:

Where

f 0 is the resonant frequency of the circuit, Hz

L - coil inductance, H

C - capacitance of the capacitor, F

We substitute L=1mH and C=1uF and get a resonant frequency of 5033 Hz for our circuit.

Now we need to determine the bandwidth of our filter. This is done as usual at a level of -3 dB, if the vertical scale is , or at a level of 0.707, if the scale is linear.

Let's increase the top of our frequency response and find two cutoff frequencies.

f 1 = 4839 Hz

f 2 = 5233 Hz

Therefore, the bandwidth Δf=f 2 – f 1 = 5233-4839=394 Hz

Well, all that remains is to find the quality factor:

Q=5033/394=12.77

Notch filters

Another type of LC circuit is the series LC circuit.

Its frequency response will look something like this:

Of course, this drawback can be eliminated by placing the inductor in a mu-metal screen, but this will only make it more expensive. Designers try to avoid inductors whenever possible. But, thanks to progress, coils are currently not used in active filters built on op-amps.

Conclusion

Filters find many applications in radio electronics. For example, in the field of telecommunications, bandpass filters are used in the audio frequency range (20 Hz-20 KHz). Data acquisition systems use low-pass filters (LPF). In musical equipment, filters suppress noise, select a certain group of frequencies for the corresponding speakers, and can also change the sound. In power supply systems, filters are often used to suppress frequencies close to the 50/60 Hz mains frequency. In industry, filters are used to compensate for cosine phi and are also used as harmonic filters.

Summary

Electrical filters are used to highlight a certain frequency range and dampen unnecessary frequencies.

Filters built on passive radio elements such as resistors, inductors and capacitors are called passive filters. Filters that contain an active radio element, such as a transistor or op-amp, are called active filters.

The steeper the decline in the frequency response characteristic, the better the selective properties of the filter.

With the participation of JEER

When working with electrical signals, it is often necessary to isolate one frequency or frequency band from them (for example, to separate noise and useful signals). Electric filters are used for such separation. Active filters, unlike passive ones, include op-amps (or other active elements, for example, transistors, vacuum tubes) and have a number of advantages. They provide better separation of passbands and attenuation, and it is relatively easy to adjust unevenness in them frequency response in the region of transmission and attenuation. Also, active filter circuits typically do not use inductors. In active filter circuits, frequency characteristics are determined by frequency-dependent feedback.

Low pass filter

The low pass filter circuit is shown in Fig. 12.

Rice. 12. Active low pass filter.

The transmission coefficient of such a filter can be written as

, (5)

And
. (6)

At TO 0 >>1

Transmission coefficient
in (5) turns out to be the same as for a second-order passive filter containing all three elements ( R, L, C) (Fig. 13), for which:

Rice. 14. Frequency response and phase response of an active low-pass filter for differentQ .

If R 1 = R 3 = R And C 2 = C 4 = C(in Fig. 12), then the transmission coefficient can be written as

Amplitude and phase frequency characteristics of an active low-pass filter for different quality factors Q shown in Fig. 14 (the parameters of the electrical circuit are selected so that ω 0 = 200 rad/s). The figure shows that with increasing Q

The active low-pass filter of the first order is implemented by the circuit Fig. 15.

Rice. 15. Active low-pass filter of the first order.

The filter transmission coefficient is

.

The passive analogue of this filter is shown in Fig. 16.

Comparing these transmission coefficients, we see that for the same time constants τ’ 2 And τ the modulus of the gain of the first order active filter will be in TO 0 times more than the passive one.

Rice. 17.Simulink-active low pass filter model.

You can study the frequency response and phase response of the active filter under consideration, for example, in Simulink, using a transfer function block. For parameters electrical diagram TO R = 1, ω 0 = 200 rad/s and Q = 10 Simulink-the model with the transfer function block will look as shown in Fig. 17. Frequency response and phase response can be obtained using LTI- viewer. But in this case it is easier to use the command MATLAB freqs. Below is a listing for obtaining frequency response and phase response graphs.

w0=2e2; %natural frequency

Q=10; % quality factor

w=0:1:400; %frequency range

b=; %vector of the numerator of the transfer function:

a=; %vector of the denominator of the transfer function:

freqs(b,a,w); %calculation and construction of frequency response and phase response

Amplitude-frequency characteristics of an active low-pass filter (for τ = 1s and TO 0 = 1000) are shown in Fig. 18. The figure shows that with increasing Q the resonant nature of the amplitude-frequency characteristic is manifested.

Let's build a model of a low-pass filter in SimPowerSystems, using the op-amp block we created ( operationalamplifier), as shown in Figure 19. The operational amplifier block is nonlinear, so in the settings Simulation/ ConfigurationParametersSimulink to increase the calculation speed you need to use methods ode23tb or ode15s. It is also necessary to choose the time step wisely.

Rice. 18. Frequency response and phase response of the active low-pass filter (forτ = 1c).

Let R 1 = R 3 = R 6 = 100 Ohm, R 5 = 190 Ohm, C 2 = C 4 = 5*10 -5 F. For the case when the source frequency coincides with the natural frequency of the system ω 0 , the signal at the filter output reaches its maximum amplitude (shown in Fig. 20). The signal represents steady-state forced oscillations with the source frequency. The graph clearly shows the transient process caused by turning on the circuit at a moment in time t= 0. The graph also shows deviations of the signal from the sinusoidal shape near the extremes. In Fig. 21. An enlarged part of the previous graph is shown. These deviations can be explained by op-amp saturation (maximum permissible voltage values ​​at the op-amp output ± 15 V). It is obvious that as the amplitude of the source signal increases, the area of ​​signal distortion at the output also increases.

Rice. 19. Model of an active low-pass filter inSimPowerSystems.

Rice. 20. Signal at the output of an active low-pass filter.

Rice. 21. Fragment of the signal at the output of an active low-pass filter.

In this article we will talk about high and low pass filters, how they are characterized and their varieties.

High and low pass filters- This electrical circuits, consisting of elements that have a nonlinear frequency response - having different resistance at different frequencies.

Frequency filters can be divided into high-pass (high-pass) filters and low-pass (low-pass) filters. Why do people often say “upper” rather than “high” frequencies? Because in audio engineering low frequencies end at 2 kilohertz and high frequencies begin. And in radio engineering, 2 kilohertz is another category - sound frequency, which means “low frequency”! In audio engineering there is another concept - mid frequencies. So, mid-pass filters are usually either a combination of two low-pass and high-pass filters, or another kind of bandpass filter.

Let's repeat it again:

To characterize low- and high-pass filters, and not only filters, but any elements of radio circuits, there is a concept - amplitude-frequency response, or frequency response

Frequency filters are characterized by indicators

Cutoff frequency– this is the frequency at which the amplitude of the filter output signal decreases to a value of 0.7 from the input signal.

Filter frequency response slope is a filter characteristic that shows how sharply the amplitude of the filter’s output signal decreases when the frequency of the input signal changes. Ideally, you should strive for the maximum (vertical) decrease in frequency response.

Frequency filters are made from elements with reactance - capacitors and inductors. Reactances used in capacitor filters ( X C ) and inductors ( XL ) are related to frequency by the formulas below:

Calculation of filters before conducting experiments using special equipment (generators, spectrum analyzers and other devices) is easier to do at home in Microsoft program Excel, making a simple automatic calculation table (you must be able to work with formulas in Excel). I use this method to calculate any circuits. First, I make a table, insert the data, get a calculation, which I transfer to paper in the form of an frequency response graph, change the parameters, and again draw the frequency response points. In this method, there is no need to deploy a “laboratory of measuring instruments”; the calculation and drawing of the frequency response is carried out quickly.

It should be added that the filter calculation will then be correct when the rule is executed:

To ensure filter accuracy, it is necessary that the resistance value of the filter elements be approximately two orders of magnitude less (100 times) the resistance of the load connected to the filter output. As this difference decreases, the quality of the filter deteriorates. This is due to the fact that the load resistance affects the quality of the frequency filter. If you do not need high accuracy, then this difference can be reduced by up to 10 times.

Frequency filters are:

1. Single-element (capacitor - as a high-pass filter, or inductor - as a low-pass filter);

2. L-shaped - by appearance resemble the letter G facing the other direction;

3. T-shaped - in appearance they resemble the letter T;

4. U-shaped - in appearance they resemble the letter P;

5. Multi-link - the same L-shaped filters connected in series.

Single element high and low pass filters

Typically, single-element high- and low-pass filters are used directly in speaker systems powerful amplifiers audio frequency, to improve the sound of the audio speakers themselves.

They are connected in series with the dynamic heads. Firstly, they protect both the dynamic heads from a powerful electrical signal and the amplifier from low load resistance without loading it with extra speakers at a frequency that these speakers do not reproduce. Secondly, they make playback more pleasant to the ear.

To calculate a single-element filter, you need to know the reactance of the dynamic head coil. The calculation is made using the voltage divider formulas, which is also true for an L-shaped filter. Most often, single-element filters are selected “by ear”. To highlight high frequencies on the tweeter, a capacitor is installed in series with it, and to highlight low frequencies on a low-frequency speaker (or subwoofer), a choke (inductor) is connected in series with it. For example, with powers of the order of 20...50 Watts, it is optimal to use a 5...20 µF capacitor for tweeters, and as a choke for a low-frequency speaker, use a coil wound with enameled copper wire, 0.3...1.0 mm in diameter, on a reel from a VHS video cassette, and containing 200...1000 turns. Wide limits are indicated, because selection is an individual matter.

L-shaped filters

L-shaped high-pass or low-pass filter— a voltage divider consisting of two elements with a nonlinear frequency response. For an L-shaped filter, the circuit and all the formulas for the voltage divider apply.

L-shaped frequency filters on a capacitor and resistor

R 1 WITH X C .

The principle of operation of such a filter: a capacitor, having a low reactance at high frequencies, passes current unhindered, and at low frequencies its reactance is maximum, so no current passes through it.

From the article “Voltage Divider” we know that the values ​​of resistors can be described by the formulas:

or

X C and cutoff frequency.

R 2 to resistor resistance R 1 (X C ) corresponds to: R 2 / R 1 = 0.7 / 0.3 = 2.33 . This implies: C = 1.16 / R 2 πf , Where f – cutoff frequency of the frequency response of the filter.

R 2 voltage divider to capacitor WITH , having its own reactance X C .

The principle of operation of such a filter: the capacitor, having low reactance at high frequencies, shunts high-frequency currents to the housing, and at low frequencies its reactance is maximum, so no current passes through it.

From the article “Voltage Divider” we use the same formulas:

or

Taking the input voltage as 1 (unit), and output voltage for 0.7 (the value corresponding to the cut), knowing the reactance of the capacitor, which is equal to:

Substituting the voltage values, we find X C and cutoff frequency.

R 2 (X C ) to the resistance of the resistor R 1 corresponds to: R 2 / R 1 = 0.7 / 0.3 = 2.33 . This implies: C = 1 / (4.66 x R 1 πf) , Where f – cutoff frequency of the frequency response of the filter.

L-shaped frequency filters on an inductor and a resistor

A high-pass filter is obtained by replacing the resistor R 2 L XL .

The principle of operation of such a filter: inductance, having low reactance at low frequencies, shunts them to the housing, and at high frequencies its reactance is maximum, so no current passes through it.

Substituting the voltage values, we find XL and cutoff frequency.

As with the high-pass filter, the calculations can be done in reverse. Taking into account that the amplitude of the output voltage of the filter (as a voltage divider) at the cutoff frequency of the frequency response should be equal to 0.7 of the input voltage, it follows that the ratio of the resistor resistance R 2 (XL ) to the resistance of the resistor R 1 corresponds to: R 2 / R 1 = 0.7 / 0.3 = 2.33 . This implies: L = 1.16 R 1 / (πf) .

A low-pass filter is obtained by replacing the resistor R 1 voltage divider to inductor L , which has its own reactance XL .

The operating principle of such a filter: the inductor, having low reactance at low frequencies, passes current unhindered, and at high frequencies its reactance is maximum, so no current passes through it.

Using the same formulas from the article “Voltage Divider” and taking the input voltage as 1 (unity), and the output voltage as 0.7 (the value corresponding to the cutoff), knowing the reactance of the inductor, which is equal to:

Substituting the voltage values, we find XL and cutoff frequency.

You can do the calculations in reverse order. Taking into account that the amplitude of the output voltage of the filter (as a voltage divider) at the cutoff frequency of the frequency response should be equal to 0.7 of the input voltage, it follows that the ratio of the resistor resistance R 2 to resistor resistance R 1 (XL ) corresponds to: R 2 / R 1 = 0.7 / 0.3 = 2.33 . This implies: L = R 2 / (4.66 πf)

L-shaped frequency filters on a capacitor and inductor

A high-pass filter is obtained from an ordinary voltage divider by replacing not only the resistor R 1 to the capacitor WITH , as well as a resistor R 2 on the throttle L . Such a filter has a more significant frequency cut (steeper decline) in the frequency response than the above-mentioned filters based on R.C. or R.L. chains.

As was done earlier, we use the same calculation methods. Capacitor WITH , has its own reactance X C , and the throttle L — reactance XL :

By substituting the values ​​of various quantities - voltages, input or output resistances of filters, we can find WITH And L , frequency response cutoff frequency. You can also do the calculations in reverse order. Since there are two variable quantities - inductance and capacitance, the value of the input or output resistance of the filter is most often set as a voltage divider at the cutoff frequency of the frequency response, and based on this value, the remaining parameters are found.

A low-pass filter is obtained by replacing the resistor R 1 voltage divider to inductor L , and the resistor R 2 to the capacitor WITH .

As described earlier, the same calculation methods are used, through the voltage divider formulas and the reactance of the filter elements. In this case, we equate the value of the resistor R 1 to throttle reactance XL , A R 2 to capacitor reactance X C .

T-shaped high and low pass filters

T-shaped high- and low-pass filters are the same L-shaped filters, to which one more element is added. Thus, they are calculated in the same way as a voltage divider consisting of two elements with a nonlinear frequency response. And then, the reactance value of the third element is added to the calculated value. Another, less accurate way of calculating a T-shaped filter begins with calculating the L-shaped filter, after which the value of the “first” calculated element of the L-shaped filter is increased or decreased by half - “distributed” between two elements of the T-shaped filter. If it is a capacitor, then the value of the capacitance of the capacitors in the T-filter is doubled, and if it is a resistor or inductor, then the value of the resistance or inductance of the coils is halved. The transformation of filters is shown in the figures. The peculiarity of T-shaped filters is that, compared to L-shaped ones, their output resistance has a lower shunting effect on the radio circuits behind the filter.

U-shaped high and low pass filters

U-shaped filters are the same L-shaped filters, to which another element is added in front of the filter. Everything that has been written for T-shaped filters is true for U-shaped ones, the only difference is that compared to L-shaped ones, they slightly increase the shunting effect on the radio circuits in front of the filter.

As in the case of T-shaped filters, to calculate U-shaped filters, voltage divider formulas are used, with the addition of an additional shunt resistance of the first filter element. Another, less accurate method of calculating a U-shaped filter begins with calculating the L-shaped filter, after which the value of the “last” calculated element of the L-shaped filter is increased or decreased by half - “distributed” between two elements of the U-shaped filter. In contrast to the T-shaped filter, if it is a capacitor, then the value of the capacitance of the capacitors in the P-filter is halved, and if it is a resistor or inductor, then the value of the resistance or inductance of the coils is doubled.

Due to the fact that the manufacture of inductors (chokes) requires certain efforts, and sometimes additional space for their placement, it is more profitable to manufacture filters from capacitors and resistors, without the use of inductors. This is especially true on audio frequencies. Thus, high-pass filters are usually made T-shaped, and low-pass filters are made U-shaped. There are also mid-pass filters, which, as a rule, are made L-shaped (from two capacitors).

Bandpass Resonant Filters

Band-pass resonant frequency filters are designed to isolate or reject (cut out) a certain frequency band. Resonant frequency filters can consist of one, two, or three oscillatory circuits tuned to a specific frequency. Resonant filters have the steepest rise (or fall) in the frequency response compared to other (non-resonant) filters. Band-pass resonant frequency filters can be single-element - with one circuit, L-shaped - with two circuits, T and U-shaped - with three circuits, multi-element - with four or more circuits.

The figure shows a diagram of a T-shaped bandpass resonant filter designed to isolate a certain frequency. It consists of three oscillatory circuits. C 1 L 1 And C 3 L 3 – series oscillatory circuits, at the resonant frequency have low resistance to the flowing current, and at other frequencies, on the contrary, they have high resistance. Parallel circuit C 2 L 2 on the contrary, it has high resistance at the resonant frequency, while having low resistance at other frequencies. To expand the bandwidth of such a filter, they reduce the quality factor of the circuits, changing the design of the inductors, detuning the circuits “right, left” to a frequency slightly different from the central resonant one, parallel to the circuit C 2 L 2 connect a resistor.

The following figure shows a diagram of a T-shaped notch resonant filter designed to suppress a specific frequency. It, like the previous filter, consists of three oscillatory circuits, but the principle of frequency selection for such a filter is different. C 1 L 1 And C 3 L 3 – parallel oscillatory circuits, at the resonant frequency have a large resistance to the flowing current, and at other frequencies - small. Parallel circuit C 2 L 2 on the contrary, it has low resistance at the resonant frequency, but has high resistance at other frequencies. Thus, if the previous filter selects the resonant frequency and suppresses the remaining frequencies, then this filter freely passes all frequencies except the resonant frequency.

The procedure for calculating bandpass resonant filters is based on the same voltage divider, where the LC circuit with its characteristic resistance acts as a single element. How an oscillatory circuit is calculated, its resonant frequency, quality factor and characteristic (wave) impedance are determined, you can find in the article