3rd order filter circuits. Analog measuring devices. What is filter order and cutoff slope

Description

Any filter, in essence, does to the signal spectrum what Rodin does to marble. But unlike the sculptor’s work, the idea does not belong to the filter, but to you and me.

For obvious reasons, we are most familiar with one area of ​​​​application of filters - separating the spectrum of sound signals for their subsequent reproduction by dynamic heads (often we say “speakers”, but today the material is serious, so we will also approach the terms with the utmost rigor). But this area of ​​using filters is probably still not the main one, and it is absolutely certain that it is not the first in historical terms. Let's not forget that electronics was once called radio electronics, and its original task was to serve the needs of radio transmission and radio reception. And even in those childhood years of radio, when signals of a continuous spectrum were not transmitted, and radio broadcasting was still called radiotelegraphy, there was a need to increase the noise immunity of the channel, and this problem was solved through the use of filters in receiving devices. On the transmitting side, filters were used to limit the spectrum of the modulated signal, which also improved transmission reliability. In the end, the cornerstone of all radio technology of those times, the resonant circuit, is nothing more than a special case of a bandpass filter. Therefore, we can say that all radio technology began with a filter.

Of course, the first filters were passive; they consisted of coils and capacitors, and with the help of resistors it was possible to obtain standardized characteristics. But they all had a common drawback - their characteristics depended on the impedance of the circuit behind them, that is, the load circuit. In the simplest cases, the load impedance could be kept high enough that this influence could be neglected, in other cases the interaction of the filter and the load had to be taken into account (by the way, calculations were often carried out even without a slide rule, just in a column). It was possible to get rid of the influence of load impedance, this curse of passive filters, with the advent of active filters.

Initially, it was intended to devote this material entirely to passive filters; in practice, installers have to calculate and manufacture them on their own much more often than active ones. But logic demanded that we still start with the active ones. Oddly enough, because they are simpler, no matter what it might seem at first glance at the illustrations provided.

I want to be understood correctly: information about active filters is not intended to serve solely as a guide to their manufacture; such a need does not always arise. Much more often there is a need to understand how existing filters work (mainly as part of amplifiers) and why they do not always work as we would like. And here, indeed, the thought of manual work may come. Schematic diagrams of active filters

In the simplest case, an active filter is a passive filter loaded onto an element with unity gain and high input impedance - either an emitter follower or an operational amplifier operating in follower mode, that is, with unity gain. (You can also implement a cathode follower on a lamp, but, with your permission, I will not touch on lamps; if anyone is interested, please refer to the relevant literature). In theory, it is not forbidden to construct an active filter of any order in this way. Since the currents in the input circuits of the repeater are very small, it would seem that the filter elements can be chosen to be very compact. Is that all? Imagine that the filter load is a 100 ohm resistor, you want to make a first order low pass filter consisting of a single coil, at a frequency of 100 Hz. What should the coil rating be? Answer: 159 mH. How compact is this? And the main thing is that the ohmic resistance of such a coil can be quite comparable to the load (100 Ohms). Therefore, we had to forget about inductors in active filter circuits; there was simply no other way out.

First order filter

For first-order filters (Fig. 1), I will give two options for the circuit implementation of active filters - with an op-amp and with an emitter follower on an n-p-n transistor, and you yourself, if necessary, will choose which will be easier for you to work with. Why n-p-n? Because there are more of them, and because, other things being equal, in production they turn out somewhat “better”. The simulation was carried out for the KT315G transistor - probably the only semiconductor device, the price of which until recently was exactly the same as a quarter of a century ago - 40 kopecks. In fact, you can use any npn transistor whose gain (h21e) is not much lower than 100.

Rice. 1. First order high pass filters

6.5.2.1. Low pass filters.

A low-pass filter is a circuit that transmits low-frequency signals without changes, and at high frequencies ensures attenuation of the signals and their phase lag relative to the input signals.

Passive First Order Low Pass Filters

Figure 2.25 shows the circuit of a simple first-order low-pass RC filter. The transmission coefficient in complex form can be expressed by the formula:

. (2.45)
Rice. 2.25 From here we obtain formulas for the frequency response and phase response

, . (2.46)

Putting this we get the expression for the cutoff frequency ωSR

The phase shift at this frequency is – 450.
| K | = 1 = 0 dB at low frequencies f<< fCR .
At high frequencies f >> fС P according to formula (2.46), | K | ≈ 1/ (ωRC),
those. the transmission coefficient is inversely proportional to the frequency. When the frequency increases by a factor of 10, the gain decreases by a factor of 10, i.e., it decreases by 20 dB per decade or by 6 dB per octave. | K | = 1/√2 = -3dB at f= fCP .
To reduce the gain more quickly, n low-pass filters can be connected in series. When several low-pass filters are connected in series, the cutoff frequency is approximately determined as

. (2.48)

For the case of n filters with equal cutoff frequencies

At input frequency fVX>> fSR for the circuit (Fig. 2.25) we get

. (2.50)

From 2.50 it is clear that the low-pass filter can act as an integrating link.
For an alternating voltage containing a constant component, the output voltage can be represented as

, (2.51)

Where is the average value
A low pass filter can act as an average detector.
To implement a general approach to describing filters, it is necessary to normalize the complex variable r.

. (2.52)

For filter fig. 2.25 we get P = r RC and

I use the transfer function to estimate the amplitude of the output signal versus frequency, we get

. (2.54)

The low-pass filter transfer function in general can be written as

, (2.55)

Where c1, s2 ,…, sn are positive real coefficients.
The order of the filter is determined by the maximum power of the variable P. To implement the filter, it is necessary to factorize the denominator polynomial. If among the roots of the polynomial there are complex ones, in this case the polynomial should be written as a product of second-order factors

Where Ai And bi are positive real coefficients. For odd orders of the polynomial, the coefficient b1 equal to zero.

Active low-pass filters of the first order

The simple filter shown in Fig. 2.26 has a disadvantage: the properties of the filter depend on the load. To eliminate this drawback, the filter must be supplemented with an impedance converter. The filter circuit with an impedance converter is shown in Fig. 2.27. The constant signal transmission coefficient can be set by selecting the values ​​of resistors R2 and R3.

To simplify the low-pass filter circuit, you can use an RC circuit to feedback the op-amp. A similar filter is shown in Fig. 2.27.

Rice. 2.26 Fig. 2.27

The filter transfer function (Fig. 2.27) has the form

. (2.58)

To calculate the filter, you need to set the cutoff frequency fSR (ω SR), constant signal transmission coefficient K0 (for the circuit in Fig. 2.27 it must be specified with a minus sign) and capacitance of capacitor C1. Equating the coefficients of the resulting transfer function to the coefficients of expression 2.56 for a first-order filter, we obtain

And . (2.59)

Second order passive low pass filter

Based on expression (2.56), we write in general form the transfer function of the second-order low-pass filter

. (2.60)

Such a transfer function cannot be realized using passive RC circuits. Such a filter can be implemented using inductors. In Fig. Figure 2.28 shows a second-order passive low-pass filter circuit.
The filter transfer function has the form
. (2.61)
You can calculate the filter using the formulas
Rice. 2.28
And . (2.62)
For example, for a second-order low-pass filter of the Butterworth type with coefficients a1= 1.414 and b1 = 1.000, setting the cutoff frequency fSR= 10 Hz and capacitance C = 10 μF, from (2.62) we obtain R = 2.25 kOhm and L = 25.3 H.
Such filters are inconvenient to implement due to too high inductance. A given transfer function can be realized with the help of an operational amplifier with appropriate RC circuits, which eliminates inductance.

Second-order active low-pass filters

An example of an active second-order low-pass filter is a filter with complex negative feedback, the circuit of which is shown in Fig. 2.29.
The transfer function of this filter has the form

Rice. 2.29
To calculate the filter, you can write

,
, (2.63)

When calculating the circuit, it is better to set the capacitance values ​​of the capacitors and calculate the required resistance values.

.
, (2.64)
.

In order for the value of resistance R2 to be valid, the condition must be met

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> LPF1)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> HPF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> PF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> RF)

4th order Butterworth filter

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> LPF1)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> HPF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> PF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> RF)

Chebyshev filter 3rd order

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> LPF1)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> HPF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> PF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> RF)

Chebyshev filter 4 orders

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> LPF1)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> HPF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> PF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> RF)

Bessel filter 3rd order

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> LPF1)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> HPF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> PF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> RF)

Bessel filter 4th order

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> LPF1)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> HPF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> PF)

CONVERTING THE FREQUENCY PROPERTIES OF DF (LPF --> RF)

Analyze the influence of errors in setting the digital low-pass filter coefficients on the frequency response (by changing one of the coefficients b j). Describe the nature of the change in frequency response. Draw a conclusion about the effect of changing one of the coefficients on the behavior of the filter.

We will analyze the influence of errors in setting the digital low-pass filter coefficients on the frequency response using the example of a 4th order Bessel filter.

Let us choose the deviation value of the coefficients ε equal to –1.5%, so that the maximum deviation of the frequency response is about 10%.

The frequency response of an “ideal” filter and filters with changed coefficients by the value ε is shown in the figure:

AND

The figure shows that the greatest influence on the frequency response is exerted by changes in the coefficients b 1 and b 2 (their value exceeds the value of other coefficients). Using a negative value of ε, we note that positive coefficients reduce the amplitude in the lower part of the spectrum, while negative coefficients increase it. For a positive value of ε, everything happens the other way around.

Quantize the coefficients of the digital filter by such a number of binary digits that the maximum deviation of the frequency response from the original is about 10 - 20%. Sketch the frequency response and describe the nature of its change.

By changing the number of digits of the fractional part of the coefficients b j Note that the maximum deviation of the frequency response from the original one does not exceed 20% when n≥3.

Type of frequency response at different n shown in the pictures:

n =3, maximum frequency response deviation =19.7%

n =4, maximum frequency response deviation =13.2%

n =5, maximum frequency response deviation =5.8%

n =6, maximum frequency response deviation =1.7%

Thus, it can be noted that increasing the bit depth when quantizing filter coefficients leads to the fact that the frequency response of the filter tends more and more to the original one. However, it should be noted that this complicates the physical realizability of the filter.

Quantization at different n can be seen in the figure:

B. Uspensky

A simple method for separating cascades based on frequency is to install separating capacitors or integrating RC circuits. However, there is often a need for filters with steeper slopes than the RC chain. Such a need always exists when it is necessary to separate a useful signal from interference that is close in frequency.

The question arises: is it possible, by connecting cascade integrating RC chains, to obtain, for example, a complex low-pass filter (LPF) with a characteristic close to an ideal rectangular one, as in Fig. 1.

Rice. 1. Ideal low-pass frequency response

There is a simple answer to this question: even if you separate individual RC sections with buffer amplifiers, you still cannot make one steep bend out of many smooth bends in the frequency response. Currently, in the frequency range 0...0.1 MHz, a similar problem is solved using active RC filters that do not contain inductances.

The integrated operational amplifier (op-amp) has proven to be a very useful element for implementing active RC filters. The lower the frequency range, the more pronounced the advantages of active filters are from the point of view of microminiaturization of electronic equipment, since even at very low frequencies (up to 0.001 Hz) it is possible to use resistors and capacitors of not too large values.

Table 1

Active filters provide the implementation of frequency characteristics of all types: low and high frequencies, bandpass with one tuning element (equivalent to a single LC circuit), bandpass with several associated tuning elements, notch, phase filters and a number of other special characteristics.

The creation of active filters begins with the selection, using graphs or functional tables, of the type of frequency response that will provide the desired suppression of interference relative to a single level at the required frequency, which differs by a specified number of times from the passband boundary or from the average frequency for the resonant filter. Let us recall that the passband of the low-pass filter extends in frequency from 0 to the cutoff frequency fgr, and that of the high-frequency filter (HPF) - from fgr to infinity. When constructing filters, the Butterworth, Chebyshev and Bessel functions are most widely used. Unlike others, the characteristic of the Chebyshev filter in the passband oscillates (pulsates) around a given level within established limits, expressed in decibels.

The degree to which the characteristics of a particular filter approach the ideal depends on the order of the mathematical function (the higher the order, the closer). As a rule, filters of no more than 10th order are used. Increasing the order makes it difficult to tune the filter and worsens the stability of its parameters. The maximum quality factor of the active filter reaches several hundred at frequencies up to 1 kHz.

One of the most common structures of cascade filters is a multi-loop feedback element, built on the basis of an inverting op-amp, which is taken as ideal in calculations. The second order link is shown in Fig. 2.

Rice. 2. Second order filter structure:

The values ​​of C1, C2 for the low-pass filter and R1, R2 for the high-pass filter are then determined by multiplying or dividing C0 and R0 by the coefficients from the table. 2 by rule:
C1 = m1С0, R1 = R0/m1
C2 = m2C0, R2 = R0/m2.

The third-order links of the low-pass filter and the high-pass filter are shown in Fig. 3.

Rice. 3. Third order filter structure:
a - low frequencies; b - high frequencies

In the passband, the link transmission coefficient is 0.5. We define the elements according to the same rule:
С1 = m1С0, R1 = R0/m1 С2 = m2С0, R2 = R0/m2 С3 = m3С0, R3 = R0/m3.

The odds table looks like this.

Table 2

The order of the filter must be determined by calculation, specifying the ratio Uout/Uin at a frequency f outside the passband at a known cutoff frequency fgr. For the Butterworth filter there is a dependence

For illustration in Fig. Figure 4 compares the performance of three sixth-order low-pass filters with the attenuation performance of an RC circuit. All devices have the same fgr value.

Rice. 4. Comparison of sixth-order low-pass filter characteristics:
1- Bessel filter; 2 - Butterrort filter; 3 - Chebyshev filter (ripple 0.5 dB)

A bandpass active filter can be built using one op-amp according to the circuit in Fig. 5.

Rice. 5. Bandpass filter

Let's look at a numerical example. Let it be necessary to construct a selective filter with a resonant frequency F0 = 10 Hz and a quality factor Q = 100.

Its band is in the range of 9.95...10.05 Hz. At the resonant frequency, the transmission coefficient is B0 = 10. Let us set the capacitance of the capacitor C = 1 μF. Then, according to the formulas for the filter in question:

The device remains operational if you exclude R3 and use an op-amp with a gain exactly equal to 2Q 2. But then the quality factor depends on the properties of the op-amp and will be unstable. Therefore, the gain of the op-amp at the resonant frequency should significantly exceed 2Q 2 = 20,000 at a frequency of 10 Hz. If the op amp gain exceeds 200,000 at 10 Hz, you can increase R3 by 10% to achieve the design Q value. Not every op-amp has a gain of 20,000 at a frequency of 10 Hz, much less 200,000. For example, the K140UD7 op-amp is not suitable for such a filter; you will need KM551UD1A (B).

Using a low-pass filter and a high-pass filter connected in cascade, a bandpass filter is obtained (Fig. 6).

Rice. 6. Band pass filter

The steepness of the slopes of the characteristic of such a filter is determined by the order of the selected low-pass filters and high-pass filters. By differentiating the boundary frequencies of high-quality high-pass filters and low-pass filters, it is possible to expand the passband, but at the same time the uniformity of the transmission coefficient within the band deteriorates. It is of interest to obtain a flat amplitude-frequency response in the passband.

Mutual detuning of several resonant bandpass filters (PFs), each of which can be constructed according to the circuit in Fig. 5 gives a flat frequency response while increasing selectivity. In this case, one of the known functions is selected to implement the specified requirements for the frequency response, and then the low-frequency function is converted into a bandpass function to determine the quality factor Qp and the resonant frequency fp of each link. The links are connected in series, and the unevenness of the characteristics in the passband and selectivity improve with an increase in the number of cascades of resonant PFs.

To simplify the methodology, create cascade PFs in Table. Figure 3 shows the optimal values ​​of the frequency band delta fр (at a level of -3 dB) and the average frequency fp of the resonant sections, expressed through the total frequency band delta f (at a level of -3 dB) and the average frequency f0 of the composite filter.

Table 3

The exact values ​​of the average frequency and level limits - 3 dB are best selected experimentally, adjusting the quality factor.

Using the example of low-pass filters, high-pass filters and pass-pass filters, we saw that the requirements for the gain or bandwidth of an op-amp can be excessively high. Then you should move on to second-order links on two or three op-amps. In Fig. 7 shows an interesting second-order filter that combines the functions of three filters; from the output and DA1 we will receive a low-pass filter signal, from output DA2 - a high-pass filter signal, and from output DA3 - a PF signal.

Rice. 7. Second order active filter

The cut-off frequencies of the low-pass filter, high-pass filter and the central frequency of the PF are the same. The quality factor is also the same for all filters.

All filters can be adjusted by simultaneously changing R1, R2 or C1, C2. Regardless of this, the quality factor can be adjusted using R4. The finiteness of the op-amp gain determines the true quality factor Q = Q0(1 +2Q0/K).

It is necessary to select an op-amp with a gain K >> 2Q0 at the cutoff frequency. This condition is much less categorical than for filters on a single op-amp. Consequently, using three op-amps of relatively low quality, it is possible to assemble a filter with the best characteristics.

A band-stop (notch) filter is sometimes necessary to cut out narrow-band interference, such as the mains frequency or its harmonics. Using, for example, four-pole low-pass filters and Butterworth high-pass filters with cutoff frequencies of 25 Hz and 100 Hz (Fig. 8) and a separate op-amp adder, we obtain a filter for a frequency of 50 Hz with a quality factor Q = 5 and a rejection depth of -24 dB.

Rice. 8. Band-stop filter

The advantage of such a filter is that its response in the passband - below 25 Hz and above 100 Hz - is perfectly flat.

Like a bandpass filter, a notch filter can be assembled on a single op-amp. Unfortunately, the characteristics of such filters are not stable. Therefore, we recommend using a gyrator filter on two op-amps (Fig. 9).

Rice. 9. Notch gyrator filter

The resonant circuit on the DA2 amplifier is not prone to oscillation. When choosing resistances, you should maintain the ratio R1/R2 = R3/2R4. By setting the capacitance of capacitor C2, changing the capacitance of capacitor C1, you can adjust the filter to the required frequency

Within small limits, the quality factor can be adjusted by adjusting resistor R5. Using this circuit, it is possible to obtain a rejection depth of up to 40 dB, however, the amplitude of the input signal should be reduced to maintain the linearity of the gyrator on the DA2 element.

In the filters described above, the gain and phase shift depended on the frequency of the input signal. There are active filter circuits in which the gain remains constant and the phase shift depends on frequency. Such circuits are called phase filters. They are used for phase correction and delay of signals without distortion.

The simplest first-order phase filter is shown in Fig. 10.

Rice. 10 First order phase filter

At low frequencies, when capacitor C does not work, the transmission coefficient is +1, and at high frequencies -1. Only the phase of the output signal changes. This circuit can be successfully used as a phase shifter. By changing the resistance of resistor R, you can adjust the phase shift of the input sinusoidal signal at the output.

There are also phase links of the second order. By combining them in cascade, high-order phase filters are built. For example, to delay an input signal with a frequency spectrum of 0...1 kHz for a time of 2 ms, a seventh-order phase filter is required, the parameters of which are determined from the tables.

It should be noted that any deviation of the ratings of the RC elements used from the calculated ones leads to a deterioration in the filter parameters. Therefore, it is advisable to use precise or selected resistors, and create non-standard values ​​by connecting several capacitors in parallel. Electrolytic capacitors should not be used. In addition to the amplification requirements, the op-amp must have a high input impedance, significantly exceeding the resistance of the filter resistors. If this cannot be ensured, connect an op-amp repeater before the input of the inverting amplifier.

The domestic industry produces hybrid integrated circuits of the K298 series, which includes sixth-order high- and low-pass RC filters based on unity-gain amplifiers (repeaters). The filters have 21 cutoff frequency ratings from 100 to 10,000 Hz with a deviation of no more than ±3%. Designation of filters K298FN1...21 and K298FV1...21.

The principles of filter design are not limited to the examples given. Less common are active RC filters without lumped capacitances and inductances, which use the inertial properties of op-amps. Extremely high quality factors, up to 1000 at frequencies up to 100 kHz, are provided by synchronous filters with switched capacitors. Finally, charge-coupled device semiconductor technology is used to create active filters on charge-transfer devices. Such a high-pass filter 528FV1 with a cutoff frequency of 820...940 Hz is available as part of the 528 series; The dynamic low-pass filter 1111FN1 is one of the new developments.

Literature
Graham J., Toby J., Huelsman L. Design and application of operational amplifiers. - M.: Mir, 1974, p. 510.
Marchais J. Operational amplifiers and their application. - L.: Energy, 1974, p. 215.
Gareth P. Analog devices for microprocessors and mini-computers. - M.: Mir, 1981, p. 268.
Titze U., Schenk K. Semiconductor circuitry. - M. Mir, 1982, p. 512.
Horowitz P., Hill W. The Art of Circuit Design, vol. 1. - M. Mir, 1983, p. 598.
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• Tutorial

Brief Introduction

I continue to write spam on the topic of operational amplifiers. In this article I will try to give an overview of one of the most important topics related to op amps. So, welcome active filters.

Topic overview

You may have already come across RC, LC and RLC filter models. They are quite suitable for most tasks. But for some applications, it is very important to have filters with flatter bandwidth characteristics and steeper slopes. This is where we need active filters.
To refresh your memory, let me remind you what filters are:
Low Pass Filter(LPF) - passes a signal that is below a certain frequency (also called the cutoff frequency). Wikipedia
High Pass Filter(HPF) - passes a signal above the cutoff frequency. Wikipedia
Bandpass Filter- passes only a certain range of frequencies. Wikipedia
Notch Filter- delays only a certain frequency range. Wikipedia
Well, a little more lyrics. Look at the amplitude-frequency response (AFC) of the high-pass filter. Don’t look for anything interesting on this graph yet, but just pay attention to the areas and their names:

The most commonplace examples of active filters can be seen in the “Integrators and differentiators” section. But in this article we will not touch these circuits, because they are not very effective.

Selecting a filter

Let's assume that you have already decided on the frequency you want to filter. Now you need to decide on the type of filter. More precisely, you need to choose its characteristics. In other words, how the filter will “behave.”
The main characteristics are:
Butterward filter- has the flattest characteristic in the passband, but has a smooth roll-off.
Chebyshev filter- has the steepest roll-off, but it has the most uneven characteristics in the passband.
Bessel filter- has a good phase-frequency response and quite a “decent” roll-off. Considered the best choice if there is no specific task.

Some more information

Let's assume you completed this task. And now you can safely begin the calculations.
There are several calculation methods. Let's not complicate things and use the simplest. And the simplest is the “tabular” method. Tables can be found in the relevant literature. So that you don’t have to search for a long time, I will quote from Horowitz and Hill “The Art of Circuit Design”.
For low pass filter:

Let's just say that you could find and read all this in literature. Let's move on specifically to filter design.

Calculation

In this section I will try to briefly go over all types of filters.
So, task #1. Construct a second-order low-pass filter with a cutoff frequency of 150 Hz according to the Butterward characteristic.
Let's get started. If we have a filter of the nth even order, this means that it will have n/2 opamps. In this task - one.
Low pass filter circuit:


For this type of calculation it is taken into account that R1 = R2, C1 = C2.
Let's look at the sign. We see that K = 1.586. We'll need this a little later.
For a low pass filter:
, where, of course,
is the cutoff frequency.
Having done the calculation, we get . Now let's start selecting the elements. We decided on the op-amp - “ideal” in the amount of 1 piece. From the previous equality we can assume that it does not matter to us which element we choose “first”. Let's start with the resistor. It is best that its resistance value be in the range from 2 kOhm to 500 kOhm. By eye, let it be 11 kOhm. Accordingly, the capacitance of the capacitor will become equal to 0.1 µF. For feedback resistors the value R we take it arbitrarily. I usually take 10 kOhm. Then, for the upper value we take K from the table. Therefore, the lower one will have a resistance value R= 10 kOhm, and the top 5.8 kOhm.
Let's collect and simulate the frequency response.

Task #2. Construct a fourth-order high-pass filter with a cutoff frequency of 800 Hz using the Bessel characteristic.
Let's decide. Since it is a fourth-order filter, there will be two op-amps in the circuit. Everything here is not difficult at all. We simply cascade 2 high-pass filter circuits.
The filter itself looks like this:


A fourth-order filter looks like this:


Now the calculation. As you can see, for the fourth-order filter we have as many as 2 values TO. It is logical that the first is intended for the first cascade, the second - for the second. Values TO are equal to 1.432 and 1.606, respectively. The table was for low-pass filters (!). To calculate the high-pass filter, you need to change something. Odds TO remain the same in any case. For the Bessel and Chebyshev characteristics the parameter changes
- normalizing frequency. It will now be equal to:

For Chebyshev and Bessel filters, both for low frequencies and for high frequencies, the same formula is valid:

Please note that for each individual cascade you will have to count separately.
For the first cascade:

Let WITH= 0.01 µF, then R= 28.5 kOhm. Feedback resistors: lower, as usual, 10 kOhm; upper - 840 Ohms.
For the second cascade:

Let us leave the capacitance of the capacitor unchanged. Once C = 0.01 µF, then R= 32 kOhm.
We are building the frequency response.

To create bandpass or notch filters, you can cascade a low-pass filter and a high-pass filter. But these types are often not used due to poor characteristics.
For bandpass and notch filters, you can also use the “table method”, but the characteristics are slightly different.
I’ll just give you a sign and explain it a little. In order not to stretch it too much, the values ​​​​are taken immediately for a fourth-order bandpass filter.

a1 And b1- calculated coefficients. Q- quality factor. This is a new option. The higher the value of the quality factor, the more “sharp” the decline will be. Δf- range of transmitted frequencies, and sampling is at a level of -3 dB. Coefficient α - another calculated coefficient. It can be found using formulas that are quite easy to find on the Internet.
Okay, that's enough. Now the work task.
Task #3. Construct a fourth-order bandpass filter using the Butterward characteristic with a center frequency of 10 kHz, a bandwidth of transmitted frequencies of 1 kHz, and a gain at the center frequency point equal to 1.
Let's go. Fourth order filter. That means two op-amps. I’ll give you a typical diagram with calculation elements right away.


For the first filter, the center frequency is defined as:

For the second filter:

Specifically in our case, again from the table, we determine that the quality factor Q= 10. Calculate the quality factor for the filter. Moreover, it is worth noting that the quality factor of both will be equal.

Gain correction for center frequency region:

The final stage is the calculation of components.
Let the capacitor be 10 nF. Then, for the first filter:



In the same order as (1) we find R22 = R5 = 43.5 kOhm, R12 = R4= 15.4 kOhm, R32 = R6= 54.2 Ohm. Just keep in mind that for the second filter we use
And finally, frequency response.

The next stop is band-stop filters or notch filters.
There are several variations here. Probably the simplest is the Wien-Robinson Filter. The typical circuit is also a 4th order filter.


Our last task.
Task #4. Construct a notch filter with a central frequency of 90 Hz, quality factor Q= 2 and gain in the passband equal to 1.
First of all, we randomly select the capacitance of the capacitor. Let's say C = 100 nF.
Let's determine the value R6 = R7 = R:

It is logical that by “playing” with these resistors, we can change the frequency range of our filter.
Next, we need to determine intermediate coefficients. We find them through quality factor.


Let's choose a resistor arbitrarily R2. In this particular case, it is best for it to be 30 kOhm.
Now we can find resistors that will regulate the gain in the passband.


And finally, you need to randomly select R5 = 2R1. In my circuit, these resistors have a value of 40 kOhm and 20 kOhm, respectively.
Actually, frequency response:

Almost the end

For those interested in learning a little more, I can recommend reading Horowitz and Hill’s “The Art of Circuit Design.”
Also, D. Johnson “A handbook of active filters”.