Yuri Sadikov
Moscow
The article presents the results of work on creating a device that is a set of active filters for building high-quality three-band low-frequency amplifiers of the HiFi and HiEnd classes.
In the process of preliminary studies of the total frequency response of a three-band amplifier built using three second-order active filters, it turned out that this characteristic has a very high unevenness at any filter junction frequencies. At the same time, it is very critical to the accuracy of filter settings. Even with a small mismatch, the unevenness of the total frequency response can be 10...15 dB!
MASTER KIT produces the NM2116 kit, from which you can assemble a set of filters, built on the basis of two filters and a subtractive adder, which does not have the above-mentioned disadvantages. The developed device is insensitive to the parameters of the cutoff frequencies of individual filters and at the same time provides a highly linear total frequency response.
The main elements of modern high-quality sound reproducing equipment are acoustic systems (AS).
The simplest and cheapest are single-way speakers that contain one loudspeaker. Such acoustic systems are not capable of operating with high quality in a wide frequency range due to the use of a single loudspeaker (loudspeaker head - GG). When reproducing different frequencies, different requirements are placed on the GG. At low frequencies (LF), the speaker must have a large and rigid cone, a low resonant frequency and have a long stroke (to pump a large volume of air). And at high frequencies (HF), on the contrary, you need a small, lightweight but solid diffuser with a small stroke. It is almost impossible to combine all these characteristics in one loudspeaker (despite numerous attempts), so a single loudspeaker has high frequency unevenness. In addition, in wideband loudspeakers there is an intermodulation effect, which manifests itself in the modulation of high-frequency components of an audio signal by low-frequency ones. As a result, the sound picture is disrupted. The traditional solution to this problem is to divide the reproduced frequency range into subranges and build acoustic systems based on several speakers for each selected frequency subrange.
Passive and active electrical isolation filters
To reduce the level of intermodulation distortion, electrical isolation filters are installed in front of the loudspeakers. These filters also perform the function of distributing the energy of the audio signal between the GG. They are designed for a specific crossover frequency, beyond which the filter provides a selected amount of attenuation, expressed in decibels per octave. The slope of the attenuation of the separating filter depends on the design of its construction. The first order filter provides an attenuation of 6 dB/oct, the second order - 12 dB/oct, and the third order - 18 dB/oct. Most often, second-order filters are used in speakers. Filters of higher orders are rarely used in speakers due to the complex implementation of the exact values of the elements and the lack of need to have higher attenuation slopes.
The filter separation frequency depends on the parameters of the GG used and on the properties of hearing. The best choice of crossover frequency is at which each GG speaker operates within the piston action area of the diffuser. However, in this case, the speaker must have many crossover frequencies (respectively, GG), which significantly increases its cost. It is technically justified that for high-quality sound reproduction it is enough to use three-band frequency separation. However, in practice there are 4, 5 and even 6-way speaker systems. The first (low) crossover frequency is selected in the range of 200...400 Hz, and the second (middle) crossover frequency in the range of 2500...4000 Hz.
Traditionally, filters are made using passive L, C, R elements, and are installed directly at the output of the final power amplifier (PA) in the speaker housing, according to Fig. 1.
Fig.1. Traditional performance of speakers.
However, this design has a number of disadvantages. Firstly, to ensure the required cutoff frequencies, you have to work with fairly large inductances, since two conditions must be met simultaneously - to provide the required cutoff frequency and to ensure that the filter is matched with the GG (in other words, it is impossible to reduce the inductance by increasing the capacitance included in the filter). It is advisable to wind inductors on frames without the use of ferromagnets due to the significant nonlinearity of their magnetization curve. Accordingly, air inductors are quite bulky. In addition, there is a winding error, which does not allow for an accurately calculated cutoff frequency.
The wire used to wind the coils has a finite ohmic resistance, which in turn leads to a decrease in the efficiency of the system as a whole and the conversion of part of the useful power of the PA into heat. This is especially noticeable in car amplifiers, where the supply voltage is limited to 12 V. Therefore, to build car stereo systems, GGs with reduced winding resistance (~2...4 Ohms) are often used. In such a system, the introduction of additional filter resistance of the order of 0.5 Ohm can lead to a decrease in output power by 30%...40%.
When designing a high-quality power amplifier, they try to minimize its output impedance to increase the degree of damping of the GG. The use of passive filters significantly reduces the degree of damping of the GG, since additional filter reactance is connected in series with the amplifier output. For the listener, this manifests itself in the appearance of “booming” bass.
An effective solution is to use not passive, but active electronic filters, which do not have all the listed disadvantages. Unlike passive filters, active filters are installed before the PA as shown in Fig. 2.
Fig.2. Construction of a sound-reproducing path using active filters.
Active filters are RC filters on operational amplifiers (op amps). It is easy to build active audio filters of any order and with any cutoff frequency. Such filters are calculated using tabular coefficients with a pre-selected filter type, required order and cutoff frequency.
The use of modern electronic components makes it possible to produce filters with minimal intrinsic noise levels, low power consumption, dimensions and ease of execution/replication. As a result, the use of active filters leads to an increase in the degree of damping of the GG, reduces power losses, reduces distortion and increases the efficiency of the sound reproduction path as a whole.
The disadvantages of this architecture include the need to use several power amplifiers and several pairs of wires to connect speaker systems. However, this is not critical at this time. The level of modern technology has significantly reduced the price and size of the mind. In addition, quite a lot of powerful integrated amplifiers with excellent characteristics have appeared, even for professional use. Today, there are a number of ICs with several PAs in one case (Panasonic produces the RCN311W64A-P IC with 6 power amplifiers specifically for building three-way stereo systems). In addition, the PA can be placed inside the speakers and short, large-section wires can be used to connect the speakers, and the input signal can be supplied via a thin shielded cable. However, even if it is not possible to install the PA inside the speakers, the use of multi-core connecting cables does not pose a difficult problem.
Modeling and selection of the optimal structure of active filters
When constructing a block of active filters, it was decided to use a structure consisting of a high-pass filter (HPF), a medium-frequency filter (band-pass filter, PSF) and a low-pass filter (LPF).
This circuit solution was practically implemented. A block of active filters LF, HF and PF was built. A three-channel adder was chosen as a model of a three-way speaker, providing summation of frequency components, according to Fig. 3.
Fig.3. Model of a three-channel speaker with a set of active filters and a filter filter on the PF.
When measuring the frequency response of such a system, with optimally selected cutoff frequencies, it was expected to obtain a linear dependence. But the results were far from expected. At the junction points of the filter characteristics, dips/overshoots were observed depending on the ratio of the cutoff frequencies of neighboring filters. As a result, by selecting the cutoff frequency values, it was not possible to bring the pass-through frequency response of the system to a linear form. The nonlinearity of the pass-through characteristic indicates the presence of frequency distortions in the reproduced musical arrangement. The results of the experiment are presented in Fig. 4, Fig. 5 and Fig. 6. Fig. 4 illustrates the pairing of a low-pass filter and a high-pass filter at a standard level of 0.707. As can be seen from the figure, at the junction point the resulting frequency response (shown in red) has a significant dip. When expanding the characteristics, the depth and width of the gap increases, respectively. Fig. 5 illustrates the pairing of a low-pass filter and a high-pass filter at a level of 0.93 (shift in the frequency characteristics of the filters). This dependence illustrates the minimum achievable unevenness of the pass-through frequency response by selecting the cutoff frequencies of the filters. As can be seen from the figure, the dependence is clearly not linear. In this case, the cutoff frequencies of the filters can be considered optimal for a given system. With a further shift in the frequency characteristics of the filters (matching at a level of 0.97), an overshoot appears in the pass-through frequency response at the junction point of the filter characteristics. A similar situation is shown in Fig. 6.
Fig.4. Low-pass frequency response (black), high-pass frequency response (black) and pass-through frequency response (red), matching at level 0.707.
Fig.5. Low-pass frequency response (black), high-pass frequency response (black) and pass-through frequency response (red), matching at level 0.93.
Fig.6. Low-pass frequency response (black), high-pass frequency response (black) and pass-through frequency response (red), matching at the level of 0.97 and the appearance of an overshoot.
The main reason for the nonlinearity of the pass-through frequency response is the presence of phase distortions at the boundaries of the filter cutoff frequencies.
A similar problem can be solved by constructing a mid-frequency filter not in the form of a bandpass filter, but using a subtractive adder on an op-amp. The characteristics of such a PSF are formed in accordance with the formula: Usch = Uin - Uns - Uss
The structure of such a system is shown in Fig. 7.
Fig.7. Model of a three-channel speaker with a set of active filters and a PSF on a subtractive adder.
With this method of forming a mid-frequency channel, there is no need to fine-tune adjacent filter cutoff frequencies, because The mid-frequency signal is formed by subtracting the high- and low-pass filter signals from the total signal. In addition to providing complementary frequency responses, the filters also produce complementary phase responses, which guarantees the absence of emissions and dips in the total frequency response of the entire system.
The frequency response of the mid-frequency section with cutoff frequencies Fav1 = 300 Hz and Fav2 = 3000 Hz is shown in Fig. 8. According to the fall in the frequency response, an attenuation of no more than 6 dB/oct is ensured, which, as practice shows, is quite sufficient for the practical implementation of the PSF and obtaining high-quality sound of the midrange GG.
Fig.8. Frequency response of the mid-pass filter.
The pass-through transmission coefficient of such a system with a low-pass filter, a high-pass filter and a high-pass filter on a subtracting adder turns out to be linear over the entire frequency range of 20 Hz...20 kHz, according to Fig. 9. Amplitude and phase distortions are completely absent, which ensures crystal purity of the reproduced sound signal.
Fig.9. Frequency response of a filter system with a frequency filter on a subtractive adder.
The disadvantages of such a solution include strict requirements for the accuracy of the values of resistors R1, R2, R3 (according to Fig. 10, which shows the electrical circuit of the subtracting adder) that ensure balancing of the adder. These resistors should be used within 1% accuracy tolerances. However, if problems arise with the acquisition of such resistors, you will need to balance the adder using trimming resistors instead of R1, R2.
Balancing the adder is performed using the following method. First, a low-frequency oscillation with a frequency much lower than the low-pass filter cutoff frequency, for example 100 Hz, must be applied to the input of the filter system. By changing the value of R1, it is necessary to set the minimum signal level at the output of the adder. Then an oscillation with a frequency obviously higher than the high-pass filter cutoff frequency, for example 15 kHz, is applied to the input of the filter system. By changing the value of R2, the minimum signal level at the output of the adder is again set. The setup is complete.
Fig. 10. Subtractive adder circuit.
Methodology for calculating active low-pass filters and high-pass filters
As theory shows, to filter the frequencies of the audio range, it is necessary to use Butterworth filters of no more than the second or third order, ensuring minimal unevenness in the passband.
The second-order low-pass filter circuit is shown in Fig. 11. Its calculation is made according to the formula:
where a1=1.4142 and b1=1.0 are tabular coefficients, and C1 and C2 are selected from the ratio C2/C1 greater than 4xb1/a12, and you should not choose the ratio C2/C1 much greater than the right side of the inequality.
Fig. 11. 2nd order Butterworth low pass filter circuit.
The second-order high-pass filter circuit is shown in Fig. 12. Its calculation is made using the formulas:
where C=C1=C2 (set before calculation), and a1=1.4142 and b1=1.0 are the same table coefficients.
Fig. 12. 2nd order Butterworth high-pass filter circuit.
MASTER KIT specialists have developed and studied the characteristics of such a filter unit, which has maximum functionality and minimal dimensions, which is essential when using the device in everyday life. The use of modern element base made it possible to ensure maximum quality of development.
Technical characteristics of the filter unit
The electrical circuit diagram of the active filter is shown in Fig. 13. The list of filter elements is given in the table.
The filter is made using four operational amplifiers. The op-amps are combined in one MC3403 (DA2) IC package. DA1 (LM78L09) contains a supply voltage stabilizer with corresponding filter capacitors: C1, C3 at the input and C4 at the output. An artificial midpoint is made on the resistive divider R2, R3 and capacitor C5.
The DA2.1 op amp has a buffer cascade for pairing the output and input impedances of the signal source and low-pass, high-pass and mid-range filters. A low-pass filter is assembled on op-amp DA2.2, and a high-pass filter is assembled on op-amp DA2.3. Op-amp DA2.4 performs the function of a bandpass midrange filter shaper.
The supply voltage is supplied to contacts X3 and X4, and the input signal is supplied to contacts X1, X2. The filtered output signal for the low-frequency path is removed from contacts X5, X9; with X6, X8 – HF and with X7, X10 – MF paths, respectively.
Fig. 13. Electrical circuit diagram of an active three-band filter
List of elements of an active three-band filter
| Position | Name | Note | Col. |
| C1, C4 | 0.1 µF | Designation 104 | 2 |
| C2, C10, C11, C12, C13, C14, C15 | 0.47 µF | Designation 474 | 7 |
| C3, C5 | 220 µF/16 V | Replacement 220 uF/25 V | 2 |
| C6, C8 | 1000 pF | Designation 102 | 2 |
| C7 | 22 nF | Designation 223 | 1 |
| C9 | 10 nF | Designation 103 | 1 |
| DA1 | 78L09 | 1 | |
| DA1 | MC3403 | Replacement LM324, LM2902 | 1 |
| R1…R3 | 10 kOhm | 3 | |
| R8…R12 | 10 kOhm | Tolerance no more than 1%* | 5 |
| R4…R6 | 39 kOhm | 3 | |
| R7 | 75 kOhm | - | 1 |
| DIP-14 block | 1 | ||
| Pin connector | 2 pin | 2 | |
| Pin connector | 3 pin | 2 |
The appearance of the filter is shown in Fig. 14, the printed circuit board is shown in Fig. 15, the location of the elements is shown in Fig. 16.
Structurally, the filter is made on a printed circuit board made of foil fiberglass. The design provides for installation of the board into a standard BOX-Z24A case; for this purpose, mounting holes are provided along the edges of the board with a diameter of 4 and 8 mm. The board is secured in the case with two self-tapping screws.
Fig. 14. External view of the active filter.
Fig. 15. Active filter printed circuit board.
Fig. 16. Arrangement of elements on the active filter printed circuit board.
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 the transmission and attenuation bands; in them, it is relatively easy to adjust the unevenness of the frequency response in the transmission and attenuation regions. Also, active filter circuits typically do not use inductors. In active filter circuits, frequency characteristics are determined by frequency-dependent feedback.
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 electrical circuit parameters 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.
When implementing low-pass, high-pass, and bandpass filters, the second-order Sallen-Key filter design is widely used. In Fig. 1 shows its version for a low-pass filter.
Negative feedback (voltage divider R 3 + (b-1) R 3), provides a gain equal to b. Positive feedback is generated by a capacitor WITH 2 .
Rice. 1.
The filter transfer function has the form:
where do we get from
From these equations for the coefficients of the filter transfer function b 0 , a 1 , a 2 you can calculate the values of the circuit elements.
To simplify the calculation, you can set some additional conditions, for example, choose the gain coefficient b = 1. Then ( b-1) R 3 = 0 and the resistive voltage divider in the negative feedback circuit can be eliminated, therefore the op-amp is connected according to a non-inverting follower circuit.
In this case, the filter transfer function is equal to
where do we get from
Next, you can set the resistor values R 1 and R 3 and calculate the values R 2 , WITH 1 , WITH 2. For circuit implementation, it is more convenient, however, to select from the nominal range of capacitances of capacitors the values WITH 1 and WITH 2 and then calculate R 1 and R 2 .
By swapping the resistances and capacitors in the circuit in Fig. 1. We get a high-pass filter (Fig. 2).
Rice. 2.
Its transfer function is described by the expression
To simplify the calculations, you can choose b = 1 and WITH 1 = WITH 2 = WITH. Then
A second-order bandpass filter based on the Sallen-Key circuit is shown in Fig. 3. The filter transfer function has the form:
Here is the resonant frequency of the PF, the lower and upper cutoff frequencies of the PF. Parameter b can be expressed through: . Having calculated b and sch R, can be calculated by further asking WITH or R find the value of the remaining component.
Rice. 3.
The advantage of filters according to the Sallen-Key circuit is the ability to separately adjust the quality factor of the poles and cutoff frequencies, the disadvantage is the high sensitivity of the filter parameters to changes in the parameters of the components that make up its circuit.
Less sensitive to the parameters of the components is the second-order link circuit with multi-loop negative feedback (Rauch circuit). The second-order low-pass filter section of such a filter is shown in Fig. 4.
Rice. 4.
The transfer function of such a filter has the following form:
Expressing the coefficients of the transfer function from here, we get
Coefficient b 0 defines the filter gain at zero frequency. By asking them, you can find the relationship between R 1 and R 2, then select the capacitances of the capacitors and by solving two equations for two unknowns, you can find the resistance values of the resistors.
Rice. 5.
By swapping the capacitors and resistors in the circuit in Fig. 4, we get a high-pass filter with multi-loop feedback (Fig. 5).
Multi-loop negative feedback can also be used to construct bandpass filters. The corresponding diagram is shown in Fig. 6. The transfer function of this filter is given by
To calculate the parameters of the circuit, you can use the fact that at the resonant frequency the coefficient at s 2 in the denominator of the transfer function must equal 1. Therefore, . Substituting this expression into H(s) and equating the corresponding coefficients to the coefficients of the transfer function of the designed filter, we obtain formulas for calculating the passband and gain of the filter at the resonant frequency. By choosing, for example, the values WITH and, from the formula for bandwidth one can find R 2, then using the formula to calculate R 1 and finally, from the formula for calculate R 3 .
By adding a resistive divider in the circuit in Fig. 6 between the input of the circuit and the non-inverting input of the op-amp, we obtain a notch (band-stop) filter with multi-loop negative feedback.
Rice. 6.
The transfer function of this filter is equal to
The condition for complete suppression of the signal at the resonant frequency is that the coefficient is equal to zero b 1 transfer function, for which it is necessary to select the resistor resistances from the condition: . Resistor resistance R 3 and capacitor capacity WITH are selected in the same way as for the bandpass filter discussed above (Fig. 5).
Bandpass and notch filters can also be constructed by combining low-pass filter and high-pass filter units (Fig. 7).
A bandpass filter can be obtained by cascading a low-pass filter and a high-pass filter (Fig. 7a). The lower cutoff frequency of the PF is determined by the cutoff frequency of the high-pass filter, and the upper cutoff frequency of the low-pass filter. A notch filter can be constructed by connecting a low-pass filter and a high-pass filter in parallel (Fig. 7c).
a - bandpass filter, b - frequency response of a bandpass filter, c - notch filter, d - frequency response of a notch filter
Rice. 7. Structures of bandpass and notch filters based on low-pass filters and high-pass filters
Second-order link of a filter with a general transfer function
(the so-called biquadratic link) can be implemented in several ways. One of the variants of the diagram of such a link is shown in Fig. 8.
Rice. 8.
The transfer function of the low-pass filter section of this filter is expressed through the circuit parameters as follows:
If the condition is met R 1 R 3 =R 2 R 7, then the filter will be elliptical, but if you choose R 7 =?, then we get a second-order link of Butterworth, Chebyshev or Bessel polynomial filters.
The calculation of this filter can be done, for example, as follows. Having specified the filter gain at zero frequency K 0 and values WITH 1 , WITH 2 , R 3 we find the remaining parameters of the circuit:
Usually choose WITH 1 =WITH 2, a R 3 =1/sch c C 1 . The biquad filter is not very sensitive to the inaccuracy of the elements and is easy to configure.
The transfer functions of the high-pass, band-pass and notch filter sections can be obtained by appropriately replacing the variable s in the expression for the transfer function, similar to what is done when converting a normalized low-pass prototype into the required filter.
Active filters are produced in the form of ICs by many companies, for example, AFI00/150 (National Semiconductor), LTC1562 (Linear Technology), MAX270/271 or MAX274/275 (Maxim) chips. The cutoff frequency of the filters is tunable up to several hundred kHz, the order changes up to the eighth, and it is usually possible to program the filter type. An example is the MAX270 IC, which contains two programmable sections of a second-order Chebyshev low-pass filter according to the Salen-Key circuit (Fig. 9).
Rice. 9.
The cutoff frequency of each section is set by a parallel 7-bit binary code ranging from 1 to 25 kHz.
Conclusions and results
1. When implementing filters, the principle of constructing filters by cascading connections of second-order links is widely used. As such links, second-order filter links according to the Sallen-Key scheme, filter links with multi-loop negative feedback (Rauch scheme) and so-called biquadratic links are used.
2. The advantage of filter links according to the Sallen-Key circuit is the ability to separately adjust the quality factor of the poles and cutoff frequencies, the disadvantage is the high sensitivity of the filter parameters to changes in the parameters of the components that make up its circuit. The second-order link circuit with multi-loop negative feedback (Rauch circuit) is less sensitive to the parameters of the components.
3. In scientific and technical literature, as a rule, expressions of transfer functions and calculated relationships for elements in relation to low-pass filters are given. The required ratios for high-pass, band-pass and notch filters can be obtained by appropriately replacing the variable s in the expression for the transfer function, similar to what is done when converting a normalized low-pass filter prototype into the required filter.
filter circuit operational amplifier
Bibliography.
1. Volovich G.I. Circuit design of analog and analog-digital electronic devices. 2nd ed. - M.: DODEKA-XXI, 2007. - 528 p.: ill.
2. Polonnikov D.E. Operational amplifiers: principles of construction, theory, circuit design. - M.: Energoatomizdat, 1983. - 216 p.
3. Peyton A. J., Walsh V. Analog electronics on operational amplifiers. M.: BINOM, 1994. - 352 p.
4. Lam G. Analog and digital filters: calculation and implementation / G. Lam; lane from English Levina V.L. [and etc.]. - M.: Mir, 1982. - 592 p.
As the filter order increases, its filtering properties improve. A second-order filter is quite simply implemented on one op-amp. To implement low-pass, high-pass, and bandpass filters, the second-order Sallen-Key filter circuit has been widely used. In Fig. 17 shows its version for a low-pass filter. Negative feedback generated using a voltage divider R 3 , ( – 1)R 3, provides a gain equal to . Positive feedback is due to the presence of a capacitor WITH 2. The filter transfer function has the form:
Fig. 17. Second order active low pass filter
The calculation of the circuit is greatly simplified if you set some additional conditions from the very beginning. You can choose the gain = 1. Then ( – 1) R 3 = 0, and the resistive voltage divider in the negative feedback circuit can be eliminated. The op-amp turns out to be connected according to a non-inverting follower circuit. In the simplest case, it can even be replaced by an emitter follower on a composite transistor. When = 1, the filter transfer function takes the form:
Assuming that the capacitances of the capacitors WITH 1 and WITH 2 are selected, we get for the given values A 1 and b 1 (see (13)):
K 0 = 1,
.
To values R 1 and R 2 were valid, the condition must be met
.
The calculations can be simplified by putting R 1 =R 2 =R And WITH 1 =WITH 2 =WITH. In this case, to implement filters of various types, it is necessary to change the value of the coefficient . The filter transfer function will have the form
.
From here, taking into account formula (13), we obtain
,
.
From the last relationship it is clear that the coefficient determines the quality factor of the poles and does not affect the cutoff frequency. The value of in this case determines the type of filter.
By swapping resistances and capacitors we get high pass filter(Fig. 18). Its transfer function has the form:
Rice. 18. Second-order active high-pass filter
To simplify the calculations, let’s set = 1 and WITH 1 =WITH 2 =WITH. In this case we obtain the following formulas:
K besk = 1,R 1 = 2/ c Ca 1 , R 2 =a 1 /2 s Cb 1 .
If the frequency response of a second-order filter is not steep enough, a higher-order filter should be used. To do this, links representing first and second order filters are connected in series. In this case, the frequency response of the filter sections is multiplied (on a logarithmic scale - added). However, keep in mind that connecting, for example, two second-order Butterworth filters in series will not result in a fourth-order Butterworth filter. The resulting filter will have a different cutoff frequency and a different frequency response. Therefore, it is necessary to set the coefficients of the filter sections such that the result of multiplying their frequency characteristics corresponds to the desired type of filter.
Bandpass filter second order can be implemented based on the Sallen-Key scheme, as shown in Fig. 19. The filter transfer function has the form:
|
|
Rice. 19. Second order bandpass filter circuit
Equating the coefficients of this expression with the coefficients of the transfer function (18), we obtain formulas for calculating filter parameters:
f p = 1/2 R.C.; K p =/(3 –); Q = 1/(3 –).
The disadvantage of the circuit is that the gain at the resonant frequency K p and quality factor Q are not independent from each other. The advantage of the circuit is that its quality factor varies depending on , while the resonant frequency does not depend on the coefficient .
An active rejection filter can be implemented using a double T-bridge. Although the double T-bridge itself is a rejection filter, its quality factor is only 0.25. It can be increased if the bridge is included in the op-amp feedback circuit. One of the variants of such a scheme is shown in Fig. 20. High and low frequency signals pass through the double T bridge without change. For them, the filter output voltage is equal to U input At the resonant frequency, the output voltage is zero. The transfer function of the circuit in Fig. 20 looks like:
,
or considering that p = 1/ R.C.,
|
|
Using this expression, you can directly determine the required filter parameters. Setting the gain of the non-inverting amplifier to 1, we get Q=0.5. As the gain increases, the quality factor increases and tends to infinity if tends to 2.
Rice. 20. Active barrier filter with double T-bridge
Active RC filters are used at frequencies below 100 kHz. The use of positive feedback allows you to increase the quality factor of the filter pole. In this case, the filter pole can be implemented on RC elements, which are much cheaper and in this frequency range have smaller inductance dimensions. In addition, the capacitance value of the capacitor included in the active filter can be reduced, since in some cases the amplifying element allows its value to be increased. The use of capacitors with low capacitance allows you to choose their types with low losses and high stability of parameters.
When designing active filters, a filter of a given order is divided into first and second order units. The resulting frequency response is obtained by multiplying the characteristics of all links. The use of active elements (transistors, operational amplifiers) makes it possible to eliminate the influence of the links on each other and design them independently. This circumstance greatly simplifies and reduces the cost of designing and configuring active filters.
Figure 2 shows the circuit of an active RC low-pass filter of the first order on an operational amplifier. This circuit allows you to implement a gain pole at zero frequency; the values of the resistance of resistor R1 and the capacitance of capacitor C1 can set its cutoff frequency. It is the values of capacitance and resistance that will determine the bandwidth of a given active filter circuit.
In the circuit shown in Figure 2, the gain is determined by the ratio of resistors R2 and R1:
(1),and the capacitance value of capacitor C1 increases by a factor of gain plus one times due to the Miller effect.
(2),It should be noted that this method of increasing the capacitance value leads to a decrease in the dynamic range of the circuit as a whole. Therefore, this method of increasing the capacitance of a capacitor is resorted to in extreme cases. Usually they get by with an integrating RC circuit, in which a decrease in the cutoff frequency is achieved by increasing the resistance of the resistor at a constant value of the capacitor capacitance. In order to eliminate the influence of load circuits, a buffer amplifier with unity voltage gain is usually installed at the output of the RC circuit.
However, if the cutoff frequency of the low-pass filter is low enough, a large capacitor value may be required. Electrolytic capacitors, which have a significant capacitance, are not suitable for creating filters due to the wide spread of parameters and low stability. Capacitors made of ceramics with a high electrical constant ε , also do not differ in the stability of the capacitance value. Therefore, highly stable, low-capacity capacitors are used, and their value increases in the active filter circuit shown in Figure 2.
Even more common are second-order active filter circuits, which make it possible to realize a greater slope of the frequency response compared to a first-order circuit. In addition, these links allow you to adjust the pole frequency to a given value obtained by approximating the amplitude-frequency response. The most widely used scheme is the Sallen-Key scheme shown in Figure 4.
The amplitude-frequency response of this circuit is similar to the frequency response of the second-order section of a passive LC filter. Its appearance is shown in Figure 5.
The resonance frequency of the pole can be determined from the formula:
(3),
and its quality factor:
(4),
The zero frequencies are ideally equal to infinity. In a real circuit, they depend on the design of the printed circuit board and the parameters of the resistors and capacitors used.
The Sallen-Key scheme makes it possible to simplify the selection of circuit elements as much as possible. Typically, capacitors C1 and C2 are chosen to have the same capacitance. Resistors R1 and R2 choose the same resistance. First, they are set by the values of capacitances C1 and C2. As already discussed above, they try to choose minimal capacities. It is these capacitors that have the most stable characteristics. Then determine the value of R1 and R2:
(5),
Resistors R3 and R4 in the Sallen-Key circuit determine the voltage gain in the same way as in a conventional inverting amplifier circuit. In an active filter circuit, it is these elements that will determine the quality factor of the pole.
(6),
In an active RC filter circuit, the amplifier is covered by both negative and positive feedback. The depth of positive feedback is determined by the ratio of resistors R1R2 or capacitors C1C2. If the quality factor of the pole is set due to this ratio (rejecting the equality of resistances or capacitors), then the operational amplifier can be covered by 100% negative feedback and provide unity gain of the active element. This will simplify the second-order link diagram. A simplified circuit of a second order active RC filter is shown in Figure 6.
Unfortunately, with a unity gain, you can only set the same values of the resistances R1 and R2, and the required quality factor can be obtained by the ratio of the capacitances. Therefore, the calculation begins by setting the nominal value of the resistors R1 = R2 = R. Then the capacitances can be calculated as follows:
(7),
(8),
For many years now, everyone has become accustomed to using an operational amplifier as an active element. However, in some cases it may turn out that the transistor circuit will either occupy a smaller area or be more broadband. Figure 7 shows a diagram of an active low-pass filter made on a bipolar transistor.
The calculation of this circuit (elements R1, R2, C1, C2) does not differ from the calculation shown in Figure 6. The calculation of resistors R3, R4, R5 does not differ from the calculation of a conventional emitter stabilization cascade.
The first frequency filters were passive LC filters. Then, already in the 30s of the 20th century, it was noticed that feedback in amplifier stages can increase the quality factor of LC circuits of radio amplifiers. One of the most common schemes for increasing the quality factor of a parallel LC circuit is shown in Figure 1.
This feature is not widely used in LC circuits, since LC circuits allow constructive methods to provide the quality factor necessary for the implementation of most filter circuits operating at high frequencies. At the same time, positive feedback circuits used to increase the quality factor of circuits are self-exciting and usually limit the dynamic range of the output signal due to the influence of noise in the amplifier stage.
A completely different situation has developed in the low frequency region. These are mainly frequencies in the audio range (from 20 Hz to 20 kHz). In this frequency range, the dimensions of inductors and capacitors become unacceptably large. In addition, the losses of these radio-technical elements also increase, which in most cases does not allow obtaining the quality factor of the filter poles necessary to implement a given value. All this led to the need to use amplification stages.
Last file update date: 06/18/2018
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