Active frequency filters on operational amplifiers. Filters on OU microcircuits. Conclusions and results
When designing an active filter based on an op-amp, the following data must be determined in advance:
available power supplies – bipolar or unipolar;
range of transmitted and filtered frequencies;
transition frequency, i.e. the point of the characteristic at which the filter begins to operate, or the resonant frequency around which the filter characteristic is symmetrical;
the initial value of the capacitor capacitance: for high-pass filters (HPF) it should be selected from 100 pF, and for low-pass filters - from 0.1 µF.
Let's consider six options for active filter structures. On frequency characteristics ah the frequency transmission region is shaded.
1. Low pass filter (LPF). The filter circuit for a bipolar power supply is shown in Fig. P.2.1a, and for unipolar - in Fig. P.2.1b. Amplitude characteristic filter is shown in Fig. P.2.2. This type of filter is a unit gain filter.
| A |  b |
Calculation procedure
1. Select the size of capacitance C1 (according to the recommendations).
2. We count WITH 2 = 2WITH 1 .
3. Calculate the values of resistors R1 and R2:
,
Where f
For filter with unipolar power supply(Fig. 1b) WITH in = WITH out = (100…1000) WITH 1 (not critical), but R 3 = R 4 = 100 kOhm.
2. High pass filter (HPF). The filter circuit for a bipolar power supply is shown in Fig. P.2.3a, and for unipolar - in Fig. P.2.3b. The amplitude response of the filter is shown in Fig. P.2.4.
 A) |  b) |
Calculation procedure
1. Choose WITH 1 = WITH 2 (according to recommendations).
2. Calculate the value of resistor R1:
,
3. Calculate the value of resistor R2:
,
Where f– frequency of the fundamental harmonic of the filter output voltage.
WITH in = WITH out = (100…1000) ∙ WITH 1 (not critical).
3. Narrowband filter. The filter circuit for a bipolar power supply is shown in Fig. P.2.5a, and for unipolar - in Fig. P.2.5b. The amplitude response of the filter is shown in Fig. P.2.6.
 A) |  b) |
The quality factor of such a filter Q= 10, which allows us to obtain the transmission coefficient k= 10 because
.
It is not advisable to choose a higher quality factor, since the product of the gain and the bandwidth of the operational amplifier can easily be achieved even with k= 20 dB. At least a 40 dB bandwidth must be provided above the peak resonant frequency. The slew rate of the operational amplifier's output voltage must be sufficient to ensure that the amplitude of the output voltage at the resonant frequency reaches the required level.
Calculation procedure
1. Choose WITH 1 = WITH 2 .
,
Where f
3. We count R 2 = R 1/19 and R 3 = 19 · R 1 .
For a filter with single-polar power supply (Fig. 3b) WITH in = WITH out = (100…1000) ∙ WITH 1 (not critical).
4. Wideband filter. The filter circuit for a bipolar power supply is shown in Fig. 7a, and for unipolar – in Fig. P.2.7b. The amplitude response of the filter is shown in Fig. P.2.8. The start and end frequencies of the passband must differ by at least five times.
 A) |  b) |
This is nothing more than a cascade of Salen-Key high- and low-pass filters. The high-pass filter runs first, so its output energy, which tends to infinite frequency, passes through the low-pass filter.
Calculation procedure
1. Using section 2, we calculate a high-pass filter for the lower limit of the bandwidth.
2. Using section 1, we calculate the low-chat filter for the upper limit of the bandwidth.
For a filter with single-polar power supply (Fig. 3b) WITH in = WITH out = (100…1000) ∙ WITH 1 (not critical).
5. Filter plug. The filter circuit for a bipolar power supply is shown in Fig. 9a, and for unipolar – in Fig. P.2.9b. The amplitude response of the filter is shown in Fig. P.2.10. The start and end frequencies of the passband must differ by at least five times.
 A) |  b) |
In such a scheme, the quality factor Q= 10. It can be adjusted independently of the resonant frequency by changing R1 and R2. The quality factor depends on the resistor that sets the resonant frequency as follows:
.
With this filter circuit topology, the transmission coefficient is 1.
The only problem is the amplitude of the common mode noise of the lower amplifier in the case of a single-pole supply.
Calculation procedure
1. Choose WITH 1 = WITH 2 .
2. Calculate the resistor values:
,
Where f– input voltage frequency.
3. We count R 1 = R 2 = 20 · R 3 .
For a filter with single-polar power supply (Fig. 3b) WITH in = WITH out = (100…1000) ∙ WITH 1 (not critical), R 5 = R 6 = 100 kOhm.
5. Band-stop filter. The filter circuit for a bipolar power supply is shown in Fig. P.2.11a, and for unipolar - in Fig. P.2.11b. The amplitude response of the filter is shown in Fig. P.2.12. The start and end frequencies of the passband must differ by at least fifty times.
 A) |  b) |
 Rice. P.2.12 |
In this case, cascading is not possible since the filter characteristics do not overlap, as in the case of a wideband filter
1. Using Section 2, we calculate the high-pass filter for the lower limit of the upper passband.
2. Using Section 1, we calculate the low-pass filter for the upper limit of the lower passband.
For a filter with single-polar power supply (Fig. 3b) WITH in = WITH out =
= (100…1000) ∙ WITH 1 (not critical). R 3 = R 4 = R 5 = 100 k.
Appendix 3
ACTIVE RECTIFIERS
Often it is necessary to isolate a signal component of one polarity (half-wave rectification), or determine the absolute value of the signal (full-wave rectification). Such circuits can be implemented using diode-resistive circuits. However, the large voltage drop across the diode at forward bias (0.5–1 V) and the nonlinearity of their current-voltage characteristics will introduce significant errors, especially at low input signal levels. The use of op-amps can significantly reduce the influence of the characteristics of real diodes.
Circuits of non-inverting half-wave rectifiers are shown in Fig. P.3.1. ( And out > 0 – fig. P.3.1a; And out< 0 – рис. П.3.1б) Диод VD2 необходим для повышения быстродействия схем за счет замыкания выхода ОУ на землю. Поэтому следует использовать такие ОУ, которые допускают короткоезамыкание выхода в течение длительного времени. При отсутствии этого диода в режиме отсечки ОУ будет входить в состояние ограничения сигнала на уровне напряжения питания.
 A |  b |
 Rice. P.3.2 |
The circuit of a full-wave active rectifier is shown in Fig. P.3.2. In such a circuit, an inverting connection of the op-amp is used and the same input resistances are provided for both half-waves of the output voltage. The circuit consists of an adder (DA2) and a half-wave rectifier based on an op-amp (DA1).
Let's consider the operating modes of op-amp DA1. With a positive input voltage, DA1 works as an inverting amplifier - voltage And 2 is negative, so diode VD1 is open and diode VD2 is closed. As a result And 1 = –And input When the input voltage is negative, And 2 becomes positive, and diode VD1 is turned off and VD2 is unlocked. The negative feedback circuit is closed, as a result of which the summation point remains at zero potential. Since diode VD1 is locked, the voltage And 1 is also zero.
Active filters are called electronic amplifiers containing R.C.-circuits, with the help of which the amplifier is given certain selective properties.
The use of amplifying elements distinguishes active filters from filters based on passive elements.
The advantages of active filters primarily include:
The ability to amplify a signal lying in the filter passband;
The ability to refuse the use of such low-tech elements as inductors, the use of which is incompatible with the methods of integral technology;
Easy to set up;
Small mass and volume, which weakly depend on the bandwidth, which is especially important when developing devices operating in the low-frequency region;
Simplicity of cascade inclusion when constructing high-order filters.
However, active filters have the following disadvantages that limit their scope of application:
Inability to use in power circuits, for example as filters in secondary power supplies;
Need for use additional source energy intended to power the active elements of the amplifier;
Let's consider the general principles of using op-amps with frequency-dependent feedback loops to form devices with different frequency properties.
Low and High Pass Filters
The simplest active filters of low and high frequencies of the first order are, respectively, integrating (Figures 3.13, 3.14) and differentiating (Figures 3.16, 3.17) amplifiers. In them, the main element that determines the frequency response of the amplifier is a capacitor included in the feedback circuit.
The transfer functions of the simplest filters are first order equations, that's why filters are called first order filters. The slope of the logarithmic frequency response (LAFC) outside the passband for first-order filters is only -20 dB/dec, which indicates poor selective properties such filters.
To improve selectivity, it is necessary either to increase the order of the filter transfer function by introducing additional R.C.-circuits, or sequentially connect several identical active filters.
In practice, op-amps with feedback circuits are most often used as filters, the operation of which is described by second-order equations. If it is necessary to increase the selectivity of the system, several second-order filters are included sequentially(for example, to obtain a fourth-order low-pass filter, two second-order low-pass filters are connected sequentially, to obtain a sixth-order low-pass filter, three second-order low-pass filters are connected in series, etc.).
Active filters of low and high frequencies of the second order are shown in Figure 3.28, A, b. For them, with appropriate selection of resistor and capacitor values, the LFC decline outside the passband is 40 dB/dec. Moreover, as can be seen from Figure 3.28, the transition from a low-pass filter to a high-pass filter is carried out by replacing resistors with capacitors, and vice versa.
a b
Figure 3.28 - Low-pass filter ( A) and high-pass filter ( b) second order on an operational amplifier
The transfer function of a second-order low-pass filter is described by the expression
and the second-order high-pass filter - by the expression
The cutoff frequencies of second-order filters are respectively equal to:
; (3.40)
. (3.41)
Last time wide use we obtained active low-pass filters and second-order high-pass filters implemented on voltage followers (the maximum value of the voltage gain for such filters within the passband is 1). The circuits of these filters are shown in Figure 3.29, A(LPF) and 3.29, b(HPF).
a b
Figure 3.29 - Low-pass filter ( A) and high-pass filter ( b) second order on voltage followers
The sequence of calculation of filter elements based on repeaters is as follows:
a) using the graphs (Figure 3.30), select the appropriate filter characteristic (taking into account the required selectivity) and determine the number of poles required to obtain the desired attenuation;
b) select a suitable filter circuit from the repeater circuits (Figure 3.29);
c) using the data in Table 3.2, perform the necessary recalculation of the parameters of the filter elements.
Table 3.2 gives the capacitance values (in farads) for the repeater circuit depending on the number of filter poles. In this case, to obtain a filter, for example, a fourth order, a cascade connection of two identical repeaters is used, but the elements of the first cascade are calculated as for a filter with two poles, and the elements of the second cascade as for a filter with four poles.
Figure 3.30 - Amplitude-frequency characteristics of low-pass filter (left) and high-pass filter (right) Butterworth
Table 3.2 - Capacitor values (farads)
| Number of poles | Bessel filter | Butterworth filter | ||
| C 1 | WITH 2 | C 1 | WITH 2 | |
| 0,9066 | 0,6799 | 1,414 | 0,7071 | |
| 0,7351 1,0120 | 0,6746 0,3900 | 1,082 2,613 | 0,9241 0,3825 | |
| 0,6352 0,7225 1,0730 | 0,6098 0,4835 0,2561 | 1,035 1,414 3,863 | 0,9660 0,7071 0,2588 | |
| 0,5673 0,6090 0,7257 1,1160 | 0,5539 0,4861 0,3590 0,1857 | 1,091 1,202 1,800 5,125 | 0,9809 0,8313 0,5557 0,1950 | |
| 0,5172 0,5412 0,5999 0,7326 1,1510 | 0,5092 0,4682 0,3896 0,2792 0,1437 | 1,012 1,122 1,414 2,202 6,389 | 0,9874 0,8908 0,7071 0,4540 0,1563 |
Figure 3.31 shows the procedure for calculating filter circuits using repeaters using the example of a two-pole low-pass filter (left) and a Butterworth high-pass filter (right) with a cutoff frequency f in= 1 kHz.
The component values taken from Table 3.2 for the low-pass filter circuit are normalized for a frequency of 1 rad/s with a resistor resistance of 1 Ohm and capacitor capacity in farads. The capacitances of the filter capacitors are recalculated by frequency by dividing the capacitance values taken from the table by the cutoff frequency in radians (2p f in). The filter components are recalculated by multiplying the resistance values by a suitable coefficient (for example, 10 4) and dividing the capacitance values by the same coefficient. As a result, we obtain the following values for the parameters of the low-pass filter elements: WITH 1 = 0.0225 µF, WITH 2 = 0.0112 µF, R 1 = R 2 = 10 kOhm.
The component values taken from Table 3.2 for the high-pass filter circuit are normalized for a frequency of 1 rad/s with a capacitor capacity of 1 F and resistor resistance in ohms, the reciprocal of the capacitance values. The capacitances of the filter capacitors are recalculated by frequency by dividing the capacitance values by the cutoff frequency in radians (2p f n). The filter components are recalculated by multiplying the resistance values by a suitable coefficient (for example, 14.1 10 3) and dividing the capacitance values by the same coefficient. As a result, we obtain the following values for the parameters of the high-pass filter elements: WITH 1 = WITH 2 = 0.0113 µF, R 1 = 10 kOhm, R 2 = 20 kOhm.
Bandpass and notch filters
simplest band pass filter can be obtained by combining low-pass and high-pass filters (e.g., integrator and differentiator). An example of such a circuit is shown in Figure 3.32, A, and its logarithmic frequency response is in Figure 3.32, b.
The filter cutoff frequencies are determined from the expressions:
Figure 3.31 - Sequence of calculation of low-pass filter (left) and high-pass filter (right)
For measurement technology and signal processing technology, three types of PF circuits are of interest:
- multi-loop filter feedback - used for quality factors up to 10 and differs favorably from other circuits in that it has only one operational amplifier;
- biquad resonator- is a more complex electrical filter, performed on three op-amps and providing a quality factor of up to 200;
- switched filter- provides a quality factor of up to 1000, necessary for the selection of narrow-band signals.
A
Figure 3.32 - Circuit and logarithmic frequency response of a bandpass filter
Quality factor Q in all cases is determined by the following relation
Where f 0 - average frequency of the passband;
D f- bandwidth at -3 dB (that is, at 0.707 K U max).
Frequency response of bandpass filters for various values Q are shown in Figure 3.33.
 |
Figure 3.33 - Frequency response of bandpass filters at different Q values
Figure 3.34 shows the circuit of a bandpass filter with multi-loop feedback (MFMOS) and the type of its frequency response.
Figure 3.34 - Bandpass filter with multi-loop feedback
Resistor values R1, R2 And R3 PFMOS at a given capacitor capacity WITH= 1 µF, selected taking into account the required quality factor Q and mid frequency f 0 by formulas:
, (3.46)
To obtain maximum filter stability, the calculation is carried out for unity gain at frequency f 0 .
Second order bandpass filter can be performed according to the scheme shown in Figure 3.45.
Figure 3.45 - Second-order bandpass filter
The quasi-resonant frequency of the second order PF (at which the filter transmission coefficient is maximum) can be found from the expression
. (3.49)
Notch filter can be obtained based on the PFMOS circuit if a non-inverting adder is connected to its output (Figure 3.46). In such a scheme, allocated at frequency f 0, the signal from the output of the inverting PFMOS, the voltage gain of which is equal to unity, is supplied to one of the inputs of the non-inverting adder. The input broadband signal is fed to the second input of the adder also without amplification and without phase change. As a result of the addition of two signals in antiphase, the signal is suppressed in the region of the notch frequency f 0, that is, the required type of frequency response for the notch filter is provided.
Figure 3.46 - Notch filter based on the PFMOS circuit
It should be noted that only individual examples of constructing active filter circuits are considered above. In practice, circuits based on a Wien bridge or a double T-bridge are also widely used.
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.
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 the active filter low frequencies 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 Fig. 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 are ± 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.
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 a set NM2116, 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 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 speaker systems are not capable of high quality operate in a wide frequency range due to the use of one 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 subbands 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. Best choice separation frequency - 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 high quality amplifier power 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 based on operational amplifiers(OU). It's easy to build active filters audio frequencies 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.
Use of modern electronic components allows the production of filters with minimal values of self-noise levels, low power consumption, dimensions and ease of execution/repetition. 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. Level modern technologies significantly reduced the price and size of the PA. In addition, there have been quite a lot powerful amplifiers in an integral design with excellent performance, even for professional use. Today there are a number of ICs with several PAs in one housing ( Panasonic produces IC RCN311W64A-P 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 filter high frequency(HPF), mid-pass filter (band-pass filter, BPF) and 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 on the filter filter.
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 values of the cutoff frequencies, 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 low-pass filters and high-pass filters according to standard level 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. 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 SCH 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 electrical diagram subtracting adder) providing 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.
Low-pass filter circuit second order 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 provide maximum quality 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 |
Appearance 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 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.
Bandpass filters are used in many areas of electronics. They are especially widely used in radio reception and radio transmission circuits, in particular in resonant circuits. However, even for low frequencies, an active bandpass filter is an effective means of isolating intermediate frequency signals. For these filters, the most widely used active element is the operational amplifier (op-amp).
Op-amp-based bandpass filters easy to design and build as it requires minimal components. In addition to this, they provide very high level productivity.
What is a bandpass filter
As the name suggests, a bandpass filter filters all frequencies, allowing only frequencies within a certain range to pass through. All frequencies outside this frequency range are weakened.
There are two main parameters that define the characteristics of a bandpass filter: the passband, where the filter allows signals to pass through, and the attenuation band, where the signals are attenuated.
Ideal band pass filter has a flat passband (gain and no attenuation of the signal throughout the entire passband) and complete attenuation outside the passband. Additionally, the transition out of the passband is absolutely abrupt.
But in practice it is impossible to create a perfect bandpass filter. A real filter is unable to completely reject all frequencies outside the desired frequency range. In particular, there is an area in close proximity to the edge of a given range where the signal is partially attenuated, but not completely filtered out. This area is called the filter slope, and is measured in dB attenuation per octave. As a rule, when designing, they strive to make this roll-off as narrow as possible, which makes it possible to obtain a filter as close as possible to the specified parameters.
Bandpass Filter Calculation
Calculating a bandpass filter can be very complex even when using op amps. Nevertheless, it is possible to slightly simplify the calculation methodology, and at the same time maintain the performance of the op-amp bandpass filter at an acceptable level.
This circuit and calculation method represent a good balance between performance and simplicity of filter design.
The figure shows that in addition to the operational amplifier, the circuit also contains two capacitors and three resistors.
An example of a simplified calculation of bandpass filter elements using an op-amp
Input data:
- Resonant frequency f = 20Hz.
- Quality factor Q = 10.
- Transfer coefficient Ho = 5
Since fmax – fmin = f / Q = 2Hz,
then the bandwidth will be fmax = 21 Hz, fmin = 19 Hz.
We will assume that C1=C2=C=1uF
Then the resistor resistances can be calculated using the following formulas:
In our case we get the following results:
R1 = 10 / (5*2*3.14*20*0.000001) = 15.9 kOhm
R2 = 10 / ((2*10*10-5)*2*3.14*20*0.000001) = 408 Ohm
R3 = 2*10 / (2*3.14*20*0.000001) = 159.2 kOhm
In a circuit with one operational amplifier, it is desirable that the transmission coefficient does not exceed 5 and the quality factor is no more than 10. To obtain a high-quality filter, the parameters of resistors and capacitors should correspond as closely as possible to the calculated values.