Ac low-frequency filter two speakers circuit. Electrical passive filters

In order to reduce intermodulation distortion during sound reproduction, the loudspeakers of Hi-Fi systems are composed of low-frequency, mid-frequency and high-frequency dynamic heads. They are connected to the amplifier outputs through crossover filters, which are combinations of LC low-pass and high-pass filters.

Below is a method for calculating a three-band crossover filter using the most common scheme.

Frequency response of a three-way loudspeaker crossover filter in general view shown in Fig. 1. Here: N is the relative voltage level on the voice coils of the heads: fн and fв - the lower and upper limiting frequencies of the band reproduced by the loudspeaker; fр1 and fр2 are crossover frequencies.

Ideally, the output power at the crossover frequencies should be distributed equally between the two heads. This condition is met if, at the crossover frequency, the relative voltage level supplied to the corresponding head decreases by 3 dB compared to the level in the middle part of its operating frequency band.

Crossover frequencies should be chosen outside the area of ​​greatest sensitivity of the ear (1... 3 kHz). If this condition is not met, due to the difference in the phases of the oscillations emitted by the two heads at the interface frequency simultaneously, a “split” sound may be noticeable. The first crossover frequency usually lies in the frequency range 400... 800 Hz, and the second - 4... 6 kHz. In this case, the low-frequency head will reproduce frequencies in the range fн...fp1. mid-frequency - in the range fp1...fр2 and high-frequency - in the range fр2...fв.

One of the common variants of the electrical circuit diagram of a three-way loudspeaker is shown in Fig. 2. Here: B1 is a low-frequency dynamic head connected to the amplifier output through a low-pass filter L1C1; B2 is a mid-frequency head connected to the amplifier output through a bandpass filter formed by high-pass filters C2L3 and low-pass filters L2C3. The signal is fed to the high-frequency head B3 through high-pass filters C2L3 and C4L4.

The capacitances of capacitors and inductances of coils are calculated based on the nominal resistance of the loudspeaker heads. Since the nominal resistances of the heads and the nominal capacitances of the capacitors form a series of discrete values, and the crossover frequencies can vary within wide limits, it is convenient to carry out the calculation in this sequence. Having specified the nominal resistance of the heads, select the capacitances of the capacitors from a number of nominal capacitances (or the total capacitance of several capacitors from this row) such that the resulting crossover frequency falls within the frequency intervals indicated above.

(mospagebreak)The capacitances of filter capacitors C1...C4 for various head resistances and the corresponding crossover frequencies are given in Table 2.

It is easy to see that all capacitance values ​​can either be directly taken from the nominal series of capacitances. or received parallel connection no more than two capacitors (see Table 1).

After the capacitor capacitances are selected, the inductance of the coils in millihenries is determined using the formulas:

In both formulas: Zg-in ohms; fp1, fр2 - in hertz.

Since the impedance of the head is a frequency-dependent quantity, the nominal resistance Zg indicated in the head’s passport is usually taken for calculation; it corresponds to the minimum value of the impedance of the head in the frequency range above the main resonance frequency to the upper limit frequency of the operating band. It should be borne in mind that the actual nominal resistance of different head samples of the same type may differ from the rated value by ±20%.

In some cases, radio amateurs have to use existing dynamic heads with a nominal impedance that differs from the nominal impedances of the low-frequency and high-frequency heads as high-frequency heads. In this case, resistance matching is carried out by connecting high-frequency head B3 and capacitor C4 to different terminals of coil L4 (Fig. 2), i.e., this filter coil simultaneously plays the role of a matching autotransformer. The coils can be wound on round wooden, plastic or cardboard frames with getinax cheeks. The lower cheek should be made square; This makes it convenient to attach it to the base - the getinax board, on which the capacitors and coils are mounted. The board is secured with screws to the bottom of the speaker box. To avoid additional nonlinear distortions, the coils must be made without cores made of magnetic materials.

Example of filter calculation.

As a low-frequency loudspeaker head, a 6GD-2 dynamic head is used, the nominal impedance of which is Zg = 8 Ohms. as a mid-frequency - 4GD-4 with the same value of Zg and as a high-frequency - ZGD-15, for which Zg = 6.5 Ohm. According to table. 2 with Zg=8 Ohm and capacitance C1=C2=20 µF fp1=700 Hz, and with capacitance C3=C4=3 µF fp2=4.8 kHz. In the filter, you can use MBGO capacitors with standard capacitances (C3 and C4 are made up of two capacitors).

Using the above formulas we find: L1=L3=2.56 mg; L2=L4=0.375 mH (for autotransformer L4 this is the inductance value between pins 1-3).

Autotransformer transformation ratio

In Fig. Figure 3 shows the dependence of the voltage level on the voice coils of the heads on the frequency for a three-way system corresponding to the calculation example. The amplitude-frequency characteristics of the low-frequency, mid-frequency and high-frequency regions of the filter are designated LF, MF and HF, respectively. At crossover frequencies, the filter attenuation is 3.5 dB (with a recommended attenuation of 3 dB).

The deviation is explained by the difference in the impedances of the heads and capacitors of the capacitors from the given (nominal) values ​​and the inductances of the coils from those obtained by calculation. The slope of the LF and MF curves is 9 dB per octave and the HF curve is 11 dB per octave. The HF curve corresponds to the uncoordinated activation of loudspeaker 1 GD-3 (at points 1-3). As you can see, in this case the filter introduces additional frequency distortion.

Note from the authors:

In the given calculation method, it is assumed that the average sound pressure at the same input electrical power for all heads has approximately the same value. If the sound pressure created by any head is noticeably higher, then in order to equalize the frequency response of the loudspeaker according to the sound pressure, it is recommended to connect this head to the filter through a voltage divider, the input impedance of which should be equal to the nominal resistance of the heads accepted in the calculation.

RADIO N 9, 1977, p.37-38 E. FROLOV, Moscow

Crossover filters with a flat frequency response have a number of advantages over other types of filters, and are currently the most used in HI-FI class speakers. Therefore, only this type of filter will be considered in the calculation methodology. The essence of the calculation is that the isolation filters are first calculated from the condition of an active load and a voltage source with an infinitesimal output resistance (which is true for modern audio amplifiers). Then measures are taken to reduce the influence of amplitude-frequency and phase-frequency distortions of loudspeakers and the complex nature of their input impedance on the characteristics of the filters.

The calculation of crossover filters begins with determining their order and finding the parameters of the elements of the prototype low-pass ladder filter.

A prototype filter is a low-pass ladder filter, the values ​​of which elements are normalized relative to a unit cutoff frequency and a unit active load. Having calculated the elements of a low-pass filter of a certain order at a real frequency and a real value of load resistance, it is possible, by applying frequency conversion, to determine the circuit and calculate the values ​​of the elements of a high-pass filter and a band-pass filter of the corresponding order. The normalized values ​​of the elements of a prototype filter operating from a voltage source are determined by expanding its output conductivity into a continued fraction. The normalized values ​​of prototype filter elements for calculating “all-pass type with flat frequency response” separation filters of the 1st...6th order are summarized in the table:

Figure 1 shows a diagram of a sixth-order prototype filter. Lower order prototype filter circuits are formed by discarding the corresponding elements − α (starting with large ones) - for example, a 1st order prototype filter consists of one inductance α 1 and loads R n.

Rice. 1. Circuit diagram of a one-way loaded prototype low-pass filter of the 6th order

The value of the real parameters of the elements corresponding to the selected order of isolation filters, load resistance R n(Ohm) and cutoff frequency f d(Hz) are calculated as follows:

a) for a low-pass filter:

every element α -inductance the prototype filter is converted into real inductance (H), calculated by the formula:

L=αR n/ 2πf d

every element α -capacity the prototype filter is converted into a real capacity (F), calculated by the formula:

C=α/ 2πf dR n

b) for a high pass filter:

every element α -inductance the prototype filter is replaced real capacity calculated by the formula:

C= 1/ 2πf dαR n

every element α -capacity The prototype filter is replaced by real inductance, calculated by the formula:

L=R n/ 2πf dα

c) for a bandpass filter:

every element α -inductance is replaced by a sequential circuit consisting of real L And C -elements calculated using formulas

L=αR n/ 2π (f d 2 -f d 1 )

Where f d 2 And f d 1 – lower and upper cutoff frequencies of the bandpass filter, respectively,

C= 1/ 4π 2 f 0 2 L

Where f 0 =√f d 1 f d 2 – average frequency of the bandpass filter.

Each element α - the capacitance is replaced by a parallel circuit consisting of real L And C-elements calculated using the formulas:

С=α/ 2π(f d 2 -f d 1 )R n,

L= 1/ 4π 2 f 0 2 C

Example. It is required to calculate the values ​​of the elements of separate filters for a three-way speaker system.

We select second-order separation filters. Let the selected crossover frequencies be: between the low-frequency and mid-frequency channel f d 1 =500 Hz, between mid-frequency and high-frequency f d 2 =5000 Hz. Speaker impedance DC: low-frequency and mid-frequency - 8 Ohms, high-frequency - 16 Ohms.

Rice. 2. An example of calculating crossover filters for a three-way speaker A) Frequency response of loudspeakers without filters; b) Frequency response of loudspeakers with filters, matching and correction circuits; V) total frequency response of the speakers on the working axis and when the microphone is displaced at an angle of ±10° in the vertical plane

The amplitude-frequency characteristics of loudspeakers, measured in an anechoic chamber on the working axis of the speaker at a distance of 1 m, are shown in Fig. 2, a) (low-frequency loudspeaker 100GD-1, mid-frequency 30GD-8, high frequency 10GD-43).

Let's calculate the low pass filter:

The value of the normalized parameters of the elements is determined from the table: α 1 =2,0, α 2 =0,5.

From Fig. 1 we determine the circuit of the prototype low-pass filter: the filter consists of inductance α 1 , containers α 2 and loads R n.

The values ​​of real elements of low-pass filters are found using the expressions and :

L 1 LF=αR n/ 2πf d 1 =2.0 8.0/(2 3.14 500)=5.1 mH,

C 1 LF=α/ 2πf d 1 R n=0.5/(2·3.14·500·8.0)=20 µF.

The values ​​of the bandpass filter elements (for a mid-frequency loudspeaker) are determined in accordance with the expressions ...:

L 1 midrange=α 1 R n / 2π (f d 2 -f d 1 )=2.0 8.0/2 3.14(5000-500)=0.566 mH(HF side)

WITH 1 midrange= 1/ 4π 2 f 0 2 L 1 MF =1/4 3.14 2 5000 500 5.66 10 -4 =18 µF(bass side)

WITH 2 midrange=α 2 / 2π(f d 2 -f d 1 )R n=0.5/2·3.14(5000-500)·8.0=2.2 µF(HF side)

L 2 midrange= 1/ 4π 2 f 0 2 C 2 SF =1/4 3.14 2 5000 500 2.2 10 -6 =4.6 mH(bass side)

The values ​​of the high-pass filter elements are determined in accordance with the expressions and:

C 1 HF= 1/ 2πf d 2 α 1 R n=1/(2·3.14·5000·2.0·16)=1.00 µF,

L 2 HF=R n/ 2πf d 2 α 2 =16/(2·3.14·5000·2.0)=0.25 mH.

To match the filters with the input impedance of the loudspeakers, a special matching circuit can be used. In the absence of this circuit, the input impedance of the loudspeaker affects the frequency response and phase response of the crossover filters. The parameters of the elements of the matching circuit connected in parallel with the loudspeaker are found from the condition:

Y c(s )+ Y GR(s )=1/ R E,

Where Y c(s ) – conductivity of the matching circuit, Y GR(s ) – input conductance of the loudspeaker, R E– electrical resistance of the loudspeaker at direct current.

The matching circuit diagram is shown in Fig. 3. The circuit is dual to the equivalent loudspeaker circuit. The values ​​of the circuit elements are determined as follows:

R K 1 = R E,

C K 1 = L V.C./ R E 2

R K=R E 2 / R ES = Q ES R E/ Q M.S.

C K=L CES/ R E 2 =1/ Q ES R E 2π f s,

L K=C MESR E 2 =Q ES R E/2π f s,

Where L V.C.– voice coil inductance, f s, C MES, L CES, R ES– electromechanical parameters of the loudspeaker.

To compensate for the input impedance of a low-frequency loudspeaker, a simplified circuit consisting of series-connected resistors is used R K1 and containers C K1. This is because the mechanical resonance of the loudspeaker does not affect the low-pass filter characteristics and only compensates for the inductive nature of the loudspeaker input impedance. The feasibility of connecting a complete matching circuit to high-frequency and mid-frequency loudspeakers is justified if resonant frequency speaker is near the cutoff frequency of the high-pass filter or the low cutoff frequency of the band-pass filter. In the event that the cutoff frequencies of the filters are significantly higher than the resonant frequencies of the loudspeakers, the inclusion of a simplified circuit is sufficient.

Fig.3. Matching circuit diagram to compensate for the complex nature of the loudspeaker input impedance

The influence of the input complex impedance of loudspeakers can be considered using the example of second-order separation filters for high and low frequencies (Fig. 4).

Rice. 4. Electric equivalent circuit loudspeaker with 2nd order crossover filters: a – with a low-pass filter; b – with a high-pass filter; (1 – filter; 2 – loudspeaker)

The parameters of the low-frequency loudspeaker are selected in such a way that its frequency response corresponds to the Butterworth approximation, i.e. full quality factor Q ts =0.707. The cutoff frequency of the low-pass filter is selected to be 10 times the resonant frequency of the loudspeaker f d =10 f s. The voice coil inductance is selected from the condition: Q VC=0.1, where Q V.C.– quality factor of the voice coil, defined as:

Q V.C.=L VC 2 π f s/ R E,

Where fs – resonant frequency of the loudspeaker, R E – voice coil DC resistance, L VC– voice coil inductance.

Meaning Q V.C.=0.1 corresponds to the average value of the voice coil inductance of powerful low-frequency loudspeakers. As a result, we can assume that the inductance of the voice coil L V.C. and active resistance R E connected in parallel to the filter tank C 1 and form a wide maximum in the frequency response of the input resistance in the region of the filter cutoff frequency, followed by a sharp dip (Fig. 5a). The corresponding changes in the frequency response of the voltage filter consist of a slight increase in the frequency response at frequency f ≈ 2 f s(due to the inductance of the voice coil) and a smooth dip, followed by a sharp peak in the frequency response due to the resonance of the circuit formed by the inductance of the voice coil and the capacitance of the crossover filter. Corresponding changes in frequency response and Z BX after turning on the matching circuit from a series-connected resistor and capacitor are shown in Fig. 5, a (curves 2, 4, 6). The inclusion of a matching circuit brings the nature of the input impedance of the loudspeaker closer to the active one and the frequency response of the isolation filter in voltage to the desired one. However, due to the influence of the voice coil inductance, the frequency response in terms of sound pressure differs from the desired one (curve 4), therefore, even after the matching circuit, a slight adjustment of the filter elements and the matching circuit is sometimes required.

Rice. 5 Frequency response and input impedance of 2nd order isolation filters loaded onto a loudspeaker: a) low-pass filter; b) high pass filter;

1. Voltage frequency response at the filter output without a matching circuit;
2. Voltage frequency response at the output of the filter with a matching circuit;
3. Frequency response for sound pressure without a matching circuit;
4. Frequency response for sound pressure with a matching circuit;
5. input impedance of a filter with a loudspeaker without a matching circuit;
6. input impedance of a filter with a loudspeaker with a matching network.

In the case of a high-pass filter, the influence of the complex nature of the input impedance of the loudspeaker on the input impedance and frequency response of the filter is of a different nature. If the cutoff frequency of the high-pass filter is near the speaker resonance frequency f s(a case sometimes encountered in filters for midrange loudspeakers, but practically impossible for high-frequency loudspeakers), the input impedance of a high-pass filter with a loudspeaker without a matching network may have a deep dip due to the fact that at the loudspeaker's resonance frequency f s its input impedance increases significantly and is purely active. The filter appears to be idling, due to a sharp increase in load resistance, and its input resistance is determined by the elements connected in series C 1 , L 1 . A more common situation is when the cutoff frequency of the high-pass filter f d significantly higher than the resonance frequency of the loudspeaker f s. Figure 5b shows an example of the influence of the input impedance of a loudspeaker and its compensation on the frequency response of a high-pass filter in terms of voltage and sound pressure. The filter cutoff frequency is chosen significantly higher than the loudspeaker resonance frequency f d≈8 f s, speaker parameters Q T.S.=1,5 , Q MS=10, Q VC =0.08. A rise in the frequency response in sound pressure and voltage in the high-frequency region, accompanied by a drop in input impedance, is explained by the influence of the voice coil inductance L V.C.. For more high frequencies ah the frequency response drops, and the input impedance increases due to an increase in the inductive reactance of the voice coil.

Curves 2, 4, 6 in Fig. 5b show the influence of the matching R.C.-chains.

The output impedance of the high-pass crossover filter, which increases with decreasing frequency, affects the electrical quality factor of the loudspeaker, increasing it, and accordingly increases the total quality factor and the shape of the frequency response in terms of sound pressure. In other words, there is an effect of “dedamping” of the loudspeaker. To achieve this, you need to select the slope of the filter’s frequency response and the cutoff frequency of the high-pass filter f d>> f s so that at the resonance frequency f s the signal attenuation was at least 20 dB.

When calculating the isolation filters in the example discussed above, it was assumed that the nature of the load is active, so we will calculate the matching circuits that compensate for the complex nature of the input impedance of the loudspeaker.

Crossover frequency of low-frequency and mid-frequency channels f d 1 is selected approximately two octaves above the resonant frequency of the mid-range loudspeaker, and the crossover frequency of the mid-frequency and high-frequency channels f d 2 – two octaves above the resonant frequency of the tweeter. In addition, it can be assumed that the inductance of the voice coil of a high-frequency loudspeaker is negligible in the operating frequency range and can be ignored (this is true for most high-frequency loudspeakers). In this case, you can limit yourself to using a simplified matching circuit for low-frequency and mid-frequency loudspeakers.

Example. Measured (or determined from the input impedance frequency response curve) voice coil inductances: woofer L V.C.=3·10 -3 G=3 mH, midrange speaker L VC =0.5·10 -3 G=0.5 mH. Then the value of the elements of the compensating circuits is calculated using the formulas and:

for LF: R K 1 – R π =8 Ohm; WITH K1 = L V.C./ R 2 E =3·10 -3 /64=47 µF

for midrange: R' K 1 = R E-8 Ohm; WITH' K1 = L V.C./ R 2 E =0.5·10 -3 /64=8.0 µF.

There is a peak in the frequency response of a mid-frequency loudspeaker, which increases the unevenness of the total frequency response of the speaker (Fig. 2a); in this case, it is advisable to turn on the amplitude corrector. The rejecting element (Fig. 6) is used to correct the peaks in the frequency response of loudspeakers or the entire speaker system. This link has a purely active input impedance, equal to resistance loads R H and can therefore be connected between a filter and a loudspeaker with compensated input impedance. In the case of turning on the rejecting element at the input of the AC, the circuit can be simplified, since there is no need for elements C q, L q, R q, ensuring the active nature of the input impedance of the link. The values ​​of the elements are calculated using the formulas:

R K≈ R H(10 -0,05 N -1),

L K= R K∆ f /2π f 0 2 ,

C K =1/ L K 4 π 2 f 0 2 ,

C q= L K/ R H 2 ,

L q= C KR H 2 ,

R q= R H(1+ R H/ R K),

Where R H– loudspeaker impedance (compensated) or speaker input impedance (Ohm) in the region of the resonant frequency of the rejecting link;

∆ f – frequency band of the adjusted frequency response peak (measured by level – 3 dB), Hz;

f 0 – resonant frequency of the notch, Hz;

N – magnitude of the frequency response peak, dB.

Rice. 6. Reducing link: a) schematic diagram; b) frequency response

Let's use a rejection link connected between the filter and the mid-frequency loudspeaker with a matching circuit.

From the frequency response of a mid-frequency loudspeaker we determine ∆ f =1850 Hz, f 0 =4000 Hz, N =6 dB. Resistance of a midrange loudspeaker with a matching network R H=8 Ohm.

The values ​​of the elements of the control link are as follows:

R K≈ R H(10 -0,05 N -1)=8(10 -0.05·6 -1)=7.96 Ohm,

L K= R K∆ f /2π f 0 2 =7.96·1850/2 π (4000) 2 =0.147 mH,

C K=1/L K 4 π 2 f 0 2 =1/1.47·10 -4 (2 π 4000) 2 =11uF,

C q= L K/ R H 2 =1.47·10 -4 /64=2.3 µF,

L q= C KR H 2 =10.8·10 -6 ·64=0.7 mH,

R q= R H(1+ R H/ R K)=8(1+8/7.96)≈16.0 Ohm.

In the example under consideration, the frequency response of the high-frequency and mid-frequency loudspeaker have average levels that are approximately 6 dB and, accordingly, 3 dB higher than the frequency response of the low-frequency loudspeaker (the sound pressure was recorded when a sinusoidal voltage of the same magnitude was applied to all loudspeakers). In this case, to reduce the unevenness of the total frequency response of the speaker, it is necessary to weaken the level of mid-frequency and high-frequency components. This can be done either using a first-order corrective high-frequency link (Fig. 7), the elements of which are calculated using the formulas:

R K≈ R H(10 -0,05 N -1),

L K= R K/2π f d√(10 0,1 N -2), N≥3 dB,

Or using L-shaped passive attenuators that provide a given level of attenuation N (dB) and specified input impedance R BX(Fig. 8). The value of the attenuator elements is calculated using the formulas:

R 1 ≈ R BX(1-10 -0,05 N ),

R 2 ≈ R HR BX10 -0,05 N /(R H– R BX10 -0,05 N ).

Rice. 7. 1st order link, correcting high frequencies: a) circuit diagram; b) frequency response

Rice. 8. Passive L-shaped attenuator

As an example, let us calculate the values ​​of the attenuator elements to attenuate a high-frequency loudspeaker signal by 6 dB. Let the input impedance of the loudspeaker with the attenuator turned on be equal to the input impedance of the loudspeaker, i.e. 16 Ohm, then:

R 1 ≈16(1-10 -0.05·6)≈8.0 Ohm,R 2 ≈16·10 -0.05·6 /(1-10 -0.05·6)≈16.0 Ohm.

Let us similarly calculate the values ​​of the attenuator elements for a mid-frequency loudspeaker: R 1 =4.7 Ohm, R 2 =39 Ohm. Attenuators are switched on immediately after the loudspeakers with matching circuits.

The complete circuit of the isolation filters is shown in Fig. 9, the frequency response of the speaker with the calculated filters is shown in Fig. 2, c.

As mentioned above, even-order filters allow only one option for the polarity of loudspeaker switching; in particular, second-order filters require switching in antiphase. For the example under consideration, the low-frequency and high-frequency loudspeakers must have identical switching polarities, and the mid-frequency loudspeaker must have the opposite polarity. The requirements for the polarity of loudspeakers were discussed above on a speaker model with ideal loudspeakers. Therefore, when turning on real loudspeakers that have their own phase response ≠0 (in the case of choosing crossover frequencies near the boundary frequencies of the operating range of the loudspeakers or when the frequency response of the loudspeakers is highly uneven), the condition for matching the real phase response of the channels may not be met. Therefore, to monitor the real phase response from the sound pressure of loudspeakers with filters, it is necessary to use a phase meter with a delay line or determine the matching condition indirectly by the nature of the total frequency response of the speakers in the channel separation bands. The correct polarity for switching on the loudspeakers can be considered the one that corresponds to less unevenness of the total frequency response in the channel separation band. Accurate matching of the phase response of the separated channels while satisfying all other requirements (flat frequency response, etc.) is carried out using numerical methods for synthesizing optimal separation filters-correctors on a computer.

Fig.9. Fundamental electrical diagram Speakers with calculated isolation filters (capacitance in microfarads, inductance in millihenries, resistance in ohms).

In the development of passive isolation filters, an important role is played by their design, as well as the choice of the type of specific elements - capacitors, inductors, resistors, in particular, the relative placement of the inductors has a great influence on the characteristics of speakers with filters; if they are poorly positioned due to mutual coupling, interference is possible signal between closely spaced coils. For this reason, it is recommended to place them mutually perpendicular; only such an arrangement can minimize their influence on each other. Inductors are one of the essential components passive separation filters. Currently, many foreign companies use inductors on cores made of magnetic materials, providing high dynamic range, low level of nonlinear distortion and small dimensions of the coils. However, the design of coils with magnetic cores involves the use of special materials, so until now many developers use coils with air cores, the main disadvantages of which are large dimensions with low losses (especially in the low-frequency channel filter), as well as high copper consumption; advantages - negligible nonlinear distortions.

The air-core inductor configuration shown in Fig. 10 is optimal as it provides maximum ratio L/R , i.e. coil with a given inductance L , wound with a wire of the selected diameter, has the least resistance for a given winding configuration R or the highest quality factor compared to any other. Attitude L/R , which has the dimension of time, is related to the dimensions of the coil by the relation:

L /R=161,7alc/(6a+9l+10c);

L– in microhenry, R– in ohms, a,l,c - in millimeters.

Fig. 10. Inductor coil with an air core of optimal configuration: a) in section; b) appearance.

Design ratios for this coil configuration: a=1,5With , l=c ; coil design parameter c=√(L/R 8,66) , number of turns N=19,88√(L / c ), wire diameter in millimeters, d=0,841c/√ N , mass of wire (material - copper) in grams, q = c 3 /21, wire length in millimeters, B=187.3√Lc . If the inductor is calculated based on a wire of a given diameter, the main calculation ratios are as follows:

design parameter c = 5 √(d 4 19,88 2 L /0,841 4)=3,8 5 √(d 4 L ) , wire resistance R=L/c 2 8,66 .

Let us find, for example, the parameters of the inductor of the previously calculated low-pass filter. The coil inductance is L 1LF =5.1 mg. Resistance R coils on direct current will be determined from the permissible signal attenuation introduced by a real coil on low frequencies Oh. Let the signal weakening due to losses R in the coil is N≤1dB. Since the DC resistance of the woofer is R E=8 Ohm, then the permissible coil resistance, determined from the expression R≤R E (10 0.05 N -1), amounts to R≤0.980 ohm; then the design parameter of the coil c =√5100/0.98·8.66=24.5 mm; number of turns N=19.8√(5100/24.5)=287 turns; wire diameter d=0.841·24.5/√287=1.2 mm; wire weight q =24.5 3 /21.4≈697 g; wire length B =187.3√(85.1·24.5)≈46 m.

Another important element of passive coupling filters are capacitors. Typically, paper or film capacitors are used in filters. The most commonly used paper capacitors are domestic MBGO capacitors. The advantage of these types of capacitors is low losses, high temperature stability, the disadvantage is large dimensions, a decrease in the permissible maximum voltage at high frequencies. Currently, the filters of a number of foreign speakers use electrolytic non-polar capacitors with low internal losses, combining the advantages of the capacitors considered and free from their disadvantages.

Based on materials from the book: "High-quality acoustic systems and radiators"

(Aldoshina I.A., Voishvillo A.G.)

If you find impedance lows around 3 ohms, don't be discouraged. Some speaker models from well-known companies have a minimum of up to 2.6 Ohms. One or two models are even 2 Ohm! On the other hand, there is nothing good in such impedance “dips”. Amplifiers overheat when working under such a load if you listen to music loudly. Amplifier distortion increases in the area of ​​minimum impedance of the speaker system.

For tube triode amplifiers, minimums in the low and mid-low frequencies are especially dangerous. Moreover, if the impedance drops below 3 ohms, the output tubes may fail. The output pentodes do not break in such cases.

It is important to remember that the output impedance of the amplifier is involved in setting the speaker filter. For example, if you provide 1 dB boost to the Fc region by tuning the speakers with transistor amplifier, which has almost zero output impedance, then when these speakers are connected to a tube amplifier (typical output impedance ~2 Ohms), there will be no trace of afterburner left. The frequency response will be different. To repeat the characteristics achieved with a transistor amplifier, in the case of working with a tube apparatus, you will have to create another filter.

A listener capable of developing his own personality will eventually come to understand the value of good tube amplifiers. For this reason, I usually configure speakers with a tube amplifier, and when connecting to a transistor amplifier, I place a 10-watt low-inductance (no more than 4-8 uN) resistor with a resistance of 2 ohms in series with the speaker.

If you have a transistor amplifier, but do not exclude the possibility of purchasing tube equipment in the future, then connect your speakers to the amplifier output through the above resistors during setup and subsequent operation. Then, when switching to tube amplifier, you don’t need to configure the speaker again, just connect to it directly, without resistors.

For those who cannot get a generator, I recommend finding a test CD with tracks containing test signals for evaluating the frequency response. In this case, you will not be able to smoothly change the frequency of the test signal and will miss the point of the deepest drop in impedance in the region of its decline. However, even a rough estimate of the impedance frequency response will be useful. For a rough estimate, pseudonoise signals in third-octave bands are even more convenient than sinusoidal ones. Such signals are on the test CD of the magazine "Salon AV" (#07 from 2002).

As a last resort, you can avoid impedance measurements by limiting the return boost at the filter cutoff frequency to 1 dB. Under this condition, the impedance is unlikely to drop by more than 20%. For example, for a 4 Ohm speaker this corresponds to a minimum of 3.2 Ohms, which is acceptable.

Please note that you will have to “catch” the parameters of the filter elements needed for the desired frequency response correction yourself. Preliminary calculation of test filters is necessary so as not to miss “by a kilometer” initially.

You can add resistors to a simple low-midrange filter of the head for some manipulations with the frequency response, which may be required when tuning your speakers.

If the average sound pressure level of this speaker is higher than the corresponding parameter of the HF head, it is necessary to connect a resistor in series with the speaker. Options for inclusion are shown in Fig. 6a and 6b.

The amount of required reduction in the output of the bass-midrange head, expressed in dB, will be denoted by the symbol N. Then:

Where Rd is the average impedance of the speaker.

Instead of calculations, you can use the following information:

Table 1

Where V ac is the effective value of the voltage at the output of the amplifier. V d - the same on the speaker. V d is less than V c, due to the attenuation of the signal by resistor R 1. In addition, N = N HF - N LF, where N LF and N HF are the sound pressure levels developed, respectively, by the LF and HF heads. These levels are averaged over the bands reproduced by the LF and HF heads. Naturally, N LF and N HF are measured in dB.

An example of a quick estimate of the required value of R1:

For N = 1 dB; R1 = Rd (1.1 - 1) = 0.1 Rd.

For N = 2 dB; R1 = Rd (1.25 - 1) = 0.25 Rd.

For N = 6 dB; R1 = Rd (2 - 1) = Rd.

More specific example:

Rd = 8 Ohm, N = 4 dB.
R1 = 8 ohms (1.6 - 1) = 4.8 ohms.

How to calculate the power of R1?

Let R d be the rated power of the LF-MF loudspeaker, PR 1 be the permissible power dissipated by R 1. Then:

You should not make it difficult to remove heat from R1, that is, you do not need to wrap it with electrical tape, fill it with hot-melt adhesive, etc.

Features of filter pre-calculation with R1:

For the circuit in Fig. 6b, the values ​​of L 1 and C 1 are calculated for an imaginary speaker, the total resistance of which is R Σ = R 1 + R d. In this case, L 1 is greater, and C 1 is less than that of a filter without R 1 .

For the circuit in Fig. 6a - everything is the other way around: the introduction of R 1 into the circuit requires a decrease in L 1 and an increase in C 1. It is easier to calculate the filter according to the scheme in Fig. 6b. Use this exact scheme.

Additional correction of frequency response using a resistor:

If, to improve the frequency response uniformity, it is necessary to reduce the filter’s suppression of signals above the cutoff frequency, you can use the circuit shown in Fig. 7.

The use of R 2 in this case leads to a decrease in returns in F s. Above Fc, the return, on the contrary, increases compared to a filter without R2. If it is necessary to restore the frequency response close to the original one (measured without R 2), L 1 should be reduced and C 1 increased in the same proportion. In practice, the range of R 2 is within: R 2 ~= (0.1-1) * R d.

Frequency response correction:

The simplest case: on a fairly uniform characteristic there is a zone of increased output (“presence”) in the mid-frequency region. You can use a corrector in the form of a resonant circuit (Fig. 8).

At resonance frequency

The circuit has a certain impedance value, in accordance with the value of which the signal at the speaker is attenuated. Outside the resonance frequency, attenuation is reduced so that the circuit can selectively suppress presence. You can roughly calculate the values ​​of L 2 and C 2 depending on F p and the degree of suppression N 2 (in dB) as follows:

It’s convenient to use Table 1. I’ll draw it differently:

Example. It is necessary to suppress the "presence" with a central frequency of 1600 Hz. Speaker impedance - 8 Ohms. Suppression level: 4 dB.

The specific shape of a loudspeaker's frequency response may require more complex correction. Examples in Fig. 9.

The case in Fig. 9a is the simplest. It is easy to select the parameters of the correction contour, since the “presence” has a “mirror” shape to the possible filter characteristic.

In Fig. Figure 9b shows another possible option. It can be seen that the simplest circuit allows you to “exchange” one large “hump” into two small ones with a small dip in the frequency response in addition. In such cases, you must first increase L2 and decrease C2. This will expand the suppression band to the required limits. Then you should bypass the circuit with resistor R 3, as shown in Fig. 10. The value of R3 is selected based on the required degree of suppression of the signal supplied to the speaker in the band determined by the circuit parameters. R 3 = R d (Δ - 1)

Example: We need to suppress the signal by 2 dB. Speaker - 8 Ohm. Refer to Table 1. R 3 = 8 ohms (1.25 - 1) = 2 ohms.

How correction occurs in this case is shown in Fig. 9th century

A combination of two problems is quite typical for modern loudspeakers: “presence” in the region of 1000-2000 Hz and some excess of the upper mids. A possible type of frequency response is shown in Fig. 11a.

The correction method that is most free from harmful “side” effects requires a slight complication of the contour. The corrector is shown in Fig. 12.

The resonance of the L 2, C 2 circuit is needed, as usual, to suppress “presence”. Below Fp, the signal passes almost losslessly to the speaker through L 2. Above F p the signal goes through C 2 and is attenuated by resistor R 4 .

The corrector is optimized in several stages. Since the introduction of R 4 weakens the resonance of the L 2 , C 2 circuit, you should initially choose L 2 more and C 2 less. This will provide excess suppression on Fp, which is normalized after the introduction of R4. R 3 = R d (Δ - 1), where "Δ" is the amount of suppression of signals above F p. "Δ" is selected in accordance with the excess of the upper middle, referring to Table 1. The stages of correction are roughly illustrated in Fig. 11b.

In rare cases, a reverse effect on the frequency response slope using a correction circuit is required. It is clear that for this R 4 must move to the chain L 2. Scheme in Fig. 13.

The problematic frequency response and its correction for this case are shown in Fig. 14.

With a certain combination of L2, C2 and R4 values, the equalizer may not have much suppression at Fp. An example when just such a correction is necessary is shown in Fig. 15.

If necessary, you can use a second-order filter and a correction circuit together. Switching options are shown in Fig. 16.

With the same element values, option a) provides greater output at mid frequencies and at the cutoff frequency. In principle, by selecting the values ​​of the elements, you can almost equalize the frequency response of the speakers for both filter options. For some reasons that would take a long time to talk about, I advise you to use option a) more often. Sometimes very pronounced “presence” requires the use of option b). The joint operation of the filter and corrector is illustrated in Fig. 17.

Let's look at filters for high-frequency speakers.

For HF heads, much more often than for LF speakers, we use a first-order filter, that is, simply a capacitor connected in series with the loudspeaker. The fact that such a simple filter introduces a noticeable tilt into the frequency response of the speaker does not have such a detrimental effect on the sound as in the case of the woofer. Firstly, often this slope is partially compensated by a smooth complementary (mutually complementary) slope of the frequency response of the woofer in the same frequency region. Secondly, some “dip” in the lower-high region (3-6 kHz) is quite acceptable based on the results of subjective examinations. The possible frequency response of a tweeter without a filter, with a filter, and together with a woofer is shown in Fig. 18.

Don't be afraid to experiment with connecting a tweeter out of phase with a woofer. Sometimes this is one of the few ways to achieve good sound. The most likely results of changing the polarity of the HF head are shown in Fig. 19

Take a block of marble and cut off everything unnecessary from it...

Auguste Rodin

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.

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 a transistor n-p-n type, and you yourself, on occasion, will choose what 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, the gain of which (h21e) is not much lower than 100.

Rice. 1. First order high pass filters

The resistor in the emitter circuit (R1 in Fig. 1) sets the collector current; for most transistors it is recommended to select it approximately equal to 1 mA or slightly less. The cutoff frequency of the filter is determined by the capacitance of the input capacitor C2 and the total resistance of resistors R2 and R3 connected in parallel. In our case, this resistance is 105 kOhm. You just need to make sure that it is significantly less than the resistance in the emitter circuit (R1), multiplied by the h21e indicator - in our case it is approximately 1200 kOhm (in reality, with a range of h21e values ​​from 50 to 250 - from 600 kOhm to 4 MOhm) . The output capacitor is added, as they say, “for the sake of order” - if the filter load is the input stage of the amplifier, there, as a rule, there is already a capacitor to decouple the input for DC voltage.

The op-amp filter circuit here (as well as in the following) uses the TL082C model, since this operational amplifier is very often used to build filters. However, you can take almost any op-amp from those that work normally with single-supply power, preferably with an input to field effect transistors. Here, too, the cutoff frequency is determined by the ratio of the capacitance of the input capacitor C2 and the resistance of parallel-connected resistors R3, R4. (Why connected in parallel? Because from the point of view alternating current power plus and minus are the same thing.) The ratio of resistors R3, R4 determines the midpoint; if they differ slightly, this is not a tragedy, it only means that the maximum amplitude signal will begin to be limited on one side a little earlier. The filter is designed for a cutoff frequency of 100 Hz. To lower it, you need to increase either the value of resistors R3, R4, or the capacitance C2. That is, the denomination changes in inverse proportion to the first power of frequency.

In the low-pass filter circuits (Fig. 2) there are a couple more parts, since the input voltage divider is not used as an element of the frequency-dependent circuit and a separating capacitance is added. To lower the filter cutoff frequency, you need to increase the input resistor (R5).

Rice. 2. First order low pass filters

The separating capacitance has a serious rating, so it will be difficult to do without an electrolyte (although you can limit yourself to a 4.7 µF film capacitor). It should be taken into account that the separating capacitance together with C2 forms a divider, and the smaller it is, the higher the signal attenuation. As a result, the cutoff frequency also shifts somewhat. In some cases, you can do without a coupling capacitor - if, for example, the source is the output of another filter stage. In general, the desire to get rid of bulky coupling capacitors was probably the main reason for the transition from unipolar to bipolar power supply.

In Fig. Figures 3 and 4 show the frequency characteristics of the high-pass and low-pass filters, the circuits of which we have just examined.

Rice. 3. Characteristics of first order HF filters

Rice. 4. Characteristics of first-order low-pass filters

It is very likely that you already have two questions. First: why are we so busy studying first-order filters, when they are not suitable for subwoofers at all, and for separating the bands of front acoustics, if you believe the author’s statements, they are, to put it mildly, not often used? And second: why didn’t the author mention either Butterworth or his namesakes - Linkwitz, Bessel, Chebyshev, in the end? I won’t answer the first question for now, but a little later everything will become clear to you. I'll move on to the second one right away. Butterworth and his colleagues determined the characteristics of filters from the second order and higher, and the frequency and phase characteristics of first order filters are always the same.

So, second order filters, with a nominal roll-off slope of 12 dB/oct. Such filters are commonly made using op-amps. You can, of course, get by with transistors, but in order for the circuit to work accurately, you have to take a lot of things into account, and as a result, the simplicity turns out to be purely imaginary. A certain number of circuit implementation options for such filters are known. I won’t even say which one, since any listing may always be incomplete. And it won’t give us much, since it hardly makes sense for us to really delve into the theory of active filters. Moreover, for the most part, only two circuit implementations are involved in the construction of amplifier filters, one might even say one and a half. Let's start with the one that is “whole”. This is the so-called Sallen-Key filter.

Rice. 5. Second order high pass filter

Here, as always, the cutoff frequency is determined by the values ​​of the capacitors and resistors, in this case - C1, C2, R3, R4, R5. Please note that for the Butterworth filter (finally!) the resistor value in the circuit is feedback(R5) should be half the value of the resistor connected to ground. As usual, resistors R3 and R4 are connected to ground in parallel, and their total value is 50 kOhm.

Now a few words aside. If your filter is not tunable, there will be no problems with selecting resistors. But if you need to smoothly change the cutoff frequency of the filter, you need to simultaneously change two resistors (we have three of them, but in amplifiers the power supply is bipolar, and there is one resistor R3, the same value as our two R3, R4, connected in parallel). Dual variable resistors of different values ​​are produced especially for such purposes, but they are more expensive and there are not so many of them. In addition, it is possible to develop a filter with very similar characteristics, but in which both resistors will be the same, and the capacitances C1 and C2 will be different. But it's troublesome. Now let's see what happens if we take a filter designed for medium frequency (330 Hz) and start changing only one resistor - the one to ground. (Fig. 6).

Rice. 6. Rebuilding the high-pass filter

Agree, we have seen something similar many times in graphs in amplifier tests.

The low-pass filter circuit is similar to the mirror image of the high-pass filter: there is a capacitor in the feedback, and resistors in the horizontal shelf of the letter “T”. (Fig. 7).

Rice. 7. Second order low pass filter

As with the first order low pass filter, a coupling capacitor (C3) is added. The size of the resistors in the local ground circuit (R3, R4) affects the amount of attenuation introduced by the filter. Given the nominal value indicated on the diagram, the attenuation is about 1.3 dB, I think this can be tolerated. As always, the cutoff frequency is inversely proportional to the value of the resistors (R5, R6). For a Butterworth filter, the value of the feedback capacitor (C2) must be twice that of C1. Since the values ​​of resistors R5 and R6 are the same, almost any dual trimming resistor is suitable for smooth adjustment of the cutoff frequency - this is why in many amplifiers the characteristics of low-pass filters are more stable than the characteristics of high-pass filters.

In Fig. Figure 8 shows the amplitude-frequency characteristics of second-order filters.

Rice. 8. Characteristics of second order filters

Now we can return to the question that remained unanswered. We “passed” the first-order filter circuit because active filters are created mainly by cascading basic links. So a series connection of first and second order filters will give the third order, a chain of two second order filters will give the fourth, and so on. Therefore, I will give only two variants of circuits: a third-order high-pass filter and a fourth-order low-pass filter. Characteristic type - Butterworth, cutoff frequency - the same 100 Hz. (Fig. 9).

Rice. 9. Third order high pass filter

I foresee a question: why did the values ​​of resistors R3, R4, R5 suddenly change? Why shouldn't they change? If in each “half” of the circuit the level of -3 dB corresponded to a frequency of 100 Hz, then the combined action of both parts of the circuit will lead to the fact that the decline at a frequency of 100 Hz will already be 6 dB. But we didn’t agree that way. So the best thing to do is to give a methodology for choosing denominations - for now only for Butterworth filters.

1. Using a known filter cutoff frequency, set one of the characteristic values ​​(R or C) and calculate the second value using the relationship:

Fc = 1/(2?pRC) (1.1)

Since the range of capacitor ratings is usually narrower, it is most reasonable to set the base value of the capacitance C (in farads), and from this determine the base value R (Ohm). But if you, for example, have a pair of 22 nF capacitors and several 47 nF capacitors, no one is stopping you from taking both of them - but in different parts of the filter, if it is composite.

2. For a first-order filter, formula (1.1) immediately gives the resistor value. (In our particular case, we get 72.4 kOhm, round to the nearest standard value, we get 75 kOhm.) For a basic second-order filter, you determine the starting value of R in the same way, but in order to get the actual resistor values, you will need to use the table. Then the value of the resistor in the feedback circuit is determined as

and the value of the resistor going to ground will be equal to

The ones and twos in parentheses indicate the lines related to the first and second stages of the fourth-order filter. You can check: the product of two coefficients in one line is equal to one - these are, indeed, reciprocals. However, we agreed not to delve into the theory of filters.

The calculation of the values ​​of the defining components of the low-pass filter is carried out in a similar way and according to the same table. The only difference is that in the general case you will have to dance from a convenient resistor value, and select the capacitor values ​​​​from the table. The capacitor in the feedback circuit is defined as

and the capacitor connecting the op-amp input to ground is like

Using our newly acquired knowledge, we draw a fourth-order low-pass filter, which can already be used to work with a subwoofer (Fig. 10). This time in the diagram I show the calculated values ​​of the capacities, without rounding to the standard value. This is so that you can check yourself if you wish.

Rice. 10. Fourth order low pass filter

I still haven’t said a word about phase characteristics, and I was right - this is a separate issue, we’ll deal with it separately. Next time, you understand, we are just getting started...

Rice. 11. Characteristics of third and fourth order filters

Prepared based on materials from the magazine "Avtozvuk", April 2009.www.avtozvuk.com

Now that we have accumulated a certain amount of material, we can move on to the phase. It must be said from the very beginning that the concept of phase was introduced a long time ago to serve the needs of electrical engineering.

When the signal is a pure sine (although the degree of purity varies) of a fixed frequency, then it is quite natural to represent it in the form of a rotating vector, determined, as is known, by the amplitude (modulus) and phase (argument). For sound signal, in which the sines are present only in the form of expansion, the concept of phase is no longer so clear. However, it is no less useful - if only because sound waves from different sources are added vectorially. Now let's see what the phase-frequency characteristics (PFC) of filters up to the fourth order inclusive look like. The numbering of the figures will remain continuous, from the previous issue.

We begin, therefore, with Fig. 12 and 13.

You can immediately notice interesting patterns.

1. Any filter “twists” the phase by an angle that is a multiple of?/4, more precisely, by an amount (n?)/4, where n is the order of the filter.

2. The phase response of the low-pass filter always starts from 0 degrees.

3. The phase response of the high-pass filter always comes at 360 degrees.

Last point You can clarify: the “destination point” of the phase response of the high-pass filter is a multiple of 360 degrees; if the filter order is higher than fourth, then with increasing frequency the phase of the high-pass filter will tend to 720 degrees, that is, 4? ?, if above the eighth - to 6? etc. But for us this is pure mathematics, which has a very distant relationship to practice.

From a joint consideration of the listed three points, it is easy to conclude that the phase response characteristics of the high-pass and low-pass filters coincide only for the fourth, eighth, etc. orders, and the validity of this statement for fourth-order filters is clearly confirmed by the graph in Fig. 13. However, it does not follow from this fact that the fourth-order filter is “the best”, just as, by the way, the opposite does not follow. In general, it’s too early to draw conclusions.

The phase characteristics of filters do not depend on the method of implementation - they are active or passive, and even on the physical nature of the filter. Therefore, we will not specifically focus on the phase response characteristics of passive filters; for the most part, they are no different from those that we have already seen. By the way, filters are among the so-called minimum-phase circuits - their amplitude-frequency and phase-frequency characteristics are strictly interconnected. Non-minimum phase links include, for example, a delay line.

It is quite obvious (if there are graphs) that the higher the order of the filter, the steeper its phase response drops. How is the steepness of any function characterized? Its derivative. The frequency derivative of the phase response has a special name - group delay time (GDT). The phase must be taken in radians, and the frequency must be taken not as vibrational (in hertz), but as angular, in radians per second. Then the derivative will receive the dimension of time, which explains (albeit partially) its name. The group delay characteristics of high-pass and low-pass filters of the same type are no different. This is what the group delay graphs look like for Butterworth filters from the first to the fourth order (Fig. 14).

Here the difference between filters of different orders seems especially noticeable. The maximum (in amplitude) group delay value for a fourth-order filter is approximately four times greater than that of a first-order filter and twice that of a second-order filter. There are statements that according to this parameter, a fourth-order filter is just four times worse than a first-order filter. For a high-pass filter - perhaps. But for a low-pass filter, the disadvantages of a high group delay are not so significant in comparison with the advantages of a high frequency response slope.

For further discussion, it will be useful for us to imagine what the phase response “over the air” of an electrodynamic head looks like, that is, how the radiation phase depends on frequency.

A remarkable picture (Fig. 15): at first glance it looks like a filter, but, on the other hand, it is not a filter at all - the phase drops all the time, and with an increasing steepness. I won’t let on any unnecessary mystery: this is what the delay line phase response looks like. Experienced people will say: of course, the delay is caused by the travel of the sound wave from the emitter to the microphone. And experienced people will make a mistake: my microphone was installed along the head flange; Even if we take into account the position of the so-called center of radiation, this can cause an error of 3 - 4 cm (for this particular head). And here, if you estimate, the delay is almost half a meter. And, in fact, why shouldn’t there be a delay? Just imagine such a signal at the output of the amplifier: nothing, nothing, and suddenly a sine - as it should be, from the origin and with maximum slope. (For example, I don’t need to imagine anything, I have this written down on one of the measuring CDs, we check the polarity using this signal.) It is clear that the current will not flow through the voice coil immediately, it still has some kind of inductance. But these are minor things. The main thing is that sound pressure is volumetric velocity, that is, the diffuser must first accelerate, and only then sound will appear. For the delay value, it is probably possible to derive a formula; it will probably include the mass of the “movement”, the force factor and, possibly, the ohmic resistance of the coil. By the way, I obtained similar results on different equipment: both on the Bruel & Kjaer analog phase meter, and on the MLSSA and Clio digital complexes. I know for sure that mid-frequency drivers have less delay than bass drivers, and tweeters have less delay than both of them. Surprisingly, I have not seen any references to such results in the literature.

Why did I bring this instructive graph? And then, if this is really the case as I see it, then many discussions about the properties of filters lose practical meaning. Although I will still present them, and you can decide for yourself whether all of them are worth adopting.

Passive filter circuits

I think few people will be surprised if I say that there are much fewer circuit implementations of passive filters than active filters. I would say there are about two and a half. That is, if elliptic filters are put into a separate class of circuits, you get three, if you don’t do this, then two. Moreover, in 90% of cases in acoustics, so-called parallel filters are used. Therefore, we will not start with them.

Serial filters, unlike parallel ones, do not exist “in parts” - here is a low-pass filter, and there is a high-pass filter. This means you cannot connect them to different amplifiers. In addition, in terms of their characteristics, these are first-order filters. And by the way, the ubiquitous Mr. Small justified that first-order filters for acoustic applications are unsuitable, no matter what orthodox audiophiles (on the one hand) and supporters of the every possible reduction in the cost of acoustic products (on the other) say. However, series filters have one advantage: the sum of their output voltages is always equal to unity. This is what the circuit of a two-band sequential filter looks like (Fig. 16).

In this case, the values ​​correspond to a cutoff frequency of 2000 Hz. It is easy to understand that the sum of the voltages across the loads is always exactly equal to the input voltage. This feature of the serial filter is used when “preparing” signals for their further processing by the processor (in particular, in Dolby Pro Logic). In the next graph you see the frequency response of the filter (Fig. 17).

You can believe that its phase response and group delay graphs are exactly the same as those of any first-order filter. A three-band sequential filter is also known to science. Its diagram is in Fig. 18.

The values ​​shown in the diagram correspond to the same crossover frequency (2000 Hz) between the tweeter (HF) and the midrange driver and the frequency of 100 Hz - the crossover frequency between the midrange and low-frequency heads. It is clear that a three-band series filter has the same property: the sum of the voltages at its output is exactly equal to the voltage at the input. In the following figure (Fig. 19), which shows a set of characteristics of this filter, you can see that the slope of the tweeter filter in the range of 50 - 200 Hz is higher than 6 dB/oct., since its band here overlaps not only with the midrange band , but also to the woofer head band. This is what parallel filters cannot do - their overlap of bands inevitably brings surprises, and always unpleasant ones.

The parameters of the sequential filter are calculated in exactly the same way as the values ​​of the first order filters. The dependence is still the same (see formula 1.1). It is most convenient to introduce the so-called time constant; through the filter cutoff frequency it is expressed as TO = 1/(2?Fc).

C = TO/RL (2.1), and

L = TO*RL (2.2).

(Here RL is the load impedance, in this case 4 ohms).

If, as in the second case, you have a three-band filter, then there will be two crossover frequencies and two time constants.

Probably, the most technically savvy of you have already noticed that I slightly “distorted” the cards and replaced the real load impedance (that is, the speaker) with an ohmic “equivalent” of 4 Ohms. In reality, of course, there is no equivalent. In fact, even a forcibly inhibited voice coil, from the point of view of an impedance meter, looks like active and inductive reactance connected in series. And when the coil is mobile, the inductance increases at a high frequency, and near the resonance frequency of the head, its ohmic resistance seems to increase, sometimes ten times or more. There are very few programs that can take into account such features of a real head; I personally know of three. But we in no way set out to learn how to work, say, in software environment Linearx. Our task is different - to understand the main features of filters. Therefore, we will, in the old fashioned way, simulate the presence of a head with a resistive equivalent, and specifically with a nominal value of 4 Ohms. If in your case the load has a different impedance, then all the impedances included in the passive filter circuit must be proportionally changed. That is, inductance is proportional, and capacitance is inversely proportional to the load resistance.

(After reading this in a draft, the editor-in-chief said: “What, sequential filters are the Klondike, let’s dig into it somehow.” I agree. Klondike. I had to promise that we’ll dig into it separately and specifically in one of the upcoming issues.)

The most widely used parallel filters are also called “ladder” filters. I think it will be clear to everyone where this name comes from after you look at the generalized filter circuit (Fig. 20).

To get a fourth-order low-pass filter, you need to replace all horizontal “bars” in this circuit with inductances, and all vertical ones with capacitors. Accordingly, to build a high-pass filter you need to do the opposite. Lower order filters are obtained by discarding one or more elements, starting with the last one. Higher order filters are obtained in a similar way, only by increasing the number of elements. But we will agree: there are no filters higher than the fourth order for us. As we will see later, along with the increase in filter steepness, their shortcomings also deepen, so such an agreement is not something seditious. To complete the presentation, it would be necessary to say one more thing. There is an alternative option for constructing passive filters, where the first element is always a resistor rather than a reactive element. Such circuits are used when it is necessary to normalize the input impedance of the filter (for example, operational amplifiers “do not like” loads less than 50 Ohms). But in our case, an extra resistor means unjustified power losses, so “our” filters begin with reactivity. Unless, of course, you need to specifically reduce the signal level.

The most complex bandpass filter in design is obtained if in a generalized circuit each horizontal element is replaced with a series connection of capacitance and inductance (in any sequence), and each vertical element must be replaced with parallel connected ones - also capacitance and inductance. Probably, I will still give such a “scary” diagram (Fig. 21).

There is one more little trick. If you need an asymmetrical “bandpass” (bandpass filter), in which, say, the high-pass filter is of the fourth order, and the low-pass filter is of the second, then the unnecessary parts from the above circuit (that is, one capacitor and one coil) must certainly be removed from “ tail" of the circuit, and not vice versa. Otherwise, you will get somewhat unexpected effects from changing the nature of the loading of the previous filter cascades.

We did not have time to get acquainted with elliptical filters. Well, then, next time we’ll start with them.

Prepared based on materials from the magazine "Avtozvuk", May 2009.www.avtozvuk.com

That is, not really at all. The fact is that the schematics of passive filters are quite diverse. We immediately disowned filters with a normalizing resistor at the input, since they are almost never used in acoustics, unless, of course, you count those cases when the head (tweeter or midrange driver) needs to be “depressed” by exactly 6 dB. Why six? Because in such filters (they are also called dual-loaded), the value of the input resistor is chosen to be the same as the load impedance, say, 4 Ohms, and in the passband such a filter will provide an attenuation of 6 dB. In addition, double-loaded filters are P-type and T-type. To imagine a P-type filter, it is enough to discard the first element (Z1) in the generalized filter diagram (Fig. 20, No. 5/2009). The first element of such a filter is connected to ground, and if there is no input resistor in the filter circuit (single-loaded filter), then this element does not create a filtering effect, but only loads the signal source. (Try the source, that is, the amplifier, to connect to a capacitor of several hundred microfarads, and then write to me whether its protection has worked or not. Just in case, write post restante; it’s better not to litter those giving such advice with addresses.) Therefore, we use P-filters We don't consider it either. In total, as is easy to imagine, we are dealing with one fourth of the circuit implementations of passive filters.

Elliptic filters stand apart because they have an extra element and an extra root of the polynomial equation. Moreover, the roots of this equation are distributed in the complex plane not in a circle (like Butterworth, say), but in an ellipse. In order not to operate with concepts that probably make no sense to clarify here, we will call elliptic filters (like all others) by the name of the scientist who described their properties. So…

Cauer filter circuits

There are two known circuit implementations of Cauer filters - for a high-pass filter and a low-pass filter (Fig. 1).

Those that are designated by odd numbers are called standard, the other two are called dual. Why is this and not otherwise? Maybe because in standard circuits the additional element is a capacitance, and dual circuits differ from a conventional filter by the presence of additional inductance. By the way, not every circuit obtained in this way is an elliptic filter; if everything is done according to science, the relationships between the elements must be strictly observed.

The Cauer filter has a fair number of shortcomings. As always, secondly, let’s think positively about them. After all, Kauer has a plus, which in other cases can outweigh everything. Such a filter provides deep signal suppression at the tuning frequency of the resonant circuit (L1-C3, L2-C4, L4-C5, L6-C8 in diagrams 1 - 4). In particular, if it is necessary to provide filtering near the resonance frequency of the head, then only Cauer filters can cope with this task. It is quite troublesome to count them manually, but in simulator programs there are, as a rule, special sections dedicated to passive filters. True, it is not a fact that there will be single-load filters there. However, in my opinion, there will be no great harm if you take a Chebyshev or Butterworth filter circuit, and additional element calculate the resonance frequency using the well-known formula:

Fр = 1/(2 ? (LC)^1/2), whence

C = 1/(4 ? ^2 Fр ^2 L) (3.1)

Required condition: the resonant frequency must be outside the transparency band of the filter, that is, for a high-pass filter - below the cutoff frequency, for a low-pass filter - above the cutoff frequency of the “original” filter. From a practical point of view, high-pass filters of this type are of greatest interest - it happens that it is desirable to limit the band of a mid-range driver or tweeter as low as possible, excluding, however, its operation near the resonance frequency of the head. For unification, I present a high-pass filter circuit for our favorite frequency of 100 Hz (Fig. 2).

The ratings of the elements look a little wild (especially the capacitance of 2196 μF - the resonance frequency is 48 Hz), but as soon as you move to higher frequencies, the ratings will change in inverse proportion to the square of the frequency, that is, quickly.

Types of filters, pros and cons

As already mentioned, the characteristics of filters are determined by a certain polynomial (polynomial) of the appropriate order. Since mathematics describes a certain number of special categories of polynomials, there can be exactly the same number of types of filters. Even more, in fact, since in acoustics it was also customary to give special names to some categories of filters. Since there are polynomials of Butterworth, Legendre, Gauss, Chebyshev (advice: write and pronounce the name of Pafnutiy Lvovich through “e”, as it should be - this is the most easy way to show the thoroughness of one’s own education), Bessel, etc., then there are filters bearing all these names. In addition, Bessel polynomials have been studied intermittently for almost a hundred years, so a German, like the corresponding filters, will name them by the name of his compatriot, and an Englishman will most likely remember Thomson. A special article is Linkwitz filters. Their author (vivacious and cheerful) proposed a certain category of high-pass and low-pass filters, the sum of the output voltages of which would give an even frequency dependence. The point is this: if at the junction point the drop in the output voltage of each filter is 3 dB, then in terms of power (voltage squared) the total characteristic will be straightforward, and in terms of voltage at the junction point a hump of 3 dB will appear. Linkwitz suggested matching filters at a level of -6 dB. In particular, second-order Linkwitz filters are the same as Butterworth filters, only for the high-pass filter they have a cutoff frequency 1.414 times higher than for the low-pass filter. (The coupling frequency is exactly between them, that is, 1.189 times higher than the Butterworth low-pass filter with the same ratings.) So when I encounter an amplifier in which the tunable filters are specified as Linkwitz filters, I understand that the authors of the design and writers of the specification did not were familiar with each other. However, let's return to the events of 25 - 30 years ago. Richard Small also took part in the general celebration of filter construction, who proposed combining Linkwitz filters (for convenience, no less) with series filters, which also provide an even voltage characteristic, and calling them all filters DC voltage(constant voltage design). This is despite the fact that neither then, nor, it seems, now, is it really established whether a flat voltage or power characteristic is preferable. One of the authors even calculated intermediate polynomial coefficients, so that filters corresponding to these “compromise” polynomials should have produced a 1.5-dB voltage hump at the junction point and a power dip of the same magnitude. One of additional requirements to the filter designs was that the phase-frequency characteristics of the low-pass and high-pass filters must be either identical or diverge by 180 degrees - which means that if the polarity of one of the links is changed, an identical phase characteristic will again be obtained. As a result, among other things, it is possible to minimize the area of ​​overlapping stripes.

It is possible that all these mind games turned out to be very useful in the development of multi-band compressors, expanders and other processor systems. But it’s difficult to use them in acoustics, to put it mildly. Firstly, it is not the voltages that are added up, but the sound pressures, which are related to the voltage through a tricky phase-frequency characteristic (Fig. 15, No. 5/2009), so not only their phases can vary arbitrarily, but also the slope of the phase dependence will certainly be different (unless it occurred to you to separate heads of the same type into stripes). Secondly, voltage and power are related to sound pressure and acoustic power through the efficiency of the heads, and they also do not have to be the same. Therefore, it seems to me that the focus should not be on pairing filters by bands, but on the filters’ own characteristics.

What characteristics (from an acoustics perspective) determine the quality of filters? Some filters provide a smooth frequency response in the transparency band, while for others the roll-off begins long before the cutoff frequency is reached, but even after it the slope of the roll-off slowly reaches the desired value; for others, a hump (“notch”) is observed on the approach to the cutoff frequency, after which a sharp decline begins with a slope even slightly higher than the “nominal” one. From these positions, the quality of filters is characterized by “smoothness of the frequency response” and “selectivity”. The phase difference for a filter of a given order is a fixed value (this was discussed in the last issue), but the phase change can be either gradual or rapid, accompanied by a significant increase in the group delay time. This property of the filter is characterized by phase smoothness. Well, and the quality of the transition process, that is, the reaction to stepwise influence (Step Response). The low-pass filter processes the transition from level to level (though with a delay), but the transition process may be accompanied by an overshoot and an oscillatory process. With a high-pass filter, the step response is always a sharp peak (without delay) with a return to zero dc, but the zero-crossing and subsequent oscillations are similar to what would be seen with a low-pass filter of the same type.

In my opinion (my opinion may not be controversial, those who want to argue can enter into correspondence, even not on demand), for acoustic purposes three types of filters are quite sufficient: Butterworth, Bessel and Chebyshev, especially since the latter type actually combines a whole group of filters with different magnitudes of “teeth”. In terms of smoothness of the frequency response in the transparency band, Butterworth filters are unrivaled - their frequency response is called the characteristic of the greatest smoothness. And then, if we take the Bessel - Butterworth - Chebyshev series, then in this series there is an increase in selectivity with a simultaneous decrease in the smoothness of the phase and the quality of the transition process (Fig. 3, 4).

It is clearly seen that Bessel’s frequency response is the smoothest, while Chebyshev’s is the most “decisive”. The phase-frequency response of the Bessel filter is also the smoothest, while that of the Chebyshev filter is the most “angular”. For generality, I also present the characteristics of the Cauer filter, the diagram of which was shown just above (Fig. 5).

Notice how at the resonance point (48 Hz, as promised), the phase abruptly changes by 180 degrees. Of course, at this frequency the signal suppression should be highest. But in any case, the concepts of “phase smoothness” and “Cauer filter” are in no way compatible.

Now let's see what the transient response of four types of filters looks like (all are low-pass filters with a cutoff frequency of 100 Hz) (Fig. 6).

The Bessel filter, like all others, has a third order, but it has virtually no overshoot. The largest emissions are found in Chebyshev and Cauer, and in the latter the oscillatory process is longer. The magnitude of the overshoot increases as the filter order increases and, accordingly, falls as it decreases. For illustration, I present the transient characteristics of the second-order Butterworth and Chebyshev filters (there are no problems with Bessel) (Fig. 7).

In addition, I came across a table showing the dependence of the flop value on the order of the Butterworth filter, which I also decided to present (Table 1).

This is one of the reasons why it is hardly worth getting carried away with Butterworth filters above the fourth order and Chebyshev filters above the third, as well as Cauer filters. A distinctive feature of the latter is its extremely high sensitivity to the spread of element parameters. In my experience, the percent selection accuracy of parts can be defined as 5/n, where n is the order of the filter. That is, when working with a fourth-order filter, you must be prepared for the fact that the nominal value of the parts will have to be selected with an accuracy of 1% (for Cauer - 0.25%!).

And now it’s time to move on to the selection of parts. Electrolytes, of course, should be avoided due to their instability, although if the capacitance count is hundreds of microfarads, there is no other choice. Capacities, of course, will have to be selected and assembled from several capacitors. If desired, you can find electrolytes with low leakage, low terminal resistance and a real capacity spread of no worse than +20/-0%. Coils, of course, are better “coreless”; if you can’t do without a core, I prefer ferrites.

To select denominations, I suggest using the following table. All filters are designed for a cutoff frequency of 100 Hz (-3 dB) and a load rating of 4 ohms. To get the nominal values ​​for your project, you need to recalculate each of the elements using simple formulas:

A = At ​​Zs 100/(4*Fc) (3.2),

where At is the corresponding table value, Zs is the nominal impedance of the dynamic head, and Fc, as always, is the calculated cutoff frequency. Attention: inductance ratings are given in millihenry (and not in henry), capacitance ratings are in microfarads (and not in farads). There is less science, more convenience (Table 2).

We have another interesting topic ahead - frequency correction in passive filters, but we will look at it in the next lesson.

In the last chapter of the series, we took a first look at passive filter circuits. True, not really.

Chebyshev frequency response of third order

Third order Butterworth frequency response

Bessel frequency response of third order

Third order Bessel phase response

Third-order Butterworth phase response

Chebyshev phase response characteristic of the third order

Frequency response of a third-order Cauer filter

Phase response response of a third-order Cauer filter

Bessel transient response

Cowher step response

Chebyshev transition characteristic

Butterworth step response

Prepared based on materials from the magazine "Avtozvuk", July 2009.www.avtozvuk.com

The devices and circuits that make up passive filters (of course, if they are filters of the appropriate level) can be divided into three groups: attenuators, frequency correction devices and what English-speaking citizens call miscellaneous, simply put, “miscellaneous”.

Attenuators

At first this may seem surprising, but an attenuator is an indispensable attribute of multi-band acoustics, because heads for different bands not only do not always have, but also should not have the same sensitivity. IN otherwise freedom of maneuver for frequency correction will be reduced to zero. The fact is that in a passive correction system, in order to correct a failure, you need to “settle” the head in the main band and “release” where the failure was. In addition, in residential areas it is often desirable for the tweeter to slightly “overplay” the midbass or midrange and bass in volume. At the same time, “downsetting” the bass speaker is expensive in any sense - a whole group of powerful resistors is required, and a fair portion of the amplifier’s energy is spent on warming up the said group. In practice, it is considered optimal when the output of the midrange driver is several (2 - 5) decibels higher than that of the bass, and that of the tweeter is the same amount higher than that of the midrange head. So you can’t do without attenuators.

As you know, electrical engineering operates with complex quantities, and not with decibels, so today we will only partially use them. Therefore, for your convenience, I provide a table for converting the attenuation indicator (dB) into the transmittance of the device.

So, if you need to "sag" the head by 4 dB, the transmittance N of the attenuator should be equal to 0.631. The simplest option is a series attenuator - as the name implies, it is installed in series with the load. If ZL is the average head impedance in the region of interest, then the value RS of the series attenuator is determined by the formula:

RS = ZL * (1 - N)/N (4.1)

As ZL you can take the “nominal” 4 Ohms. If we, with the best of intentions, install a series attenuator directly in front of the head (the Chinese, as a rule, do this), then the load impedance for the filter will increase, and the cutoff frequency of the low-pass filter will increase, and the cutoff frequency of the high-pass filter will decrease. But that is not all.

For example, take a 3 dB attenuator operating at 4 ohms. The resistor value according to formula (4.1) will be equal to 1.66 Ohms. In Fig. 1 and 2 are what you get when using a 100 Hz high pass filter, as well as a 4000 Hz low pass filter.

Blue curves in Fig. 1 and 2 - frequency characteristics without an attenuator, red - frequency response with a series attenuator turned on after the corresponding filter. The green curve corresponds to the inclusion of the attenuator before the filter. The only side effect is a frequency shift of 10 - 15% in minus and plus for the high-pass filter and low-pass filter, respectively. So in most cases the series attenuator should be installed before the filter.

To avoid drift of the cutoff frequency when the attenuator is turned on, devices were invented that in our country are called L-shaped attenuators, and in the rest of the world, where the alphabet does not contain the magical letter “G” that is so necessary in everyday life, they are called L-Pad. Such an attenuator consists of two resistors, one of them, RS, is connected in series with the load, the second, Rp, is connected in parallel. They are calculated like this:

RS = ZL * (1 - N), (4.2)

Rp = ZL * N/(1 - N) (4.3)

For example, we take the same 3 dB attenuation. The resistor values ​​turned out to be as shown in the diagram (ZL again 4 Ohms).

Rice. 3. L-shaped attenuator circuit

Here the attenuator is shown along with the 4 kHz high pass filter. (For uniformity, all filters today are of the Butterworth type.) In Fig. 4 you see the usual set of characteristics. The blue curve is without an attenuator, the red curve is with the attenuator turned on before the filter, and the green curve is with the attenuator turned on after the filter.

As you can see, the red curve has a lower quality factor, and the cutoff frequency is shifted down (for a low-pass filter it will shift up by the same 10%). So there is no need to be clever - it is better to turn on the L-Pad exactly as shown in the previous figure, directly in front of the head. However, under certain circumstances, you can use the rearrangement - without changing the denominations, you can correct the area where the bands separate. But this is already aerobatics... And now let’s move on to “miscellaneous things”.

Other common schemes

Most often found in our crossovers is a head impedance correction circuit, usually called a Zobel circuit after the famous researcher of filter characteristics. It is a serial RC circuit connected in parallel with the load. According to classical formulas

C = Le/R 2 e (4.5), where

Le = [(Z 2 L - R 2 e)/2?pFo] 1/2 (4.6).

Here ZL is the load impedance at the frequency Fo of interest. As a rule, for the ZL parameter, without further ado, they choose the nominal impedance of the head, in our case, 4 Ohms. I would advise looking for the value of R using the following formula:

R = k * Re (4.4a).

Here the coefficient k = 1.2 - 1.3, it’s still impossible to select resistors more accurately.

In Fig. 5 you can see four frequency characteristics. Blue is the usual characteristic of a Butterworth filter loaded with a 4 ohm resistor. Red curve - this characteristic is obtained if the voice coil is represented as a series connection of a 3.3 Ohm resistor and an inductance of 0.25 mH (such parameters are typical for a relatively light midbass). Feel the difference, as they say. The black color shows how the frequency response of the filter will look if the developer does not simplify his life, and determines the filter parameters using formulas 4.4 - 4.6, based on the total impedance of the coil - with the specified parameters of the coil, the total impedance will be 7.10 Ohms (4 kHz). Finally, the green curve is the frequency response obtained using a Zobel circuit, the elements of which are determined by formulas (4.4a) and (4.5). The discrepancy between the green and blue curves does not exceed 0.6 dB in the frequency range 0.4 - 0.5 of the cutoff frequency (in our example it is 4 kHz). In Fig. 6 you see a diagram of the corresponding filter with “Zobel”.

By the way, when you find a resistor with a nominal value of 3.9 Ohms (less often - 3.6 or 4.2 Ohms) in the crossover, you can say with minimal probability of error that a Zobel circuit is involved in the filter circuit. But there are other circuit solutions that lead to the appearance of an “extra” element in the filter circuit.

Of course, I am referring to the so-called “strange” filters, which are distinguished by the presence of an additional resistor in the filter ground circuit. The already well-known 4 kHz low-pass filter can be represented in this form (Fig. 7).

Resistor R1 with a nominal value of 0.01 Ohm can be considered as the resistance of the capacitor leads and connecting tracks. But if the resistor value becomes significant (that is, comparable to the load rating), you will get a “strange” filter. We will change resistor R1 in the range from 0.01 to 4.01 Ohms in 1 Ohm increments. The resulting family of frequency characteristics can be seen in Fig. 8.

The upper curve (in the area of ​​the inflection point) is the usual Butterworth characteristic. As the resistor value increases, the filter cutoff frequency shifts down (up to 3 kHz at R1 = 4 Ohms). But the slope of the decline changes slightly, at least within the band limited to the -15 dB level - and it is precisely this region that is of practical importance. Below this level the roll-off slope will tend to be 6 dB/oct., but this is not that important. (Please note that the vertical scale of the graph has been changed, so the decline appears steeper.) Now let’s see how the phase-frequency response changes depending on the resistor value (Fig. 9).

The behavior of the phase response graph changes starting from 6 kHz (that is, from 1.5 cutoff frequencies). By using a "strange" filter, the mutual phase of the radiation from adjacent heads can be smoothly adjusted to achieve the desired shape of the overall frequency response.

Now, in accordance with the laws of the genre, we will take a break, promising that next time it will be even more interesting.

Rice. 1. Frequency response of a serial attenuator (HPF)

Rice. 2. The same for the low-pass filter

Rice. 4. Frequency characteristics of the L-shaped attenuator

Rice. 5. Frequency characteristics of a filter with a Zobel circuit

Rice. 6. Filter circuit with Zobel circuit

Rice. 7. “Strange” filter circuit

Rice. 8. Amplitude-frequency characteristics of the “strange” filter

Rice. 9. Phase-frequency characteristics of the “strange” filter

Prepared based on materials from the magazine "Avtozvuk", August 2009.www.avtozvuk.com

As promised, today we will finally take a closer look at frequency correction circuits.

In my writings, I have argued more than once or twice that passive filters can do many things that active filters cannot do. He asserted indiscriminately, without proving his rightness in any way and without explaining anything. But really, what can’t active filters do? They solve their main task - “cutting off the unnecessary” - quite successfully. And although it is precisely because of their versatility that active filters, as a rule, have Butterworth characteristics (if they are performed correctly at all), Butterworth filters, as I hope you have already understood, in most cases represent an optimal compromise between the shape of the amplitude and phase frequency characteristics , as well as the quality of the transition process. And the ability to smoothly adjust the frequency generally compensates for too much. In terms of level matching, active systems certainly outperform any attenuators. And there is only one area in which active filters lose - frequency correction.

In some cases, a parametric equalizer can be useful. But analog equalizers often lack either frequency range, or Q-tuning limits, or both. Multiband parametrics, as a rule, have both in abundance, but they add noise to the path. In addition, these toys are expensive and rare in our industry. Digital parametric equalizers are ideal if they have a central frequency tuning step of 1/12 octave, and we don’t seem to have those either. Parameters with 1/6 octave steps are partially suitable, provided that they have a sufficiently wide range of available quality values. So it turns out that only passive corrective devices best suit the assigned tasks. By the way, high-quality studio monitors often do this: bi-amping/tri-amping with active filtering and passive correction devices.

High frequency correction

At higher frequencies, as a rule, a rise in the frequency response is required; it lowers itself without any correctors. A chain consisting of a capacitor and a resistor connected in parallel is also called a horn circuit (since horn emitters very rarely do without it), and in modern (not our) literature it is often called simply a circuit. Naturally, in passive system To raise the frequency response in some area, you must first lower it in all others. The resistor value is selected using the usual formula for a series attenuator, which was given in the previous series. For convenience, I will still give it again:

RS = ZL (1 - N)/N (4.1)

Here, as always, N is the attenuator transmittance, ZL is the load impedance.

I choose the capacitor value using the formula:

C = 1/(2 ? F05 RS), (5.1)

where F05 is the frequency at which the attenuator action needs to be “halved”.

No one will forbid you to turn on more than one “circuit” in series in order to avoid “saturation” in the frequency response (Fig. 1).

As an example, I took the same second-order Butterworth high-pass filter for which in the last chapter we determined the resistor value Rs = 1.65 Ohms for 3 dB attenuation (Fig. 2).

This double circuit allows you to raise the “tail” of the frequency response (20 kHz) by 2 dB.

It would probably be useful to recall that multiplying the number of elements also multiplies errors due to the uncertainty of the load impedance characteristics and the spread of element values. So I wouldn’t recommend messing with three or more step circuits.

Frequency response peak suppressor

In foreign literature, this corrective chain is called peak stopper network or simply stopper network. It already consists of three elements - a capacitor, a coil and a resistor connected in parallel. It seems like a small complication, but the formulas for calculating the parameters of such a circuit turn out to be noticeably more cumbersome.

The value of Rs is determined by the same formula for a series attenuator, in which this time we will change one of the notations:

RS = ZL (1 - N0)/N0 (5.2).

Here N0 is the transmission coefficient of the circuit at the center frequency of the peak. Let's say, if the peak height is 4 dB, then the transmission coefficient is 0.631 (see table from the last chapter). Let us denote as Y0 the value of the reactance of the coil and capacitor at the resonance frequency F0, that is, at the frequency where the center of the peak in the frequency response of the speaker that we need to suppress falls. If Y0 is known to us, then the values ​​of capacitance and inductance will be determined using the known formulas:

C = 1/(2 ? F0 x Y0) (5.3)

L = Y0 /(2 ? F0) (5.4).

Now we need to set two more frequency values ​​FL and FH - below and above the central frequency, where the transmission coefficient has the value N. N > N0, say, if N0 was set as 0.631, the N parameter can be equal to 0.75 or 0.8 . The specific value of N is determined from the frequency response graph of a particular speaker. Another subtlety concerns the choice of FH and FL values. Since the correcting circuit in theory has a symmetrical frequency response shape, then the selected values ​​must satisfy the condition:

(FH x FL)1/2 = F0 (5.5).

Now we finally have all the data to determine the Y0 parameter.

Y0 = (FH - FL)/F0 sqr (1/(N2/(1 - N)2/ZL2 - 1/R2)) (5.6).

The formula looks scary, but I warned you. May you be encouraged by the knowledge that we will no longer encounter more cumbersome expressions. The multiplier in front of the radical is the relative bandwidth of the correction device, that is, a value inversely proportional to the quality factor. The higher the quality factor, the (at the same central frequency F0) the inductance will be smaller and the capacitance will be larger. Therefore, with a high quality factor of the peaks, a double “ambush” arises: with an increase in the central frequency, the inductance becomes too small, and it can be difficult to manufacture it with the appropriate tolerance (±5%); As the frequency decreases, the required capacitance increases to such values ​​that it is necessary to “parallel” a certain number of capacitors.

As an example, let's calculate a corrector circuit with these parameters. F0 = 1000 Hz, FH = 1100 Hz, FL = 910 Hz, N0 = 0.631, N = 0.794. This is what happens (Fig. 3).

And here is what the frequency response of our circuit will look like (Fig. 4). With a purely resistive load (blue curve), we get almost exactly what we expected. In the presence of head inductance (red curve), the corrective frequency response becomes asymmetrical.

The characteristics of such a corrector depend little on whether it is placed before or after the high-pass filter or low-pass filter. In the next two graphs (Fig. 5 and 6), the red curve corresponds to turning on the corrector before the corresponding filter, the blue curve corresponds to turning it on after the filter.

Compensation scheme for dip in frequency response

What was said regarding the high-frequency correction circuit also applies to the dip compensation circuit: in order to raise the frequency response in one section, you must first lower it in all others. The circuit consists of the same three elements Rs, L and C, with the only difference being that the reactive elements are connected in series. At the resonance frequency they bypass a resistor, which acts as a series attenuator outside the resonance zone.

The approach to determining the parameters of elements is exactly the same as in the case of a peak suppressor. We must know the central frequency F0, as well as the transmittance coefficients N0 and N. In this case, N0 has the meaning of the transmittance coefficient of the circuit outside the correction region (N0, like N, is less than one). N is the transmittance coefficient at the points of the frequency response corresponding to the frequencies FH and FL. The values ​​of the frequencies FH, FL must meet the same condition, that is, if you see an asymmetrical dip in the real frequency response of the head, for these frequencies you must choose compromise values ​​so that condition (5.5) is approximately met. By the way, although this is not explicitly stated anywhere, it is most practical to choose the N level in such a way that its value in decibels corresponds to half of the N0 level. This is exactly what we did in the example of the previous section, N0 and N corresponded to levels of -4 and -2 dB.

The resistor value is determined by the same formula (5.2). The values ​​of capacitance C and inductance L will be related to the value of reactive impedance Y0 at the resonance frequency F0 by the same dependencies (5.3), (5.4). And only the formula for calculating Y0 will be slightly different:

Y0 = F0/(FH-FL) sqr (1/(N2/(1 - N)2/ZL2 - 1/R2)) (5.7).

As promised, this formula is no more cumbersome than equality (5.6). Moreover, (5.7) differs from (5.6) in the inverse value of the factor before the expression for the root. That is, as the quality factor of the correction circuit increases, Y0 increases, which means that the value of the required inductance L increases and the value of capacitance C decreases. In this regard, only one problem arises: with a sufficiently low central frequency F0, the required value of inductance forces the use of coils with cores, and then There are problems of our own, which probably make no sense to dwell on here.

For example, we take a circuit with exactly the same parameters as for the peak suppressor circuit. Namely: F0 = 1000 Hz, FH = 1100 Hz, FL = 910 Hz, N0 = 0.631, N = 0.794. The values ​​obtained are as shown in the diagram (Fig. 7).

Please note that the inductance of the coil here is almost twenty times greater than for the peak suppressor circuit, and the capacitance is the same amount less. Frequency response of the circuit we calculated (Fig. 8).

In the presence of load inductance (0.25 mH), the efficiency of the series attenuator (Rs resistor) decreases with increasing frequency (red curve), and a rise appears at high frequencies.

The dip compensation circuit can be installed on either side of the filter (Fig. 9 and 10). But we must remember that when the compensator is installed after the high-pass or low-pass filter (blue curve in Fig. 9 and 10), the quality factor of the filter increases and the cutoff frequency increases. So, in the case of the high-pass filter, the cutoff frequency moved from 4 to 5 kHz, and the cutoff frequency of the low-pass filter decreased from 250 to 185 Hz.

This concludes the series dedicated to passive filters. Of course, many questions were left out of our research, but, in the end, we have a general technical, not a scientific journal. And, in my personal opinion, the information provided within the series will be sufficient to solve most practical problems. For those who would like to receive additional information, you may find it helpful to refer to the following resources. First: http://www.educypedia.be/electronics/electronicaopening.htm. This is an educational site, it leads to other sites dedicated to specific issues. In particular, a lot of useful information on filters (active and passive, with calculation programs) can be found here: http://sim.okawa-denshi.jp/en/. In general, this resource will be useful to those who have decided to engage in engineering activities. They say that such people are appearing now...

Rice. 1. Double RF circuit diagram

Rice. 2. Frequency response of a double correction circuit

Rice. 3. Peak suppressor circuit

Rice. 4. Frequency characteristics of the peak suppression circuit

Rice. 5. Frequency characteristics of the corrector together with a high-pass filter

Rice. 6. Frequency characteristics of the corrector together with a low-pass filter

Rice. 7. Failure compensation scheme

Rice. 8. Frequency characteristics of the sag compensation circuit

Rice. 9. Frequency characteristics of the circuit together with a high-pass filter

Rice. 10. Frequency characteristics of the circuit together with a low-pass filter

Prepared based on materials from the magazine "Avtozvuk", October 2009.www.avtozvuk.com

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

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

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

Let's repeat it again:

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

Frequency filters are characterized by indicators

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

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

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

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

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

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

Frequency filters are:

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

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

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

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

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

Single element high and low pass filters

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

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

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

L-shaped filters

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

L-shaped frequency filters on a capacitor and resistor

R 1 WITH X C .

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

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

or

X C and cutoff frequency.

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

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

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

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

or

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

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

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

L-shaped frequency filters on an inductor and a resistor

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

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

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

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

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

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

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

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

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

L-shaped frequency filters on a capacitor and inductor

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

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

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

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

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

T-shaped high and low pass filters

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

U-shaped high and low pass filters

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

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

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

Bandpass Resonant Filters

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

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

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

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