AC low-pass filter two speaker 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 outputs of amplifiers through crossover filters, which are combinations of LC filters of low and high frequencies.

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

The frequency response of the crossover filter of a three-way loudspeaker is generally shown in fig. 1. Here: N is the relative voltage level on the voice coils of the heads: fn and fv are the lower and upper boundary frequencies of the band reproduced by the loudspeaker; fр1 and fр2 - section frequencies.

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

The crossover frequencies should be chosen outside the region of the 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 crossover frequency at the same time, a "bifurcation" of the 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 fn ... fp1. mid-frequency - in the range fp1 ... fp2 and high-frequency - in the range fp2 ... fv.

One of the common options for the electrical circuit diagram of a three-way loudspeaker is shown in fig. 2. Here: B1 - low-frequency dynamic head connected to the output of the amplifier through the low-pass filter L1C1; B2 - mid-range 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 the high-pass filters C2L3 and C4L4.

The calculation of capacitances of capacitors and inductances of coils is carried out on the basis of the nominal resistance of the loudspeaker heads. Since the nominal resistances of the heads and the nominal capacitances of the capacitors form series of discrete values, and the crossover frequencies can vary over a wide range, it is convenient to calculate in this sequence. Given the nominal resistance of the heads, the capacitances of the capacitors are selected from a series of nominal capacitances (or the total capacitance of several capacitors from this series) so that the resulting crossover frequency falls within the above frequency intervals.

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

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

After the capacitances of the capacitors are selected, the inductances of the coils are determined in millihenries according to the formulas:

In both formulas: Zg-in ohms; fp1, fp2 - in hertz.

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

In some cases, radio amateurs have to use existing dynamic heads with a different nominal impedance 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 the high-frequency head B3 and capacitor C4 to different terminals of the L4 coil (Fig. 2), i.e. this filter coil simultaneously plays the role of a matching autotransformer. Coils can be wound on round wooden, plastic or cardboard frames with getinaks cheeks. The lower cheek should be made square; so it is convenient to attach it to the base - a getinax board, on which capacitors and coils are attached. The board is fixed with screws to the bottom of the loudspeaker box. In order to avoid additional non-linear distortions, the coils must be made without cores made of magnetic materials.

Filter calculation example.

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

According to the above formulas, we find: L1=L3=2.56 mg; L2=L4=0.375mH (for an autotransformer, L4 is the value of the inductance between terminals 1-3).

Autotransformer transformation ratio

On fig. 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 between the total resistances of the heads and capacitances of capacitors from the given (nominal) values ​​and the inductances of the coils from those obtained by calculation. The steepness of the decline of the bass and midrange curves is 9 dB per octave and the high frequency curve is 11 dB per octave. The HF curve corresponds to the uncoordinated inclusion of the loudspeaker 1 GD-3 (at points 1-3). As you can see, in this case the filter introduces additional frequency distortions.

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 generated by any head is noticeably greater, then in order to equalize the frequency response of the loudspeaker in terms of 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 impedance of the heads adopted 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 filters of other types, and are currently the most used in HI-FI class speakers. Therefore, only this type of filters will be considered in the calculation procedure. The essence of the calculation is that, first, crossover filters are calculated from the active load condition and a voltage source with an infinitely small output impedance (which is true for modern audio frequency 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 filters.

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

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

Figure 1 shows the scheme of the sixth-order prototype filter. The prototype filter schemes of lower orders are formed by discarding the corresponding elements − α (starting with large ones) - for example, a 1st order prototype filter consists of a single inductor α 1 and loads R n.

Rice. one. Scheme of a unilaterally loaded prototype low-pass filter of the 6th order

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

a) for the low pass filter:

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

L=αR n/ 2πf d

each element α -capacity prototype filter translates to real capacity(F), calculated by the formula:

C=α/ 2πf dR n

b) for the high pass filter:

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

C= 1/ 2πf daR n

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

L=R n/ 2πf dα

c) for a band pass filter:

each element α -inductance is replaced by a series circuit consisting of real L and C -elements calculated by formulas

L=αR n/ 2π (f d 2 -f d 1 )

where f d 2 and f d 1 are the 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 is the average frequency of the band pass filter.

Each element α -capacitance is replaced by a parallel circuit consisting of real L and C-elements calculated by 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.

We select second-order crossover filters. Let the selected values ​​of the crossover frequencies be: between the low-frequency and mid-frequency channels f d 1 =500 Hz, between mid and high frequencies f d 2 =5000 Hz. Speaker impedance at DC: low-frequency and mid-frequency - 8 ohms, high-frequency - 16 ohms.

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

The amplitude-frequency characteristics of the loudspeakers, measured in a muted 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, medium frequency 30GD-8, high frequency 10GD-43).

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 an inductance α 1 , capacity α 2 and loads R n.

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

L 1 bass=αR n/ 2πf d 1 \u003d 2.0 8.0 / (2 3.14 500) \u003d 5.1 mH,

C 1 bass=α/ 2πf d 1 R n\u003d 0.5 / (2 3.14 500 8.0) \u003d 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 \u003d 1 / 4 3.14 2 5000 500 5.66 10 -4 \u003d 18 μF(LF side)

WITH 2 MF=α 2 / 2π(f d 2 -f d 1 )R n\u003d 0.5 / 2 3.14 (5000-500) 8.0 \u003d 2.2 μF(HF side)

L 2 MF= 1/ 4π 2 f 0 2 C 2 MF \u003d 1 / 4 3.14 2 5000 500 2.2 10 -6 \u003d 4.6 mH(LF side)

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

C 1 HF= 1/ 2πf d 2 α 1 R n\u003d 1 / (2 3.14 5000 2.0 16) \u003d 1.00 μF,

L 2 HF=R n/ 2πf d 2 α 2 \u003d 16 / (2 3.14 5000 2.0) \u003d 0.25 mH.

To match the filters with the input complex 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 crossover filters. The parameters of the elements of the matching circuit, connected in parallel to 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 ) is the 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 loudspeaker's equivalent circuitry. The values ​​of the circuit elements are determined as follows:

R K 1 = R E,

C K 1 = L VC/ R E 2

R K=R E 2 / R ES= Q ES R E / Q MS,

C K=L CES / R E 2 \u003d 1 / Q ES R E 2π f s,

L K=C MESR E 2 =Q ES R E /2π f s,

where L VC is the inductance of the voice coil, f s, C MES, L CES, R ES– electromechanical parameters of the loudspeaker.

To compensate for the input impedance of a subwoofer, a simplified circuit is used, consisting of resistors connected in series. R K1 and containers C K1. This is because the mechanical resonance of the loudspeaker does not affect the performance of the low-pass filter and only compensates for the inductive nature of the loudspeaker input impedance. It is advisable to connect a full matching circuit to high-frequency and mid-range loudspeakers if the resonant frequency of the loudspeaker is near the cutoff frequency of the high-pass filter or the lower cutoff frequency of the bandpass 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 to compensate for the complex nature of loudspeaker input impedance

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

Rice. 4. The electrical equivalent circuit of a 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 bass loudspeaker are chosen 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 set to 10 times the resonant frequency of the loudspeaker f d=10 f s. The inductance of the voice coil is selected from the condition: Q VC=0.1, where Q VC- the quality factor of the voice coil, defined as:

Q VC=L VC2 π f s/ R E,

where fs is the resonant frequency of the loudspeaker, R E is the resistance of the voice coil at direct current, LVC is the inductance of the voice coil.

Meaning Q VC=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 VC and active resistance R E connected in parallel with the filter tank C 1 and form a wide maximum of 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 filter by voltage consist in a slight rise in the frequency response at a frequency f ≈ 2 f s(due to voice coil inductance) and a smooth dip followed by a sharp frequency response peak due to circuit resonance created by voice coil inductance and crossover capacitance. The corresponding changes in frequency response and Z BX after turning on the matching circuit of a series-connected resistor and capacitor are shown in Fig. 5a (curves 2, 4, 6). The inclusion of a matching circuit brings the character of the input impedance of the loudspeaker closer to the active one and the frequency response of the crossover voltage filter to the desired one. However, due to the influence of the inductance of the voice coil, 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 separation filters loaded on a loudspeaker: a) low-pass filter; b) high pass filter;

1. Frequency response for voltage at the filter output without a matching circuit;
2. AFC voltage at the output of the filter with a matching circuit;
3. Frequency response for sound pressure without matching circuit;
4. AFC 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 circuit.

In the case of a high-pass filter, the effect of the complex nature of the input impedance of the loudspeaker on the input impedance and the frequency response of the filter is different. If the cutoff frequency of the high pass filter is near the resonance frequency of the speaker f s(a case sometimes found in filters for mid-range speakers, but almost impossible for high-frequency speakers), the input impedance of a high-pass filter with a speaker without a matching circuit can have a deep dip due to the fact that at the speaker's resonance frequency f s its input impedance increases significantly and has a purely active character. The filter turns out to be at idle, due to a sharp increase in load resistance and its input resistance is determined by the elements connected in series C 1 , L 1 . The most common situation is when the cutoff frequency of the high-pass filter f d well above the loudspeaker resonance frequency f s. Figure 5b shows an example of the influence of the loudspeaker input impedance and its compensation on the frequency response of the high-pass filter in terms of voltage and sound pressure. The filter cutoff frequency is chosen well above the loudspeaker resonance frequency f d≈8 f s, speaker settings Q TS=1,5 , Q MS=10, Q VC=0.08. The rise in the frequency response in terms of sound pressure and voltage in the high-frequency region, accompanied by a dip in the input resistance, is explained by the influence of the inductance of the voice coil L VC. For more high frequencies ah, the frequency response drops, and the input impedance grows due to the increase in the inductive resistance of the voice coil.

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

The output impedance of the crossover high-pass 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 "de-damping" the loudspeaker. To avoid this, it is necessary to choose the slope of the slope of the frequency response of the filter and the cutoff frequency of the high-pass filter f d>> f s so that at the resonance frequency f s signal attenuation was at least 20 dB.

When calculating crossover filters in the example discussed above, it was assumed that the nature of the load is active, therefore, we 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 selected approximately two octaves above the resonant frequency of the mid-range loudspeaker, and the crossover frequency of the mid-range and high-frequency channels f d 2 – two octaves above the resonant frequency of the tweeter. In addition, it can be assumed that the voice coil inductance of a high-frequency loudspeaker is negligible in the operating frequency range and can be neglected (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 frequency response curve of the input impedance) voice coil inductances: subwoofer L VC=3 10 -3 G=3 mH, midrange speaker L VC \u003d 0.5 10 -3 G \u003d 0.5 mH. Then the value of the elements of the compensating circuits is calculated by the formulas and:

for bass: R K 1 – R π =8 ohm; WITH K1 = L VC / R 2E=310 -3 / 64 = 47 uF

for MF: R' K 1 = R E-8 ohm; WITH' K1 = L VC / R 2E=0.510 -3 / 64 = 8.0 uF.

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

R K≈ R H(10 -0,05 N -1),

L K= R K∆ f /2π f 0 2 ,

C K =1/ L K4 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 AC input impedance (Ohm) in the region of the resonant frequency of the rejecting link;

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

f 0 is the resonant frequency of the notch, Hz;

N – magnitude of the frequency response peak, dB.

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

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

From the frequency response of the mid-frequency loudspeaker, we determine ∆ f =1850 Hz, f 0 =4000 Hz, N =6 dB. Midrange speaker impedance with matching circuit R H=8 ohm.

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

R K≈ R H(10 -0,05 N -1) \u003d 8 (10 -0.05 6 -1) \u003d 7.96 Ohm,

L K= R K∆ f /2π f 0 2 =7.96 1850/2 π (4000) 2 \u003d 0.147 mH,

C K=1/L K4 π 2 f 0 2 \u003d 1 / 1.47 10 -4 (2 π 4000) 2 = 11uF,

C q= L K/ R H 2 \u003d 1.47 10 -4 / 64 \u003d 2.3 μF,

L q= C KR H 2 \u003d 10.8 10 -6 64 \u003d 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 loudspeakers have average levels approximately 6 dB and, accordingly, 3 dB higher than the frequency response of the low-frequency loudspeaker (the sound pressure was recorded when the 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 speakers, it is necessary to attenuate the level of mid-frequency and high-frequency components. This can be done either with the help of a first-order corrective high-frequency link (Fig. 7), the elements of which are calculated by the formulas:

R K≈ R H(10 -0,05 N -1),

L K= R K/2π f d√(10 0,1 N -2), N≥3dB,

Or with the help of 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 by 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. eight. Passive L-shaped attenuator

For example, let's calculate the values ​​of the attenuator elements for attenuating the signal of a high-frequency loudspeaker by 6 dB. Let the input impedance of the loudspeaker with the attenuator 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.

Similarly, we calculate the values ​​of the attenuator elements for a mid-range loudspeaker: R 1 \u003d 4.7 Ohm, R 2 \u003d 39 Ohm. The attenuators are switched on immediately after the loudspeakers with matching circuits.

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

As mentioned above, even-order filters allow only one option for the polarity of switching on loudspeakers, in particular, second-order filters require switching in antiphase. For this example, the low-frequency and high-frequency loudspeakers must have identical switching polarity, and the mid-frequency one must be reversed. The speaker polarity requirements have been discussed above for speaker models with ideal speakers. Therefore, when real loudspeakers with their own PFC ≠ 0 are turned on (in the case of choosing crossover frequencies near the boundary frequencies of the loudspeaker operating range or with a large frequency response unevenness of the loudspeakers), the condition for matching the real PFC of the channels may not be observed. Therefore, to control the real phase response by 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 of switching on the loudspeakers can be considered the one that corresponds to the lesser unevenness of the total frequency response in the channel separation band. The exact matching of the PFC of the shared channels, while satisfying all other requirements (flat frequency response, etc.), is carried out by numerical methods for the synthesis of optimal separation filters-correctors on a computer.

Fig.9. principled circuit diagram Speakers with calculated crossover filters (capacitances in microfarads, inductances in millihenries, resistances in ohms).

In the development of passive crossover filters, their design plays an important role, as well as the choice of the type of specific elements - capacitors, inductors, resistors, in particular, the mutual placement of inductors has a great influence on the characteristics of speakers with filters; signal between closely spaced coils. For this reason, it is recommended to place them mutually perpendicular, only such an arrangement allows minimizing their influence on each other. Inductors are one of the critical components passive separation filters. At present, many foreign firms use inductors on cores made of magnetic materials, which provide a large dynamic range, low level of non-linear distortion and small dimensions of the coils. However, the design of coils with magnetic cores is associated with the use of special materials, therefore, 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 are negligible non-linear distortions.

The configuration of the air-core inductor shown in Figure 10 is optimal as it provides the maximum ratio L/R , i.e. coil with a given inductance L , wound with a wire of a selected diameter, has the lowest 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. An air-core inductor 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 , the mass of the wire (material - copper) in grams, q = c 3 /21, wire length in millimeters, B=187.3√Lc . In the event that the inductor is calculated based on a wire of a given diameter, the main design 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's find, for example, the parameters of the inductor of the previously calculated low-pass filter. The inductance of the coil is L 1LF = 5.1 mg. Resistance R DC coils are determined from the allowable attenuation of the signal introduced by a real coil at low frequencies. Let signal attenuation due to losses R in the coil is N≤1dB. Since the DC resistance of a subwoofer is R E\u003d 8 Ohm, then the permissible coil resistance, determined from the expression R≤R E (10 0.05 N -1), is R≤0.980 ohm; then the design parameter of the coil c \u003d √ 5100 / 0.98 8.66 \u003d 24.5 mm; number of turns N\u003d 19.8 √ (5100 / 24.5) \u003d 287 turns; wire diameter d\u003d 0.841 24.5 / √287 \u003d 1.2 mm; wire weight q \u003d 24.5 3 / 21.4≈697 g; wire length B \u003d 187.3 √ (85.1 24.5) ≈ 46 m.

Capacitors are another important element of passive crossover filters. Typically, paper or film capacitors are used in filters. From paper, the most used domestic capacitors are MBGO. The advantage of these types of capacitors is low losses, high temperature stability, the disadvantage is large dimensions, a decrease in the allowable maximum voltage at high frequencies. At present, in the filters of a number of foreign ASs, electrolytic non-polar capacitors with low internal losses are used, which combine the advantages of the considered capacitors and are free from their shortcomings.

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

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

If you find impedance minima around 3 ohms, don't be discouraged. Some models of speakers from well-known companies have a minimum of up to 2.6 ohms. One - two models even 2 Ohm! On the other hand, there is nothing good in such "dips" of impedance. Amplifiers overheat when driving this load if you listen to music loudly. Amplifier distortions grow in the region of minima of the acoustic system impedance.

For tube triode amplifiers, minima in the low and mid-low frequencies are especially dangerous. However, if the impedance drops below 3 ohms, the output lamps may fail. Output pentodes in such cases do not break.

It is important to remember that the output impedance of the amplifier is involved in setting the filter of the speaker system. For example, if you provide afterburner by 1 dB of the Fc region by setting up speakers with a transistor amplifier, which has almost zero output impedance, then when these acoustic systems are connected to a tube amplifier (typical output impedance ~ 2 Ohm), there will be no afterburner. AFC will be different. To repeat the characteristics achieved with a transistor amplifier, in the case of working with a tube device, you will have to create another filter.

The listener capable of self-development eventually comes to understand the value of good tube amplifiers. For this reason, I usually set up speakers with a tube amplifier, and when connected to a transistor amplifier, I put a 10-watt low-inductance (no more than 4-8 uH) 2 ohm resistor in series with the speakers.

If you have a transistor amplifier, but do not exclude the possibility of acquiring tube technology in the future, then connect your speakers to the amplifier output through the above resistors during setup and subsequent operation. Then, when switching to a tube amplifier, you will not need to reconfigure the speakers, just connect to it directly, without resistors.

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

In an extreme case, impedance measurements can be dispensed with by limiting the recoil boost at the filter cutoff frequency to 1 dB. Under this condition, the impedance is unlikely to drop 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 necessary for the desired frequency response correction yourself. A preliminary calculation of test filters is needed so that initially you do not miss "a kilometer".

Resistors can be added to the simple low-mid filter of the head for some frequency response manipulation that may be required when setting up your speakers.

If the average sound pressure level of this speaker is higher than the corresponding parameter of the tweeter, a resistor must be connected in series with the speaker. Switching options - in Fig. 6a and 6b.

The value of the required reduction in the output of the LF-MF head, expressed in dB, is denoted by the symbol N. Then:

Where Rd is the average value of the speaker impedance.

You can use the following information instead of calculations:

Table 1

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

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

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 \u003d 8 Ohm, N \u003d 4 dB.
R1 = 8 ohms (1.6 - 1) = 4.8 ohms.

How to calculate the power R1?

Let R d - nameplate power of the LF-MF loudspeaker, PR 1 - allowable power dissipated by R 1. Then:

It should not be difficult to remove heat from R 1, that is, it is not necessary to wrap it with electrical tape, fill it with hot glue, etc.

Features of filter precalculation with R1:

For the diagram 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 \u003d 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 diagram in Fig. 6a - the opposite is true: the introduction of R 1 into the scheme requires a decrease in L 1 and an increase in C 1 . It is easier to calculate the filter according to the scheme of Fig. 6b. Please use this schema.

Additional frequency response correction with a resistor:

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

The use of R 2 in this case leads to a decrease in returns in F s. Above F c, the return, on the contrary, increases in comparison with the filter without R 2 . If it is necessary to restore a frequency response close to the original (measured without R 2), you should reduce L 1 and increase C 1 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 sufficiently uniform characteristic, there is a zone of increased feedback ("presence") in the mid-range. You can apply a corrector in the form of a resonant circuit (Fig. 8).

at the resonance frequency

The circuit has some impedance value, according to the value of which the signal on the speaker is attenuated. Outside the resonant frequency, the attenuation decreases so the circuit can selectively suppress "presence". Approximately 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 is convenient to use table 1. I will draw it differently:

Example. It is necessary to suppress the "presence" with a center frequency of 1600 Hz. Speaker impedance - 8 ohms. Degree of suppression: 4 dB.

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

The case in Fig. 9a is the simplest. It is easy to choose the parameters of the corrective circuit, since the "presence" has a "mirror" shape to the possible filter characteristic.

On Fig. 9b shows another possible variant. It can be seen that the simplest circuit allows you to "exchange" one large "hump" for two small ones with a slight drop in frequency response to boot. In such cases, you must first increase L 2 and reduce C 2. This will expand the suppression bandwidth to the desired limits. Then you should shunt 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 applied to the speaker in the band determined by the circuit parameters. R 3 \u003d R d (Δ - 1)

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

How the correction takes place in this case is shown in Fig. 9th century

For modern loudspeakers, a combination of two problems is quite characteristic: "presence" in the region of 1000-2000 Hz and some excess of the upper middle. A possible type of frequency response is shown in Fig. 11a.

The method of correction 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 circuit L 2 , C 2 is needed, as usual, to suppress the "presence". Below Fp, the signal passes almost without loss 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 circuit L 2 , C 2 , then initially you should choose L 2 more and C 2 less. This will provide excessive suppression on F p , which is normalized after the introduction of R 4 . R 3 = R d (Δ - 1), where "Δ" is the amount of signal suppression above F p . "Δ" is selected in accordance with the excess of the upper middle, referring to Table 1. The stages of correction are conventionally illustrated in Fig. 11b.

In rare cases, feedback on the slope of the frequency response is required using a corrective circuit. It is clear that for this R 4 must move to the chain L 2 . The scheme in Fig. thirteen.

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 corrector may not have a special suppression on Fp. An example of when such a correction is needed is shown in Fig. 15.

If necessary, you can use a second-order filter and a corrective contour together. Switching options - in Fig. sixteen.

With the same values ​​of the elements, option a) provides a greater return at medium 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 are long to talk about, I advise you to use option a) more often. Sometimes a very pronounced "presence" requires the use of option b). The joint operation of the filter and the corrector is illustrated in Fig. 17.

Consider filters for tweeters.

For tweeters, much more often than for woofers, we apply a first-order filter, that is, just a capacitor connected in series with the loudspeaker. The fact that such a simple filter introduces a noticeable slope in the frequency response of the speaker does not have such a detrimental effect on the sound as in the case of a woofer. Firstly, this slope is often partially compensated by a smooth complementary (mutually complementary) slope of the frequency response of the woofer in the same frequency region. Secondly, some "failure" in the area of ​​the lower top (3-6 kHz) is quite acceptable according to the results of subjective examinations. The possible course of the frequency response of a tweeter without a filter, with a filter and together with a woofer is shown in Fig. eighteen.

You should not be afraid to experiment with connecting a tweeter in antiphase with a woofer. Sometimes this is one of the few ways to get good sound. The most likely results of a RF head reversal are shown in Fig. nineteen

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

Auguste Rodin

Any filter, in essence, does the same with the signal spectrum that Rodin does with marble. But unlike the work of a sculptor, 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 audio signals for subsequent playback by their dynamic heads (often we say “speakers”, but today the material is serious, so we will also approach the terms with all severity). But this area of ​​​​use of filters is probably still not the main one, and it is definitely not the first in historical terms. Let's not forget that electronics was once called radio electronics, and its initial task was to serve the needs of radio transmission and radio reception. And even in those childhood years of radio, when continuous spectrum signals were not transmitted, and broadcasting was also called radio telegraphy, 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 the reliability of the transmission. In the end, the cornerstone of all radio engineering of those times, the resonant circuit is nothing more than a special case of a bandpass filter. Therefore, we can say that all radio engineering 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 normalized characteristics. But they all had a common drawback - their characteristics depended on the impedance of the circuit that stands behind them, that is, the load circuit. In the simplest cases, the load impedance could be kept high enough so 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, the calculations were often carried out even without a slide rule, just in a column). It was possible to get rid of the influence of the load impedance, this curse of passive filters, with the advent of active filters.

Initially, it was supposed to devote this material entirely to passive filters; in the practice of installers, they have to be calculated and manufactured on their own incomparably more often than active ones. But logic required that we still start with active ones. Oddly enough, because they are simpler, no matter what it seems at first glance at the illustrations given.

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 appear. 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 the way we would like. And here, indeed, the thought of manual work can come.

Schematic diagrams of active filters

In its simplest form, an active filter is a passive filter loaded onto a unity gain element with high input impedance, either an emitter follower or an op amp operating in follower mode, that is, with unity gain. (You can also implement a cathode follower on a lamp, but I, with your permission, will not touch the lamps, if anyone is interested, refer to the relevant literature). In theory, it is not forbidden to build an active filter of any order in this way. Since the currents in the input circuits of the follower are very small, it would seem that the filter elements can be chosen 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 be the value of the coil? Answer: 159 mH. What a compactness. And most importantly, the ohmic resistance of such a coil can be quite comparable with 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 an n-p-n type transistor, and you yourself, on occasion, choose what it 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 to be somewhat “better”. The simulation was carried out for the KT315G transistor, probably the only semiconductor device, the price of which until recently was exactly the same as a quarter of a century ago - 40 kopecks. In fact, you can use any npn transistor, whose gain (h21e) is not much lower than 100.

Rice. 1. First order high pass filters

The resistor in the emitter circuit (R1 in Fig. 1) sets the collector current, for most transistors it is recommended to choose 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 the resistors R2 and R3 connected in parallel. In our case, this resistance is 105 kOhm. It is only necessary to ensure that it is significantly less than the resistance in the emitter circuit (R1), multiplied by the h21e index - in our case it is about 1200 kOhm (in fact, with a spread of h21e values ​​​​from 50 to 250 - from 600 kOhm to 4 MΩ) . The output capacitor is added, as they say, “for order” - if the input stage of the amplifier is the load of the filter, as a rule, there is already a capacitor there to decouple the input with a constant voltage.

In the op-amp filter circuit here (as in the following), the TL082C model is used, since this operational amplifier is very often used to build filters. However, you can take almost any op-amp from those that normally work with unipolar power supply, 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 the resistors R3, R4 connected in parallel. (Why connected in parallel? Because from the point of view of alternating current, plus power and minus are the same.) The ratio of resistors R3, R4 determines the midpoint, if they differ slightly, this is not a tragedy, it only means that the signal is maximum amplitudes 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, it is necessary to increase either the value of the resistors R3, R4, or the capacitance C2. That is, the nominal value changes inversely with the first power of the frequency.

In the low-pass filter circuits (Fig. 2), there are a couple of parts more, since the input voltage divider is not used as an element of a frequency-dependent circuit and a separating capacitance is added. To lower the cutoff frequency of the filter, 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 microfarad film capacitor). It should be taken into account that the separating capacitance together with C2 form 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 decoupling capacitor - if, for example, the source is the output of another filter stage. In general, the desire to get rid of bulky isolation capacitors was probably the main reason for the transition from unipolar to bipolar power supply.

On fig. Figures 3 and 4 show the frequency responses of the high and low pass filters we have just reviewed.

Rice. 3. Characteristics of first-order high-pass 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 closely engaged in the study of first-order filters, when they are not suitable for subwoofers at all, and if you believe the author’s statements, they are not often applicable to separate the bands of frontal acoustics? And second: why did the author not mention either Butterworth or his namesakes - Linkwitz, Bessel, Chebyshev, after all? I will not answer the first question yet, a little later everything will become clear to you. I'm going straight to the second one. Butterworth et al. have determined the characteristics of filters from the second order and above, and the frequency and phase response of the first order filters is always the same.

So, second-order filters, with a nominal 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 into account a lot of things, and as a result, the simplicity turns out to be purely imaginary. A certain number of options for the circuit implementation of such filters are known. I will not even say which one, since any enumeration can always be incomplete. Yes, and it will give us little, 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 can even say that 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 capacitors and resistors, in this case - C1, C2, R3, R4, R5. Please note that for the Butterworth filter (well, finally!) The value of the resistor in the feedback circuit (R5) must be half the value of the resistor connected to the ground. As usual, resistors R3 and R4 are connected to the "ground" in parallel, and their total value is 50 kOhm.

Now a few words, as it were, to the side. If your filter is not tunable, there will be no problems with the selection of resistors. But if you need to smoothly change the cutoff frequency of the filter, you need to change two resistors at the same time (we have three of them, but the power supply in the amplifiers is bipolar, and there is one resistor R3, the value is the same as our two R3, R4 connected in parallel). Especially for such purposes, dual variable resistors of different denominations are produced, but they are also more expensive, and there are not so many of them. In addition, it is possible to design a filter with very similar characteristics, but in which both resistors will be the same, but the capacitances C1 and C2 will be different. But it's troublesome. And now let's see what happens if we take a filter designed for an average frequency (330 Hz) and start changing only one resistor - the one that is in the "ground". (Fig. 6).

Rice. 6. Rebuilding the high-pass filter

Agree, we have repeatedly seen something similar on the 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 value of the resistors in the local ground circuit (R3, R4) affects the amount of attenuation introduced by the filter. With 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 value of the resistors R5, R6 is the same, almost any dual tuning resistor is suitable for smooth adjustment of the cutoff frequency - this is why in many amplifiers the low-pass filter characteristics are more stable than the high-frequency filter characteristics.

On fig. 8 shows the frequency response of the second order filters.

Rice. 8. Characteristics of second-order filters

Now we can return to the question that remained unanswered. We “passed through” the first-order filter scheme because active filters are created mainly by cascading basic links. So that serial connection filters of the first and second order will give the third order, a chain of two filters of the second order will give the fourth and so on. Therefore, I will give only two options for 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 the question: why did the values ​​​​of the resistors R3, R4, R5 suddenly change? Why wouldn't they change? If in each "half" of the circuit the frequency 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. And we didn't agree on that. So the best thing to do is to give a method for choosing denominations - so far only for Butterworth filters.

1. Based on the known filter cutoff frequency, set one of the characteristic ratings (R or C) and calculate the second rating using the dependence:

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

Since the range of capacitor values ​​is usually narrower, it is most reasonable to set the base capacitance value C (in farads), and from it 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 bothers you to take both those and these - but in different parts of the filter, if it is composite.

2. For a first-order filter, formula (1.1) immediately gives the value of the resistor. (In our particular case, we get 72.4 kΩ, rounded up to the nearest standard value, we get 75 kΩ.) For a basic second-order filter, you determine the starting value of R in the same way, but in order to get the actual values ​​\u200b\u200bof the resistors, 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 the "ground" will be equal to

The one and two in parentheses denote 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 go into the theory of filters.

The calculation of the ratings 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 ​​according to the table. The capacitor in the feedback circuit is defined as

and the capacitor connecting the input of the op-amp to the "ground", as

Using the newly acquired knowledge, we draw a fourth-order low-pass filter, which can already be applied to work with a subwoofer (Fig. 10). In the diagram, this time I give the calculated values ​​\u200b\u200bof the capacities, without rounding to the standard denomination. This is so that you can check yourself if you wish.

Rice. 10. 4th order low pass filter

I have not yet said a word about the phase characteristics, and I did the right thing - this is a separate issue, we will deal with it separately. Next time, you get the idea, we're just getting started...

Rice. 11. Characteristics of filters of the third and fourth order

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

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

When the signal is a pure sine (although the degree of purity is different) of a fixed frequency, then it is quite natural to represent it as a rotating vector, determined, as you know, by amplitude (modulus) and phase (argument). For sound signal, in which sines are present only in the form of decomposition, the concept of phase is no longer so clear. However, it is no less useful - if only because sound waves from different sources add up vectorially. And now let's see how the phase-frequency characteristics (PFC) of filters look up to the fourth order inclusive. The numbering of figures will be kept through, from the last issue.

Let's start with Fig. 12 and 13.

At once it is possible to notice curious regularities.

1. Any filter "twists" the phase by an angle that is a multiple of?/4, more precisely, by (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 PFC of the high pass filter always comes in 360 degrees.

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

From the joint consideration of the above three points, it is easy to conclude that the PFCs of the high-pass and low-pass filters are the same only for the fourth, eighth, etc. orders, and the validity of this statement for filters of the fourth order 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”, as, by the way, the opposite does not follow either. In general, it is too early to draw conclusions.

The phase characteristics of filters do not depend on the implementation method - they are active or passive, and even on the physical nature of the filter. Therefore, we will not specifically focus on the phase response 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. The non-minimum-phase links include, for example, the delay line.

It is quite obvious (in the presence of graphs) that the higher the order of the filter, the steeper its PFC falls. And the steepness of any function is characterized by what? Its derivative. The derivative of the PFC with respect to frequency has a special name - the group delay time (GDT). The phase must be taken in radians, and the frequency - not oscillatory (in hertz), but 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 the same type of high-pass and low-pass filters are no different. This is how group delay graphs look for Butterworth filters from the first order to the fourth (Fig. 14).

Here the difference between filters of different orders seems to be especially noticeable. The maximum (in amplitude) group delay value for the fourth order filter is approximately four times greater than that of the first order filter and twice as large as that of the second order filter. There are statements that in this parameter the fourth-order filter is just four times worse than the first-order filter. For the high-frequency 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 presentation, it will be useful for us to imagine how the PFC looks like “in the air” of an electrodynamic head, 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 increasing steepness. I will not let in too much mystery: this is how the phase response of the delay line looks like. Experienced people will say: of course, the delay is due to the path of the sound wave from the emitter to the microphone. And experienced people will be mistaken: my microphone was installed on 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 it (delay) should not be? 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 steepness. (For example, I don’t need to imagine anything, I have this recorded on one of the measuring CDs, we check the polarity using this signal.) It is clear that the current through the voice coil will not flow immediately, it still has some kind of inductance. But these are trifles. The main thing is that sound pressure is volumetric velocity, that is, the diffuser must first accelerate, and only then will sound appear. For the magnitude of the delay, it is probably possible to derive a formula, for sure, the mass of the “movement”, the force factor and, possibly, the ohmic resistance of the coil will appear there. By the way, I got 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 midrange speakers have less delay than bassists, and tweeters have less delay than those and these. Surprisingly, I have not seen references to such results in the literature.

Why did I bring this instructive graph? And then, that if things really are exactly as I see them, then many arguments about the properties of filters lose their practical meaning. Although I will still state them, and you yourself will decide whether all of them are worth adopting.

Passive Filter Circuits

I think few people will be surprised if I declare that there are much fewer circuit implementations of passive filters than active filters. I would say that there are about two and a half. That is, if elliptic filters are displayed in a separate class of circuits, it will turn out three, if this is not done, then two. Moreover, in 90% of cases, so-called parallel filters are used in acoustics. 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 that you will not be able to connect them to different amplifiers. In addition, according to their characteristics, these are first-order filters. And by the way, the still 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 all-round reduction in the cost of acoustic products (on the other) say. However, series filters have one plus: the sum of their output voltages is always equal to one. Here is what the circuit of a two-band series filter looks like (Fig. 16).

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

You can believe that the PFC 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 scheme in Fig. eighteen.

The ratings shown in the diagram correspond to the same frequency of the section (2000 Hz) between the tweeter (HF) and the midrange and the frequency of 100 Hz - the section between the midrange and woofers. 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 the set of characteristics of this filter, you can see that the slope of the tweeter filter rolloff in the range of 50 - 200 Hz is higher than 6 dB / oct., since its band here is superimposed not only on the midrange band , but also on the bass head band. That's what parallel filters can't do - their overlap of bands inevitably brings surprises, and always - joyless ones.

The series filter parameters are calculated in exactly the same way as the ratings 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; it is expressed in terms of the filter cutoff frequency 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 tech-savvy of you have already noticed that I "jumped" the cards a bit and replaced the actual load impedance (i.e. speaker) with an ohmic "equivalent" of 4 ohms. In reality, of course, there is no equivalent. In fact, even a forcibly retarded voice coil from the point of view of an impedance meter looks like active and inductive resistance connected in series. And when the coil has mobility, the inductance increases at a high frequency, and near the resonance frequency of the head, its ohmic resistance seems to increase, it happens ten times or more. There are very few programs that can take into account such features of a real head, I personally know three. But we in no way set ourselves the goal of learning how to work, say, in the Linearx software environment. Our task is different - to deal with the main features of the filters. Therefore, we will imitate the presence of a head in the old fashioned way 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 are you, sequential filters - this is Klondike, let's dig somehow." I agree. Klondike. I had to promise that in one of the upcoming issues we will separately and specially dig.)

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 take a look at the generalized filter circuit (Fig. 20).

To get a fourth-order low-pass filter, it is necessary to replace all horizontal "bars" in this circuit with inductances, and all vertical ones with capacitances. 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 with you: there are no filters above the fourth order for us. As we will see later, along with the increase in the steepness of the filter, their disadvantages deepen, so this arrangement is not something seditious. For the sake of completeness, one more thing needs to be said. There is an alternative option for constructing passive filters, where the first element is always a resistor, not a reactive element. Such circuits are used when it is required to normalize the input impedance of the filter (for example, operational amplifiers "do not like" a load of less than 50 ohms). But in our case, an extra resistor is an unjustified power loss, so "our" filters start with reactivity. Unless, of course, it is required to specifically reduce the signal level.

The most complex band-pass filter is obtained if in the generalized circuit each horizontal element is replaced by a series connection of capacitance and inductance (in any sequence), and each vertical element must be replaced by parallel connected ones - also by capacitance and inductance. Probably, I will nevertheless give such a “terrible” scheme (Fig. 21).

There is one more little trick. If you need an unbalanced "bandpass" (bandpass filter), in which, say, the high-pass filter has a fourth order, and the low-pass filter has a second order, then the extra details from the above circuit (that is, one capacitor and one coil) must be removed without fail with " tail" of the scheme, and not vice versa. Otherwise, you will get somewhat unexpected effects from changing the nature of the loading of the previous filter stages.

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

Prepared based on the materials of 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, we count those cases when the head (tweeter or midrange) needs to be “sedimented” by exactly 6 dB. Why six? Because in such filters (they are also called bi-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 attenuate by 6 dB. In addition, two-loaded filters are of P-type and T-type. To imagine a P-type filter, it is enough to discard the first element (Z1) in the generalized filter circuit (Fig. 20, No. 5/2009). The first element of such a filter is connected to the 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 to connect the source, that is, the amplifier, to a capacitor of several hundred microfarads, and then write to me - did the protection work for it or not. Just in case, write on demand, it’s better not to litter those who give such advice with addresses.) Therefore, we P-filters also not considered. In total, as it is easy to imagine, we are dealing with one fourth of the circuit implementations of passive filters.

Elliptic filters stand apart, at least 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 (as in Butterworth, say), but in an ellipse. In order not to operate with concepts, which probably do not make 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 schemes

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

Those that I have indicated with odd numbers are called standard, the other two are called dual. Why so, and not otherwise? Maybe because in standard circuits an additional element is a capacitance, and dual circuits differ from a conventional filter in the presence of an additional inductance. By the way, not every scheme obtained in this way is an elliptical filter, if everything is done according to science, it is necessary to strictly observe the relationships between the elements.

The Cauer filter has a fair amount of drawbacks. As always, secondly, let's think positively about them. After all, Cauer has a plus, which in other cases is able to outweigh everything. Such a filter provides deep suppression of the signal 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 required to provide filtering near the resonance frequency of the head, then only Cauer filters can cope with such a task. It is quite troublesome to count them manually, however, in simulator programs, as a rule, there are special sections dedicated to passive filters. True, it is not a fact that there are single-loaded filters there. However, in my opinion, there will be no great harm if you take the Chebyshev or Butterworth filter circuit, and calculate the additional element from the resonance frequency using the well-known formula:

Fr \u003d 1 / (2? (LC) ^ 1/2), whence

C = 1/(4 ? ^2 Fр ^2 L) (3.1)

Mandatory condition: the resonant frequency must be outside the filter transparency band, that is, for the high-pass filter - below the cutoff frequency, for the 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 the greatest interest - it happens that it is desirable to limit the band of a midrange or tweeter as low as possible, excluding, however, its operation near the resonance frequency of the head. For unification, I give the high-pass filter circuit for our favorite frequency of 100 Hz (Fig. 2).

The element ratings look a little wild (especially the capacitance of 2196 uF - the resonance frequency is 48 Hz), but as soon as you move to higher frequencies, the ratings will change inversely with the square of the frequency, that is, quickly.

Filter types, pros and cons

As already mentioned, the filter characteristics are determined by a certain polynomial (polynomial) of the corresponding order. Since a certain number of special categories of polynomials are described in mathematics, there can be exactly the same number of filter types. In fact, even more, because in acoustics it was also customary to give certain categories of filters special names. Since there are polynomials of Butterworth, Legendre, Gauss, Chebyshev (advice: write and pronounce the name of Pafnuty Lvovich through "e", as it should be - this is the most easy way to show the solidity of one's own education), Bessel, and so on, then there are filters that bear 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 (alive and peppy) 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 interface point, the output voltage drop of each filter is 3 dB, then the total characteristic will be straightforward in terms of power (voltage square), and a 3 dB hump will appear in voltage at the junction point. Linkwitz suggested that the filters be matched at -6 dB. In particular, second-order Linkwitz filters are the same Butterworth filters, only for the high-pass filter they have a cutoff frequency 1.414 times higher than for the low-pass filter. (The crossover frequency is exactly in between, i.e. 1.189 times higher than a Butterworth low-pass filter of the same rating.) So when I come across an amplifier in which tunable filters are specified as Linkwitz filters, I understand that the authors of the design and compilers of the specification did not were familiar with each other. However, let us return to the events of 25-30 years ago. Richard Small also took part in the general triumph of filter engineering, who proposed combining Linkwitz filters (for convenience, not otherwise) with series filters, which also provide a flat voltage characteristic, and call everything together constant voltage design. This is despite the fact that neither then, nor, it seems, and now, it was not really established whether a flat voltage or power characteristic is preferable. One of the authors even calculated intermediate polynomial coefficients, so that the filters corresponding to these “compromise” polynomials should have given a 1.5 dB voltage hump at the junction point and a power dip of the same magnitude. One of additional requirements to the designs of the filters was that the phase-frequency characteristics of the LF and HF filters must either be identical or diverge by 180 degrees - which means that when the polarity of switching on one of the links is reversed, an identical phase characteristic will again be obtained. As a result, among other things, it is possible to minimize the overlap area of ​​the bands.

It is possible that all these mind games turned out to be very useful in the development of multiband compressors, expanders and other processor systems. But in acoustics it is difficult to apply them, to put it mildly. Firstly, it is not the voltages that add up, but the sound pressures, which are related to the voltage through a tricky phase-frequency characteristic (Fig. 15, No. 5/2009), so that not only their phases can arbitrarily differ, but the steepness of the phase dependence will most likely be different (unless it occurred to you to breed heads of the same type in 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 it is necessary to put at the forefront not the conjugation of filters by bands, but their own characteristics of the filters.

What characteristics (acoustically speaking) determine the quality of filters? Some filters provide a smooth frequency response in the transparency band, while for others, the decline begins long before the cutoff frequency is reached, but even after it the slope of the decline slowly reaches the desired value, for others, a hump ("tooth") is observed on the approach to the cutoff frequency, after which a sharp decline begins with a steepness even slightly higher than the "nominal". From these positions, the quality of filters is characterized by "smoothness of frequency response" and "selectivity". The phase difference for a filter of this order is fixed (this was discussed in the last issue), but the phase change can be either gradual or fast, accompanied by a significant increase in the group delay. This filter property is characterized by phase smoothness. Well, the quality of the transient process, that is, the reaction to the step action (Step Response). The low-pass filter works out 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 (no delay) with a return to zero DC, but the roll-over and subsequent oscillation is similar to what you would see with a low-pass filter of the same type.

In my opinion (my opinion may not be disputable, those who wish to argue can enter into correspondence, even not on demand), three types of filters are quite enough for acoustic purposes: Butterworth, Bessel and Chebyshev, especially since the last type actually combines a whole group of filters with different magnitudes of "teeth". In terms of the smoothness of the frequency response in the transparency band, Butterworth filters are beyond competition - 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 characteristic of the Bessel filter is also the smoothest, while that of the Chebyshev filter is the most “angular”. For generality, I also give the characteristics of the Cauer filter, the circuit of which was shown a little higher (Fig. 5).

Notice how at the resonance point (48 Hz as promised) the phase jumps by 180 degrees. Of course, at this frequency, signal suppression should be the highest. But in any case, the concepts of "phase smoothness" and "Cauer filter" do not combine in any way.

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

The Bessel filter, like all others, has a third order, but it has practically no outlier. Chebyshev and Cauer have the largest emissions, and the latter has a longer oscillatory process. The magnitude of the outlier increases with increasing filter order and, accordingly, decreases as it decreases. To illustrate, I give the transient responses of the Butterworth and Chebyshev second-order filters (there are no problems with Bessel) (Fig. 7).

In addition, I came across a plate of the dependence of the magnitude of the transfer on the order of the Butterworth filter, which I also decided to bring (tab. 1).

This is one of the reasons why it is hardly worth getting carried away with Butterworth filters of order higher than the fourth and Chebyshev - higher than 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 percentage accuracy of detail matching can be defined as 5/n, where n is the order of the filter. That is, when working with a fourth-order filter, you should be prepared for the fact that the denomination of 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 details. Electrolytes, of course, should be avoided due to their instability, although if the capacitance count is hundreds of microfarads, there is no other choice. Capacitances, of course, will have to be selected and recruited from several capacitors. If you wish, you can find electrolytes with low leakage, low lead resistance and a real capacity spread of no worse than +20/-0%. Coils, of course, are better "heartless", if there is no way without a core, I prefer ferrites.

For the selection of denominations, I suggest using the following table. All filters are rated for 100Hz (-3dB) cutoff and 4 ohm load ratings. To get the denominations 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 driver, and Fc, as always, is the calculated cutoff frequency. Attention: inductance ratings are given in millihenries (not henries), capacitance ratings are in microfarads (not farads). There is less scientific imagery, more convenience (tab. 2).

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

In the last chapter of the series, we first got acquainted with passive filter circuits. True, not really.

Chebyshev frequency response of the third order

Butterworth frequency response of the third order

Bessel frequency response of the third order

PFC of Bessel of the third order

PFC Butterworth of the third order

Chebyshev PFC of the third order

Frequency response of the Cauer filter of the third order

PFC of the Cauer filter of the third order

Bessel step response

Cauer step response

Chebyshev transient response

Butterworth transient response

Prepared based on the materials of the magazine "Avtozvuk", July 2009www.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 it may seem surprising, but the attenuator is an indispensable attribute of multiband acoustics, because heads for different bands not only do not always have, but should not have the same sensitivity. Otherwise, the freedom of maneuver for frequency correction will be reduced to zero. The fact is that in the passive correction system, in order to correct the dip, it is necessary to "siege" the head in the main band and "release" where the dip was. In addition, in residential areas it is often desirable for the tweeter to “replay” the midbass or midrange and bass a little in volume. At the same time, "upsetting" the bass speaker is expensive in any sense - a whole group of powerful resistors is required, and a fair part of the amplifier's energy is spent on heating up the mentioned group. In practice, it is considered optimal when the return of the midrange is several (2 - 5) decibels higher than that of the bass, and the tweeter is as much 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 decibels, so we will only partially use them today. Therefore, for your convenience, I give a table for converting the attenuation index (dB) into the transmittance of the device.

So, if you need to "reset" the head by 4 dB, the N attenuator transmittance 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 series attenuator value RS is given by:

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

As ZL, you can take the "nominal" 4 ohms. If we, in good faith, put a series attenuator right in front of the head (the Chinese tend to do this), then the load impedance for the filter will increase, and the frequency of the cutoff of the low-pass will increase, and the high-pass filter will decrease. But that's not all.

Take for example a 3 dB attenuator operating at 4 ohms. The resistor value according to the formula (4.1) will be equal to 1.66 ohms. On 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 serial attenuator connected after the corresponding filter. The green curve corresponds to the inclusion of an attenuator before the filter. The only side effect is a frequency shift of 10 - 15% minus and plus for the high-pass filter and low-pass filter, respectively. So in most cases the serial attenuator should be placed before the filter.

To avoid the drift of the cutoff frequency when the attenuator is turned on, devices were invented that we call L-shaped attenuators, and in the rest of the world, where the alphabet does not contain the magic letter “G” 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, take the same 3 dB attenuation. The resistor values ​​turned out to be as shown in the diagram (ZL is again 4 ohms).

Rice. 3. Scheme of the L-shaped attenuator

Here the attenuator is shown along with a 4 kHz high-pass filter. (For consistency, 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 one is with an attenuator before the filter, and the green one is with an attenuator 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 no need to be smart - 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, the permutation can be used - without changing the denominations, correct the area of ​​separation of the bands. But this is already aerobatics ... And now let's move on to "different".

Other common schemes

Most commonly found in our crossovers is a head impedance correction circuit, commonly referred to as a Zobel circuit, after the renowned researcher on filter characteristics. It is a series RC circuit connected in parallel with the load. According to the 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 parameter ZL, without further ado, choose the nominal impedance of the head, in our case, 4 ohms. I would advise the value of R to look for the following formula:

R = k * Re (4.4a).

Here the coefficient k \u003d 1.2 - 1.3, anyway, resistors cannot be selected more accurately.

On fig. 5 you can see four frequency responses. Blue is the usual characteristic of a Butterworth filter loaded with a 4 ohm resistor. Red curve - such a 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 like if the developer does not simplify his life, and the filter parameters are determined by formulas 4.4 - 4.6, based on the total impedance of the coil - with the specified coil parameters, the total impedance will be 7.10 Ohm (4 kHz). Finally, the green curve is the frequency response obtained using the Zobel chain, the elements of which are determined by formulas (4.4a) and (4.5). The divergence of the green and blue curves does not exceed 0.6 dB in the frequency range of 0.4 - 0.5 of the cutoff frequency (in our example it is 4 kHz). On 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 a minimum probability of error that the 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 (Strange Filters), which are distinguished by the presence of an additional resistor in the ground circuit of the filter. The already well-known low-pass filter at 4 kHz 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 value of the resistor becomes significant (that is, comparable to the value of the load), you get a "strange" filter. We will change the resistor R1 in the range from 0.01 to 4.01 ohms in 1 ohm increments. The resulting family of frequency responses can be seen in Fig. eight.

The upper curve (near the inflection point) is the usual Butterworth characteristic. As the value of the resistor increases, the cutoff frequency of the filter shifts down (up to 3 kHz at R1 = 4 ohms). But the slope of the roll-off does not change much, at least within the -15 dB band - and it is this region that is of practical importance. Below this level, the slope of the roll-off will tend to be 6 dB/oct, but this is not so important. (Note that the vertical scale of the graph has been changed, so the rolloff seems steeper.) Now let's see how the phase response changes depending on the value of the resistor (Fig. 9).

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

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

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

Rice. 2. Same for LPF

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

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

Rice. 6. Diagram of a filter with a Zobel circuit

Rice. 7. Scheme of the "strange" filter

Rice. 8. The amplitude-frequency characteristics of the "strange" filter

Rice. 9. Phase characteristics of the "strange" filter

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

As promised, today we will finally come to grips with frequency correction circuits.

In my writings, I have repeatedly argued that passive filters can do a lot of things that active filters cannot do. He asserted indiscriminately, without proving his innocence and without explaining anything. But really, what can't active filters do? Their main task - "to cut off the excess" - they solve quite successfully. And although, precisely because of their versatility, active filters, as a rule, have Butterworth characteristics (if they are done correctly at all), Butterworth filters, as I hope you have already understood, in most cases represent the best compromise between the shape of the amplitude and phase characteristics. , as well as the quality of the transition process. And the possibility of smooth frequency tuning generally compensates for too much. In terms of level matching, active systems, of course, outperform any attenuators. And there is only one article on 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 with a margin, but they add noise to the path. In addition, these toys are expensive and rare in our industry. Digital parametric EQs are ideal if they have a 1/12 octave center frequency step, which we don't seem to have either. Parametrics with 1/6 octave steps are partially suitable and provided that they have a sufficiently wide range of available Q values. So it turns out that only passive corrective devices correspond to the tasks set to the greatest extent. By the way, high quality studio monitors often do this: bi-amping/tri-amping with active filtering and passive equalizers.

Treble correction

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

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

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

The value of the capacitor I choose according to 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 include more than one "loop" in series to avoid "saturation" in the frequency response (Fig. 1).

For example, I took the same second-order Butterworth RF filter, for which in the last chapter we determined the resistor value Rs = 1.65 ohms for 3 dB attenuation (Fig. 2).

Such a double circuit allows you to raise the "tail" of the frequency response (20 kHz) by 2 dB.

It is probably useful to recall that multiplying the number of elements also multiplies errors due to the uncertainty of the load impedance characteristic and the variation in element ratings. So I would not advise you to get involved with three or more stepped circuits.

Peak suppressor on frequency response

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

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

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

Here N0 is the gain of the circuit at the center frequency of the peak. Let's say if the peak height is 4 dB, then the gain is 0.631 (see the table from the last chapter). We 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 on the frequency response of the speaker, which we need to suppress, falls. If Y0 is known to us, then the values ​​​​of capacitance and inductance will be determined by the known formulas:

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

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

Now we need to specify two more frequencies FL and FH - below and above the center frequency, where the gain has a value of N. N > N0, say, if N0 was set as 0.631, the parameter N 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 corrective circuit in theory has a symmetrical shape of the frequency response, 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 parameter Y0.

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

The formula looks scary, but I warned you. Let you be encouraged by the information that we will no longer meet more cumbersome expressions. The factor in front of the radical is the relative bandwidth of the corrective device, that is, a value inversely proportional to the quality factor. The higher the quality factor, the (at the same center frequency F0) the inductance will be less, and the capacitance will be greater. Therefore, with a high quality factor of the peaks, a double “ambush” arises: with an increase in the center frequency, the inductance becomes too small, and it can be difficult to manufacture it with a proper 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 such parameters. F0 = 1000 Hz, FH = 1100 Hz, FL = 910 Hz, N0 = 0.631, N = 0.794. Here's what happens (Fig. 3).

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

The characteristics of such a corrector do not depend much on whether it is placed before or after the high-pass filter or the low-pass filter. In the next two graphs (Fig. 5 and 6), the red curve corresponds to the inclusion of the corrector before the corresponding filter, the blue curve corresponds to its inclusion after the filter.

Compensation scheme for the dip in the frequency response

What was said about the high-frequency corrective circuit also applies to the dip compensation circuit: in order to raise the frequency response in some area, you must first lower it on all others. The circuit consists of the same three elements Rs, L and C, with the only difference that the reactive elements are connected in series. At the resonant frequency, they shunt a resistor, which acts as a series attenuator outside the resonant zone.

The approach to determining the parameters of the elements is exactly the same as in the case of the peak suppressor. We need to know the center frequency F0, as well as the transmittances N0 and N. In this case, N0 has the meaning of the circuit transmittance outside the correction region (N0, like N, is less than one). N is the transmittance at the frequency response points corresponding to the frequencies FH and FL. The frequency values ​​FH, FL must meet the same condition, that is, if you see an asymmetric dip on the real frequency response of the head, you must choose compromise values ​​​​for these frequencies so that condition (5.5) is approximately observed. 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 decibel value 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 value of the resistor is determined by the same formula (5.2). The values ​​of capacitance C and inductance L will be related to the reactive impedance Y0 at the resonance frequency F0 by the same dependences (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) by the reciprocal of the factor in front of the expression for the root. That is, with an increase in the quality factor of the characteristic of the corrective circuit, Y0 increases, which means that the value of the required inductance L increases and the value of capacitance C decreases. In this regard, there is only one problem: at a sufficiently low center frequency F0, the required inductance value forces the use of coils with cores, and there there are problems of their own, to dwell on which, probably, it makes no sense.

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 denominations are obtained as shown in the diagram (Fig. 7).

Note that the coil inductance here is almost twenty times greater than for the peak suppressor circuit, and the capacitance is the same amount less. AFC of the circuit calculated by us (Fig. 8).

In the presence of a load inductance (0.25 mH), the efficiency of the series attenuator (resistor Rs) 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 has moved from 4 to 5 kHz, and the cutoff frequency of the low-pass filter has decreased from 250 to 185 Hz.

On this, the series dedicated to passive filters will be considered finished. 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 given within the series will be enough to solve most practical problems. For those who would like more information, the following resources may be helpful. 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 about 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 decide to engage in engineering activities. They say they are now...

Rice. 1. Diagram of a double RF circuit

Rice. 2. Frequency response of a double corrective circuit

Rice. 3. Scheme of the peak suppressor

Rice. 4. Frequency characteristics of the peak suppression circuit

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

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

Rice. 7. Dip Compensation Circuit

Rice. 8. Frequency characteristics of the dip compensation circuit

Rice. 9. Frequency characteristics of the circuit in conjunction with the high-pass filter

Rice. 10. Frequency response of the circuit in conjunction with a low-pass filter

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

In this article, we will talk about the high and low pass filter, as well as their varieties.

High and low pass filters- these are electrical circuits consisting of elements with a non-linear frequency response - having different resistance at different frequencies.

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

Let's repeat one more time:

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

Frequency filters are characterized by indicators

Cutoff frequency is the frequency at which the filter output falls off to 0.7 of the input.

Filter Slope is a characteristic of the filter, showing how sharply the amplitude of the output signal of the filter decreases with a change in the frequency of the input signal. Ideally, one should strive for the maximum (vertical) drop in the frequency response.

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

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 the simplest automatic calculation plate (you must be able to work with formulas in Excel). I use this method to calculate any circuits. First, I make a plate, substitute the data, get a calculation, which I transfer to paper in the form of a frequency response graph, change the parameters, and again draw frequency response points. In this way, there is no need to deploy a “laboratory of measuring instruments”, the calculation and drawing of the frequency response is done quickly.

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

To ensure the accuracy of the filter, it is necessary that the value of the resistance of the filter elements be approximately two orders of magnitude less (100 times) than the resistance of the load connected to the filter output. As this difference decreases, the quality of the filter deteriorates. This is due to the fact that the load resistance affects the quality of the frequency filter. If you do not need high accuracy, then this difference can be reduced 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 - according to appearance resemble the letter G, facing the other way;

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

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

5. Multilink - the same L-shaped filters connected in series.

Single element high and low pass filters

As a rule, single-element high and low pass filters are used directly in acoustic systems powerful audio frequency amplifiers, to improve the sound of the sound "columns" themselves.

They are connected in series with dynamic heads. Firstly, they protect both dynamic heads from powerful electrical signal, and the amplifier from low load resistance without loading it with extra speakers, at the 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 coil reactance of the dynamic head. The calculation is made according to the voltage divider formulas, which is also true for the 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, at powers of the order of 20 ... 50 watts, it is optimal to use a 5 ... 20 microfarad capacitor for tweeters, and use a coil wound with copper enameled wire with a diameter of 0.3 ... 1.0 mm on a reel from a VHS video cassette as a woofer choke, and containing 200 ... 1000 turns. Wide limits are indicated, because selection is an individual matter.

L-shaped filters

L-shaped filter of high or low frequencies- voltage divider, consisting of two elements with a non-linear frequency response. For the L-shaped filter, the circuit and all the formulas of the voltage divider apply.

L-shaped frequency filters on a capacitor and a resistor

R1 WITH X C .

The principle of operation of such a filter: the capacitor, having a low reactance at high frequencies, passes the current freely, and at low frequencies its reactance is maximum, so the current does not pass 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.

R2 to the resistance of the resistor R1 (X C ) corresponds to: R 2 / R 1 \u003d 0.7 / 0.3 \u003d 2.33 . This implies: C \u003d 1.16 / R 2 πf , where f is the cutoff frequency of the filter frequency response.

R2 capacitor voltage divider WITH , which has its own reactance X C .

The principle of operation of such a filter: the capacitor, having a low reactance at high frequencies, shunts high-frequency currents to the case, and at low frequencies its reactance is maximum, so the current does not pass through it.

From the Voltage Divider article, we use the same formulas:

or

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

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

R2 (X C ) to the resistance of the resistor R1 corresponds to: R 2 / R 1 \u003d 0.7 / 0.3 \u003d 2.33 . This implies: C \u003d 1 / (4.66 x R 1 πf) , where f is the cutoff frequency of the filter frequency response.

L-shaped frequency filters on the inductor and resistor

The high-pass filter is obtained by replacing the resistor R2 L X L .

The principle of operation of such a filter: the inductance, having a low reactance at low frequencies, shunts them to the case, and at high frequencies its reactance is maximum, so the current does not pass through it.

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

As in the case of the high-pass filter, the calculations can be done in reverse order. 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 the input voltage, it follows that the ratio of the resistance of the resistor R2 (X L ) to the resistance of the resistor R1 corresponds to: R 2 / R 1 \u003d 0.7 / 0.3 \u003d 2.33 . This implies: L = 1.16 R 1 / (πf) .

The low-pass filter is obtained by replacing the resistor R1 voltage divider on the inductor L , which has its own reactance X L .

The principle of operation of such a filter: the inductor, having a low reactance at low frequencies, passes current freely, and at high frequencies its reactance is maximum, so the current does not pass through it.

Using the same formulas from the article "Voltage divider" and taking the input voltage as 1 (one), 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 stress values, we find X L and cutoff frequency.

You can do the calculations in reverse order. 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 the input voltage, it follows that the ratio of the resistance of the resistor R2 to the resistance of the resistor R1 (X L ) corresponds to: R 2 / R 1 \u003d 0.7 / 0.3 \u003d 2.33 . This implies: L \u003d R 2 / (4.66 πf)

L-shaped frequency filters on the capacitor and inductor

The high-pass filter is obtained from an ordinary voltage divider by replacing not only the resistor R1 to the capacitor WITH , as well as a resistor R2 on the throttle L . Such a filter has a more significant frequency cut (steeper rolloff) of the frequency response than the above filters on RC or RL chains.

As before, 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 filter resistances, we can find WITH and L , the cutoff frequency of the frequency response. You can also do the calculations in reverse order. Since there are two variables - 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.

The low-pass filter is obtained by replacing the resistor R1 voltage divider on the inductor L , and the resistor R2 to the capacitor WITH .

As described earlier, the same calculation methods are used, through the voltage divider formulas and the reactances of the filter elements. In this case, we equate the value of the resistor R1 to inductor reactance X L , a R2 to the reactance of the capacitor X C .

T - shaped filters of high and low frequencies

T-shaped filters of high and low frequencies, these 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 non-linear frequency response. And then, the value of the reactance of the third element is added to the calculated value. Another, less accurate way of calculating the T-shaped filter begins with the calculation of the L-shaped filter, after which the value of the "first" calculated element of the L-shaped filter is increased, or halved - "distributed" to 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. A feature of T-shaped filters is that, compared to L-shaped ones, their output impedance has a smaller shunt effect on the radio circuits behind the filter.

U - shaped filters of high and low frequencies

U-shaped filters are the same L-shaped filters, to which one more 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 somewhat increase the shunting effect on the radio circuit in front of the filter.

As in the case of T-shaped filters, voltage divider formulas are used to calculate U-shaped filters, with the addition of an additional shunt resistance of the first filter element. Another, less accurate way of calculating the U-shaped filter begins with the calculation of the L-shaped filter, after which the value of the “last” calculated element of the L-shaped filter is increased or halved - it is “distributed” to 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 some effort, 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 at audio frequencies. So, high-pass filters are usually made T-shaped, and low-pass filters are made U-shaped. There are also mid-range filters, which, as a rule, are made L-shaped (from two capacitors).

Bandpass resonant filters

Bandpass resonant frequency filters - designed to highlight, or reject (cut) 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 frequency response compared to other (non-resonant) filters. Bandpass resonant frequency filters can be single-element - with one circuit, L-shaped - with two circuits, T and U-shaped - with three circuits, multi-link - 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 they have a small resistance to the flowing current, and at other frequencies, on the contrary, they have a large one. Parallel circuit C 2 L 2 on the contrary, it has a large resistance at the resonant frequency, having a small resistance at other frequencies. To expand the bandwidth of such a filter, the quality factor of the circuits is reduced by changing the design of the inductors, detuning the “right, left” circuit by 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 schematic of a T-notch resonant filter designed to suppress a specific frequency. It, like the previous filter, consists of three oscillatory circuits, but the principle of selecting frequencies for such a filter is different. C 1 L 1 and C 3 L 3 - parallel oscillatory circuits, at the resonant frequency they have a large resistance to the flowing current, and at other frequencies - a small one. Parallel circuit C 2 L 2 on the contrary, it has a small resistance at the resonant frequency, having a large 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 impedance acts as a single element. How the oscillatory circuit is calculated, its resonant frequency, quality factor and characteristic (wave) resistance can be found in the article