Application Note 27A June 1988 A Simple Method of Designing Multiple Order All Pole Bandpass Filters by Cascading 2nd Order Sections Nello Sevastopoulos Richard Markell INTRODUCTION DESIGNING BANDPASS FILTERS Filter design, be it active, passive, or switched capacitor, is traditionally a mathematically intensive pursuit. There are many architectures and design methods to choose from. Two methods of high order bandpass filter design are discussed herein. These methods allow the filter designer to simplify the mathematical design process and allow LTC’s switched capacitor filters (LTC®1059, LTC1060, LTC1061,LTC1064) to be utilized as high quality bandpass filters. Table 1 was developed to enable anyone to design Butterworth bandpass filters. We will discuss the tables in more detail later in this paper, but let’s first design a filter. AN27A is the first of a series of application notes from LTC concerning our universal filter family. Additional notes in the series will discuss notch, lowpass and highpass filters implemented with the universal switched capacitor filter. An addition to this note will extend the treatment of bandpass filters to the elliptic or Cauer forms. This note will first present a finished design example and proceed to present the design methodology, which relies on tabular simplification of traditional filter design techniques. A 4th order 2kHz Butterworth bandpass filter with a –3dB bandwidth equal to 200Hz is required as shown in Figure 1. Noting that (foBP/BW) = 10/1 we can go directly to Table 1 for our normalized center frequencies. From Table 1 under 4th order Butterworth bandpass filters, we go to (foBP/BW) = 10. L, LT, LTC, LTM, Linear Technology and the Linear logo are registered trademarks of Linear Technology Corporation. All other trademarks are the property of their respective owners. 0 –3 200Hz –10 GAIN (dB) The first method consists of the traditional cascading of non-identical 2nd order bandpass sections to form the familiar Butterworth and Chebyshev bandpass filters. The second method consists of cascading identical 2nd order bandpass sections. This approach, although “nontextbook,” enables the hardware to be simple and the mathematics to be straightforward. Both methods will be described here. EXAMPLE 1—DESIGN –20 –30 –40 –50 –60 0.5k 0.7k 1k 2k 3k 4k 5k FREQUENCY (Hz) 10k AN27A F01 Figure 1. 4th Order Butterworth BP Filter, foPB = 2kHz an28f AN27A-1 Application Note 27A We find fo1 = 0.965 and fo2 = 1.036 (both normalized to foBP = 1). To find our desired actual center frequencies, we must multiply by foBP = 2kHz to obtain fo1 = 1.930kHz and fo2 = 2.072kHz. R42 all belong to the second of two 2nd order sections in our example. Our requirements are shown in the following table: The Qs are Q1 = Q2 = 14.2 which is read directly from Table 1. Also available from the table is K, which is the product of each individual bandpass gain HoBP. To put it another way, the value of K is the gain required to make the gain, H, of the overall filter equal to 1 at foBP. Our filter parameters are highlighted in the following table: foBP fo1 fo2 Qs K 2kHz 1.93kHz 2.072kHz Q1 = Q2 = 14.2 2.03 SECTION 2 fo1 = 1.93kHz Q1 = 14.2 HoBP1 = 1 fo2 = 2.072kHz Q2 = 14.2 HoBP2 = 2.03 Note that HoBP1 × HoBP2 = K and this is the reason for choosing HoBP2 = 2.03. For this example we choose the fo = fCLK 50 R2 mode, R4 so we will tie the 50/100/Hold pin on the SCF chip to V+, generally (5V to 7V). We choose 100kHz as our clock and calculate resistor values. Choosing the nearest 1% resistor values we can implement the filter using Figure 3’s topology and the resistor values listed below. HARDWARE IMPLEMENTATION Universal switched capacitor filters are simple to implement. A bandpass filter can be built from the traditional state-variable filter topology. Figure 2 shows this topology for both switched capacitor and active operational amplifier implementations. Our example requires four resistors for each 2nd order section. So eight resistors are required to build our filter. We start with two 2nd order sections (1 LTC1060, 2/3 LTC1061 or 1/2 LTC1064), Figure 3. R11 = 147k R21 = 10k R31 = 147k R41 = 10.7k R12 = 71.5k R22 = 10.7k R32 = 147k R42 = 10k Our design is now complete. We have only to generate a TTL or CMOS compatible clock at 100kHz, which we feed to the clock pin of the switched capacitor filter, and we should be “on the air.” We associate resistors as belonging to 2nd order sections, so R1x belongs to the x section. Thus R12, R22, R33 and STATE VARIABLE SCF SECTION 1 ACTIVE (OP-AMP) STATE VARIABLE R4 R4 R3 R2 VIN R1 – HP S – + + AGND C R2 BP LP VIN ∫ ∫ R1 C – R R5 – + R – + 1/2 LTC1060 1/3 LTC1061 1/4 LTC1064 + R6 AN27A F02 3/4 LTC1014 f R2 R3 R2 fo = CLK | Q= | HoHP = –R2 / R1 HoBP = –R3 / R1 HoLP = –R4 / R1 100(50) R4 R2 R4 1 R2 fO = 2πRC R4 MODE 3 Figure 2. Switched Capacitor vs Active RC State Variable Topology an28f AN27A-2 Application Note 27A R41 R42 R31 R32 R21 VIN R11 HP – + + S1A – – BP R22 LP ∫ + fCLK 50 LP BP ∫ – 1/2 LTC1060 fo = – + ∫ S1A HP – R12 ∫ BP OUTPUT AN27A F03 1/2 LTC1060 R3 R2 R2 ;Q= HoBP = –R3 /R1 R2 R4 R4 Figure 3. Two 2nd Order Sections Cascaded to Form 4th Order BP Filiter DESIGNING BANDPASS FILTERS—THEORY BEHIND THE DESIGN Traditionally, bandpass filters have been designed by laborious calculations requiring some time to complete. At the present time programs for various personal or laboratory computers are often used. In either case, no small amount of time and/or money is involved to evaluate, and later test, a filter design. Many designers have inquired as to the feasibility of cascading 2nd order bandpass sections of relatively low Q to obtain more selective, higher Q, filters. This approach is ideally suited to the LTC family of switched capacitor filters (LTC1059, LTC1060, LTC1061 and LTC1064). The clock to center frequency ratio accuracy of a typical “Mode 1” design with non “A” parts is better than 1% in a design that simply requires three resistors of 1% tolerance or better. Also, no expensive high precision film capacitors are required as in the active op amp state variable design. We present here an approach for designing bandpass filters using the LTC1059, LTC1060, LTC1061 or the LTC1064 which many designers have “on the air” in days instead of weeks. with the required characteristics (generally the Qs are too high). We wish to explore here the use of cascaded identical 2nd order sections for building high Q bandpass filters. For a 2nd order bandpass filter Q= 1– G2 f/ fo × 2 G 1– ( f / fo ) (1) Where Q is the required filter quality factor f is the frequency where the filter should have gain, G, expressed in Volts/V. fo is at the filter center frequency. Unity gain is assumed at fo. Example 2—Design We wish to design a 2nd order BP filter to pass 150Hz and to attenuate 60Hz by 50dB. The required Q may be calculated from Equation (1): So,Q = ( 1– 3.162×10 –3 3.162×10 –3 ) 2 × 60 / 150 1– ( 60 / 150 ) 2 = 150.7 Cascading Identical 2Nd Order Bandpass Sections This very high Q dictates a –3dB bandwidth of 1Hz. When we want to detect single frequency tones and simultaneously reject signals in close proximity, simple 2nd order bandpass filters often do the job. However, there are cases where a 2nd order section cannot be implemented Although the universal switched capacitor filters can realize such high Qs, their guaranteed center frequency accuracy of ±0.3%, although impressive, is not enough to pass the 150Hz signal without gain error. According to the an28f AN27A-3 Application Note 27A previous equation, the gain at 150Hz will be 1 ±26%; the rejection, however, at 60Hz will remain at –50dB. The gain inaccuracy can be corrected by tuning resistor R4 when mode 3, Figure 2, is used. Also, if only detection of the signal is sought, the gain inaccuracy could be acceptable. This high Q problem can be solved by cascading two identical 2nd order bandpass sections. To achieve a gain, G, at frequency f the required Q of each 2nd order section is: Q= 1– G f / fo × 2 G 1– ( f / fo ) (2) The gain at each bandpass section is assumed unity. In order to obtain 50dB attenuation at 60Hz, and still pass 150Hz, we will use two identical 2nd order sections. We can calculate the required Q for each of two 2nd order sections from Equation (2): So,Q = 1– 3.162×10 –3 3.162×10 –3 × 60 / 150 1– ( 60 / 150 ) 2 = 8.5!! With two identical 2nd order sections each with a potential error in center frequency, fo, of ±0.3% the gain error at 150Hz is 1 ±0.26%. If lower cost (non “A” versions of LTC1060 and LTC1064) 2nd order bandpass sections are used with an fo tolerance of ±0.8%, the gain error at 150Hz is 1 ±1.8%! The benefits of lower Q sections are therefore obvious. Hardware Implementation Mode 1 Operation of LTC1060, LTC1061, LTC1064 As previously discussed, we associate resistors with each 2nd order section, so R1x belongs to x section. Thus R12, R22 and R23 belong to the second of the two 2nd order sections, Figure 4. Each section has the same requirements as shown: fo1 = fo2 = 150Hz Q1 = Q = 8.5 HoBP1 = HoBP2 = 1 Note that we could get gain out of our BP filter structure by letting the product of the HoBP terms be >1 (within the performance limits of the filter itself). For our example using the LTC1060 we will use fo1 = fo2 = fCLK/100. So we input a 15kHz clock and tie the 50/100/ Hold pin to mid-supplies (ground for ±5V supplies). We can implement this filter using the two sections of an LTC1060 filter operated in mode 1. Mode 1 is the fastest operating mode of the switched capacitor filters. It provides Lowpass, Bandpass and Notch outputs. Each 2nd order section will perform approximately as shown in Figure 5, curve (a). Implementation in mode 1 is simple as only three resistors are required per section. Since we are cascading identical sections, the calculations are also simple. R31 R21 VIN R32 S N – R11 – + + AGND fo = – 1/2 LTC1060 BP R22 LP R12 ∫ ∫ N – + + AGND S – – LP BP ∫ ∫ BP OUTPUT AN27A F04 1/2 LTC1060 fCLK –R3 R3 H = Q= 100 oBP R1 R2 Figure 4. LTC1060 as BP Filter Operating in Mode 1 an28f AN27A-4 Application Note 27A We can calculate the resistor values from the indicated formulas and then choose 1% values. (Note that we let our minimum value be 20k.) The required values are: R11 = R12 = 169k R21 = R22 = 20k R31 = R32 = 169k Our design is complete. The performance of two 2nd order sections cascaded versus one 2nd order section is shown in Figure 5, curve (b). We must, however, generate a TTL or CMOS clock at 15kHz to operate the filter. Mode 2 Operation of LTC1060 Family Suppose that we have no 15kHz clock source readily available. We can use what is referred to as mode 2, which allows the input clock frequency to be less than 50:1 or If we wish to operate our previous filter from a television crystal at 14.318MHz we could divide this frequency by 1000 to give us a clock of 14.318kHz. We could then set up our mode 2 filter as shown in Figure 6. We can calculate the resistor values from the formulas shown and then choose 1% values. The required values are: R11, R12 = 162k R21, R22 = 20k R31, R32 = 162k R41, R42 = 205k Cascading More Than Two Identical 2Nd Order Bp Sections If more than two identical bandpass sections (2nd order) are cascaded, the required Q of each section may be shown to be: 0 BANDPASS GAIN (dB) 100:1 [fCLK/fo = 50 or 100]. This still depends on the connection of the 50/100/Hold pin. –10 (a): ONE SECTION –20 Q= –30 –40 G1/n (b): TWO SECTIONS × ( f / fo ) 2 1– ( f / fo ) (3) where Q, G, f and fo are as previously defined and n = the number of cascaded 2nd order sections. –50 –60 1– G2/n 30 50 100 150 300 500 FREQUENCY (Hz) 1k AN27A F05 Figure 5. Cascading Two 2nd Order BP Sections for Higher Q Response R41 R42 R31 R32 R21 VIN R11 N – + + S – – BP R22 LP R12 ∫ ∫ – + + 1/2 LTC1060 fo = N S – – 1/2 LTC1060 LP BP ∫ ∫ BP OUTPUT AN27A F06 fCLK R2 R3 R2 1+ Q= 1+ R4 R2 R4 100 HoBP = R3 /R1 Figure 6. LTC1060 as BP Filter Operating in Mode 2 an28f AN27A-5 Application Note 27A The equivalent Q of the overall bandpass filter is then: Q(identical section ) (4) 1/n Figure 7 shows the passband curves for Q = 2 cascaded bandpass sections where n is the number of 2nd order sections cascaded. The benefits can be seen for two and three cascaded sections. Cascading four or more sections increases the Q, but not as rapidly. Nevertheless for designers requiring high Q bandpass filters cascading identical sections is a very real option considering the simplicity. Simple 2nd Order Bandpass Filters Gain and Phase Relations The bandpass output of each 2nd order filter section of the LTC1059, LTC1060, LTC1061 and LTC1064, closely approximates the gain and phase response of an ideal “textbook” filter. G= GAIN (dB) (2 ) – 1 N=1 –10 N=2 –15 –20 –25 –30 N=3 –35 N=4 –40 0.4 0.6 N = NUMBER OF SECOND ORDER Q = 2, BANDPASS SECTIONS 0.8 1 1.2 1.4 NORMALIZED FREQUENCY 1.6 AN27A F07 Figure 7. Frequency Response of n Cascaded Identical 2nd Order Bandpass Sections frequency and Q drift, but for system considerations, this may not be practical. BANDPASS OUTPUT HoBP GAIN (V/V) Q equiv = 0 –3 –5 0.707 HoBP (H oBP ) × ( ffo ) /Q 1/2 2 2 2 2 + ( ffo / Q ) fo – f ( fL fo fH f(LOG SCALE) ) Q= G = filter gain in Volts/V f0 = the filter’s center frequency Q = the quality coefficient of the filter HoBP = the maximum voltage gain of the filter occurring at fO fo = –3dB bandwidth of the filter Q 2 –1 1 fL = fo + +1 2Q 2Q 2 1 1 fH = fo + +1 2Q 2Q Figure 8. Bandpass Filter Parameters 0 –10 Q = 1 BANDPASS GAIN (dB) Figure 8 illustrates the above definitions. Figure 9 illustrates the bandpass gain, G, for various values of Q. This figure is very useful for estimating the filter attenuation when several identical 2nd order bandpass filters are cascaded. High Qs make the filter more selective, and at the same time, more noisy and more difficult to realize. Qs in excess of 100 can be easily realized with the universal switched capacitor filters, LTC1059, LTC1060, LTC1061 and LTC1064, and still maintain low center fo ; f = fL fH fH – fL o –20 Q=2 Q=3 Q=5 Q = 10 –30 –40 –50 Q = 20 –60 –70 –80 0.1 0.3 0.5 0.7 1 3 5 7 10 NORMALIZED FREQUENCY AN27A F09 Figure 9. Bandpass Gain as a Function of Q an28f AN27A-6 Application Note 27A used in multichannel systems where phase matching is required. By way of comparison, a state variable active bandpass filter built with 1% resistors and 1% capacitors may have center frequency variation of ±2% resulting in phase variations of ±2% resulting in phase variations of ±33.8° for Q = 20 and ±11.4° for Q = 5. The phase shift, φ, of a 2nd order bandpass filter is: f 2 – f 2 ×Q φ = – arctan o ffo The phase shift at fo is 0° or, if the filter is inverting, it is –180°. All the bandpass outputs of the LTC1059, LTC1060, LTC1061 and LTC1064 universal filters are inverting. The phase shift, especially in the vicinity of fo, depends on the value of Q, see Figure 10. By the same argument, the phase shift at a given frequency varies from device to device due to the fo tolerance. This is true especially for high Qs and in the vicinity of fo. For instance, an LTC1059A, 2nd order universal filter, has a guaranteed initial center frequency tolerance of ±0.3%. The ideal phase shift at the ideal fo should be –180°. With a Q of 20, and without trimming, the worst-case phase shift at the ideal fo will be –180° ±6.8°. With a Q of 5 the phase shift tolerance becomes –180° ±1.7°. These are important considerations when bandpass filters are Constant Q Versus Constant BW The bandpass outputs of the universal filters are “constant Q.” For instance, a 2nd order bandpass filter operating in mode 1 with a 100kHz clock (see LTC1060 data sheet) ideally has a 1kHz or 2kHz center frequency, and a –3dB bandwidth equal to (fo/Q). When the clock frequency varies, the center frequency and bandwidth will vary at the same rate. In a constant bandwidth filter, when the center frequency varies, the Q varies accordingly to maintain a constant (fo/Q) ratio. A constant bandwidth BP filter could be implemented using 2nd order switched capacitor filters but this is beyond the scope of this paper. 0 ±10° ±20° Q=1 ±30° ±40° φ ±50° Q = 10 ±60° ±70° ±80° ±90° 0 0.2 0.4 0.6 0.8 1.0 f/fo 1.2 1.4 1.6 1.8 2.0 AN27A F10 Figure 10. Phase Shift, φ, of a 2nd Order BP Filter Section (LTC1059, 1/2 LTC1060, 1/3 LTC1061) an28f AN27A-7 Application Note 27A Tables 1 through 4 were derived from textbook filter theory. They can be easily applied to the LTC filter family (LTC1059, LTC1060, LTC1061 and LTC1064) if the Qs are kept relatively low (<20) and the tuning resistors are at least 1% tolerance. These lower Q designs provide almost textbook BP filter performance using LTC’s switched capacitor filters. For higher Q implementations, tuning should be avoided and the “A” versions of the LTC1059, LTC1060, LTC1061 or LTC1064 should be specified. Also, resistor tolerances of better than 1% are a necessity. This is true for any bandwidth, BW, and any set of frequencies. The tables can now be arithmetically scaled as illustrated. A2 2BW A3 3BW A4 4BW A5 5BW Table 1 may be used to find pole positions and Qs for Butterworth bandpass filters. It should be noted that the bandpass filters in these tables are geometrically symmetrical about their center frequencies, foBP. Any frequency,f3, as shown in Figure 11 has its geometrical counterpart f4 such that: f9 f 7 ( f1,f2 ) = Under this condition, for either Butterworth or Chebyshev bandpass filters: f –f foBP ≅ 3 4 + f3 2 foPB ≅ f5 – f6 + f5 2 • • • AN27A-8 ±BW + f1f2 = foBP f1 foBP f2 f4 f6 f8 f10 AN27A F11 (BW )2 + 4( foBP )2 2 MORE GENERALLY ( fx ,fx+1) = ±nBW + (nBW )2 + 4( foBP )2 ( VALID FOR ANY fx ,fx+1PAIR,ANYBW ) Additionally, Table 1 illustrates the attenuation at the frequencies f3, f5, f7 and f9, which correspond to bandwidths 2, 3, 4 and 5 times the passband (see Figure 11). These values allow the user to get a good estimate of filter selectivity, foPB 1 >> , BW = f2 – f1 BW 2 f5 f3 ( f2 – f1) = BW FREQUENCY (LOG SCALE) f 2 f4 = oBP f3 2 Figure 11. Generalized Bandpass Butterworth Response (See Table 1) BW 0 –2 GAIN (dB) An important approximation can be made for not only the Butterworth filters in Table 1, but also for the Chebyshev filter Tables 2, 3 and 4. Treating Figure 11 (or Figure 12) as a generalized bandpass filter, the two corner frequencies f2 and f1 can be seen to be nearly arithmetically symmetrical with respect to foBP provided that: BW 0 –3 GAIN (dB) Using The Tables A2 2BW A3 3BW A4 4BW A5 5BW f9 f 7 f5 f3 f1 foBP f2 f4 f6 f8 f10 AN27A F12 FREQUENCY (LOG SCALE) f4 f3 = foPB ( f4,f3 ) = ±2BW + (2BW )2 + 4( foBP )2 2 FOR ANY ( fx ,fx–1) PAIR AND ANY CORRESPONDING BANDWIDTH (2BW, 3BW, ETC.) FOR EXAMPLE: ( f6,f5 ) = ±3BW + (3BW )2 + 4( foBP )2 2 Figure 12. Generalized 4th, 6th, and 8th Order Chebyshev Bandpass Filter with 2dB Passband Ripple (AMAX) an28f fo1 (Hz) fo2 (Hz) fo3 (Hz) fo4 (Hz) f–3dB (Hz) f–3dB (Hz) Q1 = Q2 20 1 0.982 0.965 0.932 0.885 0.836 0.693 1.018 1.036 1.073 1.125 1.195 1.442 0.975 0.951 0.905 0.847 0.781 0.500 1.025 1.051 1.105 1.180 1.281 2.000 28.3 14.2 7.1 4.3 2.9 1.5 10 20 1 1 0.650 0.979 0.958 0.917 0.866 0.805 1.539 1.022 1.044 1.091 1.155 1.242 1.000 1.000 1.000 1.000 1.000 1.000 0.500 0.975 0.951 0.905 0.847 0.781 2.000 1.025 1.051 1.105 1.180 1.281 2.2 40.0 20.0 10.0 6.1 4.1 1.0 20.0 10.0 5.0 3.0 2.0 5 10 20 1 1 3 1 1 1 2 1 1 0.809 0.990 0.981 0.962 0.938 0.907 1.237 1.010 1.019 1.039 1.066 1.103 0.636 0.977 0.955 0.912 0.858 0.795 1.574 1.023 1.047 1.097 1.166 1.259 0.500 0.975 0.951 0.905 0.847 0.781 2.000 1.025 1.051 1.105 1.180 1.281 1.1 21.6 10.8 5.4 3.3 2.2 52.3 26.2 13.1 7.9 5.4 2.9 Q3 = Q4 8th Order Butterworth Bandpass Filter Normalized to its Center Frequency, foBP = 1, and –3dB Bandwidth (BW) 3 5 1 1 1 2 1 1 Q3 6th Order Butterworth Bandpass Filter Normalized to its Center Frequency, foBP = 1, and –3dB Bandwidth (BW) 5 10 1 1 2 3 1 1 1 1 4th Order Butterworth Bandpass Filter Normalized to its Center Frequency, foBP = 1, and –3dB Bandwidth (BW) foBP (Hz) foBP/BW (Hz) Table 1. Butterworth Bandpass Filters Normalized to foBP = 1 8.00 8.00 8.05 8.15 8.48 10.14 4.00 4.01 4.03 4.07 4.18 4.79 2.03 2.03 2.04 2.07 2.07 2.28 K 0.975 0.951 0.905 0.847 0.781 0.500 0.975 0.951 0.905 0.847 0.781 0.500 0.975 0.951 0.905 0.847 0.781 0.500 0.951 0.905 0.820 0.721 0.618 0.414 0.951 0.905 0.820 0.721 0.618 0.414 0.951 0.905 0.820 0.721 0.618 0.414 f1 (Hz) f3 (Hz) –24.0 –24.0 –24.0 –24.0 –24.0 –24.0 –18.2 –18.2 –18.2 –18.2 –18.2 –18.2 –12.3 –12.3 –12.3 –12.3 –12.3 –12.3 0.928 0.861 0.744 0.618 0.500 0.303 0.928 0.861 0.744 0.618 0.500 0.303 0.928 0.861 0.744 0.618 0.500 0.303 –38.0 –38.0 –38.0 –38.0 –38.0 –38.0 –28.6 –28.6 –28.6 –28.6 –28.6 –28.6 –19.1 –19.1 –19.1 –19.1 –19.1 –19.1 0.905 0.820 0.677 0.535 0.414 0.236 0.905 0.820 0.677 0.535 0.414 0.236 0.905 0.820 0.677 0.535 0.414 0.236 –48.1 –48.1 –48.1 –48.1 –48.1 –48.1 –36.1 –36.1 –36.1 –36.1 –36.1 –36.1 –24.0 –24.0 –24.0 –24.0 –24.0 –24.0 0.883 0.781 0.618 0.469 0.351 0.193 0.883 0.781 0.618 0.469 0.351 0.193 0.883 0.781 0.618 0.469 0.351 0.193 –55.8 –55.8 –55.8 –55.8 –55.8 –55.8 –41.9 –41.9 –41.9 –41.9 –41.9 –41.9 –28.0 –28.0 –28.0 –28.0 –28.0 –28.0 GAIN AT f3 GAIN AT f5 GAIN AT f7 GAIN AT f9 (dB)-A2 f5 (Hz) (dB)-A3 f7 (dB) (dB)-A4 f9 (Hz) (dB)-A5 Application Note 27A an28f AN27A-9 AN27A-10 2 3 5 10 20 1 1 1 1 1 0.966 0.933 0.871 0.793 0.703 0.488 20 1 0.975 0.951 0.904 0.845 0.777 0.602 2 3 5 10 20 1 1 1 1 1 0.978 0.956 0.914 0.861 0.799 0.639 20 1 0.980 0.960 0.922 0.873 0.816 0.668 1.021 1.041 1.085 1.145 1.225 1.496 1.022 1.046 1.094 1.161 1.251 1.564 1.025 1.051 1.106 1.182 1.287 1.660 1.035 1.071 1.148 1.261 1.422 2.050 fo2 (Hz) 18.87 9.35 4.65 2.79 1.86 0.93 16.39 8.20 4.12 2.47 1.64 0.82 14.49 7.19 3.60 2.16 1.44 0.72 10.31 5.15 2.58 1.54 1.03 0.52 0.974 0.948 0.898 0.837 0.767 0.598 0.970 0.941 0.886 0.818 0.741 0.562 0.966 0.933 0.871 0.795 0.711 0.523 0.953 0.908 0.825 0.727 0.626 0.423 1.027 1.055 1.113 1.195 1.304 1.672 1.031 1.063 1.129 1.223 1.349 1.779 1.035 1.072 1.149 1.258 1.406 1.912 1.050 1.102 1.213 1.375 1.597 2.364 49.8 24.9 12.5 7.5 5.1 2.7 36.5 18.2 9.2 5.5 3.7 2.0 28.1 14.1 7.1 4.3 2.9 1.6 16.9 8.5 4.3 2.6 1.8 1.1 foBP/BW2** (Hz) f–3dB (Hz) f–3dB (Hz) Q1 = Q2 *foBP/BW1 – This is the ratio of the bandpass filter center frequency to the ripple bandwidth of the filter. **foBP/BW2 – This is the ratio of the bandpass filter center frequency to the –3dB filter bandwidth. 5 10 3 1 1 2 1 1 1 1 Passband Ripple AMAX = 2.0dB 1 1 Passband Ripple AMAX = 1.0dB 5 10 3 1 1 2 1 1 1 1 Passband Ripple AMAX = 0.5dB 1 1 Passband Ripple, AMAX = 0.1dB foBP (Hz) foBP/BW1* (Hz) fo1 (Hz) 5.07 5.13 5.13 5.22 5.30 6.00 3.63 3.70 3.71 3.76 3.85 4.42 2.97 2.98 3.03 3.07 3.17 3.80 2.37 2.38 2.38 2.48 2.66 3.81 K Table 2. 4th Order Chebyshev Bandpass Filter Normalized to its Center Frequency foBP = 1 0.975 0.951 0.905 0.847 0.781 0.500 0.975 0.951 0.905 0.847 0.781 0.500 0.975 0.951 0.905 0.847 0.781 0.500 0.975 0.951 0.905 0.847 0.781 0.500 f1 (Hz) 0.951 0.905 0.820 0.721 0.618 0.414 0.951 0.905 0.820 0.721 0.618 0.414 0.951 0.905 0.820 0.721 0.618 0.414 0.951 0.905 0.820 0.721 0.618 0.414 f3 (Hz) –12.7 –12.7 –12.7 –12.7 –12.7 –12.7 –10.3 –10.3 –10.3 –10.3 –10.3 –10.3 –7.9 –7.9 –7.9 –7.9 –7.9 –7.9 –3.2 –3.2 –3.2 –3.2 –3.2 –3.2 GAIN AT f3 (dB)-A2 0.928 0.861 0.744 0.618 0.500 0.303 0.928 0.861 0.744 0.618 0.500 0.303 0.928 0.861 0.744 0.618 0.500 0.303 0.928 0.861 0.744 0.618 0.500 0.303 f5 (Hz) –20.3 –20.3 –20.3 –20.3 –20.3 –20.3 –17.7 –17.7 –17.7 –17.7 –17.7 –17.7 –15.0 –15.0 –15.0 –15.0 –15.0 –15.0 –08.7 –08.7 –08.7 –08.7 –08.7 –08.7 GAIN AT f5 (dB)-A3 0.905 0.820 0.677 0.535 0.414 0.236 0.905 0.820 0.677 0.535 0.414 0.236 0.905 0.820 0.677 0.535 0.414 0.236 0.905 0.820 0.677 0.535 0.414 0.236 f7 (Hz) –25.5 –25.5 –25.5 –25.5 –25.5 –25.5 –23.0 –23.0 –23.0 –23.0 –23.0 –23.0 –20.2 –20.2 –20.2 –20.2 –20.2 –20.2 –13.6 –13.6 –13.6 –13.6 –13.6 –13.6 GAIN AT f7 (dB)-A4 0.883 0.781 0.618 0.469 0.351 0.193 0.883 0.781 0.618 0.469 0.351 0.193 0.883 0.781 0.618 0.469 0.351 0.193 0.883 0.781 0.618 0.469 0.351 0.193 –29.5 –29.5 –29.5 –29.5 –29.5 –29.5 –27.0 –27.0 –27.0 –27.0 –27.0 –27.0 –24.1 –24.1 –24.1 –24.1 –24.1 –24.1 –17.4 –17.4 –17.4 –17.4 –17.4 –17.4 GAIN AT f9 (Hz) f9 (dB)-A5 Application Note 27A an28f 20 1 0.970 0.941 0.886 0.818 0.741 0.558 10 20 1 1 0.609 0.975 0.950 0.903 0.844 0.776 20 1 0.976 0.953 0.908 0.852 0.787 0.626 10 20 1 1 0.639 0.977 0.955 0.912 0.858 0.795 1.023 1.047 1.096 1.165 1.257 1.565 1.024 1.050 1.101 1.174 1.271 1.598 1.026 1.052 1.107 1.185 1.288 1.641 1.030 1.062 1.128 1.222 1.349 1.791 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 19.61 9.71 4.83 2.91 1.94 0.97 18.18 9.17 4.59 2.74 1.83 0.91 16.95 8.55 4.29 2.57 1.72 0.86 14.49 7.19 3.60 2.16 1.44 0.72 0.975 0.950 0.902 0.843 0.775 0.609 0.973 0.947 0.897 0.834 0.763 0.593 0.971 0.943 0.890 0.824 0.750 0.574 0.966 0.933 0.871 0.795 0.711 0.523 1.026 1.053 1.109 1.187 1.291 1.642 1.028 1.056 1.115 1.199 1.310 1.687 1.030 1.060 1.123 1.213 1.333 1.741 1.035 1.072 1.149 1.258 1.406 1.912 108.5 54.3 27.2 16.5 11.1 6.0 81.0 40.5 20.3 12.3 8.3 4.5 63.8 32.0 16.1 9.7 6.6 3.6 41.3 20.6 10.4 6.3 4.3 2.4 *foBP/BW1 – This is the ratio of the bandpass filter center frequency to the ripple bandwidth of the filter. **foBP/BW2 – This is the ratio of the bandpass filter center frequency to the –3dB filter bandwidth. 3 5 1 1 1 2 1 1 Passband Ripple, AMAX = 2.0dB 5 10 1 1 2 3 1 1 1 1 Passband Ripple, AMAX = 1.0dB 3 5 1 1 1 2 1 1 Passband Ripple, AMAX = 0.5dB 5 10 1 1 2 3 1 1 1 1 Passband Ripple, AMAX = 0.1dB 54.2 27.1 13.6 8.1 5.4 2.7 40.5 20.2 10.1 6.1 4.1 2.0 32.0 16.0 8.0 4.8 3.2 1.6 20.6 10.3 5.2 3.1 2.1 1.0 foBP (Hz) foBP/BW1* (Hz) fo1 (Hz) fo2 (Hz) fo3 (Hz) foBP/BW2** (Hz) f–3dB (Hz) f–3dB (Hz) Q1 = Q2 Q = 3 Table 3. 6th Order Chebychev Bandpass Filter Normalized to its Center Frequency foBP = 1 26.0 26.0 26.2 26.7 27.4 31.7 16.4 16.4 16.4 16.7 17.1 20.1 11.4 11.8 11.8 12.0 12.5 14.8 7.3 7.3 7.4 7.5 7.9 9.9 K 0.975 0.951 0.905 0.847 0.781 0.500 0.975 0.951 0.905 0.847 0.781 0.500 0.975 0.951 0.905 0.847 0.781 0.500 0.975 0.951 0.905 0.847 0.781 0.500 0.951 0.905 0.820 0.721 0.618 0.414 0.951 0.905 0.820 0.721 0.618 0.414 0.951 0.905 0.820 0.721 0.618 0.414 0.951 0.905 0.820 0.721 0.618 0.414 –26.0 –26.0 –26.0 –26.0 –26.0 –26.0 –22.5 –22.5 –22.5 –22.5 –22.5 –22.5 –19.2 –19.2 –19.2 –19.2 –19.2 –19.2 –12.2 –12.2 –12.2 –12.2 –12.2 –12.2 0.928 0.861 0.744 0.618 0.500 0.303 0.928 0.861 0.744 0.618 0.500 0.303 0.928 0.861 0.744 0.618 0.500 0.303 0.928 0.861 0.744 0.618 0.500 0.303 –37.5 –37.5 –37.5 –37.5 –37.5 –37.5 –34.0 –34.0 –34.0 –34.0 –34.0 –34.0 –30.8 –30.8 –30.8 –30.8 –30.8 –30.8 –23.6 –23.6 –23.6 –23.6 –23.6 –23.6 0.905 0.820 0.677 0.535 0.414 0.236 0.905 0.820 0.677 0.535 0.414 0.236 0.905 0.820 0.677 0.535 0.414 0.236 0.905 0.820 0.677 0.535 0.414 0.236 –45.4 –45.4 –45.4 –45.4 –45.4 –45.4 –41.9 –41.9 –41.9 –41.9 –41.9 –41.9 –38.6 –38.6 –38.6 –38.6 –38.6 –38.6 –31.4 –31.4 –31.4 –31.4 –31.4 –31.4 0.883 0.781 0.618 0.469 0.351 0.193 0.883 0.781 0.618 0.469 0.351 0.193 0.883 0.781 0.618 0.469 0.351 0.193 0.883 0.781 0.618 0.469 0.351 0.193 –51.3 –51.3 –51.3 –51.3 –51.3 –51.3 –47.8 –47.8 –47.8 –47.8 –47.8 –47.8 –44.5 –44.5 –44.5 –44.5 –44.5 –44.5 –37.3 –37.3 –37.3 –37.3 –37.3 –37.3 GAIN AT GAIN AT GAIN AT GAIN AT f1 (Hz) f3 (Hz) f3 (dB)-A2 f5 (Hz) f5 (dB)-A3 f7 (Hz) f7 (dB)-A4 f9 (Hz) f9 (dB)-A5 Application Note 27A an28f AN27A-11 AN27A-12 2 3 5 10 20 1 1 1 1 1 0.988 0.977 0.954 0.925 0.889 0.785 20 1 0.989 0.979 0.959 0.932 0.900 0.808 2 3 5 10 20 1 1 1 1 1 0.990 0.980 0.960 0.934 0.903 0.814 20 1 0.990 0.980 0.961 0.936 0.905 0.820 1.010 1.020 1.040 1.068 1.104 1.220 1.010 1.020 1.041 1.070 1.107 1.228 1.010 1.021 1.043 1.073 1.111 1.238 1.012 1.023 1.048 1.081 1.125 1.274 0.976 0.953 0.909 0.853 0.789 0.629 0.976 0.952 0.906 0.850 0.784 0.622 0.975 0.950 0.903 0.845 0.777 0.613 0.972 0.945 0.894 0.830 0.757 0.584 1.024 1.049 1.100 1.172 1.268 1.589 1.025 1.050 1.103 1.177 1.275 1.607 1.026 1.052 1.107 1.183 1.286 1.632 1.028 1.058 1.118 1.204 1.320 1.713 19.61 9.80 4.90 2.95 1.96 0.98 18.87 9.52 4.74 2.85 1.90 0.95 18.18 9.17 4.59 2.74 1.83 0.91 16.39 8.20 4.12 2.48 1.65 0.82 0.975 0.950 0.903 0.845 0.777 0.613 0.974 0.949 0.900 0.840 0.771 0.604 0.973 0.947 0.897 0.834 0.763 0.593 0.970 0.941 0.886 0.818 0.742 0.563 1.026 1.052 1.107 1.184 1.287 1.631 1.027 1.054 1.111 1.191 1.297 1.656 1.028 1.056 1.115 1.199 1.310 1.686 1.031 1.063 1.129 1.222 1.348 1.776 *foBP/BW1 – This is the ratio of the bandpass filter center frequency to the ripple bandwidth of the filter. **foBP/BW2 – This is the ratio of the bandpass filter center frequency to the –3dB filter bandwidth. 5 10 3 1 1 2 1 1 1 1 Passband Ripple, AMAX = 2.0dB 1 1 Passband Ripple, AMAX = 1.0dB 5 10 3 1 1 2 1 1 1 1 Passband Ripple, AMAX = 0.5dB 1 1 Passband Ripple, AMAX = 0.1dB 124.0 0.951 0.905 127.7 0.905 0.820 128.1 0.847 0.721 133.2 0.781 0.618 162.8 0.500 0.414 70.0 0.975 0.951 70.0 0.951 0.905 70.0 0.905 0.820 71.5 0.847 0.721 74.3 0.781 0.618 90.1 0.500 0.414 29.8 0.975 0.951 29.8 0.951 0.905 29.9 0.905 0.820 30.5 0.847 0.721 32.1 0.781 0.618 40.6 0.500 0.414 190.0 302.0 0.975 0.951 79.0 302.0 0.951 0.905 302.0 0.905 0.820 302.0 0.847 0.721 312.6 0.781 0.618 374.8 0.500 0.414 95.4 47.9 29.0 19.6 10.6 -35.4 -35.4 -35.4 -35.4 -35.4 -35.4 -32.9 -32.9 -32.9 -32.9 -32.9 -32.9 -30.2 -30.2 -30.2 -30.2 -30.2 -30.2 -23.4 -23.4 -23.4 -23.4 -23.4 -23.4 0.928 0.861 0.744 0.618 0.500 0.303 0.928 0.861 0.744 0.618 0.500 0.303 0.928 0.861 0.744 0.618 0.500 0.303 0.928 0.861 0.744 0.618 0.500 0.303 -50.8 -50.8 -50.8 -50.8 -50.8 -50.8 -48.3 -48.3 -48.3 -48.3 -48.3 -48.3 -45.5 -45.5 -45.5 -45.5 -45.5 -45.5 -38.8 -38.8 -38.8 -38.8 -38.8 -38.8 0.905 0.820 0.677 0.535 0.414 0.236 0.905 0.820 0.677 0.535 0.414 0.236 0.905 0.820 0.677 0.535 0.414 0.236 0.905 0.820 0.677 0.535 0.414 0.236 -61.3 -61.3 -61.3 -61.3 -61.3 -61.3 -58.8 -58.8 -58.8 -58.8 -58.8 -58.8 -56.0 -56.0 -56.0 -56.0 -56.0 -56.0 -49.3 -49.3 -49.3 -49.3 -49.3 -49.3 0.883 0.781 0.618 0.469 0.351 0.193 0.883 0.781 0.618 0.469 0.351 0.193 0.883 0.781 0.618 0.469 0.351 0.193 0.883 0.781 0.618 0.469 0.351 0.193 -69.2 -69.2 -69.2 -69.2 -69.2 -69.2 -66.6 -66.6 -66.6 -66.6 -66.6 -66.6 -63.9 -63.9 -63.9 -63.9 -63.9 -63.9 -57.1 -57.1 -57.1 -57.1 -57.1 -57.1 GAIN AT GAIN AT GAIN AT GAIN AT f1 (Hz) f3 (Hz) f3 (dB)-A2 f5 (Hz) f5 (dB)-A3 f7 (Hz) f7 (dB)-A4 f9 (Hz) f9 (dB)-A5 143.0 120.0 0.975 0.951 71.7 36.0 21.8 14.8 8.0 114.0 57.1 28.7 17.4 11.8 6.4 75.7 37.9 19.1 11.6 7.9 4.4 K 39.5 19.7 11.9 7.9 4.0 59.4 29.7 14.9 8.9 6.0 3.0 47.2 23.6 11.8 7.1 4.8 2.4 31.4 15.7 7.9 4.7 3.2 1.6 foBP (Hz) foBP/BW1* (Hz) fo1 (Hz) fo2 (Hz) fo3 (Hz) fo4 (Hz) foBP/BW2** (Hz) f–3dB (Hz) f–3dB (Hz) Q1=Q2 Q3=Q4 Table 4. 8th Order Chebychev Bandpass Filter Normalized to its Center Frequency foBP = 1 Application Note 27A an28f Application Note 27A Chebyshev or Butterworth—A System Designers Confusion RdB The filter designer/mathematician is familiar with terms such as: GAIN (dB) 0 KC = tanh A 1 1 A = cosh –1 ∈ n Ripple bandwidth = 1/cosh A and AdB = 10 log [1 + ∈2(Cn2(Ω)]. –3 f1´–3dB f1ripple This is all gobbledygook (not to be confused with floobydust) to the system designer. The system designer is accustomed to –3dB bandwidths and may be tempted to use only Butterworth filters because they have the cherished –3dB bandwidths. But specs are specs and Butterworth bandpass filters are only so good. Chebyshev bandpass filters trade off ripple in the passband for somewhat steeper rolloff to the stopband. More ripple translates to a higher “Q” filter. The pain of the filter designer is sometimes tolerable to the system designer. Tables 1 through 4 are unique (we think) in that they present -3dB bandwidths for Chebyshev filters for use by system designers. Nevertheless we would be amiss to Mr. Chebyshev if we did not, at least, explain ripple bandwidth. Figure 13 shows the Chebyshev bandpass filter at frequencies near the passband. foBP f2ripple f2´–3dB AN27A F13 FREQUENCY Figure 13. Typical Chebyshev BP Filter—Close-Up of Passband It can be clearly seen that the ripple bandwidth (f1ripple – f2ripple) is the band of passband frequencies where the ripple is less than or equal to a specific value (RdB). The –3dB bandwidth is seen to be greater than the ripple bandwidth and that is the subject of much confusion on the part of the system designer. Tables 1 through 4 allow the system designer to use –3dB bandwidths to specify Chebyshev BP filters. The Chebyshev approximation to the ideal BP filter has many benefits over the Butterworth filter near the cutoff frequency. YOU CAN DESIGN WITH CHEBYSHEV FILTERS!!! We choose AMAX = 0.1dB. Now we calculate: Example 3—Design Use Table 4 to design an 8th order all pole Chebyshev bandpass filter centered at foBP = 10.2kHz with a –3dB bandwidth equal to 800Hz as shown in Figure 14. foBP fBW(–3dB) = 10.2kHz = 12.75 800Hz We can now extract from Table 4 the following line: 10 0 –3 foBP foBP/BW1 fo1(Hz) fo2(Hz) fo3(Hz) fo4(Hz) foBP/BW2 Q1=Q2 Q3=Q4 800Hz GAIN (dB) 1 –50 9.000k 12.000k foBF = 10.2kHz FREQUENCY (Hz) 10 0.977 1.023 0.945 1.058 8.20 15.7 K 37.9 29.8 Since our bandwidth ratio foBP/BW2 is not exactly on a chart line, but between two lines, we must arithmetically scale to obtain our design parameters. Our foBP/BW2 ratio lies between 8.2 and 16.39. (Remember, this is –3dB BW!) AN27A F14 Figure 14. Example 3—8th Order Chebyshev BP Filter foBP = 10.2kHz, BW = 800Hz an28f AN27A-13 Application Note 27A For a symmetrical bandpass filter the poles are symmetrical about foBP. Then: 8.2 ( fo2 – fo1) = (1.023 – 0.977) ×10.2kHz × 12.75 = 302Hz 8.2 foBP Scaling Factor = Note: 12.75 BW So our first two poles lie symmetrically about fo(10.2kHz) and are 302Hz apart: Recalling that: foBP = 12.75 and that foBP = 1, BW2(–3dB) (Because all the tables are normalized), we calculate BW2(–3dB) = .0784 Comparing the Table 4 values for AMAX = 0.1dB we note that: foBP foBP ≅ × ( Scaling Factor ) BW1(ripple) BW2(–3dB) fo2 = 10200Hz + 302Hz/2 = 10351Hz fo1 = 10200Hz – 302Hz/2 = 10049Hz For AMAX = 0.1dB, 8th order Chebyshev, this factor is approximately 0.82. For other order filters and/or different values of AMAX we can examine the corresponding chart values to find our scaling factor. The Q of these two poles is equal and is also scaled: Q1= Q2 = 15.7 × 12.75 = 24.4 8.2 So our ripple BW is: We calculate the two additional poles: BW2(–3dB) × (Scaling Factor) = BW1(ripple) 8.2 = 741Hz ( fo4 – fo3 ) = (1.058 – 0.945) ×10.2kHz × 12.75 .0784 × 0.82 = .0643 fo3 = 10200Hz – 741Hz/2 = 9830Hz fo4 = 10200Hz + 741Hz/2 = 10571Hz The Qs are: Q3 = Q4 = 37.9 × 12.75 = 58.9 8.2 Qs of this magnitude are difficult to realize no matter how the filter is realized. The filter designer should strive for Qs no greater than 20 and perhaps no greater than 10 at frequencies above 20kHz. K, for this example, is not scaled and will be equal to 29.8 from Table 4. Example 3—Frequency Response Estimation Table 4 (and also Tables 1, 2 and 3) may be used by the filter designer to obtain a good approximation to the overall shape of the bandpass filter. Referring to Figure 12 for Chebyshev filters, we may use the charts to find f3, f5, f7,…. These frequencies define the band edges at 2, 3, 4,…..times the ripple bandwidth of the Chebyshev filter. Now we can calculate f3, f5, f7,….Notice that once we find f3, f5, f7,….it does not matter where on the table our filter falls. The filter bandwidth determines f3, f5, f7,….and once we know these frequencies we can directly get our gains at these frequencies. By formula: ( fx ,fx+1) = ±nBW + (nBW )2 + 4( foBP )2 2 for our case foBP = 1 Calculating: 2BW = .1286 3BW = .1929 ±2BW + (.1286)2 + 4 2 ±3BW + (.1929)2 + 4 2 = 1.0664, 0.9378 = 1.1011, 0.9082 Example 3 specified a 10.2kHz bandpass filter with an 800Hz –3dB bandwidth. Our task, if we choose to accept it, is to convert our -3dB bandwidth to the ripple bandwidth of the filter so that we may use the tables. an28f AN27A-14 Application Note 27A Then we can denormalize to find points for our Bode plot: (f3, f4) = 0.9378 × foBP = 0.9378 × 10.2kHz = 9.566kHz 1.0664 × foBP = 1.0664 × 10.2kHz = 10.877kHz Gain = –23.4dB both f3 and f4 (f5, f6) = 0.9082 × foBP = 0.9082 × 10.2kHz = 9.264kHz 1.1011 × foBP = 1.1011 × 10.2kHz = 11.231kHz Example 3—Implementation The 10.2kHz (foBP), 8th order bandpass filter can be implemented with an LTC1064A using three sections in mode 2 and one section in mode 3. The implementation is shown briefly in Figures 15 and 16. The calculations are not shown here, but are similar to the previous hardware implementations of examples 1 and 2. Gain = –38.8dB both f5 and f6 R12 = 137k VIN R11 = 232k 1 R21 = 10k 2 R31 = 232k 3 R41 = 140k 4 5 6 5V 7 8 FROM PIN 22 R13 = 255k R43 = 85.6k 9 R33 = 549k 10 R23 = 10k 11 12 INVB INVC HPB/NB HP/NC BPB BPC LPB LPC SB LTC1064 A.GND + V SA SC V– CLK 50/100 LPA LPD BPA BPD HPA HPD INVA INVD 24 23 R22 = 10k 22 R32 = 243k TO R13 21 20 19 18 R42 = 1.02M –5V 1MHz 17 R44 = 10.5k R34 = 604k VOUT R24 = 10.2k R14 = 78.7k AN27A F15 Figure 15. LTC1064 Implementation Pinout—10.2kHz 8th Order BPF an28f Information furnished by Linear Technology Corporation is believed to be accurate and reliable. However, no responsibility is assumed for its use. Linear Technology Corporation makes no representation that the interconnection of its circuits as described herein will not infringe on existing patent rights. AN27A-15 Application Note 27A R41 = 140k R31 = 232k SECTION 1 MODE 2 fo1 = 10.351kHz Q1 = 24.4 R21 = 10k VIN – R11 = 232k INVB(1) HPB/NB(2) + + – AGND(6) BPB(3) ∫ LPB(4) ∫ SB(5) R12 = 137k R42 = 1.02M R32 = 243k R22 = 10k – INVC(24) SECTION 2 MODE 2 fo2 = 10.049kHz Q2 = 24.4 LPC(21) HPC/NC(23) + + – AGND(6) BPC(22) ∫ ∫ SC(20) R13 = 255k R43 = 85.6k 5V V+(7) R33 = 549k R23 = 10k – INVA(12) 50/100(17) HPA/NA(11) + + – BPA(10) ∫ LPA(9) 1MHz CLK(18) ∫ V–(19) AGND(6) SECTION 3 MODE 2 fo = 10.571kHz Q3 = 58.9 –5V SA(8) R14 = 78.7k R44 = 10.5k R34 = 604k R24 = 10.2k – INVD(13) HPD(14) BPD(15) ∫ + ∫ LPD(16) SECTION 4 MODE 3 fo = 9.830kHz Q = 58.9 AN27A F16 VOUT AGND(6) NUMBERS IN PARENTHESIS ARE PIN NUMBERS OF LTC1064 ALL RESISTORS 1% Figure 16. Implementation of 10.2kHz 8th Order BPF—Section by Section for LTC1064 an28f Linear Technology Corporation McCarthy Blvd., Milpitas, CA 95035-7417 AN27A-16 1630 (408) 432-1900 FAX: (408) 434-0507 www.linear.com ● ● IM/GP 988 20K • PRINTED IN USA LINEAR TECHNOLOGY CORPORATION 1988

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