DESIGN IDEAS How to Design High Order Filters with Stopband Notches Using the LTC1562 Operational Filter (Part 2) by Nello Sevastopoulos This is the second in a series of articles describing applications of the LTC1562 connected as a lowpass, highpass or bandpass filter with added stopband notches to increase selectivity. Part 1 (Linear Technology VIII:2, May 1998, pp. 28–31) described one method of coupling the four Operational Filter™ building blocks of the LTC1562 to design an 8th order lowpass filter with two stopband notches. Part 2 expands the technique of Part 1 to design an 8th order bandpass filter with two stopband notches. Throughout this series of articles, notches will be generated by first summing the input signal with a 180 degree out-of-phase signal appearing at the output(s) of the LTC1562 Operational Filter and second, by adjusting the summation gains to yield a zero sum. Part 1 showed one proprietary method of creating notches in the stopband of a lowpass filter. The essence of this method is briefly revisited in Figure 1, where two of 1/2 LTC1562 four Operational Filter sections are coupled to form a 4th order lowpass filter with one stopband notch. The notch is obtained by summing the input signal, VIN, with the output, V1A, into the inverting node of the next section of the IC. The two signals, VIN and V1A, will tend to cancel each other at a frequency where they are 180 degrees out of phase. The cancellation will be complete if the amplitudes of VIN and VIA yield equal (and opposite) currents at the summing junction of the op amp of Figure 1, that is if: RIN2 = RFF2 • (RQ1/RIN1) (1) In Figure 1, the lead capacitor CIN1 raises the frequency where a 180 degree phase shift occurs above the center frequency of the 2nd order section (fO). The resulting notch frequency is then higher than the cutoff frequency of the 4th order filter. Figure 1 can be easily modified to make the frequency of the notch lower than the center frequency of the 2nd C – C R21 R1 • • RQ1 CIN1 RIN1 (R1 = 10k; C = 159.15pF) and the gain conditions dictating Equation 1 now translate to: RIN2 = RFF2 • ( ( RQ1 CIN1 • C R1 (3) The circuit of Figure 2 can be used to build a 4th order bandpass filter with one notch below its center frequency. Such a filter can simultaneously detect a tone and reject an unwanted frequency located in the vicinity of the passband. RFF2 RIN1 1 20 CIN1 RQ2 CIN1 R21 R21 (2) RQ1 + + RQ1 1– VIN – 1 VIN fN2 = fO1 • C RIN2 RIN1 order section from which it is derived. This is useful in bandpass filters where an unwanted frequency lower than the center frequency of the filter must be rejected. This is shown in Figure 2, where the input signal is summed with output V2A instead of output V1A. The frequency of the resulting notch is: 2 V1A 19 V1B R22 2 V1A LTC1562 3 3 V2A 1 sCR1 1 sCR1 18 V2B R1, C ARE PRECISION INTERNAL COMPONENTS R1 = 10k; C = 159.15pF Figure 1. Two out of four Operational Filter sections are coupled to form a 4th order lowpass filter with one stopband notch. Linear Technology Magazine • February 1999 V2A (OTHER CONNECTIONS AS SHOWN IN FIGURE 1) Figure 2. Figure 1’s circuit modified to make the frequency of the notch lower than the center frequency of the 2nd order section from which it is derived. 31 DESIGN IDEAS 20 Table 1. Parameters of the four sections of an 8th order, 100kHz bandpass filter GAIN (dB) –20 –40 –60 –65dB BANDWIDTH –80 –100 –120 50 60 70 80 90 100 110 120 130 140 150 FREQUENCY (kHz) Figure 3. Theoretical amplitude response of 8th order, 100kHz bandpass filter The notch techniques of Figures 1 and 2 will be referred as “feedforward.” This is necessary to separate these techniques from others to be shown later, in Part 3 of this series of articles. The feedforward notch technique of Figure 2 can be advantageously combined with Figure 1 to realize sharp bandpass filters with two stopband notches: one notch below and one above the center frequency. Filters of this type can be very selective, although they are quite cumbersome to design. A step-by-step design procedure is illustrated below. A Practical Example An 8th order 100kHz bandpass filter is realized, through FilterCAD™ for Windows® (available at no charge from Linear Technology—see the “Design Tools” page in this issue), by cascading four 2nd order sections of equal Q. The –3dB band-edges are arithmetrically symmetric with respect to the filter’s 100kHz center frequency and signals below 80kHz and above 125kHz are attenuated by 60dB or more. Figure 3 shows the theoretical amplitude response and Table 1 shows the desired filter parameters, namely, the center frequencies, Qs and notch frequencies. The filter of Figure 3/Table 1 can be realized by decomposing the 8th order realization into two independent 4th order filter sections and then cascading these two 4th order sections, which is an easier task than designing an 8th order elliptic bandpass filter all at once. FilterCAD, in custom mode, 32 fO 99.9687e3 96.9964e3 103.0322e3 100.0000e3 Q 10.0000 10.0000 10.0000 10.0000 fN ——— 129.2814e3 77.3023e3 ——— should be used to perform this operation. Figure 4 and Table 2 show the filter decomposition and the cascading sequence; note the left and right notches. Figure 5 uses the LTC1562 Operational Filter to realize the filter of Figure 3 as decomposed in Figure 4. The design is split into two 4th order sections. The algorithm to calculate the external passive components is outlined below. In order to obtain a practical realization that closely approximates the theoretical one, the Q of each 2nd order section will be lowered by 15%. (Please consult the LTC1562 final data sheet.) In order to follow the long and tedious algorithm below, consider the intuitive outline: We need to calculate the following set of passive components for the first 4th order section: RIN1, CIN1, R21, RQ1, and RIN2, RFF2, R22 and RQ2. The resistors R21, RQ1, QN ——— ——— ——— ——— Type BP LPN HPN BP R22 and RQ2 are easily calculated via the expression for the center frequency, fOi, and Qi for the 2nd order section “i.” The expression for the notch, equation (2), involves the product of RIN1 • CIN1, so neither component can be calculated separately. Instead, RIN1 is calculated by considering the maximum gain (which occurs around the center frequency fO1) at either node V1A or V2A. This controls premature internal clipping. Once RIN1 is set, CIN1 is easily calculated via equation (2) for the lower band notch. Similarly, equation (3) defines the ratio of RIN2 to RFF2, so neither of these components can be calculated independently of the other. R FF2 is calculated by considering the gain factor (“GAIN”) of the 4th order filter section at the V1B output (Figure 1/ Table 2)). Once RFF2 is set, RIN2 is calculated via equation (3). 20 0 –20 GAIN (dB) 0 –40 –60 –80 –100 50 60 70 80 90 100 110 120 130 140 150 FREQUENCY (kHz) 50 60 70 80 90 100 110 120 130 140 150 FREQUENCY (kHz) Figure 4. Cascading two 4th order bandpass sections to realize the filter of Figure 3. Table 2. Filter decomposition and cascading sequence fO1 = 96.9964k Q1 = 10 fO2 = 99.9687k Q2 = 10 fO3 = 100k fN2 = 77.3k Q3 = 10 fO4 = 103.0322k Q4 = 10 fN2 = 129.2814k MH(s) = GAIN • N(s)/D(s) MH(s) = GAIN • N(s)/D(s) MGAIN = 0.2823 MGAIN = 0.1788 MN(s) = A1s(s2 + 235 • 9072 • 109) MN(s) = A1s(s2 + 659 • 83 • 109) MA1 = 62.8122 • 103 MA1 = 62.8319 • 103 Linear Technology Magazine • February 1999 DESIGN IDEAS 20 RIN1 = RQ1 • I. Calculate the passive components of the of the first 4th order section (fO1 = 96.9964kHz, Q = 8.5, fO2 = 99.9687kHz, Q = 8.5, fn2 = 77.3kHz) 1. Calculate the center frequencysetting resistor, R21: (For details, please refer to the LTC1562 data sheet.) R21 = (100kHz/fO1)2 • 10k = 10.629k (choose the closest 1% value, R21 = 10.7k (1%)) 2. Calculate the Q-setting resistor, RQ1: (For details, please refer to the LTC1562 data sheet) RQ1 = Q1 √R21 • 10k = 87.925k (choose the closest 1% value, RQ1 = 86.6k (1%)) 3. Calculate the input resistor RIN1 from the following expression(s): 3a. if fO1 ≤100kHz (for LTC1562) 1 1+ (5) ( Q12 • 1 – 0 2 2 fN2 fO1 2 ( Make sure, in either case 3a or 3b, that RIN1 is greater than R21, that is, the DC gain at pin 3 in Figure 5 is less than unity; if not set RIN1 = R21 and proceed to step 4a. The expression for RIN1 sets the gain at fO1 equal to unity at the node of maximum swing (V1A or V2A). Note that, for high Qs, the gain at fO1 is the maximum gain. If you know the spectrum of the signals that will be applied to the filter input and if internal gains higher than unity will be allowed, the value of RIN1 can be reduced to improve the input signal-to-noise ratio. 4a. Use the value of RIN1, calculated above, and calculate the value for the input capacitor CIN1 from the notch equation (2). –20 GAIN (dB) The same design method is later repeated to derive the passive components for the second 4th order section: –40 –60 –65dB BANDWIDTH –80 –100 –120 50 60 70 80 90 100 110 120 130 140 150 FREQUENCY (kHz) Figure 6. Measured amplitude response of Figure 5’s filter 4b. Recalculate the value of RIN1 after CIN1 is chosen. RIN1 = (CIN1(ideal) RIN1(ideal))/ CIN1(NPO,0402) = 96.22k Choose the closest 1% value: RIN1 = 95.3k (1%) 5. Calculate the frequency- and Qsetting resistors R22, RQ2, as done in steps 1 and 2, above. Choose the closest 1% standard 1 resistor values. (6) R21 R1 2 R22 = 10k (1%); CIN1 = fN2 • • C RQ1 RIN1 1– RQ1 = 84.5k(1%) fO12 6. Calculate the feedforward resistor, RFF2: 1 (4) (fN1 < fO1; C = 159.15pF) RIN1 = Q1 • R21 • 1 + 1/(RFF2 C) = Gain • A1; 2 2 2 1 – fN2 C = 159.15pF Q1 • CIN1 = 5.639pF. fO12 The values for parameter (Gain • Use the commercially available NPO A1) are provided by FilterCAD; they RIN1 = 95.56k type 0402 surface mount capacitor relate to the coefficients of the nuAlthough not applicable for this with the value nearest the ideal value merator of the transfer function (V1B/ example, thoroughness dictates men- of CIN1 calculated above. For instance, VIN in Figure 1); a passband AC gain of tioning the case below: for CIN1, choose an off-the-shelf 5.6pF unity is assumed (see Table 2). Please 3b. if fO1 ≥ 100kHz (for LTC1562) capacitor. note that, for a lowpass case, as in Part 1 of this article series, the value of (Gain • A1) is the DC gain of the RFF2, 357k filter and its value can be easily set CIN1, 5.6pF RIN2, 110k without software assistance. Equating the numerator of the fil16 1 INV C VIN INV B R , 84.5k ter transfer function with the values RIN1, 10.7k RQ1, 86.6k Q2 15 2 V1 C V1 B provided by FilterCAD: R22, 10k R21,10.7k 14 3 ( ( ( ( V2 C V2 B 4 5V 0.1µF 5 6 RIN3, 294k R23, 10k 7 RQ3, 84.5k 8 CIN3, 18pF V– V + LTC1562 SHDN AGND V2 A V2 D V1 A V1 D INV A INV D 13 12 –5V 0.1µF 11 10 R24, 9.53k 9 RQ4, 82.5k GAIN = 0.2823 A1 = 62.8122 • 103 A2 = (2πfN2)2 = 235.9 • 109 RIN4, 95.3k RFF4, 332k V1B s(s2 + ωN22) GAIN (A1s)(s2 + A2) (7) = = VIN (RFF2 • C) • D(s) D(s) VOUT 1562 TA03 RFF2 = 1/((Gain A1) C) = 354.35k; C = 159.15pF RFF2 = 357k(1%) Figure 5. Hardware realization of the filter in Figure 3, using all four sections of an LTC1562 Linear Technology Magazine • February 1999 33 DESIGN IDEAS VIN(RMS), fOUT = 100kHz 5 4a. Use the theoretical value for RIN3, calculated above, and calculate the value of the input capacitor CIN3 from the notch equation (2) of part 1 of this article; for convenience this is repeated below: VS = ±5V 1 f 2 R CIN3 = C • Q3 • 1 – O3 RIN3 fN42 ( 0.1 0.1 1 5 ( condition for the occurrence of a notch. For convenience, this gain condition is repeated below. RIN4 = RFF4 • RQ3 RIN3 (12) RIN4 = 95.422k; RIN4 = 95.3k(1%) (10) Experimental Results Figure 6 shows the measured amplitude response of the filter of Figure 5. Use a commercially available NPO- The values of the passive component type 0402 surface mount capacitor are as calculated above and as shown with the value nearest the ideal value in Figure 5. The measured amplitude of CIN3 calculated above. For instance, response closely approximates the 7. Solve for RIN2 by using Equation C IN3 = 18pF. ideal response as synthesized by Fil(3), which dictates the gain 4b. Recalculate the value for RIN3 terCAD. The peak frequency with condition for the occurrence of calculated in step 3a after CIN3 standard 1% resistor values and 5% the notch: is chosen. capacitor values is 100.65kHz (0.65% RIN2 = (RFF2 RQ1 CIN1)/(R1 C) = 108.785k; (R1,C) = (10k, 159.15pF) RIN3 = (CIN3(ideal) RIN3(ideal))/CIN3(NPO,0402) off). The higher frequency notch, although it shows a respectable depth RIN2 = 110k (1%) = 300.058k of 70dB, is not as well defined as the RIN3 = 294k (1%) notch below the filter’s center freII. Calculate the passive 5. Calculate the frequency- and quency, yet the –65dB bandwidth is components of the second 4th Q-setting resistors, R24 and as predicted by FilterCAD. The 10dB order section RQ4, as done in steps 1 and 2, lack of the upper band notch depth is (fO3 = 100kHz, Q3 = 8.5, fO4 = above. Choose the nearest 1% due to the finite speed of the internal 103.0322kHz, Q4 = 8.5, fn4 = standard value. op amps; they cause the practical 180 129.2814kHz) degree phase shift frequency and the R24 = 9.42k; R24 = 9.53k (1%) Except for the bandpass gain gain at V1A’s output to depart slightly RQ4 = 82.97k; RQ4 = 82.5k (1%) calculations, the algorithm will from the theoretical calculations. be the same as the lowpass 6. Calculate the feedforward For the sake of perfection, the notch design of Part 1 of this article. resistor, RFF4. First equate the depth can be easily restored by tweak1. R23 = (100kHz/fO3)2 • 10k = numerator of the 4th order filter ing the value of RQ3; the new RQ3 will 10k (1%) transfer function with the be 75k. This is shown with dashed 2. RQ3 = Q3 √R23 • 10k = 85k, values provided by FilterCAD lines in Figure 6. This, however, lowRQ3 = 84.5k (1%) (see Table 2): ers the passband gain by the ratio of 3. Calculate the input resistor RIN3 the new to the old RQ3 value, that is, from the following expression(s): ωO32 s2 + ωN42 (11) by about –1.0dB (you cannot fool VOUT s 3a. if fO3 ≤ 100kHz (for LTC1562) = = • • V1B mother nature). Depending on the D(s) RFF4 • C ωO42 2 2 (8) application, the 10dB of additional 1 + 1 – fO3 • Q32 RIN3 = Q3 • R23 • GAIN • A1s • (s2 + ωN42) notch depth for 1.5dB of passband fN42 D(s) gain loss may be a reasonable trade. The passband gain can also be cor2 ωO3 RIN3 = 302.41k 1 1 rected by lowering the values of either THEN RFF4 = • • 2 GAIN • A1 C ω N4 3b. if fO3 ≥ 100kHz (for LTC1562) pair, (RFF2, RIN2) or (RFF4, RIN4), by the GAIN = 0.1788 same amount (1.5dB). In Figure 6, 2 (9) A1 = 62.8319 • 103 fO32 the gain was restored to 0dB by chang2 1 + 1 – • Q3 RIN3 = RQ3 • 2 ing the values of RIN2, RFF2 to 93.1k fN4 and 300.1k respectively. RFF4 = 334.64k, choose RFF4 = 332k The total integrated noise was an For fO3 = 100kHz, as in the example (1%). impressively low 69µVRMS, allowing a above, either expression can be used. 7. Solve for RIN4 by using equation signal-to-noise ratio well in excess of Note that the expression for RIN3 in (1) of Part 1 of this article, 80dB. The input signal-to-noise ratio 3b, above, is the same as expression which dictates the gain can be further increased if the passfor RIN1 shown in Part 1 of this article. VOUT(RMS), fOUT = 100kHz CIN3 = 17.86pF; Figure 7. Gain linearity of Figure 5’s filter, measured at the 100kHz theoretical center frequency ( ( 34 ( ( Linear Technology Magazine • February 1999 CONTINUATIONS range. This is true provided the filter magnitude response does not change with varying input signal levels, that is, the filter gain is linear. The gain linearity measured at the 100kHz theoretical center frequency of the filter is shown in Figure 7. The gain is perfectly linear for input amplitudes up to 1.25VRMS (3.5VP-P) so an 84dB dynamic range can be claimed. The input signal, however, can reach amplitudes up to 3VRMS (8.4VP-P, 92dB SNR) with some reduction in gain linearity. The LTC1735 and LTC1736 are the latest members of Linear Technology’s family of constant frequency, N-channel high efficiency controllers. With new protection features, improved circuit operation and strong MOSFET drivers, the LTC1735 is an ideal upgrade to the LTC1435/LTC1435A for higher current applications. With the integrated VID control, the LTC1736 is ideal for CPU power applications. The high performance of these controllers with wide input range, 1% reference and tight load regulation makes them ideal for next generation designs. LTC1562-2, continued from page 10 References level is 44µVRMS over a bandwidth of 800kHz or 98dB below the maximum unclipped output. 1. Hauser, Max. “Universal Continuous-Time Filter Challenges Discrete Designs.” Linear Technology VIII:1 (February 1998), pp. 1–5 and 32. 2. Sevastopoulos, Nello. “How to Design High Order Filters with Stopband Notches Using the LTC1562 Quad Operational Filter, Part 1.” Linear Technology VIII:2 (May 1998), pp. 28-31. 3. Sevastopoulos, Nello. “How to Design High Order Filters with Stopband Notches Using the LTC1562 Quad Operational Filter, Part 2.” in the Design Ideas section of this issue of Linear Technology. 4. LTC1562 Final Data Sheet. 5. For example: Schwartz, Mischa. Information Transmission, Modulation, and Noise, fourth edition, pp. 180–192. McGraw-Hill 1990. band gain can be higher than 0dB or if internal nodes are allowed to have gains higher than 0dB. Please contact the LTC Filter Design and Applications Group for further details. The low noise behavior of the filter makes it useful in applications where the input signal has a wide voltage LTC1735/LTC1736, continued from page 6 Conclusion Acknowledgments Philip Karantzalis and Nello Sevastopoulos of LTC’s Monolithic Filter Design and Applications Group contributed to the application examples. LT1505, continued from page 25 SW, VBAT and GND in Figure 2 will help in spreading the heat and will reduce the power dissipation in conductors and MOSFETs. By doing so, the required peak power from the wall adapter can be much lower than the peak power required by the load. The wall adapter has to supply the average power only. The LT1505 can also be used in other system topologies, such as the telecom application shown in Figure 5. The circuit in Figure 5 uses the battery to supply peak power demands. Conclusion The LT1505 is a complete, singlechip battery charger solution for today’s demanding charging requirements in high performance laptop applications. The device requires a small number of external components and provides all necessary functions for battery charging and power management. High efficiency and small size allow for easy integration with the laptop circuits. Also, by adding a simple external circuit, charging can be easily controlled by the host computer, allowing for more sophisticated charging schemes. Step-Down Conversion, continued from page 30 cuitry works in the same manner as in Figure 1. Efficiency and performance are virtually the same as the LTC1649 solution, but parts count and system cost are lower. In a 3.3V to 2.5V application, the steady-state, no-load duty cycle is 76%. If the input supply drops to 3.135V (3.3V – 5%), the duty cycle requirement rises to 80% at no load, and even higher under heavy or transient load conditions. Both the LTC1649 and the LTC1430A guarantee a maximum duty cycle of greater than 90% to provide acceptable load regulation and transient response. The standard LTC1430 (not the LTC1430A) can max out as low as 83%—not high enough for 3.3V to 2.5V circuits. Applications with larger step-down ratios, such as 3.3V to 2.0V, can use the circuit in Figure 3 successfully with a standar d LTC1430. Other Applications lower cost LTC1430A replacing the LTC1649. The LTC1430A does not include the 3.3V to 5V charge pump and requires a 5V supply to drive the external MOSFET gates. The current drawn from the 5V supply depends on the gate charge of the external MOSFETs but is typically below 50mA, regardless of the load current on the 2.5V output. The drains of the Q1/Q2 pair draw the main load current from the 3.3V supply. The remaining cirLinear Technology Magazine • February 1999 35