SYNTHESIS TECHNIQUES FOR ACTIVE RC FILTERS USING OPERATIONAL AMPLIFIERS by He man i Mationg Jov_e\r United States Naval Postgraduate School 4 2 V 2 THESIS SYNTHESIS TECHNIQUES FOR ACTIVE RC FILTERS USING OPERATIONAL AMPLIFIERS by Hernani Mationg Jover June 1970 Tkl& document k<u bzzn appnjovtd ^on pabtic <*e- lexuz and 6aZz; lti> (tu&vLbution ii> unturuXzd. T136104 Synthesis Techniques for Active RC Filters Using Operational Amplifiers by Hernani Mationg Jover Lieutenant, Philippine Navy B.S., Philippine Military Academy, 1962 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN ELECTRICAL ENGINEERING from the NAVAL POSTGRADUATE SCHOOL June 1970 I 'OSTGRADUATE SCHOOL 3Y, CALIF. 93940 ABSTRACT The stimulus provided by the trend towards microminiaturization has generated considerable interest in active RC filters utilizing operational amplifiers. A comprehensive and unified presentation of the many different techniques to synthesize such filters is made along the lines of modern filter theory in this thesis. TABLE OF CONTENTS I. INTRODUCTION 15 A. DISADVANTAGES OF INDUCTORS 15 B. DEFECTS OF PASSIVE RC FILTERS 16 C. EVOLUTION OF THE ACTIVE RC FILTER 16 D. ADVANTAGES OF THE OPERATIONAL AMPLIFIER 17 E. PREVIEW AND PURPOSE 18 II. MODERN FILTER THEORY 19 A. HISTORY AND EVOLUTION 19 B. TYPES OF FILTERS 2 C. THE FILTER DESIGN PROBLEM 21 D. FILTER CHARACTERISTICS 24 1. Butterworth Class 2 5 2. Chebyshev Class 26 3. Bessel Class 33 4. Comparison of Classes 34 E. FREQUENCY TRANSFORMATIONS 34 F. FREQUENCY SCALING 39 G. THE FILTER DESIGN PROCEDURE 39 H. EXAMPLES 42 III. THE OPERATIONAL AMPLIFIER 53 A. DEFINITION 53 B. ANALYSIS 55 C. NATHAN'S MATRIX ANALYSIS 60 IV. INFINITE -GAIN SINGLE -FEEDBACK REALIZATIONS 61 A. THEORETICAL DEVELOPMENT 61 B. BRADLEY AND McCOY [1952] 66 C. MATHEWS AND SEIFERT [1955] 70 D. PAUL [1963] 81 E. HAKIM [1964] 92 F. MUIR AND ROBINSON [1968] 102 G. ADVANTAGES AND DISADVANTAGES 105 V. INFINITE -GAIN MULTIPLE -FEEDBACK REALIZATIONS 107 A. THEORETICAL DEVELOPMENT 107 B. RAUCH AND NICHOLS [1956] 110 C. BRIDG MAN AND BREN NAN [195 7] 112 D. WADHWA [19,62] 116 E. FOSTER [1965] 121 F. HOLT AND SEWELL [1965] 126 G. ADVANTAGES AND DISADVANTAGES 132 VI. INFINITE -GAIN GENERAL REALIZATION TECHNIQUES 136 A. LOVERING [1965] 136 B. BRUGLER [1966] 142 C. ADVANTAGES AND DISADVANTAGES 148 VII. INFINITE-GAIN STATE-SPACE REALIZATIONS 150 A. THEORETICAL DEVELOPMENT 150 B. KERWIN, HUELSMAN, AND NEWCOMB [1967] 155 C. HOLLENBECK [1969] 162 D. SALERNO [1969] 167 E. TOW [1969] 173 F. ADVANTAGES AND DISADVANTAGES 184 VIII. CONCLUSION 185 APPENDIX A. TOW'S FREQUENCY TRANSFORMATION PROCEDURE 187 APPENDIX B. COMPUTER PROGRAM 1 190 APPENDIX C. NATHAN'S METHOD OF MATRLX ANALYSIS 192 BIBLIOGRAPHY 195 INITIAL DISTRIBUTION LIST 199 FORM DD 1473 20 1 LIST OF TABLES I. Table 2-1. The Butterworth Polynomials 2 7 II. Table 2-2. Roots of the Butterworth Polynomials 2 7 III. Table 2-3. Polynomials for Chebyshev Filters 30 IV. Table 2-4. Roots of Chebyshev-derived Polynomials 31 V. Table 2-5. The Bessel Polynomials 35 VI. Table 2-6. Roots of the Bessel Polynomials 35 VII. Table 2-7. Percentage Overshoot and Rise Time of the 37 Three Filter Classes VIII. Table 4-1. Some Transfer Functions Realizable by 84 Figure 4-13 DC. Table 4-2. Transfer Admittance Relationship of the 84 Three Type 1 T-Networks X. Table 5-1. Normalized Capacitor Values for Butterworth 125 Filter LIST OF ILLUSTRATIONS 1. Figure 2-1. Ideal magnitude and phase characteristics of 22 the four different filter types 2. Figure 2-2. Attenuation characteristics of the four filter 23 types 3. Figure 2-3. Butterworth low-pass amplitude responses for 28 various values of n 4. Figure 2-4. A fifth-order Chebyshev approximation to a 32 low-pass filter 5. Figure 2-5. Normalized magnitude responses of various 36 fifth-order filters 6. Figure 2-6. Magnitude vs Frequency Plot of Filter 43 Transfer Function of Example 2-1 7. Figure 2-7. Magnitude vs Frequency Plot of Filter 45 Transfer Function of Example 2-2 8. Figure 2-8. Magnitude vs Frequency Plot of Filter 47 Transfer Function of Example 2-3 9. Figure 2-9. Magnitude vs Frequency Plot of Filter 48 Transfer Function of Example 2-4 10. Figure 2-10. Magnitude vs Frequency Plot of Filter 50 Transfer Function of Example 2-5 11. Figure 2-11. Magnitude vs Frequency Plot of Filter 52 Transfer Function of Example 2-6 12. Figure 3-1. Circuit model for the differential-input 54 type of operational amplifier 13. Figure 3-2. Idealized model of the differential-input 54 type of operational amplifier 14. Figure 3-3. Common circuit symbols for the 56 operational amplifier 15. Figure 3-4. Fundamental inverting amplifier circuit 56 16. Figure 3-5. Complex linear feedback circuit 59 9 Z 17. Figure 3-6. Feedback circuit with neither input to the 59 operational amplifier grounded 18. Figure 4-1. Basic single-feedback active filter 62 configuration 19. Figure 4-2 . Differential-input operational amplifier 62 using resistors for input and feedback elements 20. Figure 4-3. Single-ended operational amplifier in 65 basic single-feedback configuration 21. Figure 4-4. Active network configuration of Bradley 65 and McCoy 22. Figure 4-5. Transfer impedance networks from the table 69 of Bradley and McCoy 23. Figure 4-6. Circuit realization of Example 4-1 71 24. Figure 4-7. Active network configuration of Mathews 71 and Seifert 25. Figure 4-8. Location of poles and zeros for 2-terminal 73 RC network configuration 26. Figure 4-9. Second Foster form realization of an RC 73 admittance function 27. Figure 4-10. Circuit realization of y (s) and y (s) 78 1 2, Bl 1 IB 1 ?8 28. Figure 4-11. Circuit realization of y (s) and y 11DO (s) 1 dZ 1 IB/ 29. Figure 4-12. Circuit realization of Example 4-2 80 30. Figure 4-13. Paul's active network configuration 82 31. Figure 4-14. Type 1 T-Networks for Paul's configuration 83 32. Figure 4-15. Type 2 T-Network for Paul's configuration 87 33. Figure 4-16. Type 3 Network for Paul's configuration 87 34. Figure 4-17. Paul's circuit to simulate 89 - V (s)A.(s) = 1/(1 + s T ) o 1 10 35. Figure 4-18. Paul's circuit to simulate 89 2 2 - V (s)A, (s) = 1/(1 + k 2s T + s T ) o 1 1 36. Figure 4-19. Circuit realization of Example 4-3 91 37. Figure 4-20. Hakim's active network configuration 97 4-21. 99 38. Figure Circuit realization of y (s) and y, , D1 (s) IZdI 1 Id 1 39. Figure 4-22. Circuit realization of y (s) and y (s) 99 1 2 B2 1 1B2 40. Figure 4-23. Circuit realization of Example 4-4 101 41. Figure 4-24. Active network configuration of Muir and 103 Robinson 42. Figure 4-2 5. Basic circuit block used in triple amplifier 103 realization of transfer functions 43. Figure 4-26. Circuit realization of Example 4-5 106 44. Figure 5-1. General infinite-gain multiple-feedback 108 configuration 45. Figure 5-2. Second-order prototype of multiple- 109 feedback configuration * 46. Figure 5-3. Second-order low-pass filter network of 109 Rauch and Nichols 47. Figure 5-4. Circuit realization of Example 5-1 113 48. Figure 5-5. High-pass active network configuration 113 of Bridgman and Brennan 49. Figure 5-6. Bandpass active network configurations of 115 Bridgman and Brennan 50. Figure 5-7. Wadhwa's third-order active network 117 configuration 51. Figure 5-8. Wadhwa's configuration to simulate 119 3 2 H(s) = - a /(b s +b s +b s + 1) Q 3 2 x 52. Figure 5-9. Wadhwa's active network configuration 12 3 2 to simulate H(s) = - a,s/(b s + b s + b.s + 1) 11 53. Figure 5-10. Wadhwa's active network configuration 120 2 3 2 to simulate H(s) = - as /(b s + b s + b s + 1) c» O c* X 54. Figure 5-11. Third-order Rauch active filter 123 123 55. Figure 5-12 . Circuit realization of Example 5-3 56. Figure 5-13. Configuration of Holt and Sewell 12 7 57. Figure 5-14. Active section of Holt and Sewell 127 58. Figure 5-15. Active network of Holt and Sewell to 127 2 2 realize H(s) = (s + A s + A )/(s +bs+b) 59. Figure 5-16. Network realization of passive section 133 60. Figure 5-17. Circuit realization of Example 5-4 134 61. Figure 6-1. Lovering's active network configuration 137 62. Figure 6-2. Circuit realization of Example 6-1 141 63. Figure 6-3. Circuit realization of Example 6-2 141 64. Figure 6-4. Brugler's active network configuration 143 65. Figure 6-5. Circuit realization of Example 6-3 146 66. Figure 6-6. Circuit realization of Example 6-4 146 67. Figure 7-1. Signal flow graph of general n -order 153 open-circuit voltage transfer function 68. Figure 7-2. A summer and an integrator 154 69. Figure 7-3. Signal flow graph of general second-order 157 open-circuit voltage transfer function 70. Figure 7-4. Second-order building-block network 158 of Kerwin, Huelsman, and Newcomb 71. Figure 7-5. First-order lag and lead networks 163 72. Figure 7-6. A universal hybrid active filter 164 12 73. Figure 7-7. Schematic representation of Salerno's 169 realization of the general second-order transfer function 74.