Continuous-Time Active Filter Design (Electronic Engineering Systems)

This book presents the design of active RC filters in continuous time. Topics include-- filter fundamentals-- active elements-- realization of functions using opamps-- LC ladder filters operational transconductance amplifier circuits (OTACs)-- MOSFET-C filters. Continuous-Time Active Filter Design uses wave variables to enable the reader to better understand the introduction of more complex variables created through linear transformations of voltages and currents. Intended for undergraduate students in electrical engineering, Continuous-Time Active Filter Design provides chapters as self-contained units, including introductory material leading to active RC filters. Continuous-Time Active Filter Design 1 Preface 5 Authors 8 Contents 9 2573ch01.pdf 1 Continuous-Time Active Filter Design 18 Contents -1 Chapter 1: Filter Fundamentals 19 1.1 Introduction 19 1.2 Filter Characterization 19 1.2.1 Lumped 19 1.2.2 Linear 20 1.2.3 Continuous-Time and Discrete-Time 21 1.2.4 Time-Invariant 21 1.2.5 Finite 21 1.2.6 Passive and Active 21 1.3 Types of Filters 22 1.4 Steps in Filter Design 24 1.5 Analysis 25 1.5.1 Nodal Analysis 25 1.5.2 Network Parameters 28 1.5.2.1 One-Port Network 28 1.5.2.2 Two-Port Network 29 1.5.3 Two-Port Interconnections 32 1.5.3.1 Series–Series Connection 32 1.5.3.2 Parallel–Parallel Connection 33 1.5.3.3 Series Input–Parallel Output Connection 34 1.5.3.4 Parallel Input–Series Output Connection 34 1.5.3.5 Cascade Connection 34 1.5.4 Network Transfer Functions 35 1.6 Continuous-Time Filter Functions 37 1.6.1 Pole-Zero Locations 38 1.6.2 Frequency Response 39 1.6.3 Transient Response 40 1.6.3.1 Impulse Response 40 1.6.3.2 Step Response 41 1.6.4 Step and Frequency Response 42 1.7 Stability 44 1.7.1 Short-Circuit and Open-Circuit Stability 45 1.7.2 Absolute Stability and Potential Instability 45 1.8 Passivity Criteria for One- and Two-Port Networks 47 1.8.1 One-Ports 47 1.8.2 Two-Ports 48 1.8.3 Activity 49 1.8.4 Passivity and Stability 49 1.9 Reciprocity 50 1.10 Summary 51 References and Further Reading 51 2573ch02.pdf 1 Continuous-Time Active Filter Design 53 Contents -1 Chapter 2: The Approximation Problem 54 2.1 Introduction 54 2.2 Filter Specifications and Permitted Functions 54 2.2.1 Causality 54 2.2.2 Rational Functions 55 2.2.3 Stability 55 2.3 Formulation of the Approximation Problem 56 2.4 Approximation of the Ideal Lowpass Filter 57 2.4.1 Butterworth or Maximally Flat Approximation 58 2.4.2 Chebyshev or Equiripple Approximation 61 2.4.3 Inverse Chebyshev Approximation 64 2.4.4 Papoulis Approximation 66 2.4.5 Elliptic Function or Cauer Approximation 66 2.4.6 Selecting the Filter from Its Specifications 68 2.4.7 Amplitude Equalization 71 2.5 Filters with Linear Phase: Delays 71 2.5.1 Bessel-Thomson Delay Approximation 73 2.5.2 Other Delay Functions 77 2.5.3 Delay Equalization 78 2.6 Frequency Transformations 78 2.6.1 Lowpass-to-Lowpass Transformation 79 2.6.2 Lowpass-to-Highpass Transformation 80 2.6.3 Lowpass-to-Bandpass Transformation 81 2.6.4 Lowpass-to-Bandstop Transformation 82 2.6.5 Delay Denormalization 83 2.7 Design Tables for Passive LC Ladder Filters 83 2.7.1 Transformation of Elements 84 2.7.1.1 LC Filters 84 2.7.1.2 Active RC Filters 87 2.8 Impedance Scaling 89 2.9 Predistortion 90 2.10 Summary 91 References 92 2573ch03.pdf 1 Continuous-Time Active Filter Design 94 Contents -1 Chapter 3: Active Elements 95 3.1 Introduction 95 3.2 Ideal Controlled Sources 95 3.3 Impedance Transformation (Generalized Impedance Converters and Inverters) [1, 2] 96 3.3.1 Generalized Impedance Converters [3] 98 3.3.1.1 The Ideal Active Transformer 98 3.3.1.2 The Ideal Negative Impedance Converter 99 3.3.1.3 The Positive Impedance Converter 99 3.3.1.4 The Frequency-Dependent Negative Resistor [5] 100 3.3.2 Generalized Impedance Inverters 101 3.3.2.1 The Gyrator 101 3.3.2.2 Negative Impedance Inverter 102 3.4 Negative Resistance 102 3.5 Ideal Operational Amplifier 104 3.5.1 Operations Using the Ideal Opamp 105 3.5.1.1 Summation of Voltages 105 3.5.1.2 Integration 106 3.5.2 Realization of Some Active Elements Using Opamps 107 3.5.2.1 Realization of Controlled Sources 107 3.5.2.2 Realization of Negative-Impedance Converters 108 3.5.2.3 Gyrator Realizations 110 3.5.2.4 GIC Circuit Using Opamps 111 3.5.3 Characteristics of IC Opamps 113 3.5.3.1 Open-Loop Voltage Gain of Practical Opamps 113 3.5.3.2 Input and Output Impedances 114 3.5.3.3 Input Offset Voltage VIO 115 3.5.3.4 Input Offset Current IIO 115 3.5.3.5 Input Voltage Range VI 116 3.5.3.6 Power Supply Sensitivity DVIO /DVGG 117 3.5.3.7 Slew Rate SR 117 3.5.3.8 Short-Circuit Output Current 117 3.5.3.9 Maximum Peak-to-Peak Output Voltage Swing Vopp 117 3.5.3.10 Input Capacitance Ci 118 3.5.3.11 Common-Mode Rejection Ratio CMRR 118 3.5.3.12 Total Power Dissipation 118 3.5.3.13 Rise Time tr 118 3.5.3.14 Overshoot 118 3.5.4 Effect of the Single-Pole Compensation on the Finite Voltage Gain Controlled Sources 118 3.6 The Ideal Operational Transconductance Amplifier (OTA) 120 3.6.1 Voltage Amplification 120 3.6.2 A Voltage-Variable Resistor (VVR) 121 3.6.3 Voltage Summation 121 3.6.4 Integration 122 3.6.5 Gyrator Realization 122 3.6.6 Practical OTAs 123 3.6.7 Current Conveyor [14] 124 3.7 Summary 126 References 126 2573ch04.pdf 1 Continuous-Time Active Filter Design 127 Contents -1 Chapter 4: Realization of First-and Second-Order Functions Using Opamps 128 4.1 Introduction 128 4.2 Realization of First-Order Functions 128 4.2.1 Lowpass Circuits 129 4.2.2 Highpass Circuits 130 4.2.3 Allpass Circuits 131 4.3 The General Second-Order Filter Function 131 4.4 Sensitivity of Second-Order Filters 132 4.5 Realization of Biquadratic Functions Using SABs 135 4.5.1 Classification of SABs 136 4.5.2 A Lowpass SAB 137 4.5.3 A Highpass SAB 141 4.5.4 A Bandpass SAB 142 4.5.5 Lowpass- and Highpass-Notch Biquads 147 4.5.6 Lowpass Notch (R6 = •) 148 4.5.7 Highpass Notch (R7 = •) 150 4.5.8 An Allpass SAB 150 4.6 Realization of a Quadratic with a Positive Real Zero 153 4.7 Biquads Obtained Using the Twin-T RC Network 155 4.8 Two-Opamp Biquads 157 4.8.1 Biquads by Inductance Simulation 157 4.8.2 Two-Opamp Allpass Biquads 159 4.8.3 Selectivity Enhancement 160 4.9 Three-Opamp Biquads 162 4.9.1 The Tow-Thomas [25–27] Three-Opamp Biquad 165 4.9.2 Excess Phase and Its Compensation in Three-Opamp Biquads 166 4.9.3 The Åkerberg-Mossberg Three-Opamp Biquad [29] 167 4.10 Summary 168 References 169 2573ch05.pdf 1 Continuous-Time Active Filter Design 171 Contents -1 Chapter 5: Realization of High-Order Functions 172 5.1 Introduction 172 5.2 Selection Criteria for High-Order Function Realizations 172 5.3 Multiparameter Sensitivity 174 5.4 High-Order Function Realization Methods 175 5.5 Cascade Connection of Second-Order Sections 176 5.5.1 Pole-Zero Pairing 177 5.5.2 Cascade Sequence 179 5.5.3 Gain Distribution [5] 180 5.6 Multiple-Loop Feedback Filters 183 5.6.1 The Shifted-Companion-Form (SCF) Design Method 187 5.6.2 Follow-the-Leader Feedback Design (FLF) 189 5.7 Cascade of Biquartics 192 5.7.1 The BR Section 192 5.7.2 Effect of h on and 194 5.7.3 Cascading Biquartic Sections 196 5.7.4 Realization of Biquartic Sections 196 5.7.4.1 Design Example 197 5.7.5 Sensitivity of CBR Filters 199 5.8 Summary 201 References 201 Further Reading 202 2573ch06.pdf 1 Continuous-Time Active Filter Design 203 Contents -1 Chapter 6: Simulation of LC Ladder Filters Using Opamps 204 6.1 Introduction 204 6.2 Resistively-Terminated Lossless LC Ladder Filters 205 6.3 Methods of LC Ladder Simulation 205 6.4 The Gyrator 206 6.4.1 Gyrator Imperfections 207 6.4.2 Use of Gyrators in Filter Synthesis 209 6.5 Generalized Impedance Converter, GIC 211 6.5.1 Use of GICs in Filter Synthesis 211 6.6 FDNRs: Complex Impedance Scaling 214 6.7 Functional Simulation 216 6.7.1 Example 219 6.7.2 Bandpass Filters 220 6.7.3 Dynamic Range of LF Filters 222 6.8 Summary 223 References 223 2573ch07.pdf 1 Continuous-Time Active Filter Design 225 Contents -1 Chapter 7: Wave Active Filters 226 7.1 Introduction 226 7.2 Wave Active Filters 226 7.3 Wave Active Equivalents (WAEs) 229 7.3.1 Wave Active Equivalent of a Series-Arm Impedance 229 7.3.2 Wave Active Equivalent of a Shunt-Arm Admittance 230 7.3.3 WAEs for Equal Port Normalization Resistances 230 7.3.4 Wave Active Equivalent of the Signal Source 231 7.3.5 Wave Active Equivalent of the Terminating Resistance 232 7.3.6 WAEs of Shunt-Arm Admittances 233 7.3.7 Interconnection Rules 233 7.3.8 WAEs of Tuned Circuits 235 7.3.9 WA Simulation Example 237 7.3.10 Comments on the Wave Active Filter Approach 237 7.4 Economical Wave Active Filters 238 7.5 Sensitivity of WAFs 241 7.6 Operation of WAFs at Higher Frequencies 242 7.7 Complementary Transfer Functions [7] 244 7.8 Wave Simulation of Inductance 245 7.9 Linear Transformation Active Filters (LTA Filters) 245 7.9.1 Interconnection Rule 248 7.9.2 General Remarks on the Method 250 7.10 Summary 250 References 250 2573ch08.pdf 1 Continuous-Time Active Filter Design 252 Contents -1 Chapter 8: Single Operational Transconductance Amplifier (OTA) Filters 253 8.1 Introduction 253 8.2 Single OTA Filters Derived from Three-Admittance Model 254 8.2.1 First-Order Filter Structures 254 First-Order Filters with One or Two Passive Components 255 First-Order Filters with Three Passive Components 256 8.2.2 Lowpass Second-Order Filter with Three Passive Components 257 8.2.3 Lowpass Second-Order Filters with Four Passive Components 258 8.2.4 Bandpass Second-Order Filters with Four Passive Components 260 8.3 Second-Order Filters Derived from Four-Admittance Model 263 8.3.1 Filter Structures and Design 263 Lowpass Filter 263 Bandpass Filter 265 Other Considerations on Structure Generation 266 8.3.2 Second-Order Filters with the OTA Transposed 267 Highpass Filter 267 Lowpass Filter 269 Bandpass Filters 269 8.4 Tunability of Active Filters Using Single OTA 271 8.5 OTA Nonideality Effects 271 8.5.1 Direct Analysis Using Practical OTA Macro-Model 271 8.5.2 Simple Formula Method 275 8.5.3 Reduction and Elimination of Parasitic Effects 275 8.6 OTA-C Filters Derived from Single OTA Filters 276 8.6.1 Simulated OTA Resistors and OTA-C Filters 276 8.6.2 Design Considerations of OTA-C Structures 277 8.7 Second-Order Filters Derived from Five-Admittance Model 280 8.7.1 Highpass Filter 281 8.7.2 Bandpass Filter 282 8.7.3 Lowpass Filter 284 8.7.4 Comments and Comparison 285 8.8 Summary 286 References 286 2573ch09.pdf 1 Continuous-Time Active Filter Design 290 Contents -1 Chapter 9: Two Integrator Loop OTA-C Filters 291 9.1 Introduction 291 9.2 OTA-C Building Blocks and First-Order OTA-C Filters [6, 12] 292 9.3 Two Integrator Loop Configurations and Performance 294 9.3.1 Configurations 294 9.3.2 Pole Equations 294 9.3.3 Design 295 9.3.4 Sensitivity 295 9.3.5 Tuning 295 9.3.6 Biquadratic Specifications 295 9.4 OTA-C Realizations of Distributed-Feedback (DF) Configuration 296 9.4.1 DF OTA-C Circuit and Equations 296 9.4.2 Filter Functions 298 9.4.3 Design Examples 299 9.4.4 DF OTA-C Realizations with Special Feedback Coefficients 300 9.5 OTA-C Filters Based on Summed-Feedback (SF) Configuration 302 9.5.1 SF OTA-C Realization with Arbitrary k 12 and k 11 303 Design Example of KHN OTA-C Biquad 304 9.5.2 SF OTA-C Realization with k 12 = k 11 = k 304 9.6 Biquadratic OTA-C Filters Using Lossy Integrators 305 9.6.1 Tow-Thomas OTA-C Structure 306 9.6.2 Feedback Lossy Integrator Biquad 306 9.7 Comparison of Basic OTA-C Filter Structures 307 9.7.1 Multifunctionality and Number of OTA 307 9.7.2 Sensitivity 308 9.7.3 Tunability 308 9.8 Versatile Filter Functions Based on Node Current Injection 309 9.8.1 DF Structures with Node Current Injection 310 9.8.2 SF Structures with Node Current Injection 311 9.9 Universal Biquads Using Output Summation Approach 313 9.9.1 DF-Type Universal Biquads 314 9.9.2 SF Type Universal Biquads 314 9.9.3 Universal Biquads Based on Node Current Injection and Output Summation 315 9.9.4 Comments on Universal Biquads 316 9.10 Universal Biquads Based on Canonical and TT Circuits 316 9.11 Effects and Compensation of OTA Nonidealities 317 9.11.1 General Model and Equations 317 9.11.2 Finite Impedance Effects and Compensation 320 9.11.3 Finite Bandwidth Effects and Compensation 321 9.11.4 Selection of OTA-C Filter Structures 323 9.11.5 Selection of Input and Output Methods 324 9.11.6 Dynamic Range Problem 324 9.12 Summary 325 References 326 2573ch10.pdf 1 Continuous-Time Active Filter Design 330 Contents -1 Chapter 10: OTA-C Filters Based on Ladder Simulation 331 10.1 Introduction 331 10.2 Component Substitution Method 332 10.2.1 Direct Inductor Substitution 332 OTA-C Inductors 332 Tolerance Sensitivity of Filter Function 333 Parasitic Effects on Simulated Inductor 334 Parasitic Effects on Filter Function 335 10.2.2 Application Examples of Inductor Substitution 337 OTA-C Biquad Derived from RLC Resonator Circuit 337 A Lowpass OTA-C Filter 338 10.2.3 Bruton Transformation and FDNR Simulation 339 10.3 Admittance/Impedance Simulation 342 10.3.1 General Description of the Method 342 10.3.2 Application Examples and Comparison 343 10.3.3 Partial Floating Admittance Concept 346 10.4 Signal Flow Simulation and Leapfrog Structures 347 10.4.1 Leapfrog Simulation Structures of General Ladder 347 10.4.2 OTA-C Lowpass LF Filters 350 Example 352 10.4.3 OTA-C Bandpass LF Filter Design 354 10.4.4 Partial Floating Admittance Block Diagram and OTA-C Realization 354 10.4.5 Alternative Leapfrog Structures and OTA-C Realizations 356 10.5 Equivalence of Admittance and Signal Simulation Methods 358 10.6 OTA-C Simulation of LC Ladders Using Matrix Methods 360 10.7 Coupled Biquad OTA Structures 362 10.8 Some General Practical Design Considerations 364 10.8.1 Selection of Capacitors and OTAs 364 10.8.2 Tolerance Sensitivity and Parasitic Effects 365 10.8.3 OTA Finite Impedances and Frequency-Dependent Transconductance 365 10.9 Summary 365 References 366 2573ch11.pdf 1 Continuous-Time Active Filter Design 370 Contents -1 Chapter 11: Multiple Integrator Loop Feedback OTA-C Filters 371 11.1 Introduction 371 11.2 General Design Theory of All-Pole Structures [9, 12, 28] 372 11.2.1 Multiple Loop Feedback OTA-C Model 372 11.2.2 System Equations and Transfer Function 372 11.2.3 Feedback Coefficient Matrix and Systematic Structure Generation 375 11.2.4 Filter Synthesis Procedure Based on Coefficient Matching 376 11.3 Structure Generation and Design of All-pole Filters [9, 12, 28] 377 11.3.1 First- and Second-Order Filters 377 11.3.2 Third-Order Filters 378 11.3.3 Fourth-Order Filters 379 11.3.4 Design Examples of Fourth-Order Filters 381 11.3.5 General n th-Order Architectures 382 11.3.5.1 General IFLF Configuration 382 11.3.5.2 General LF Configuration 383 11.3.6 Other Types of Realization 384 11.4 Generation and Synthesis of Transmission Zeros 385 11.4.1 Output Summation of OTA Network [12] 386 11.4.2 Input Distribution of OTA Network [12] 386 11.4.3 Universal and Special Third-Order OTA-C Filters [13] 388 11.4.3.1 IFLF and Output Summation Structure in Fig. 11.10(a) 389 11.4.3.2 IFLF and Input Distribution Structure in Fig. 11.10(b) 389 11.4.3.3 LF and Output Summation Structure in Fig. 11.10(c) 389 11.4.3.4 LF and Input Distribution Structure in Fig. 11.10(d) 390 11.4.3.5 Realization of Special Characteristics 390 11.4.3.6 Design of Elliptic Filters 390 11.4.4 General n th-Order OTA-C Filters 392 11.4.4.1 Universal IFLF Architectures [8, 10] 392 11.4.4.2 Universal LF Architectures 394 11.5 General Formulation of Sensitivity Analysis [12, 28] 395 11.5.1 General Sensitivity Relations 395 11.5.2 Sensitivities of Different Filter Structures 397 11.6 Determination of Maximum Signal Magnitude 399 11.7 Effects of OTA Frequency Response Nonidealities [10] 401 11.8 Summary 403 References 404 2573ch12.pdf 1 Continuous-Time Active Filter Design 408 Contents -1 Chapter 12: Current-Mode Filters and Other Architectures 409 12.1 Introduction 409 12.2 Current-Mode Filters Based on Single DO-OTA Model 410 12.2.1 General Model and Filter Architecture Generation 410 12.2.1.1 First-Order Filter Structures 411 12.2.1.2 Second-Order Filter Architectures 412 12.2.2 Passive Resistor and Active Resistor 412 12.2.3 Design of Second-Order Filters 413 12.2.4 Effects of DO-OTA Nonidealities 416 12.3 Current-Mode Two Integrator Loop DO-OTA-C Filters 418 12.3.1 Basic Building Blocks and First-Order Filters [15] 418 12.3.2 Current-Mode DO-OTA-C Configurations with Arbitrary k ij [15] 419 12.3.3 Current-Mode DO-OTA-C Biquadratic Architectures with k 12 = k jj 420 12.3.4 Current-Mode DO-OTA-C Biquadratic Architectures with k 12 = 1[15] 421 12.3.5 DO-OTA Nonideality Effects 423 12.3.6 Universal Current-Mode DO-OTA-C Filters 423 12.4 Current-Mode DO-OTA-C Ladder Simulation Filters 427 12.4.1 Leapfrog Simulation Structures of General Ladder 427 12.4.2 Current-Mode DO-OTA-C Lowpass LF Filters 429 12.4.3 Current-Mode DO-OTA-C Bandpass LF Filter Design 431 12.4.4 Alternative Current-Mode Leapfrog DO-OTA-C Structure 432 12.5 Current-Mode Multiple Loop Feedback DO-OTA-C Filters 433 12.5.1 Design of All-Pole Filters [23] 433 12.5.2 Realization of Transmission Zeros 437 12.5.2.1 Multiple Loop Feedback with Input Distribution The multiple current-integrator loop current-feedback model 437 12.5.2.2 Multiple Loop Feedback with Output Summation 438 12.5.2.3 Filter Structures and Design Formulas 439 12.6 Other Continuous-Time Filter Structures 441 12.6.1 Balanced Opamp-RC and OTA-C Structures 441 12.6.2 MOSFET-C Filters 442 12.6.3 OTA-C-Opamp Filter Design 444 12.6.4 Active Filters Using Current Conveyors 445 12.6.5 Log-Domain, Current Amplifier, and Integrated-RLC Filters 447 12.7 Summary 447 References 448 2573AppA.pdf 1 Continuous-Time Active Filter Design 452 Contents -1 Appendix A: A Sample of Filter Functions 453 Continuous-Time Active Filter Design......Page 1 Preface......Page 5 Authors......Page 8 Contents......Page 9 Contents......Page 0 Continuous-Time Active Filter Design......Page 18 1.2.1 Lumped......Page 19 1.2.2 Linear......Page 20 1.2.6 Passive and Active......Page 21 1.3 Types of Filters......Page 22 1.4 Steps in Filter Design......Page 24 1.5.1 Nodal Analysis......Page 25 1.5.2.1 One-Port Network......Page 28 1.5.2.2 Two-Port Network......Page 29 1.5.3.1 Series–Series Connection......Page 32 1.5.3.2 Parallel–Parallel Connection......Page 33 1.5.3.5 Cascade Connection......Page 34 1.5.4 Network Transfer Functions......Page 35 1.6 Continuous-Time Filter Functions......Page 37 1.6.1 Pole-Zero Locations......Page 38 1.6.2 Frequency Response......Page 39 1.6.3.1 Impulse Response......Page 40 1.6.3.2 Step Response......Page 41 1.6.4 Step and Frequency Response......Page 42 1.7 Stability......Page 44 1.7.2 Absolute Stability and Potential Instability......Page 45 1.8.1 One-Ports......Page 47 1.8.2 Two-Ports......Page 48 1.8.4 Passivity and Stability......Page 49 1.9 Reciprocity......Page 50 References and Further Reading......Page 51 Continuous-Time Active Filter Design......Page 53 2.2.1 Causality......Page 54 2.2.3 Stability......Page 55 2.3 Formulation of the Approximation Problem......Page 56 2.4 Approximation of the Ideal Lowpass Filter......Page 57 2.4.1 Butterworth or Maximally Flat Approximation......Page 58 2.4.2 Chebyshev or Equiripple Approximation......Page 61 2.4.3 Inverse Chebyshev Approximation......Page 64 2.4.5 Elliptic Function or Cauer Approximation......Page 66 2.4.6 Selecting the Filter from Its Specifications......Page 68 2.5 Filters with Linear Phase: Delays......Page 71 2.5.1 Bessel-Thomson Delay Approximation......Page 73 2.5.2 Other Delay Functions......Page 77 2.6 Frequency Transformations......Page 78 2.6.1 Lowpass-to-Lowpass Transformation......Page 79 2.6.2 Lowpass-to-Highpass Transformation......Page 80 2.6.3 Lowpass-to-Bandpass Transformation......Page 81 2.6.4 Lowpass-to-Bandstop Transformation......Page 82 2.7 Design Tables for Passive LC Ladder Filters......Page 83 2.7.1.1 LC Filters......Page 84 2.7.1.2 Active RC Filters......Page 87 2.8 Impedance Scaling......Page 89 2.9 Predistortion......Page 90 2.10 Summary......Page 91 References......Page 92 Continuous-Time Active Filter Design......Page 94 3.2 Ideal Controlled Sources......Page 95 3.3 Impedance Transformation (Generalized Impedance Converters and Inverters) [1, 2]......Page 96 3.3.1.1 The Ideal Active Transformer......Page 98 3.3.1.3 The Positive Impedance Converter......Page 99 3.3.1.4 The Frequency-Dependent Negative Resistor [5]......Page 100 3.3.2.1 The Gyrator......Page 101 3.4 Negative Resistance......Page 102 3.5 Ideal Operational Amplifier......Page 104 3.5.1.1 Summation of Voltages......Page 105 3.5.1.2 Integration......Page 106 3.5.2.1 Realization of Controlled Sources......Page 107 3.5.2.2 Realization of Negative-Impedance Converters......Page 108 3.5.2.3 Gyrator Realizations......Page 110 3.5.2.4 GIC Circuit Using Opamps......Page 111 3.5.3.1 Open-Loop Voltage Gain of Practical Opamps......Page 113 3.5.3.2 Input and Output Impedances......Page 114 3.5.3.4 Input Offset Current IIO......Page 115 3.5.3.5 Input Voltage Range VI......Page 116 3.5.3.9 Maximum Peak-to-Peak Output Voltage Swing Vopp......Page 117 3.5.4 Effect of the Single-Pole Compensation on the Finite Voltage Gain Controlled Sources......Page 118 3.6.1 Voltage Amplification......Page 120 3.6.3 Voltage Summation......Page 121 3.6.5 Gyrator Realization......Page 122 3.6.6 Practical OTAs......Page 123 3.6.7 Current Conveyor [14]......Page 124 References......Page 126 Continuous-Time Active Filter Design......Page 127 4.2 Realization of First-Order Functions......Page 128 4.2.1 Lowpass Circuits......Page 129 4.2.2 Highpass Circuits......Page 130 4.3 The General Second-Order Filter Function......Page 131 4.4 Sensitivity of Second-Order Filters......Page 132 4.5 Realization of Biquadratic Functions Using SABs......Page 135 4.5.1 Classification of SABs......Page 136 4.5.2 A Lowpass SAB......Page 137 4.5.3 A Highpass SAB......Page 141 4.5.4 A Bandpass SAB......Page 142 4.5.5 Lowpass- and Highpass-Notch Biquads......Page 147 4.5.6 Lowpass Notch (R6 = •)......Page 148 4.5.8 An Allpass SAB......Page 150 4.6 Realization of a Quadratic with a Positive Real Zero......Page 153 4.7 Biquads Obtained Using the Twin-T RC Network......Page 155 4.8.1 Biquads by Inductance Simulation......Page 157 4.8.2 Two-Opamp Allpass Biquads......Page 159 4.8.3 Selectivity Enhancement......Page 160 4.9 Three-Opamp Biquads......Page 162 4.9.1 The Tow-Thomas [25–27] Three-Opamp Biquad......Page 165 4.9.2 Excess Phase and Its Compensation in Three-Opamp Biquads......Page 166 4.9.3 The Åkerberg-Mossberg Three-Opamp Biquad [29]......Page 167 4.10 Summary......Page 168 References......Page 169 Continuous-Time Active Filter Design......Page 171 5.2 Selection Criteria for High-Order Function Realizations......Page 172 5.3 Multiparameter Sensitivity......Page 174 5.4 High-Order Function Realization Methods......Page 175 5.5 Cascade Connection of Second-Order Sections......Page 176 5.5.1 Pole-Zero Pairing......Page 177 5.5.2 Cascade Sequence......Page 179 5.5.3 Gain Distribution [5]......Page 180 5.6 Multiple-Loop Feedback Filters......Page 183 5.6.1 The Shifted-Companion-Form (SCF) Design Method......Page 187 5.6.2 Follow-the-Leader Feedback Design (FLF)......Page 189 5.7.1 The BR Section......Page 192 5.7.2 Effect of h on and......Page 194 5.7.4 Realization of Biquartic Sections......Page 196 5.7.4.1 Design Example......Page 197 5.7.5 Sensitivity of CBR Filters......Page 199 References......Page 201 Further Reading......Page 202 Continuous-Time Active Filter Design......Page 203 6.1 Introduction......Page 204 6.3 Methods of LC Ladder Simulation......Page 205 6.4 The Gyrator......Page 206 6.4.1 Gyrator Imperfections......Page 207 6.4.2 Use of Gyrators in Filter Synthesis......Page 209 6.5.1 Use of GICs in Filter Synthesis......Page 211 6.6 FDNRs: Complex Impedance Scaling......Page 214 6.7 Functional Simulation......Page 216 6.7.1 Example......Page 219 6.7.2 Bandpass Filters......Page 220 6.7.3 Dynamic Range of LF Filters......Page 222 References......Page 223 Continuous-Time Active Filter Design......Page 225 7.2 Wave Active Filters......Page 226 7.3.1 Wave Active Equivalent of a Series-Arm Impedance......Page 229 7.3.3 WAEs for Equal Port Normalization Resistances......Page 230 7.3.4 Wave Active Equivalent of the Signal Source......Page 231 7.3.5 Wave Active Equivalent of the Terminating Resistance......Page 232 7.3.7 Interconnection Rules......Page 233 7.3.8 WAEs of Tuned Circuits......Page 235 7.3.10 Comments on the Wave Active Filter Approach......Page 237 7.4 Economical Wave Active Filters......Page 238 7.5 Sensitivity of WAFs......Page 241 7.6 Operation of WAFs at Higher Frequencies......Page 242 7.7 Complementary Transfer Functions [7]......Page 244 7.9 Linear Transformation Active Filters (LTA Filters)......Page 245 7.9.1 Interconnection Rule......Page 248 References......Page 250 Continuous-Time Active Filter Design......Page 252 8.1 Introduction......Page 253 8.2.1 First-Order Filter Structures......Page 254 First-Order Filters with One or Two Passive Components......Page 255 First-Order Filters with Three Passive Components......Page 256 8.2.2 Lowpass Second-Order Filter with Three Passive Components......Page 257 8.2.3 Lowpass Second-Order Filters with Four Passive Components......Page 258 8.2.4 Bandpass Second-Order Filters with Four Passive Components......Page 260 Lowpass Filter......Page 263 Bandpass Filter......Page 265 Other Considerations on Structure Generation......Page 266 Highpass Filter......Page 267 Bandpass Filters......Page 269 8.5.1 Direct Analysis Using Practical OTA Macro-Model......Page 271 8.5.3 Reduction and Elimination of Parasitic Effects......Page 275 8.6.1 Simulated OTA Resistors and OTA-C Filters......Page 276 8.6.2 Design Considerations of OTA-C Structures......Page 277 8.7 Second-Order Filters Derived from Five-Admittance Model......Page 280 8.7.1 Highpass Filter......Page 281 8.7.2 Bandpass Filter......Page 282 8.7.3 Lowpass Filter......Page 284 8.7.4 Comments and Comparison......Page 285 References......Page 286 Continuous-Time Active Filter Design......Page 290 9.1 Introduction......Page 291 9.2 OTA-C Building Blocks and First-Order OTA-C Filters [6, 12]......Page 292 9.3.2 Pole Equations......Page 294 9.3.6 Biquadratic Specifications......Page 295 9.4.1 DF OTA-C Circuit and Equations......Page 296 9.4.2 Filter Functions......Page 298 9.4.3 Design Examples......Page 299 9.4.4 DF OTA-C Realizations with Special Feedback Coefficients......Page 300 9.5 OTA-C Filters Based on Summed-Feedback (SF) Configuration......Page 302 9.5.1 SF OTA-C Realization with Arbitrary k 12 and k 11......Page 303 9.5.2 SF OTA-C Realization with k 12 = k 11 = k......Page 304 9.6 Biquadratic OTA-C Filters Using Lossy Integrators......Page 305 9.6.2 Feedback Lossy Integrator Biquad......Page 306 9.7.1 Multifunctionality and Number of OTA......Page 307 9.7.3 Tunability......Page 308 9.8 Versatile Filter Functions Based on Node Current Injection......Page 309 9.8.1 DF Structures with Node Current Injection......Page 310 9.8.2 SF Structures with Node Current Injection......Page 311 9.9 Universal Biquads Using Output Summation Approach......Page 313 9.9.2 SF Type Universal Biquads......Page 314 9.9.3 Universal Biquads Based on Node Current Injection and Output Summation......Page 315 9.10 Universal Biquads Based on Canonical and TT Circuits......Page 316 9.11.1 General Model and Equations......Page 317 9.11.2 Finite Impedance Effects and Compensation......Page 320 9.11.3 Finite Bandwidth Effects and Compensation......Page 321 9.11.4 Selection of OTA-C Filter Structures......Page 323 9.11.6 Dynamic Range Problem......Page 324 9.12 Summary......Page 325 References......Page 326 Continuous-Time Active Filter Design......Page 330 10.1 Introduction......Page 331 OTA-C Inductors......Page 332 Tolerance Sensitivity of Filter Function......Page 333 Parasitic Effects on Simulated Inductor......Page 334 Parasitic Effects on Filter Function......Page 335 OTA-C Biquad Derived from RLC Resonator Circuit......Page 337 A Lowpass OTA-C Filter......Page 338 10.2.3 Bruton Transformation and FDNR Simulation......Page 339 10.3.1 General Description of the Method......Page 342 10.3.2 Application Examples and Comparison......Page 343 10.3.3 Partial Floating Admittance Concept......Page 346 10.4.1 Leapfrog Simulation Structures of General Ladder......Page 347 10.4.2 OTA-C Lowpass LF Filters......Page 350 Example......Page 352 10.4.4 Partial Floating Admittance Block Diagram and OTA-C Realization......Page 354 10.4.5 Alternative Leapfrog Structures and OTA-C Realizations......Page 356 10.5 Equivalence of Admittance and Signal Simulation Methods......Page 358 10.6 OTA-C Simulation of LC Ladders Using Matrix Methods......Page 360 10.7 Coupled Biquad OTA Structures......Page 362 10.8.1 Selection of Capacitors and OTAs......Page 364 10.9 Summary......Page 365 References......Page 366 Continuous-Time Active Filter Design......Page 370 11.1 Introduction......Page 371 11.2.2 System Equations and Transfer Function......Page 372 11.2.3 Feedback Coefficient Matrix and Systematic Structure Generation......Page 375 11.2.4 Filter Synthesis Procedure Based on Coefficient Matching......Page 376 11.3.1 First- and Second-Order Filters......Page 377 11.3.2 Third-Order Filters......Page 378 11.3.3 Fourth-Order Filters......Page 379 11.3.4 Design Examples of Fourth-Order Filters......Page 381 11.3.5.1 General IFLF Configuration......Page 382 11.3.5.2 General LF Configuration......Page 383 11.3.6 Other Types of Realization......Page 384 11.4 Generation and Synthesis of Transmission Zeros......Page 385 11.4.2 Input Distribution of OTA Network [12]......Page 386 11.4.3 Universal and Special Third-Order OTA-C Filters [13]......Page 388 11.4.3.3 LF and Output Summation Structure in Fig. 11.10(c)......Page 389 11.4.3.6 Design of Elliptic Filters......Page 390 11.4.4.1 Universal IFLF Architectures [8, 10]......Page 392 11.4.4.2 Universal LF Architectures......Page 394 11.5.1 General Sensitivity Relations......Page 395 11.5.2 Sensitivities of Different Filter Structures......Page 397 11.6 Determination of Maximum Signal Magnitude......Page 399 11.7 Effects of OTA Frequency Response Nonidealities [10]......Page 401 11.8 Summary......Page 403 References......Page 404 Continuous-Time Active Filter Design......Page 408 12.1 Introduction......Page 409 12.2.1 General Model and Filter Architecture Generation......Page 410 12.2.1.1 First-Order Filter Structures......Page 411 12.2.2 Passive Resistor and Active Resistor......Page 412 12.2.3 Design of Second-Order Filters......Page 413 12.2.4 Effects of DO-OTA Nonidealities......Page 416 12.3.1 Basic Building Blocks and First-Order Filters [15]......Page 418 12.3.2 Current-Mode DO-OTA-C Configurations with Arbitrary k ij [15]......Page 419 12.3.3 Current-Mode DO-OTA-C Biquadratic Architectures with k 12 = k jj......Page 420 12.3.4 Current-Mode DO-OTA-C Biquadratic Architectures with k 12 = 1[15]......Page 421 12.3.6 Universal Current-Mode DO-OTA-C Filters......Page 423 12.4.1 Leapfrog Simulation Structures of General Ladder......Page 427 12.4.2 Current-Mode DO-OTA-C Lowpass LF Filters......Page 429 12.4.3 Current-Mode DO-OTA-C Bandpass LF Filter Design......Page 431 12.4.4 Alternative Current-Mode Leapfrog DO-OTA-C Structure......Page 432 12.5.1 Design of All-Pole Filters [23]......Page 433 12.5.2.1 Multiple Loop Feedback with Input Distribution The multiple current-integrator loop current-feedback model......Page 437 12.5.2.2 Multiple Loop Feedback with Output Summation......Page 438 12.5.2.3 Filter Structures and Design Formulas......Page 439 12.6.1 Balanced Opamp-RC and OTA-C Structures......Page 441 12.6.2 MOSFET-C Filters......Page 442 12.6.3 OTA-C-Opamp Filter Design......Page 444 12.6.4 Active Filters Using Current Conveyors......Page 445 12.7 Summary......Page 447 References......Page 448 Continuous-Time Active Filter Design......Page 452 Appendix A: A Sample of Filter Functions......Page 453 Continuous-Time Active Filter Design takes the reader through the basics of the subject, and explores the successful and enduring opamp approaches to active filter problem solution which have been established in the past. The book then introduces the reader to state-of-the-art approaches to active filter design using devices such as the operational transconductance amplifier (OTA), and presents circuits and approaches which will take the subject into the new millenium. Intended for undergraduate and graduate students in electronic engineering, and for electronic professionals, Continuous-Time Active Filter Design presents material in a form that may be assimilated by the specialist and non-specialist reader alike. This book presents the design of active RC filters in continuous time.Topics include:ofilter fundamentalsoactive elementsorealization of functions using opampsoLC ladder filtersooperational transconductance amplifier circuits (OTACs)oMOSFET-C filtersContinuous-Time Active Filter Design uses wave variables to enable the reader to better understand the introduction of more complex variables created through linear transformations of voltages and currents.Intended for undergraduate students in electrical engineering, Continuous-Time Active Filter Design provides chapters as self-contained units, including introductory material leading to active RC filters.