Chapter 1 Filter Fundamentals 1.1 Introduction Continuous-time active filters are active networks (circuits) with characteristics that make them useful in today’s system design. Their response can be predetermined once their exci- tation is known, provided that their characteristic function is known or can be derived from their circuit diagram. Thus, it is important for the filter designer to be familiar with the con- cepts relevant to filter characterization. These useful concepts are reviewed in this chapter. For motivation, we deal with the filter characterization and the possible responses first. In order to pursue these further, we need to consider certain fundamentals; the analysis of a circuit is explained by means of the nodal method. The analysis of the circuit gives the mathematical expressions, transfer, or other functions that describe its characteristics. We examine these functions in terms of their pole-zero locations in the s-plane and use them to determine the frequency and time responses of the circuit. The concepts of stability, passivity, activity, and reciprocity, which are closely associated with the study and the design of the types of networks examined in this book, are also visited briefly. 1.2 Filter Characterization The filters examined in this book are networks that process the signal from a source before they deliver it to a load. In terms of a block diagram this is shown in Fig. 1.1. The filter network is considered here to be lumped, linear, continuous-time, time invari- ant, finite, passive, or active. These terms are clarified in the following section. 1.2.1 Lumped In lumped networks, we consider the resistance, inductance, or capacitance as symbols or simple elements concentrated within the boundaries of the corresponding physical ele- ment, the physical dimensions of which are negligible compared to the wavelength of the fields associated with the signal. This is in contrast to the distributed networks, in which the physical elements have dimensions comparable to the wavelength of the fields associated with the signal.1.2.2 Linear Consider the circuit or system shown in Fig. 1.2(a) in block diagram form, where r1(t) is the system response to the excitation e1(t). The system will be linear (LS) when its response to the excitation C1e1(t), where C1 is a constant, is also multiplied by C 1 , i.e., if it is C 1 r 1 (t), as shown in Fig. 1.2(b). This expresses the principle of proportionality. For a linear system the principle of superposition holds. This principle is stated as fol- lows: If the responses to the separate excitations C 1 e 1 (t) and C 2 e 2 ( t ) are C 1 r 1 ( t ) and C 2 r 2 ( t ), respectively, then the response to the excitation C 1 e 1 ( t ) + C 2 e 2 ( t ) will be C 1 r 1 ( t ) + C 2 r 2 ( t ), C 1 and C 2 both being constants. Some examples of linear circuits are the following: • An amplifier working in the linear region of its characteristics is a linear circuit. • A differentiator is a linear circuit. To show this, let r ( t ) be the response to the excitation e ( t ). (1.1) Then, if e ( t ) is multiplied by a constant C , we will get for the new response (1.2) • Similarly, for an integrator, the response r ( t ) to its excitation e ( t ) is: (1.3) If e ( t ) is multiplied by the constant C , the new response of the integrator will be: (1.4) FIGURE 1.1 Block diagram of a filter inserted between the signal source and the load. FIGURE 1.2 rt() de t() dt -------------= r′ t() rt() dCet()[] dt ---------------------- Cde t() dt ------------- Cr t()=== rt() e τ()τd 0 t ∫= r′ t() Ce τ()τd 0 t ∫ Ce τ()τd 0 t ∫== ''
Deliyannis, Theodore L. et al "Frontmatter" Continuous-Time Active Filter Design Boca Raton: CRC Press LLC,1999 [...]... Summation 12.5.2.3 Filter Structures and Design Formulas ©1999 CRC Press LLC 12.6 Other Continuous- Time Filter Structures 12.6.1 Balanced Opamp-RC and OTA-C Structures 12.6.2 MOSFET-C Filters 12.6.3 OTA-C-Opamp Filter Design 12.6.4 Active Filters Using Current Conveyors 12.6.5 Log-Domain, Current Amplifier, and Integrated-RLC Filters 12.7 Summary References Appendix A A Sample of Filter Functions ©1999... Sample of Filter Functions ©1999 CRC Press LLC Deliyannis, Theodore L et al "Filter Fundamentals" Continuous- Time Active Filter Design Boca Raton: CRC Press LLC,1999 Chapter 1 Filter Fundamentals 1.1 Introduction Continuous- time active filters are active networks (circuits) with characteristics that make them useful in today’s system design Their response can be predetermined once their excitation is known,... Four Passive Components 8.3 Second-Order Filters Derived from Four-Admittance Model 8.3.1 Filter Structures and Design 8.3.1.1 Lowpass Filter 8.3.1.2 Bandpass Filter 8.3.1.3 Other Considerations on Structure Generation 8.3.2 Second-Order Filters with the OTA Transposed 8.3.2.1 Highpass Filter 8.3.2.2 Lowpass Filter 8.3.2.3 Bandpass Filter 8.4 Tunability of Active Filters Using Single OTA 8.5 OTA Nonideality... • A time delayer, which introduces the time delay T to the signal, also corresponds to a linear operator, since the response to the excitation e(t) will be r (t ) = e(t – T ) 1.2.3 (1.5) Continuous- Time and Discrete -Time In a continuous- time filter, both the excitation e and the response r are continuous functions of the continuous time t, i.e., e = e(t) r = r(t) (1.6) In contrast, in a discrete -time. .. Effects 8.6 OTA-C Filters Derived from Single OTA Filters 8.6.1 Simulated OTA Resistors and OTA-C Filters ©1999 CRC Press LLC 8.6.2 Design Considerations of OY Structures Second-Ordre Filters Derived from Five-Admittance Model 8.7.1 Highpass Filter 8.7.2 Bandpass Filter 8.7.3 Lowpass Filter 8.7.4 Comments and Comparison 8.8 Summary References 8.7 Chapter 9 Two Integrator Loop OTA-C Filters 9.1 Introduction... References ©1999 CRC Press LLC Chapter 7 Wave Active Filters 7.1 Introduction 7.2 Wave Active Filters 7.3 Wave Active Equivalents (WAEs) 7.3.1 Wave Active Equivalent of a Series-Arm Impedance 7.3.2 Wave Active Equivalent of a Shunt-Arm Admittance 7.3.3 WAEs for Equal Port Normalization Resistances 7.3.4 Wave Active Equivalent of the Signal Source 7.3.5 Wave Active Equivalent of the Terminating Resistance... Current-Mode DO-OTA-C Filters 12.4 Current-Mode DO-OTA-C Ladder Simulation Filters 12.4.1 Leapfrog Simulation Structures of General Ladder 12.4.2 Current-Mode DO-OTA-C Lowpass LF Filters 12.4.3 Current-Mode DO-OTA-C Bandpass LF Filter Design 12.4.4 Alternative Current-Mode Leapfrog DO-OTA-C Structure 12.5 Current-Mode Multiple Loop Feedback DO-OTA-C Filters 12.5.1 Design of All-Pole Filters 12.5.2 Realization... Active Filter Approach 7.4 Economical Wave Active Filters 7.5 Sensitivity of WAFs 7.6 Operation of WAFs at Higher Frequencies 7.7 Complementary Transfer Functions 7.8 Wave Simulation of Inductance 7.9 Linear Transformation Active Filters (LTA Filters) 7.9.1 Interconnection Rule 7.9.2 General Remarks on the Method 7.10 Summary References Chapter 8 Single Operational Transconductance Amplifier (OTA) Filters... Introduction 8.2 Single OTA Filters Derived from Three-Admittance Model 8.2.1 First-Order Filter Structures 8.2.1.1 First-Order Filters with One or Two Passive Components 8.2.1.2 First-Order Filters with Three Passive Components 8.2.2 Lowpass Second-Order Filter with Three Passive Components 8.2.3 Lowpass Second-Order Filters with Four Passive Components 8.2.4 Bandpass Second-Order Filters with Four Passive... Feedback OTA-C Filters 11.1 Introduction 11.2 General Design Theory of All-Pole Structures 11.2.1 Multiple Loop Feedback OTA-C Model 11.2.2 System Equations and Transfer Function 11.2.3 Feedback Coefficient Matrix and Systematic Structure Generation 11.2.4 Filter Synthesis Procedure Based on Coefficient Matching 11.3 Structure Generation and Design of All-Pole Filters 11.3.1 First- and Second-Order Filters . Simulation 6.7.1Example 6.7.2Bandpass Filters 6.7.3Dynamic Range of LF Filters 6.8Summary References ωω ωω ′′ ′′ i Q′′ ′′ i ©1999 CRC Press LLC Chapter 7Wave Active Filters 7.1Introduction 7.2Wave Active Filters 7.3Wave Active. al " ;Filter Fundamentals" Continuous-Time Active Filter Design Boca Raton: CRC Press LLC,1999 ©1999 CRC Press LLC Chapter 1 Filter Fundamentals 1.1 Introduction Continuous-time. Filters 12.6.3OTA-C-Opamp Filter Design 12.6. 4Active Filters Using Current Conveyors 12.6.5Log-Domain, Current Amplifier, and Integrated-RLC Filters 12.7Summary References Appendix AA Sample of Filter Functions