EE 508 Lecture 4. Filter Concepts/Terminology Basic Properties of Electrical Circuits
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1 EE 58 Lecture 4 Filter Concepts/Terminology Basic Properties of Electrical Circuits
2 Review from Last Time Filter Design Process Establish Specifications - possibly T D (s) or H D (z) - magnitude and phase characteristics or restrictions - time domain requirements gm Specifications?? Ts Transfer Function Circuit Filter Design Strategy: Use the transfer function as an intermediate step between the Specifications and Circuit Implementation pproximation - obtain acceptable transfer functions T (s) or H (z) - possibly acceptable realizable time-domain responses Synthesis - build circuit or implement algorithm that has response close to T (s) or H (z) - actually realize T R (s) or H R (z) Filter
3 Review from Last Time If n is even and n m, Biquadratic Factorization m i n/ i=1 i Ts = n T (s) i i=1 BQi i=1 as bs i If n is odd and n m, where n-1 / i=1 i 1 T s = n T (s) m i=1 as b s a s a s+a T (s)= BQi s b s+b i i i=1 i i 1i i 1i i a s+a s+b BQi and where is a real constant and all coefficients are real (some may be ) Factorization is not unique H(z) factorizations not restricted to have m n Each biquatratic factor can be represented by any of the 6 alternative parameter sets in the numerator or denominator
4 Review from Last Time Common Filter rchitectures Cascaded Biquads T 1 (s) T (s) T k (s) T m (s) Biquad Biquad Biquad Biquad Leapfrog I 1 (s) Integrator I (s) Integrator I 3 (s) Integrator I 4 (s) Integrator I k-1 (s) Integrator I k (s) Integrator a 1 a Multiple-loop Feedback X IN α α F T 1 (s) Biquad α 1 α α k T (s) Biquad T k (s) Biquad X OUT Three classical filter architectures are shown The Cascaded Biquad and the Leapfrog approaches are most common The Cascaded Biquad structure follows directly from the Biquadratic Factorization
5 Review from Last Time Common Filter rchitectures Cascaded Biquads T 1 (s) T (s) T k (s) T m (s) Biquad Biquad Biquad Biquad 1 m T s T T T Sequence in Cascade often affect performance Different biquadratic factorizations will provide different performance lthough some attention was given to the different alternatives for biquadratic factorization, a solid general formulation of the cascade sequencing problem or the biquadratic factorization problem never evolved
6 Filter Concepts and Terminology -nd order polynomial characterization Biquadratic Factorization Op mp Modeling Stability and Instability Roll-off characteristics Distortion Dead Networks Root Characterization Scaling, normalization, and transformation
7 Gain, Bandwidth and Frequency Dependent Model of Op mps Most op amps are designed so that they behave as a first-order circuit at frequencies up to the unity gain frequency or beyond V V 1 VOUT 1s = V -V 1 1 (s) (s) 1 s = s +1 BW where = BW BW = <1 s = s+bw 1 db BW db/ decade Range of Interest ω Can usually model with a more-simplified gain expression in frequency region of interest s = s dequate model for most applications
8 Gain, Bandwidth and Effects of on closed-loop mplifiers V V 1 s = s+bw 1 = BW s = s dequate model for most applications FB FB V = 1 s 1+s OUT 1 IN 1 1 V OUT V = s V -V s s = s+bw V V s 1+ OUT IN BW BW V 1 1 (s) (s) R R 1 Basic Noninverting mplifier R =1+ R 1 db BW FB (s) db/ decade BW ω
9 Gain, Bandwidth and Effects of on closed-loop mplifiers V V 1 s = s+bw 1 = BW FB s 1+s 1 (s) (s) BW s = s db/ decade dequate model for most applications db BW FB (s) BW ω V 1 R 1 R R =1+ R 1 p CL / s-plane p OL BW Im Re Basic Noninverting mplifier
10 Gain, Bandwidth and Summary of Effects of on Basic Inverting and Noninverting mplifiers R 1 R V V 1 s = s+bw 1 Basic Noninverting mplifier R =1+ R 1 BW = BW s = s dequate model for most applications 1 (s) (s) db/ decade R 1 R Basic Inverting mplifier R = R 1 BW 1+ FB s 1+s db BW FB (s) BW ω FB s 1+ 1+s
11 Filter Concepts and Terminology -nd order polynomial characterization Biquadratic Factorization Op mp Modeling Stability and Instability Roll-off characteristics Distortion Dead Networks Root Characterization Scaling, normalization, and transformation
12 Stability and Instability True or False? n unstable circuit will oscillate False unstable circuits will either latch up or oscillate. Latch-up is often the consequence of saturating nonlinearities of circuits that have positive real axis poles chieving stability is a major goal of the filter designer False a filter is usually of little practical use if there are concerns about stability Unstable circuits are of little use in designing filters False will discuss details later
13 Theorem?: If a circuit is unstable, then if this circuit is included as a subcircuit in a larger circuit structure, the larger circuit will also be unstable. Unstable Circuit Unstable Circuit Larger Circuit Proof?: Consider First Some Related Concepts
14 Gain, Bandwidth and Consider positive feedback closed-loop amplifier V V 1 s = s+bw 1 = BW FB s 1- s 1 (s) (s) BW s = s db/ decade dequate model for most applications db BW FB (s) BW ω Im s-plane R 1 R R = 1+ R 1 p OL p CL BW / Re Feedback mplifier is Unstable!
15 Gain, Bandwidth and Summary of Effects of on Basic Inverting and Noninverting mplifiers with Positive Feedback R 1 R V V 1 s = s+bw 1 Basic Noninverting mplifier R =1+ R 1 BW = BW s = s dequate model for most applications 1 (s) (s) db/ decade R 1 R Basic Inverting mplifier R = R 1 BW 1+ FB s 1- s FB (s) db BW BW ω Both FB mplifiers are Unstable FB s s
16 Gain, Bandwidth and Summary of Effects of on Basic Inverting and Noninverting mplifiers with Positive Feedback R 1 R V V 1 s = s+bw 1 Basic Noninverting mplifier = BW s = s R 1 R Basic Inverting mplifier dequate model for most applications Is Positive Feedback bad? Both FB mplifiers are Unstable Engineers often make the assumption that positive feedback is bad and must be avoided Positive feedback in these stand-alone amplifiers resulted in unstable circuits Positive feedback is often very beneficial and should not be unilaterally avoided
17 Gain, Bandwidth and Consider Op mp with RHP Pole (Unstable Op mp) V V 1 s = s - BW 1 = BW db 1 (s) (s) db/ decade ω BW s-plane Im p OL Re BW Op mp is Unstable, dc gain is negative
18 V V 1 1 Gain, Bandwidth and Consider Op mp with RHP Pole (Unstable Op mp) VOUT V 1 = s = s - BW = BW Im s-plane p OL BW Re FB FB s s 1+s V = s V -V OUT IN 1 s = s-bw V V s 1- OUT IN BW BW V 1 R 1 R R = 1+ R 1 p CL s-plane Im p OL Re / BW Basic Noninverting mplifier Feedback mplifier is stable and performs very well! Serves as counter-example for Theorem!
19 Consider another Filter Example: V R 1 C V 1 R C V sc+sc+g = V G +V sc+v sc 1 IN OUT V sc+g = V sc+v G V 1 1 OUT 1 V OUT T s = s CR s +s - + CR1 CR 1- C R1R
20 Consider Filter Example: V R 1 C V 1 R C T s = s CR s +s - + CR1 CR 1- C R1R R X (1+)R X mplifier with gain Stable mplifier But if used in above, filter will be unstable R X (1+)R X mplifier with gain Unstable mplifier But if used in above, filter will be stable Serves as another counter example for theorem
21 Theorem: If a circuit is unstable, then if this circuit is included as a subcircuit in a larger circuit structure, the larger circuit will also be unstable. Unstable Circuit Unstable Circuit Larger Circuit Proof: This theorem is not valid though many circuit and filter designers believe it to be true!
22 Filter Concepts and Terminology X IN (s) Ts X OUT (s) X IN (z) Hz X OUT (z) Stability Issues: Is stability or instability good or bad? Often there is an impression that instability is bad - but why? Some observations: n unstable filter does not behave as a filter Unstable filter circuits are often used as waveform generators If an unstable circuit is embedded in a larger system, the larger system may be stable or it may be unstable If a stable circuit is embedded in a larger system, the larger system may be stable or it may be unstable Digital latches, RMs, etc. are unstable amplifiers Some of the best filter circuits include an embedded unstable filter Stability or Instability is neither good or bad, but it is important for the designer to be aware of the opportunities and limitations associated with this issue
23 Filter Concepts and Terminology -nd order polynomial characterization Biquadratic Factorization Op mp Modeling Stability and Instability Roll-off characteristics Distortion Dead Networks Root Characterization Scaling, normalization, and transformation
24 Single-pole roll-off characterization Magnitude Consider: Linear-Linear ωp Ts s+ω P w T jw ω j w+ω P T jω ω P P ω +ω P ω P m = - db/decade Magnitude Log-Log m = -6dB/octave ω P m 1 w ω 1 T jω tan ω P
25 Single-pole roll-off characterization Consider: Magnitude ω P ωp Ts s+ω Log-Log P w T jw ω j w+ω P ω T jω P P ω +ω P ω P m 1 w Phase 45 o 9 o ω 1 T jω tan ω P
26 Roll-off characterization t frequencies well-past a pole or zero, each LHP pole (real or complex) causes a roll-off in magnitude on a log-log axis of -db/decade and each LHP zero causes a roll-off of +db/decade t frequencies of magnitude comparable to that of a pole or zero, it is not easy to predict the roll-off in the magnitude characteristics by some simple expression
27 Filter Concepts and Terminology -nd order polynomial characterization Biquadratic Factorization Op mp Modeling Stability and Instability Roll-off characteristics Distortion Dead Networks Root Characterization Scaling, normalization, and transformation
28 Distortion in Filters Magnitude Distortion frequency dependent change in gain of a circuit (usuallybad if building amplifier but critical if building a filter) Phase Distortion a circuit has phase distortion if the phase of the transfer function is not linear with frequency Nonlinear Distortion Presence of frequency components in the outut that are not present in the input (generally considered bad in filters but necessary in many other circuits)
29 End of Lecture 4
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