EE 508 Lecture 18. Basic Biquadratic Active Filters. Second-order Bandpass Second-order Lowpass Effects of Op Amp on Filter Performance

Transcription

1 EE 58 Lecture 8 Basic ratic Active Filters Second-order Bandpass Second-order Lowpass Effects of Op Amp on Filter Performance

2 eview from Last Time Standard LP to BS Transformation map = to = ± map = to = map = to = A map = to = - B map = to = B map = to = - A T j - = - TBS j = - -

3 eview from Last Time Pole Q of BS Approximations LP to BS Transformation T j TBP j BW N Im AN BN,Q BPH LBPH BW = BN - AN,Q LP LP Im e,q BPL e LBPL Define: BW M It can be shown that M ANBN LP Q Q Q LP BSL BSH QLP For γ small, It can be shown that BS Q BS Q LP M Q BS Q BS 4 QLP Q LP Note for γ small, Q BS can get very large

4 eview from Last Time Standard LP to BS Transformation s s BW s X M Standard LP to BS transformation is a variable mapping transform Maps j axis to j axis in the s-plane Preserves basic shape of an approximation but warps frequency axis Order of BS approximation is double that of the LP Approximation Pole Q and expressions are identical to those of the LP to BP transformation Pole Q of BS approximation can get very large for narrow BW Other variable transforms exist but the standard is by far the most popular

5 eview from Last Time LP to HP Transformation (Normalized Transform) T j - THP j = - = -

6 eview from Last Time omparison of Transforms LP to BP s s + s s BW LP to BS N s BW s N T j T j TLPN j BS TBP AN j BW N BN BW L H M LP to HP T j T j LP HP s s

7 Filter Design Process Establish Specifications - possibly T D (s) or H D (z) - magnitude and phase characteristics or restrictions - time domain requirements Approximation - obtain acceptable transfer functions T A (s) or H A (z) - possibly acceptable realizable time-domain responses Synthesis - build circuit or implement algorithm that has response close to T A (s) or H A (z) - actually realize T (s) or H (z) Filter

8 Filter Design/Synthesis onsiderations There are many different filter architectures that can realize a given transfer function onsiderable effort has been focused over the years on inventing these architectures and on determining which is best suited for a given application

9 Filter Design/Synthesis onsiderations Most even-ordered designs today use one of the following three basic architectures ascaded s T (s) T (s) T k (s) T m (s) Leapfrog Ts T T Tm I (s) I (s) I 3 (s) I 4 (s) I k- (s) I k (s) Integrator Integrator Integrator Integrator Integrator Integrator a a Multiple-loop Feedback (less popular) α F α α α m X IN α T (s) T (s) T m (s) X OUT What s unique in all of these approaches?

10 Filter Design/Synthesis onsiderations Most odd-ordered designs today use one of the following three basic architectures ascaded s T (s) T (s) T k (s) T m (s) T m+ (s) First Order Leapfrog Ts T T Tm I (s) I (s) I 3 (s) I 4 (s) I k- (s) I k (s) Integrator Integrator Integrator Integrator Integrator Integrator a a Multiple-loop Feedback (less popular) X IN α F α T (s) T (s) T m (s) T m+ (s) α α α m What s unique in all of these approaches? First Order α m + X OUT

11 Filter Design/Synthesis onsiderations What s unique in all of these approaches? T(s) I(s) Integrator T m+ (s) First Order X IN α α α α k X OUT T s = a s +a s+a s +bs+b a s+a s+b Is = I Ts = s OUT k X = α i= i Most effort on synthesis can focus on synthesizing these four blocks (the summing function is often incorporated into the or Integrator) (the first-order block is much less challenging to design than the biquad) Some issues associated with their interconnections And, in many integrated structures, the biquads are made with integrators (thus, much filter design work simply focuses on the design of integrators)

12 s How many biquad filter functions are there? T(s) T s = a s +a s+a s +bs+b a s +a s+a s +bs+b, T s = a, a a a T s = a s +bs+b as T s = a s +bs+b a s +a Ts = a, a s +b s+b a s+a T s = a, a s +bs+b as Ts = a s +bs+b a s +a s Ts = a, a s +b s+b

13 Filter Design/Synthesis onsiderations eview: Second-order bandpass transfer function H H TBP j A B T BP s X IN s Q H s + s + Q BW = B- A = Q PEAK = Ts X OUT

14 Filter Design/Synthesis onsiderations There are many different filter architectures that can realize a given transfer function Will first consider second-order Bandpass filter structures X IN Ts X OUT T s s Q H s + s + Q BW = B- A = Q PEAK =

15 Filter Design/Synthesis onsiderations There are many different filter architectures that can realize a given transfer function Will first consider second-order Bandpass filter structures Example : L VOUT s Ts VIN s +s + L Second-order Bandpass Filter 3 degrees of freedom degrees of freedom for determining dimensionless transfer function (impedance values scale)? Q =? BW =?

16 Example : L VOUT s Ts VIN s +s + L L Q = BW = L an realize an arbitrary nd order bandpass function within a gain factor Simple design process (sequential but not independent control of and Q)

17 Example : K 3 VOUT K s Ts VIN -K + s +s Second-order Bandpass Filter 6 degrees of freedom (effectively 5 because dimensionless) Denote as a +K filter? Q =? BW =?

18 Example : K Equal, Equal ealization VOUT K s Ts VIN 4-K s +s +? Q =? BW =?

19 Example : K Equal, Equal ealization VOUT K s Ts VIN 4-K s +s + 3 degrees of freedom (effectively since dimensionless) Q = 4-K 4-K BW = an satisfy arbitrary nd =order BP constraints within a gain factor with this circuit Very simple circuit structure Independent control of and Q but requires tuning more than one component an actually move poles in HP by making K >4

20 Example : Unity Gain, Equal VOUT s Ts VIN s +s + +? Q =? BW =?

21 Example : Unity Gain, Equal VOUT s Ts VIN s +s + + Q + BW =

22 Example 3: -K VOUT K s Ts VIN +K s +s K +K Second-order Bandpass Filter 5 degrees of freedom (4 effective since dimensionless) Denote as a -K filter? Q =? BW =?

23 Example 3: -K Equal, Equal ealization VOUT K s Ts VIN +K 3 s +s + +K +K 3 degrees of freedom? Q =? BW =?

24 Example 3: -K Equal, Equal ealization VOUT K s Ts VIN +K 3 s +s + +K +K +K 3 degrees of freedom ( effective since dimensionless) +K Q = 3 3 BW = +K an satisfy arbitrary nd =order BP constraints within a gain factor with this circuit Very simple circuit structure Simple design process (sequential but not independent control of and Q, requires tuning of more than component if s used)

25 Observation: 3 K -K These are often termed Sallen and Key filters Sallen and Key introduced a host of filter structures Sallen and Key structures comprised of summers, network, and finite gain amplifiers These filters were really ahead of their time and appeared long before practical implementations were available 955

26 Example 4: Second-order Bandpass Filter VOUT s Ts VIN s +s degrees of freedom (3 effective since dimensionless) Denote as a bridged T feedback structure

27 Example 4: Equal implementation VOUT s Ts VIN s +s + 3 degrees of freedom ( effective since dimensionless)? Q =? BW =?

28 Example 4: Equal implementation VOUT s Ts VIN s +s + Q = BW = Simple circuit structure More tedious design/calibration process for and Q (iterative if is fixed) esistor ratio is 4Q

29 Example 4: Some variants of the bridged-t feedback structure K K

30 Are there more nd order bandpass filter structures? Yes, many other nd -order bandpass filter structures exist But, if we ask the question differently Are there more nd -order bandpass filter structures comprised of one amplifier and four passive components? Yes, but not too many more Are there more nd -order bandpass filter structures comprised of one amplifier, two capacitors, and three resistors? Yes, but not too many more Similar comments can be made about nd -order LP, BP, and B T s H s + s + Q T s Hs s + s + Q T s Hs s + s + Q Similar comments can be made about full nd -order biquadratic function T s s + s + H s + s + Q N N QN

31 End of Lecture 8