LECTURE : Butterworh & Chebeyshev BP Filters Part : Series and Parallel RLC Circuits On NOT Again. RLC Admittance/Impedance Transfer Functions EXAMPLE : Series RLC. H(s) I out (s) V in (s) Y in (s) R Ls Cs L s s R L s LC exhibits a BP characteristic. Why? Consider s 0,s, and s j LC. EXAMPLE : Parallel RLC.
Lecture 0 Sp 8 R. A. DeCarlo H(s) V out (s) I in (s) Z in (s) Cs R Ls C s s RC s LC exhibits a BP characteristic, again. OBSERVATION: Both transfer functions have the same general structure: H BP (s) s as b THE QUESTION: Rather than study individual circuits, study the transfer function H BP (s) s as b that is common to a large number of circuits. The general TF analysis then applies to ALL circuits with such a transfer function and avoids repetitive calculations. Assumption: H BP (s) has complex poles.
Lecture 0 Sp 8 3 R. A. DeCarlo Part : The Four Faces of H BP (s) s as b.. FACE BOOK EQUATION PAGES (I), (II), (III), & (IV) (I) H BP (s) ( s σ p jω d ) s σ p jω d ( ) (explicit complex-poles) (II) ( s σ p ) ω d s σ p s ω p (equivalent rational forms) (III) s ω p Q p s ω p (relationship of pole-frequency,ω p, & circuit/pole-q, Q p selectivity) (IV) s B ω s ω p (B ω 3dB Bandwidth ω p Q p σ p )
Lecture 0 Sp 8 4 R. A. DeCarlo. PICTURE BOOK PAGES & 3. DEFINITION & NOTATION BOOK (a) Poles: σ p jω d (b) Pole frequency: ω p σ p ω d (often but not always the peak frequency) (c) Peak Frequency: ω m, i.e., the frequency at which the magnitude peaks. (We need to show it is ω p for the parallel and series RLC, but NOT for all BP circuits.) (d) Peak Gain: H m, i.e., the peak value of the magnitude response defined as H m H BP ( ). Equivalently H m max ω H BP ( jω ). jω m
Lecture 0 Sp 8 5 R. A. DeCarlo (e) Half Power Frequencies, ω and ω, i.e., the frequencies at which the gain is H m H BP ( jω ) H BP ( jω ), 3dB down from the peak. Note: Power [Gain] H m H m Half Power (f) Bandwidth Half-Power Bandwidth B ω ω ω. 4. THE WHERE O WHERE ARE WE GOING BOOK (i) Show ω m ω p. (ii) Show B ω! ω ω σ p ω m ω m ω m Q p Q cir Q (iii) H m K K K Q σ p B ω ω m (iv) Thus, we have the filtering attribute forms of H BP (s) : H BP (s) s ω p Q p s ω p s B ω s ω m (v) Show ω,ω σ p σ p ω m
Lecture 0 Sp 8 6 R. A. DeCarlo (vi) Show for high Q, ω,ω ω m B ω (approximation) Part 3. The Derivations Step. Given H BP (s) s σ p s ω p, set s jω, and in which case H BP ( jω ) Kjω ω p ω jσ p ω H BP ( jω ) K ω ω p σ p j ω K 4σ p ω ω p ω Step. Derivation of maximum value and peak frequency: H m max ω H BP ( jω ) K min ω 4σ p ω ω p ω K 4σ p Thus at ω ω p, we have a maximum, i.e., ω m ω p with a peak value of H m K. σ p
Lecture 0 Sp 8 7 R. A. DeCarlo Step 3. Derivation of formulas for half power frequencies. (i) By Definition: H m H BP ( jω i ) K ( ) K 4σ p 4σ p ω i ω p ω i (ii) Upon months of meditation on a Sunday Afternoon. This equation requires that 4σ p ω i ω p (iii) Judiciously taking square roots, there appears a quadratic equation in ω i : ω i 0 ω i ± σ p ω i ω p (iv) Solving using the quadratic formula: Thus ω,ω σ p 4σ p 4ω p σ p σ p ω m ω i ± σ p ω i ω m 0 if and only if ω, ±σ p σ p ω m
Lecture 0 Sp 8 8 R. A. DeCarlo Exact Half Power Frequency Formula: ω, ±σ p σ p ω m Step 4. Bandwidth: from step 3, ( ) ( σ p σ ) p ω σ m p. B ω ω ω σ p σ p ω m Consequence: H m K K. σ p B ω Step 5. Approximate Half Power Frequency Formula when Q 6 : (i) B ω! ω p Q implies ω p QB ω Qσ p (ii) If Q 6, ω p σ p σ p ω p. (iii) σ p ω p ω p 44 ω p ω p 44 ω p Conclusion: if Q ω m σ p ω m B ω 6, then ω, ω m B ω
Lecture 0 Sp 8 9 R. A. DeCarlo The Active BP Circuit WORKSHEET ACTIVE BP DESIGN has Transfer function H cir (s) V out (s) V in (s) s R C s R C R C s R R C C Design Objective: Compute circuit parameters so that ω m 8000 rad/s and B ω 500 rad/s: for some K H BP (s) s B ω s ω m s ω m Q s ω m s s
Lecture 0 Sp 8 0 R. A. DeCarlo Part. Normalized Design: Frequency scale down so that ω m,norm : K f : H BP,norm (s) H BP K ω m ( K f s) H BP ( s) s s Q s K 8000 s s s Part. Simplification and Normalized Parameter Design: Denominators/poles make up the normalized design: s s s R C R C s R R C C. Simplification: normalize capacitance: C C F.. Design Conditions: s s s R s R R (a) R R Ω (b) R R R R Ω
Lecture 0 Sp 8 R. A. DeCarlo (c) Unfortunately (more like a BP-filter amplifier) H m H ( jω m ) R C R C R C R R Design Specs : Final capacitor values are to be C, final C, final 0. µf. Part 3. Frequency and magnitude scaling. K f. Find K m. C new C old K m K f K m C old C new K f 0. 0 6 It follows that C new C new 0. µf and R new K m R old Ω, R new K m R old kω Remark: The ratio of R to R is very large and unrealistic in general.