Lecture 28. Passive HP Filter Design

Transcription

1 Lecture 28. Paive HP Filter Deign STRATEGY: Convert HP pec to Equivalent NLP pec. Deign an appropriate 3dB NLP tranfer function. Realize the 3dB NLP tranfer function a a circuit. Convert the 3dB NLP circuit realization to a HP circuit realization OR convert the 3dB NLP circuit to a NLP circuit and then convert to a HP circuit. Cutomary Filter Deign Proce: all filter deign require converion to equivalent NLP pec, where a NLP filter i deigned and converted back into the requiite HP (or other type of) filter. Quetion: Profeor Ray. How do we do the converion? Anwer: I haven t read that far in my note. But God bleed the broken road that led you traight to Approximately Racal Flat. High Pa Spec: (, A max ), (ω HP, A min ), > ω HP : STOPBAND: 0 ω ω HP in which db lo A min TRANSITION BAND: ω HP ω PASSBAND: ω in which db lo A max

2 REMARK: Oberve that in the tranition band: ω HP ω chp. 2. CONVERSION TO NLP SPECS: The converion i achieved by a frequency tranformation on the HP pec: : (i) j j (ii) jω HP jω HP! Ω > (iii) j j 0 (iv) j0 j0 CONCLUSION : Equivalent NLP Spec: (, A max ), A min ω HP.

3 Concluion: the HP axi i tranformed to a NLP axi in ω. QUESTION: Prof Ray. Doe thi work for tranfer function? Anwer: Let check it out. EXAMPLE. Converion of a HP tranfer function to a NLP tranfer function uing above tranformation. Thi i to illutrate the validity of the proce but we would NOT actually do thi in a filter deign. H HP () Q + ω 2 php H NLP () H HP Q 2 + Q + + ω 2 p ω php 2 + Q 2 + ω 2 php STUDENT COMMENT: Profeor Ray. Thi look like the Sallen and Key LP circuit tranfer function. Can we do that crazy frequency tranformation to the Sallen and Key circuit? ANSWER: Ye, but not today or in thi coure. 3. Step in Butterworth HP Filter Deign: Given the high Pa Spec: (, A max ), (ω HP, A min ), > ω HP : Step. Generate equivalent NLP pec: (, A max ) Step 2. Compute filter order, n. In MATLAB:, A min ω HP.

4 buttord(,wphp/whp,amax,amin, ) Example 2: Suppoe n 2 for ome A max, A min, ω,hp, and ω p,hp Step 3. Determine the tranfer function. In our cae, find n-th order Butterworth tranfer function H 3dBNLP () uing: (a) [z,p,k] buttap(n) (b) num k*poly(z) (c) den poly(p) Example 2 continued: If n 2, H 3dBNLP () Step 4. Realize H 3dBNLP () a a paive circuit. Example 2 continued: If n 2, the circuit below work. Set R Ω without lo of generality. Matching coefficient we have: H cir () LC 2 + L + LC H 3dBNLP () Hence L 2 H and C 2 F.

5 Step 5. Frequency cale the circuit obtained in tep 4 by K f Ω c,min to produce a NLP circuit. 2n 0 0.A max Example 2 continued: R Ω, L NLP K f 2 H and C NLP 2 F. K f Step 6. Generate the o-called normalized HP filter (NHP); NHP mean that the A max pec i met at. (a) Here one ue the NLP to NHP tranformation: (b) Thi NLP to NHP change inductor to capacitor and capacitor to inductor: (i) L NLP L NLP L NLP, i.e., an inductor for the NLP C NHP circuit become a capacitor in the NHP circuit with value C NHP L NLP. (ii) the admittance C NLP C NLP C NLP L NHP, i.e., a capacitor of the NLP circuit become an inductor in the NHP circuit with value L NHP C NLP. Example 2 Continued: R Ω, C NHP L NLP K f L NHP C NLP K f 2 H. 2 F and O Student O Student, DRAW CIRCUIT, be prudent.

6 Step 7. Frequency and magnitude cale the NHP circuit of tep 6. Here K f 2 ω p. K m i choen according to the intruction in the problem tatement. Draw and label your final circuit. Example 2 Continued: R, final K m Ω, C HP K f K m K f 2 2 F and L HP K m K f K f 2 2 H. Example 3. A third order Butterworth HP circuit i to have ω chp 0 4 rad/. The ource and load reitance of the 3 rd NLP prototype below are to be 0 Ω in the final deign. Step. Compute tranfer function H cir () V out L L 2 C V in L L L + L 2 + C L L 2 C + 2 L L 2 C Step 2. Match Coefficient L L L + L 2 + C L L 2 C + 2 L L 2 C Simplifying aumption when circuit i ymmetric a above for Butterworth filter only: L L 2 L will maintain olvability. Hence

7 Equating coefficient: CHECK : 2L + C L 2 C L 2 + 2L + C L 2 C + 2 L 2 C 2 2 L L H and 2 L 2 C 2 C C 2 F Step 3. Compute 3dBNHP circuit for which the HP gain i 3dB down at the normalized frequency Ω. Here: which change the inductor to capacitor and the capacitor to inductor with different value. Step 4. Frequency cale by K f ω chp and magnitude cale appropriately.