Analog Filter Synthesis

Transcription

1 6 Aalog Filter Sythesis Nam Pham Aubur Uiversity Bogda M. Wilamowsi Aubur Uiversity 6. Itrodutio Methods to Sythesize Low-Pass Filter...6- Butterworth Low-Pass Filter Chebyshev Low-Pass Filter Iverse Chebyshev Low-Pass Filter Cauer Ellipti Low-Pass Filter 6.3 Frequey Trasformatios...6- Frequey Trasformatios; Low-Pass to High-Pass Frequey Trasformatios; Low-Pass to Bad-Pass Frequey Trasformatios; Low-Pass to Bad-Stop Frequey Trasformatio; Low-Pass to Multiple Bad-Pass 6.4 Summary ad Colusio Referees Itrodutio Aalog filters are essetial i may differet systems that eletrial egieers are required to desig i their egieerig areer. Filters are widely used i ommuiatio tehology as well as i other appliatios. Although we disuss ad tal a lot about digital systems owadays, these systems always otai oe or more aalog filters iterally or as the iterfae with the aalog world [SV]. There are may differet types of filters suh as Butterworth filter, Chebyshev filter, iverse Chebyshev filter, Cauer ellipti filter, et. The harateristi resposes of these filters are differet. The Butterworth filter is flat i the stop-bad but does ot have a sharp trasitio from the pass-bad to the stop-bad while the Chebyshev filter has a sharp trasitio from the pass-bad to the stop-bad but it has the ripples i the pass-bad. Oppositely, the iverse Chebyshev filter wors almost the same way as the Chebyshev filter but it does have the ripple i the stop-bad istead of the pass-bad. The Cauer filter has ripples i both pass-bad ad stop-bad; however, it has lower order [W, KAS89]. The aalog filter is a broad topi ad this hapter will fous more o the methodology of sythesizig aalog filters oly (Figures 6. ad 6.). Setio 6. will preset methods to sythesize four differet types of these low-pass filters. The we will go through desig example of a low-pass filter that has 3 db atteuatio i the pass-bad, 3 db atteuatio i the stop-bad, the pass-bad frequey at Hz, ad the stop-bad frequey at 3 Hz to see four differet results orrespodig to four differet sythesizig methods. 6. Methods to Sythesize Low-Pass Filter 6.. Butterworth Low-Pass Filter ω p pass-bad frequey ω s stop-bad frequey α p atteuatio i pass-bad α s atteuatio i stop-bad 6- K47_C6.idd 6// 5:5:4 PM

2 6- Fudametals of Idustrial Eletrois 4 4 Magitude Magitude FIGURE 6. Butterworth filter (left), Chebyshev filter (right). AQ 4 4 Magitude Magitude FIGURE 6. Iverse Chebyshev filter (left), Cauer ellipti filter (right). Butterworth respose (Figure 6.3): T( jω) + ( ω / ω ) There are three basi steps to sythesize ay type of low-pass filters. The first step is alulatig the order of a low-pass filter. The seod step is alulatig poles ad zeros of a low-pass filter. The third step is desig iruits to meet pole ad zero loatios; however, this part is aother topi of aalog filters, so it will be ot be overed i this wor [W9, WG5, WLS9]. All steps to desig Butterworth low-pass filter. Step : Calulate order of filter: / α / / αs p log[( )( )] log( ω / ω ) s p ( eeds to be roudup to iteger value) K47_C6.idd 6// 5:5:46 PM

3 Aalog Filter Sythesis 6-3 α p ω p ω s db α s FIGURE 6.3 Butterworth filter harateristi. Step : Calulate pole ad zero loatios: Agle if is odd: Agle if is eve: 8 Ω ± ;,,, Ω ± ;,,, Normalized pole loatios: a os( Ω); b ± si( Ω); ( ω ) ω ( ω ω ) [( ) / ( )] / p s αs / αp / /( 4) ; Q a Step 3: Desig iruits to meet pole ad zero loatios (ot overed i this wor) (Figure 6.4). Example: Step : Calulate order of filter: log[( )( )] log(3 / ) Step : Calulate pole ad zero loatios Normalized values of poles ad ω ad Q: Normalized values of zeros oe. 3/ 3/ / i i i i K47_C6.idd 3 6// 5:5:55 PM

4 6-4 Fudametals of Idustrial Eletrois s -plae 4 Magitude Phase FIGURE 6.4 Pole-zero loatios, magitude respose, ad phase of Butterworth filter. 6.. Chebyshev Low-Pass Filter ω p pass-bad frequey ω s stop-bad frequey α p atteuatio i pass-bad α s atteuatio i stop-bad Chebyshev respose (Figure 6.5): Step : Calulate order of filter: α C T( jω) / ( + ε ( ω)) αs / p / / l[ 4 * ( ) / ( )] ( eeds to be roudup to iteger value) / log[( ω / ω ) + (( ω / ω ) ) ] s p s p T 6 (jω) Is here Is here Frequeies at whih C Frequeies at whih C / + ε FIGURE 6.5 Chebyshev filter harateristi. K47_C6.idd 4 6// 5:6: PM

5 Aalog Filter Sythesis 6-5 Step : Calulate pole ad zero loatios: 9 ( ) 8 Ω α ε p / sih ( / ε) γ / ; a sih( γ)os( Ω); b osh( γ)si( Ω); ω a + b ; Q K ω a Step 3: Desig iruits to meet pole ad zero loatios (ot overed i this wor) (Figure 6.6). Example: Step : Calulate order of filter: 3/ 3/ / l[4 *( ) / ( )] log[(3/ ) + ((3 / ) ) ] / Step : Calulate pole ad zero loatios Normalized values of poles ad ω ad Q: i i Normalized values of zeros oe. s-plae 3 x 4 α Magitude x 9 8 Phase FIGURE 6.6 Pole-zero loatios, magitude respose, ad phase of Chebyshev filter. K47_C6.idd 5 6// 5:6:8 PM

6 6-6 Fudametals of Idustrial Eletrois 6..3 Iverse Chebyshev Low-Pass Filter ω p pass-bad frequey ω s stop-bad frequey α p atteuatio i pass-bad α s atteuatio i stop-bad Iverse Chebyshev respose (Figure 6.7): T IC ( jω) ε C( / ω) + ε C ( / ω) The method to desig the iverse Chebyshev low-pass filter is almost the same as the Chebyshev lowpass filter. It is just slightly differet. Step : Calulate order of filter order of the Chebyshev filter Step : Calulate pole ad zero loatios: P i, i i < p a + b fid zeros ω 3 5 os[ Π * i / ( )] ; :,, Notes: two ojugate poles o the imagiary axis. Step 3: Desig iruits to meet pole ad zero loatios (ot overed i this wor) (Figure 6.8). Example: Step : Calulate order of filter: 3/ 3/ / l[4 *( ) / ( )] / log[(3/ ) + ((3 / ) ) ] Gai + Passbad Stopbad FIGURE 6.7 Iverse Chebyshev filter harateristi. K47_C6.idd 6 6// 5:6:3 PM

7 Aalog Filter Sythesis 6-7 s-plae x 4 Magitude x 9 8 Phase FIGURE 6.8 Pole-zero loatios, magitude respose, ad phase of iverse Chebyshev filter. Step : Calulate pole ad zero loatios Normalized values of poles ad ω ad Q: i i Normalized values of zeros: 3.464i 3.464i i 3.464i Cauer Ellipti Low-Pass Filter Cauer ellipti respose (Figure 6.9): T( jw) + ε R ( w, L) To desig the Cauer ellipti filter is more ompliated tha desigig three previous filters. I order to alulate the trasfer futio of this filter, a mathemati proess is summarized as below. Although the low-pass Cauer ellipti filter has ripples i both stop-bad ad pass-bad, it has lower order tha the three previous filters (Figure 6.). That is the advatage of the Cauer ellipti filter: p ω ω s (6.) q. 5( ) ( + ) (6.) (6.3) K47_C6.idd 7 6// 5:6: PM

8 6-8 Fudametals of Idustrial Eletrois G + ε G + ε L i FIGURE 6.9 Cauer ellipti filter harateristi. s-plae a 4 x Magitude x 9 Phase FIGURE 6. Pole-zero loatios, magitude respose, ad phase of Cauer ellipti filter. q q + q + 5q + 5 q (6.4) αs D. α (6.5) p D log( 6 ) log( / q) (6.6). 5α p + Λ l (6.7). 5α p K47_C6.idd 8 6// 5:6:9 PM

9 Aalog Filter Sythesis 6-9 σ / 4 q m m( m+ ) q m + m m m ( ) sih[( ) Λ] (6.8) + ( ) q osh( mλ) m ω + σ ( ) + σ (6.9) Ω i / 4 q m m( m+ ) q m + πµ m m m mπµ ( ) sih(( ) / ) (6.) + ( ) q osh m i for odd µ i for eve i,,..., r (6.) Ωi Vi ( Ωi ) (6.) A (6.3) Ω i B i ( σvi ) + ( Ωiω) ( + σ Ω ) B i σvi + σ Ω i i (6.4) (6.5) H σ r Bi A i i r. 5αp i i B A i for odd for eve (6.6) Example:.973. This filter is the seod low pass filter. Normalized values of poles ad ω ad Q: Normalized values of zeros: i i i 4.854i i 4.854i K47_C6.idd 9 6// 5:6:46 PM

10 6- Fudametals of Idustrial Eletrois 6.3 Frequey Trasformatios Four typial methods of derivig a low-pass trasfer futio that satisfies a set of give speifiatios are preseted. However, there are a lot of appliatios i the real world of desigig, whih require ot oly the low-pass filters but also the bad-pass filters, high-pass filters, ad bad-rejetio filters. A desiger a desig ay type of filters by desigig a low-pass filter first. Whe a low-pass filter is ahieved, the desired filter a be derived by frequey trasformatio. I other words, the uderstadig of methods to desig a low-pass filter is the basi but ot the trivial tas Frequey Trasformatios Low-Pass to High-Pass Z( s) S ; j ; Ω Ω Ω s jω ω ω frequey of low-pass passbad frequey of high-pass passbad Frequey trasformatio trasforms the pass-bad of the low-pass, etered aroud Ω, ito that of the high-pass, etered aroud ω (Figure 6.). Similarly, it trasforms the low-pass stop-bad that is etered aroud Ω ito that of the high pass, etered aroud ω. Cosequetly, the frequey trasformatio futio Z(s) has a zero i the eter of the pass-bad of the high-pass (at ω ) ad a pole i the eter of the high-pass, stop-bad (at ω ) [SV]: T( S) S ω + ( ω S / Q) + ω ω s T( s) ( / s ) + ( ω / Qs) + ω ( / ω ) + ( s/ Q) + s ω T(S): low-pass trasfer futio; T(s): high-pass trasfer futio Frequey Trasformatios Low-Pass to Bad-Pass s + ω ω( s + ω ) ω ω Z( s) S Ω Bs Bω s Bω ; ω ω ω ; B ω ω Frequey trasformatio trasforms the pass-bad of the low-pass, etered aroud Ω, ito that of the bad-pass, etered aroud ω ω. Similarly, it trasforms the low-pass stop-bad that is etered aroud Ω ito that of the bad-pass, etered aroud ω (Figure 6.). Cosequetly, Ω ω s S FIGURE 6. Frequey trasformatios low-pass to high-pass. K47_C6.idd 6// 5:6:5 PM

11 Aalog Filter Sythesis 6- Ω B Ω ω Ω ω ω ω FIGURE 6. Frequey trasformatios low-pass to bad-pass. the frequey trasformatio futio Z(s) has zeros i the eter of the pass-bad of the bad-pass (at ω ± ω ) ad poles i the eter of the bad-pass, stop-bad (at ω ad ω ) [SV]: T( S) S ω + ( ω S/ Q) + ω T( s) s B ω 4 3 s + ( ω Bs / Q) + ( ω + B ω ) s + ( ω Bω s/ Q) + ω 4 T(S): low-pass trasfer futio; T(s): bad-pass trasfer futio Frequey Trasformatios Low-Pass to Bad-Stop Bs B Z( s) S ω Ω ; ω ω ω ; B ω ω s + ω ω ω Frequey trasformatio trasforms the pass-bad of the low-pass, etered aroud Ω, ito that of the bad-stop, etered aroud ω ad ω (Figure 6.3). Similarly, it trasforms the low-pass, stop-bad that is etered aroud Ω ito that of the bad-stop, etered aroud ω ω. Cosequetly, the frequey trasformatio futio Z(s) has zeros i the eter of the pass-bad of the bad-stop (at ω ad ω ) ad poles i the eter of the bad-stop, stop-bad (at ω ± ω ) [SV]: Ω B Ω ω Ω ω ω ω FIGURE 6.3 Frequey trasformatios low-pass to bad-stop. K47_C6.idd 6// 5:6:57 PM

12 6- Fudametals of Idustrial Eletrois T( S) S ω + ( ω S/ Q) + ω T( s) ω ω s 4 + ω ω s + ω ω s + ( ω Bs / Q) + ( ω ω + B ) s + ( ω Bω s/ Q) + ω ω Frequey Trasformatio Low-Pass to Multiple Bad-Pass Frequey trasformatio trasforms the pass-bad of the low-pass, etered aroud Ω, ito that of the multiple bad-pass, etered aroud ω ad ω ω z. Similarly, it trasforms the low-pass, stop-bad that is etered aroud Ω ito that of the multiple bad-pass, etered aroud ω ω p ad ω. Cosequetly, the frequey trasformatio futio Z(s) has zeros i the eter of the passbad of multiple bad-pass ad at ω z (at ω ad ω ±ω z ) ad poles i the eter of the bad-stop of multiple pass-bad (at ω ±ω ad ω ) [SV] (Figure 6.4): s( s + ωz) Z( s) S B( s + ω ) P ω( ω ωz) Ω B( ω ω ) P Trasfer futios from the low-pass frequey S to the frequey s of other types of filters are reogized ad a be writte uder the followig form: H( s + ωz)( s + ωz) ( s + ωz) Z( s) ( s + ω )( s + ω ) ( s + ω p p p) Or Ω( ω) H( ω ω z)( ω ω z) ( ω ω z) ( ω ω )( ω ω ) ( ω ω p ) p p Z(s) has zeros where the desired filter has pass-bads ad poles where it has stop-bads. The futio Z(s) is alled Foster Reatae futio. For example, we a write the trasfer futio of the filter (Figure 6.5) as Hs( s + ωz) Z( s) ( s + ω ) p or Hω( ω ωz) Ω( ω) ( ω ω ) p Ω ω ω ω 3 Ω ω Ω ω z ω p ω z FIGURE 6.4 Frequey trasformatio low-pass to multiple bad-pass. K47_C6.idd 6// 5:7:7 PM

13 Aalog Filter Sythesis 6-3 db 3 db Hz ω p 4 Hz ω z 6 Hz FIGURE 6.5 Frequey trasformatio by foster reatae futio. The trasfer futio has zeros at ω, ω ω z ad poles at ω ω p ad ω. At orer frequeies ω Hz, ω 4 Hz, ω 3 6 Hz, the values of Ω (ω) are equal to,, ad, respetively. Therefore, the trasformatio Ω (ω) a be rewritte ito multi-equatios orrespodig to ω ω, ω, ω 3. Three equatios with three uows always have solutios: Hω( ω ω Z ) ω ω P Hω( ω ω Z ) ω ω P Hω3( ω3 ω Z ) ω ω 3 P ωz ω p 8 ; so the Foster Trasfer Futio is S H 3 3 ( / 3) s + ( / 3) s s Summary ad Colusio Aalog filters have bee used broadly i ommuiatio. Uderstadig the methods to sythesizig aalog filters is extremely importat ad is the basi step to desig aalog filters. Four differet sythesizig methods were preseted, eah method will result i differet harateristis of filters. Besides that, this hapter also preseted steps to desig other types of filters from the low-pass filter by writig the frequey trasfer futio. Referees [SV] R. Shauma ad M.E. Va Valeburg, Aalog Filter Desig, Oxford Uiversity Press, Oxford, U.K.,. [W] S. Wider, Aalog ad Digital Filter Desig, Newes, Wobur, MA,. K47_C6.idd 3 6// 5:7:5 PM

14 6-4 Fudametals of Idustrial Eletrois [KAS89] M.R. Kobe, J. Ramirez-Agulo, ad E. Sahez-Sieio, FIESTA-A filter eduatioal sythesis teahig aid, IEEE Tras. Edu., 3(3), 8 86, August 989. [W9] B.M. Wilamowsi, A filter sythesis teahig-aid, i: Proeedigs of the Roy Moutai ASEE Setio Meetig, Golde, CO, April 6, 99. [WG5] B.M. Wilamowsi ad R. Gottiparthy, Ative ad passive filter desig with MATLAB, It. J. Eg. Edu., (4), 56 57, 5. [WLS9] B.M. Wilamowsi, S.F. Legowsi, ad J.W. Steadma, Persoal omputer support for teahig aalog filter aalysis ad desig ourses, IEEE Tras. Edu., E-35(4), 35 36, 99. K47_C6.idd 4 6// 5:7:5 PM