Filters. Passive Filters

Transcription

1 Filter Paive Filter A filter i a iruit that i deigned to pa ignal with deired frequenie and rejet or attenuate other. A filter i a paive filter if it onit of only paive element, L, and C. A lowpa filter pae low frequenie and top high frequenie. A highpa filter pae high frequenie and rejet low. A bandpa filter pae frequenie within a frequeny band and blok or attenuate frequenie outide the band. A bandtop (band rejet) filter pae frequenie outide a frequeny band and blok or attenuate frequenie within the band.

2 Filter epone Ideal Filter Claifiation Low pa filter High pa filter Band pa filter Band rejet filter

3 Paive t order Lowpa Filter H( j) L jl j L jl L iruit Plot H(j) veru : Ideal Atual H( j ) H( j) max L H LPF ( j) j 3

4 Paive t order Lowpa Filter H( j) ( jc) ( jc) jc C iruit Plot H(j) veru : Ideal Atual H( j ) H( j) max C H LPF ( j) j 4

5 Paive t order Highpa Filter L H( j) jl j L j jl jl j C H( j) jc j ( jc) jc j 5

6 Example Effet of Loading on Filter Performane H () L K ( ) L / L, K L HPF unloaded Z H, Z L H eq HPF ( loaded ) eq L Zeq HPF ( loaded ) LL L L L LL L L K L LL L ( ) L L L K L K L ; ( loaded ) ( unloaded ) ( unloaded ) L K L L, L 6

7 Let = L =500 and L=5.3mH Example ont`d Then f (unloaded)=5khz For L = and f (loaded)= f =7.5 khz Effet of Loading on Filter Performane Change the utoff frequeny Change the gain in the paband 7

8 Paive nd order Lowpa Filter H() C L C C LC ( LC) ( ) ( ) L LC o H nd orderlpf o () o LC 8

9 9 Paive Bandpa Filter ) ( ) ( ) ( ) ( ) ( H LC L L LC C C C L H BPF o o LC B L Serie LC iruit

10 Paive Bandpa Filter Parallel LC iruit Zeq LC H(), Zeq L C Z L C eq LC L C LC L C LC ( C ) H() L C L C C LC o ( ) ( ) ( ) H BPF () LC o B C 0

11 Paive Bandpa Filter Frequeny repone harateriti: Only of the 5 harateriti of a bandpa filter an be peified independently. The other are related a follow: o B Q o

12 Example Deign a bandpa filter with a enter frequeny of khz and a bandwidth of 500 Hz. Ue a 50 reitor. B 500Hz C C 500Hz 500Hz 50.7 F o khz= L LC C 4.97 mh ( khz) ( khz).7 F 3 0 Note: Q o The Q i o low in order to ahieve the peified bandwidth.

13 Paive (Bandtop) Bandrejet Filter Serie LC iruit H() L C LC L C C LC ( LC) L LC o ( ) ( ) o H BF () LC o B L 3

14 Paive (Bandtop) Bandrejet Filter Parallel LC iruit L C H(), Zeq L C Z L C H() eq L C LC L C L C C LC LC L C o o ( ) ( ) ( ) ( ) ( ) H BF () LC o B C 4

15 Example Determine what type of filter i hown. Calulate the orner or utoff frequeny. Take = k, L = H, and C = μf. 5

16 Example ont d = k, L = H, C = μf Note: o rad/ 500 rad/ 6 LC 0 C 0 Qp L

17 Example For the iruit hown, obtain the tranfer funtion Vo(ω)/Vi(ω). Identify the type of filter the iruit repreent and determine the orner frequeny. Take = 00 =, L = mh. Thi i a highpa filter. 7

18 Example ont d 8

19 Example 3 If thi bandtop filter i to rejet a 00-Hz inuoid while paing other frequenie, alulate the value of L and C. Take = 50 and the bandwidth a 00 Hz. B 00Hz 00 rad/ B 50 L L B 00 rad/ H f 400 Hz 400 rad/ o 0 o LC C.66μF L 400 rad/ H 0 9

20 Example 4 Deign thi bandpa filter with a lower utoff frequeny of 0. khz and an upper utoff frequeny of 0.3 khz. Take = 0 k. Calulate L, C, and Q. B (0.3 0.) khz 400 rad/ (0.3 0.) khz Auming high Q : o 40.4 krad/ o 40.4 krad/ Q 0 B 400 rad/ 0k B L 5.96 H Too large!! L B 400 rad/ Q C oc oq 40.4 krad/ 00 k 3.9pF 0

21 Example 4 ont d Take = 0. B (0.3 0.) khz 400 rad/ (0.3 0.) khz o 40.4 krad/ Q 0 B 400 rad/ Auming high Q : o 40.4 krad/ L C mh B 400 rad/ 3 Q rad/00 o 3.9nF

22 Example 5 Ue a 500 nf apaitor to deign thi band-rejet filter. The filter ha a enter frequeny of 4 khz and a quality fator of 50. o L 3.66 mh LC 3 9 0C 40 rad/ 5000 F Q 50 Q oc 3.98k C o rad/ F

23 Ative Filter Paive Filter Diadvantage: Maximum gain in the pa band i no amplifiation in the pa band! Adding a load to a paive filter hange the filter harateriti, like the utoff frequeny. Paive band pa and band rejet filter require the ue of indutor, whih are large, otly, and an introdue tray eletromagneti field effet. No obviou method to make the filter harateriti more ideal. 3

24 Ative Filter Ative filter Baed on iruit with op amp, whih are ative devie, ine they require an external power upply ( V CC ). Eliminate all of the diadvantage of paive filter: Can reate a pa band gain > ; Can add load without hanging the filter harateriti; All four type of filter an be reated uing op amp, reitor, and apaitor no indutor; Can aade imple (firt- and eond-order) filter to reate higher order filter that are more nearly ideal. 4

25 Firt-Order Ative Lowpa Filter Find the tranfer funtion: V () o Z f C H() V() Z i i C C C C K C C K 5

26 Example Deign an ative low pa filter with a gain of -5 and a utoff frequeny of khz. Ue a 0 nf apaitor. 9 C (000) rad/ 5.96 k (000)(0 0 ) 5, k 6

27 Firt-Order Ative Highpa Filter Find the tranfer funtion: V Z o( ) f H ( ) V ( ) Z i C C i C C C For K = - K 7

28 Example Deign an ative high pa filter with a gain of -0 and a utoff frequeny of 500 rad/. Ue a 0. F apaitor. C 500rad/ 0k (500rad/)(0.F) 0 00k 8

29 Example Deign a firt-order highpa op amp filter whoe tranfer funtion i given by H( ) 4000 Ue apaitane of any value. But you only have 0 k reitor available - no indutor! The tranfer funtion of an inverting op amp i Thu, H() Z () f Z () i H() Therefore, Z () 0k f Z ( ) where 0 k and C 5nF i C 9

30 Firt-Order Ative Bandpa Filter = 30

31 Firt-Order Ative Bandpa Filter Give Give Give K 3

32 Firt-Order Ative Bandpa Filter H () H () H () H () BPF LPF HPF Inv i ( ) f i ( ) K 0 if f So thi aading tehnique only produe broadband filter. 3

33 Example Deign an ative band pa filter with a paband gain of and utoff frequenie of 500 Hz and 50 khz. Ue. nf apaitor. Lowpa filter: L (50, 000Hz)(.nF).447 k Highpa filter: H (500Hz)(.nF) 44.7 k Gain: i f 0k 5k 33

34 Firt-Order Ative Bandtop Filter Give Give K Give LPF Summing Amp. HPF 34

35 35 Firt-Order Ative Bandtop Filter 0 0 if ) ( ) ( ] )[ ( )] ( ) ( )[ ( ) ( i f i f HPF LPF Inv BF K H H H H So again, thi tehnique only produe broadband filter. Find the tranfer funtion:

36 Example Deign a noth filter between 0 krad/ and 00 krad/ with a gain of K = 5. You have only 0 k reitor available. Lowpa filter: CL 0krad/ 0k 5nF Highpa filter: CH nf 00krad/ 0k Gain: f 5 i 0k i 50k f 36

37 Example ont d Note: Due to the fat that the lowpa and highpa filter ued in thi example go down with only 0 db/de, attenuation in the topband might be limited. 37

38 Seond-Order Op-Amp Filter vi Bandpa Filter H vo 3 j C 3 j C( j) C Exerie: Try to derive thi tranfer funtion., Q C, B, H( ) K C 3 Q C Advantage: The bandwidth, thu Q fator, i entirely ontrolled by. 38

39 Example nd -order Bandpa Filter Deign for 3kHz, K=, Q=0: 39

40 Bandtop Filter vi H vo k ( j) C j( kc ) ( j) C k H k C C ( kc ) 0 C H 0 db C 40

41 Lowpa Filter v i v o Let C C C and H k j(3 kc ) ( j) C H H k 3 0 k 0 log k db 0 40 db/de 4

42 Saling There are two way of aling a iruit: magnitude or impedane aling, and frequeny aling. Both are ueful in aling repone and iruit element to value within the pratial range. Magnitude aling leave the frequeny repone of a iruit unaltered. Frequeny aling hift the frequeny repone up or down the frequeny petrum. 4

43 Magnitude Saling Magnitude aling i the proe of inreaing all impedane in a network by the ame fator, the frequeny repone remaining unhanged. Z k Z m Z k Z L m L Z k Z C m C k m jl k jl m km jc jc L C k m k L m C k m The primed variable are the new value and the unprimed variable are the old value. Conider the erie or parallel LC iruit. We now have o o LC LC k LC/ k m m 43

44 Frequeny Saling Frequeny aling i the proe of hifting the frequeny repone of a network up or down the frequeny axi while leaving the impedane the ame. k f f f j k L jl j k C jc L C L/ k C/ k f f Again, if we onider the erie or parallel LC iruit, for the reonant frequeny k f k LC LC o f o B k B f L/ k / f C kf 44

45 Magnitude and Frequeny Saling If a iruit i aled in magnitude and frequeny at the ame time, then L C k k k m m f C kk m f L f k Bk B f 45

46 Example edeign the ative low pa filter to ue a 00 pf apaitor. Ue magnitude aling: C k m C k m C 0nF 50 C 00pF 50(383.8 ) 59.9 k 50(5,95.5 ) k 46

47 Example edeign the ative low pa filter, whih wa deigned for khz, o that the utoff frequeny i 7.5 khz, uing a 50 pf apaitor. Ue frequeny aling firt, then magnitude aling: 7500 kf kf C C 0nF C km 5.3 kfkm kfc (7.5)(50pF) C 50 pf; 5.3(383.8 ) 6.98k ; k 47

48 Example 3 The fourth-order Butterworth lowpa filter i deigned uh that the utoff frequeny i ω = rad/. Sale the iruit for a utoff frequeny of 50 khz uing 0-k reitor. m km L L k f C C kk 3 ' 500 Frequeny aling: k f 0 3 ' 00 Magnitude aling: km 0 m 4 f 5 k k f 48

49 mh 49

50 Example 4 The third-order Butterworth filter i normalized to ω =. Sale the iruit to a utoff frequeny of 0 khz. Ue 5 nf apaitor. ' 0 k f 0 C C ' k k m f 4 C 0 km Ck 9 4 ' f ' km.06k 3 4 km 0 H L' L 33.77mH 4 k 3 0 C ' 5nF f 50