Prof. Paolo Colantonio a.a

Transcription

1 Pro. Paolo olantono a.a. 3 4

2 Let s consder a two ports network o Two ports Network o L For passve network (.e. wthout nternal sources or actve devces), a general representaton can be made by a sutable matrx relatng and across the external ports: Z matrx Y matrx Z Z Z Z o o H matrx H H o H H Y Y o Y Y o BD (or Transmsson) matrx B o D o Note: Other representatons are avalable (e.g. S parameters) Pro. Paolo olantono 7

3 For a generc two ports network, the ollowng electrcal parameters are typcally dened: o Two ports Network o L nput resstance oltage gan o Output resstance o o o urrent gan o o L L Power gan o P o o P P The power gan s usually expressed n decbel db PdB P Sometmes also the voltage gan s expressed n decbel db P o log log P o log,, db Pro. Paolo olantono 3 7

4 For a cascade o networks: db db 5 db, 5 45 db PdB Power gan (lnear) Power gan (db) 3 3,5 3,,, 3 Pro. Paolo olantono 4 7

5 The requency response o a network s dented by the varaton o ts gan (voltage, current or power) vs. requency, both n terms o ampltude and phase sht. Example Z Z Z Z Z o o Transer uncton The requency response s characterzed by the Bode dagrams db phase log log Pro. Paolo olantono 5 7

6 o o o j Denng the resonatng requency phase tan j << m( ) j m( ) = j >> m( ) phase e( ) dB phase 45 e( ) phase e( ) Pro. Paolo olantono 6 7

7 db 3 = slope 6dB/octave db/decade log() phase( ) 9 45 = log() =. = Hgh Pass nput sgnal N req OUT Output sgnal tme tme mpltude mpltude Freq Pro. Paolo olantono 7 7 Freq

8 o o o j j j phase Denng the resonatng requency tan j << m( ) m( ) = j >> m( ) j phase e( ).77 3dB phase 45 e( ) phase 9 e( ) Pro. Paolo olantono 8 7

9 db 3 slope 6dB/octave db/decade phase( ) 45 = log() 9 = log() =. = Low Pass N req OUT mpltude mpltude nput sgnal Output sgnal tme tme Freq Pro. Paolo olantono 9 7 Freq

10 L o o jl L j db 3 slope 6dB/octave db/decade phase( ) 45 = log() 9 = log() =. = Pro. Paolo olantono 7

11 L o o jl jl L j db 3 = slope 6dB/octave db/decade log() phase( ) 9 45 = log() =. = Pro. Paolo olantono 7

12 rcuts usng and L technques have smlar characterstcs Pro. Paolo olantono 7

13 Straght lne approxmatons Pro. Paolo olantono 3 7

14 reatng more detaled Bode dagrams Pro. Paolo olantono 4 7

15 B B db 3 db 3 db 3 log() = log() = log() = = phase( ) 9 phase( ) phase( ) = log() 9 = log() = = log() 45 9 << c Band pass lter Pro. Paolo olantono 5 7

16 Band Pass Band Stop N OUT N OUT H L req L H req Output sgnal tme tme tme tme Pro. Paolo olantono 6 7 nput sgnal mpltude L H L H L H Freq Freq Freq L H mpltude mpltude mpltude nput sgnal Output sgnal

17 L L j j L Denng the resonatng requency L L nd the parameter Q (qualty actor) Q L L jq phase Q tan Q Pro. Paolo olantono 7 7

18 The requency behavor o the transer uncton s reported n the ollowng ampltude and phase plots, 9,8 Q=/ 6,6,4 Q= phase( ) 3-3 Q=/, Q=,,,5,,5,,5 3, / -6 Q= -9,,5,,5,,5 3, / Q= The resonatng phenomenon s hghlghted n the ampltude plot and the requency behavor s strctly dependng on the value o Q, that s also reerred as resonatng coecent. Pro. Paolo olantono 8 7

19 The crcut bandwdth s gven by the requences or those s reduced by 3dB (.e. =,5 Max) Q Q 4 Q Q Q >> (typcally) Q L H Q Q Q H L BW Pro. Paolo olantono 9 7

20 The current n the elements s gven by G jq G jq jq L G jq jq t the resonatng requency = The current lowng nto s the same o the external source G The current lowng nto L and s the same n ampltude (derent sgn) and Q tmes larger than The average energy stored n the nductor and capactor s gven by: T Q L LL t dt LL L G T 4 4 Q T Q Q v t dt G G T Q Q Pro. Paolo olantono 7

21 t the resonatng requency = L LQ G 4 LQ G 4 P G Power dsspated n the resstor ememberng that Q L Q L G The qualty actor can be also expressed as Q Energy stored Power Loss Pro. Paolo olantono 7

22 L L The voltage across each component and the current can be easly determned n requency doman jl j j jl j L jl jl j Denng the resonatng requency L nd the parameter Q (qualty actor) Q L L jq The same consderaton as or the parallel L case can be appled. Pro. Paolo olantono 7

23 Seres L The mpedance s gven by Z jl j the magntude o the reactance o the nductor and capactor are equal, the magnary part s zero, and the mpedance s smply. Ths occurs when Ths stuaton s reerred to as resonance The requency at whch s occurs s the resonant requency n the seres resonant crcut, the mpedance s at a mnmum at resonance whle the current s at a maxmum The seres L crcut s an acceptor crcut L L L Q B B L Hz Pro. Paolo olantono 3 7

24 Parallel L The admttance s gven by Y jl j the magntude o the reactance o the nductor and capactor are equal, the magnary part s zero, and theadmttances smply /. Ths occurs when L L Ths stuaton s reerred to as resonance The requency at whch s occurs s the resonant requency n the parallel resonant crcut, the mpedance s at a maxmum at resonance whle the current s at a mnmum The parallel L crcut s a rejector crcut Q B L B Hz Pro. Paolo olantono 4 7

25 The networks consdered earler are rst order or sngle pole lters. These have a maxmum roll o o 6 db/octave they also produce a maxmum o 9 phase sht ombnng multple stages can produce lters wth a greater ultmate roll o rates ( db, 8 db, etc.) but such lters have a very sot knee n deal lter would have constant gan and zero phase sht or requences wthn ts pass band, and zero gan or requences outsde ths range (ts stop band) eal lters do not have these dealzed characterstcs Pro. Paolo olantono 5 7

26 The use o combnaton o nductors and capactors can produce very hgh perormance lters. ommon orms nclude: Butterworth optmsed or a lat response hebyshev optmsed or a sharp knee Bessel optmsed or ts phase response The use o nductors s nconvenent snce they are expensve, bulky and suer rom greater losses than other passve components. ctually, actve lters are realzed by usng operatonal amplers Pro. Paolo olantono 6 7

27 ll crcuts have losses (.e., ntrnscally resstors) ll crcuts have stray capactance and stray nductance These unntended elements can dramatcally aect crcut operaton and have to be careully and properly taken nto account s adds an unntended low pass lter L s adds an unntended low pass lter s produces an unntended resonant crcut and can produce nstablty Pro. Paolo olantono 7 7