EE Lecture Small Sgnal Analss Small Sgnal Analss o BJT Ampler
Revew rom Last Lecture Comparson o Gans or MOSFET and BJT Crcuts N (t) A B BJT CC Q R EE OUT R CQ t DQ R = CQ R =, SS + T = -, t =5m R CQ A - =-8 B 5m t N (t) A MOSFET M M DD R SS SS OUT R DQ R 4 A = - 4 - DQ M SS T Observe A B >>A M Due to eponental-law rather than square-law model T
Revew rom Last Lecture Operaton wth Small-Sgnal nputs Analss procedure or these smple crcuts was ver tedous Ths approach wll be unmanageable or even modestl more complcated crcuts Faster analss method s needed!
Revew rom Last Lecture Small-Sgnal Analss Basng (voltage or current) NSS or NSS Nonlnear Crcut OUTSS OUTSS Nonlnear Analss Map Nonlnear crcut to lnear small-sgnal crcut NSS or NSS Lnear Small Sgnal Crcut OUTSS OUTSS Lnear Analss Wll commt net several lectures to developng ths approach Analss wll be MUCH smpler, aster, and provde sgncantl more nsght Applcable to man elds o engneerng
Revew rom Last Lecture Small-Sgnal Prncple Regon around Q-Pont =() Y Q Q-pont X Q Relatonshp s nearl lnear n a small enough regon around Q-pont Regon o lneart s oten qute large Lnear relatonshp ma be derent or derent Q-ponts
Revew rom Last Lecture Small-Sgnal Prncple Q-pont ss =() Y Q ss X Q Devce Behaves Lnearl n Neghborhood o Q-Pont Can be characterzed n terms o a small-sgnal coordnate sstem
Revew rom Last Lecture Arbtrar Nonlnear One-Port Nonlnear One-Port SS = v SS SS SS de = = Q v SS Q Q-pont v SS de = Q den = Q Lnear model o the nonlnear devce at the Q-pont
Revew rom Last Lecture Arbtrar Nonlnear One-Port Nonlnear One-Port -Termnal Nonlnear Devce (v) = Q Lnear small-sgnal model: A Small Sgnal Equvalent Crcut: The small-sgnal model o ths -termnal electrcal network s a resstor o value / or a conductor o value One small-sgnal parameter characterzes ths one-port but t s dependent on Q- pont Ths apples to ANY nonlnear one-port that s derentable at a Q-pont (e.g. a dode)
Small-Sgnal Prncple Goal wth small sgnal model s to predct perormance o crcut or devce n the vcnt o an operatng pont (Q-pont) Wll be etended to unctons o two and three varables (e.g. BJTs and MOSFETs)
Soluton or the eample o the prevous lecture was based upon solvng the nonlnear crcut or OUT and then lnearzng the soluton b dong a Talor s seres epanson Soluton o nonlnear equatons ver nvolved wth two or more nonlnear devces Talor s seres lnearzaton can get ver tedous multple nonlnear devces are present Standard Approach to small-sgnal analss o nonlnear networks. Solve nonlnear network. Lnearze soluton Alternatve Approach to small-sgnal analss o nonlnear networks.lnearze nonlnear devces (all). Obtan small-sgnal model rom lnearzed devce models. Replace all devces wth small-sgnal equvalent 4.Solve lnear small-sgnal network
Alternatve Approach to small-sgnal analss o nonlnear networks Nonlnear Network dc Equvalent Network Q-pont alues or small-sgnal parameters Small-sgnal (lnear) equvalent network Small-sgnal output Total output (good appromaton)
Lnearzed nonlnear devces Nonlnear Devce Lnearzed Small-sgnal Devce Ths termnolog wll be used n THS course to emphasze derence between nonlnear model and lnearzed small sgnal model
Eample: t wll be shown that the nonlnear crcut has the lnearzed small-sgnal network gven R DD OUT M OUT N M R N SS Nonlnear network Lnearzed smallsgnal network
Lnearzed Crcut Elements Must obtan the lnearzed crcut element or ALL lnear and nonlnear crcut elements DC DC AC C Large R L Small L Large C Small AC (Wll also gve models that are usuall used or Q-pont calculatons : Smpled dc models)
Small-sgnal and smpled dc equvalent elements Element ss equvalent Smpled dc equvalent dc oltage Source DC DC ac oltage Source AC AC dc Current Source DC DC ac Current Source AC AC Resstor R R R
Small-sgnal and smpled dc equvalent elements Element ss equvalent Smpled dc equvalent C Large Capactors C Small C L Large nductors L Small L Dodes Smpled MOS transstors (MOSFET (enhancement or depleton), JFET) Smpled Smpled
Small-sgnal and smpled dc equvalent elements Element ss equvalent Smpled dc equvalent Bpolar Transstors Smpled Smpled Dependent Sources (Lnear) O =A N O =R T N O =A N O =G T N
Eample: Obtan the small-sgnal equvalent crcut DD R C R OUT NSS R C s large R R OUT R OUT N R //R N R
Eample: Obtan the small-sgnal equvalent crcut R DD OUT M N SS R OUT N M OUT N M R
Eample: Obtan the small-sgnal equvalent crcut DD C R R 4 R5 R 7 C DD OUT Q M R L NSS R R C R 6 C 4 SS C,C, C large C 4 small R R 4 R5 R 7 Q M N R R R 6 C 4 OUT R L N R //R Q R 4 R 5 C 4 R 6 M R 7 OUT R L
How s the small-sgnal equvalent crcut obtaned rom the nonlnear crcut? What s the small-sgnal equvalent o the MOSFET, BJT, and dode?
Small-Sgnal Dode Model -Termnal Nonlnearl Devce () = Q A Small Sgnal Equvalent Crcut Thus, or the dode Rd - D D Q
Small-Sgnal Dode Model For the dode Rd - D D Q = e D S D t D = Se D Q t Q D t DQ t R t d= DQ
Eample o dode crcut where small-sgnal dode model s useul REF REF R R R R X X R R D D D D D D R D R D oltage Reerence Small-sgnal model o oltage Reerence (useul or compensaton when parastc Cs ncluded)
Small-Sgnal Model o BJT and MOSFET Consder 4-termnal network 4-Termnal Devce,,,,,, Dene Q Q Q v v v Q Q Q Small sgnal model s that whch represents the relatonshp between the small sgnal voltages and the small sgnal currents
Small-Sgnal Model o 4-Termnal Network,,,,,, g g g 4-Termnal Devce Small sgnal model s that whch represents the relatonshp between the small sgnal voltages and the small sgnal currents For small sgnals, ths relatonshp should be lnear Can be thought o as a change n coordnate sstems rom the large sgnal coordnate sstem to the small-sgnal coordnate sstem
Recall or a uncton o one varable Talor s Seres Epanson about the pont...! ) ( () - s small
Recall or a uncton o one varable () - s small we dene the small sgnal varables as
Recall or a uncton o one varable () - s small we dene the small sgnal varables as Then Ths relatonshp s lnear! X Q
Consder now a uncton o n varables (,... ) ( ) n we dene the small sgnal varables as X {,,... n } The multvarate Talor s seres epanson around the pont ( ) - n k k k k X s gven b nn, k j j - j k - k.. (H.O.T.)! j k Truncatng ater rst-order terms, we obtan the appromaton where n k k - k k
Multvarate Talors Seres Epanson (,... ) ( ) Lnearzed appromaton Ths can be epressed as n - - n k k k k where ss ss n k a k ss k n k a k k - a ss k k k k k k - k k
n the more general orm, the multvarate Talor s seres epanson can be epressed as http://www.chem.mtu.edu/~tbco/cm46/talor.html
Consder 4-termnal network 4-Termnal Devce,,,,,, Nonlnear network characterzed b unctons each unctons o varables
Consder now unctons each unctons o varables,,,,,, Dene Q Q Q Q n what ollows, we wll use seres epanson. Q as an epanson pont n a Talor s
,,,,,, Consder now unctons each unctons o varables Q Q Q Q Dene Q Q Q Q Q Q,,,,,,,,,, Q Q Q Q Q Q Q,,,,,, Q Q Q The multvarate Talors Seres epanson o, around the operatng pont. when truncated ater rst-order terms, can be epressed as: Q Consder rst the uncton or equvalentl as:
Make the ollowng dentons Q Q Q Q,,,,,, Q Q Q Q,, Q,, Q,, Q Q Q Q Q Q v v v repeatng rom prevous slde: t thus ollows that v v v Ths s a lnear relatonshp between the small sgnal electrcal varables
,, v v v Small Sgnal Model Development Nonlnear Model,,,, Lnear Model at (alt. small sgnal model) Q Etendng ths approach to the two nonlnear unctons and where v v v v v v j, j, Q
Small Sgnal Model Development Nonlnear Model,, v v v,,,, Lnear Model at (alt. small sgnal model) Q v v v v v v where j, j, Q
Small Sgnal Model v v v v v v v v v where j, j, Q Ths s a small-sgnal model o a 4-termnal network and t s lnear 9 small-sgnal parameters characterze the lnear 4-termnal network Small-sgnal model parameters dependent upon Q-pont!
A small-sgnal equvalent crcut o a 4-termnal nonlnear network (equvalent crcut because has eactl the same port equatons) j, j, Q Equvalent crcut s not unque Equvalent crcut s a three-port network
4-termnal small-sgnal network summar 4-Termnal Devce Small sgnal model:,,,,,, v v v v v v v v v j, j, Q
Consder -termnal network Small-Sgnal Model -Termnal Devce,, Dene Q Q v v Q Q Small sgnal model s that whch represents the relatonshp between the small sgnal voltages and the small sgnal currents
Consder -termnal network Small-Sgnal Model 4-Termnal Devce g,, g,, g,, j, j, Q
Consder -termnal network Small-Sgnal Model -Termnal Devce,, g g j, Q Q j Q A Small Sgnal Equvalent Crcut 4 small-sgnal parameters characterze ths -termnal (two-port) lnear network Small sgnal parameters dependent upon Q-pont
-Termnal Devce -termnal small-sgnal network summar,, Small sgnal model: j, j Q
Consder -termnal network Small-Sgnal Model -Termnal Devce Dene Q v Q Small sgnal model s that whch represents the relatonshp between the small sgnal voltages and the small sgnal currents
Consder -termnal network Small-Sgnal Model 4-Termnal Devce g,, g,, g,, j, j, Q
Consder -termnal network Small-Sgnal Model -Termnal Devce Q Q A Small Sgnal Equvalent Crcut Ths was actuall developed earler!
Lnearzed nonlnear devces Nonlnear Devce Lnearzed Small-sgnal Devce
How s the small-sgnal equvalent crcut obtaned rom the nonlnear crcut? What s the small-sgnal equvalent o the MOSFET, BJT, and dode?
Small Sgnal Model o MOSFET -termnal devce 4-termnal devce MOSFET s actuall a 4-termnal devce but or man applcatons acceptable predctons o perormance can be obtaned b treatng t as a -termnal devce b neglectng the bulk termnal n ths course, we have been treatng t as a -termnal devce and n ths lecture wll develop the small-sgnal model b treatng t as a -termnal devce When treated as a 4-termnal devce, the bulk voltage ntroduces one addtonal term to the small sgnal model whch s oten ether neglgbl small or has no eect on crcut perormance (wll develop 4-termnal ss model later)
Small Sgnal Model o MOSFET G Large Sgnal Model D Trode Regon -termnal devce Saturaton Regon GS6 GS5 GS4 GS Cuto Regon GS T W L W μc L DS μc D OX GS T DS GS T DS GS T OX GS T DS GS T DS GS T GS GS DS MOSFET s usuall operated n saturaton regon n lnear applcatons where a small-sgnal model s needed so wll develop the small-sgnal model n the saturaton regon
Small Sgnal Model o MOSFET,,, G GS DS, D GS DS G W =μc L D OX GS T DS Small-sgnal model: G GS j Q, j Q G DS Q D GS Q D DS Q
Small Sgnal Model o MOSFET Small-sgnal model: G W =μc L D OX GS T DS G? GS Q G? DS Q D? GS Q D? DS Q
Small Sgnal Model o MOSFET,, G W =μc L D OX GS T DS Small-sgnal model: G GS Q G DS Q W W D μc μc L GS L Q Q W μc OX GSQ T L W D μc OX GS T DQ L OX GS T DS OX GSQ T DSQ DS Q Q
Small Sgnal Model o MOSFET G W =μc L D OX GS T DS W μc L OX GSQ T DQ G GS DS D GS DS G D S GS GS An equvalent crcut
End o Lecture