EE 330 Lecture 23. Small Signal Analysis Small Signal Modelling

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1 EE 330 Lecure 23 Small Signal Analysis Small Signal Modelling

2 Exam 2 Friday March 9 Exam 3 Friday April 13 Review Session for Exam 2: 6:00 p.m. on Thursday March 8 in Room Sweeney 1116

3 Review from Las Lecure Small-Signal Operaion Oupu Range -poin Inpu Range If slope is seep, oupu range can be much larger han inpu range This can be viewed as volage gain in he circui Nonlinear circui behaves as a linear circui near -poin wih small-signal inpus IN IN

4 Review from Las Lecure Small signal operaion of nonlinear circuis IN = IN + M sinω IN Nonlinear Circui =? M is small IN M -M Small signal conceps ofen apply when building amplifiers If small signal conceps do no apply, usually he amplifier will no perform well Small signal operaion is usually synonymous wih locally linear Small signal operaion is relaive o an operaing poin

5 Review from Las Lecure Consider he following MOSFET and BJT Circuis BJT MOSFET CC R DD R 1 1 M 1 IN () IN () EE SS Assume BJT operaing in FA region, MOSFET operaing in Sauraion Assume same quiescen oupu volage and same resisor R 1 Noe archiecure is same for BJT and MOSFET circuis One of he mos widely used amplifier archiecures

6 Review from Las Lecure Small signal analysis using nonlinear models R DD IN By selecing appropriae value of SS, M 1 will operae in he sauraion region Assume M 1 operaing in sauraion region M IN M 1 SS IN = IN + M sinω M is small - M = -I R OUT DD D μc W 2L 2 OX I - - D IN SS T μc W I - - 2L μc W OX R OUT DD IN SS T 2L μc W 2 OX sin +[ - - ] OUT DD M IN SS T R 2L 2 OX D IN SS T

7 Review from Las Lecure R DD Small signal analysis example Assume M 1 operaing in sauraion region IN = IN + M sinω IN M 1 SS μc W 2 μc W OX OX - - R - - R sin OUT DD IN SS T IN SS T M 2L L uiescen Oupu μc W OX R OUT DD IN SS T 2L OUT OUT M ss olage Gain OX A sin A μc W L A ( ) OUT OUT IN IN IN SS T - - R

8 IN Small signal analysis example R DD M 1 SS Assume M 1 operaing in sauraion region When IN = IN, he -poin soluion: μc W 2 D IN SS T 2L μc W OX = R OUT DD IN SS T 2L IN = IN + M sinω OX I - - Near he -poin, small signals have linear relaionship: A ( ) A sin OUTsmall OUT OUT IN IN M A OUTsmall INsmall A μc W L OX IN SS T - - R

9 Small signal analysis example R DD A sin OUT OUT M IN = IN + M sinω M 1 IN M IN -M SS DD O M =0 SS

10 Small signal analysis example R DD DD IN M -M A sin OUT OUT M IN = IN + M sinω OX A IN SS T IN M 1 SS μc W L - - R O M SS

11 DD Small signal analysis example R DD IN M -M A sin OUT OUT M IN = IN + M sinω OX A IN SS T IN M 1 SS μc W L - - R O M SS

12 IN Small signal analysis example R DD M 1 IN M -M A sin OUT OUT M μc W L IN = IN + M sinω OX A IN SS T - - R SS DD O M SS Serious Disorion occurs if signal is oo large or -poin non-opimal Here clipping occurs for high

13 IN Small signal analysis example R DD M 1 SS IN M -M DD A sin OUT OUT M μc W L IN = IN + M sinω OX A IN SS T - - R O M SS Serious Disorion occurs if signal is oo large or -poin non-opimal Here clipping occurs for low

14 Small signal analysis example DD IN = IN + M sinω IN R M 1 SS A sin A OUT OUT M μc W L OX IN SS T - - R Bu his expression gives lile insigh ino how large he gain is! Can he gain be made arbirarily large by simply making R large? Observe increasing R wih W,L, and SS fixed will change -poin Difficul o answer his quesion wih he informaion provided!

15 Small signal analysis example IN R DD M 1 SS A sin IN = IN + M sinω OUT OUT M A μc W L OX - - R IN SS T Bu recall: Thus, subsiuing from he expression for I D we obain A v μc W I - - 2L 2 OX D IN SS T 2I R D 2I R D IN SS T GS T

16 Small signal analysis example IN = IN + M sinω R DD M 1 A v 2I R D - GS T IN SS Small signal volage gain is wice he uiescen volage across R divided by GS - T Making I D R oo big or oo small will limi signal swing Can make A large by making GS - T small This analysis which required linearizaion of a nonlinear oupu volage is quie edious. This approach becomes unwieldy for even slighly more complicaed circuis A much easier approach based upon he developmen of small signal models will provide he same resuls, provide more insigh ino boh analysis and design, and resul in a dramaic reducion in compuaional requiremens

17 IN = IN + M sinω IN Small signal analysis example (Consider wha was negleced in he previous analysis) R DD M 1 SS recall A sin OUT OUT M However, here are invariably small errors in his analsis A sin + ε OUT OUT M To see he effecs of he approximaions consider again μcoxrw 2 M sin OUT DD GS T 2L μc 2 2 OXRW 2 M sin 2 GS sin OUT DD T M GS T 2L μc 2 1 cos2 2 OXRW 2 GS sin OUT DD M GS 2 T M T 2L 2 μc 2 OXRW M μcoxw μcoxrw 2 OUT DD + GST sin cos 2 2 GS -T -T RM M 2L L 4L Noe presence of second harmonic disorion erm!

18 IN = IN + M sinω IN R Small signal analysis example DD M 1 SS Nonlinear disorion erm A sin OUT OUT M A sin + ε OUT OUT M 2 μc 2 OXRW M μcoxw μcoxrw 2 OUT DD + GS GS sin cos 2 2 -T -T R M M 2L L 4L μcoxrw 2 OUTDC OUT M 4L μcoxw A GS-T R L A μc RW 4L OX 2 M A A sin cos 2 OUT OUTDC M 2 M

19 IN = IN + M sinω IN R Small signal analysis example DD M 1 SS Nonlinear disorion erm μcoxw A GS-T R L sin cos 2 A A OUT OUTDC M 2 M A μc RW 4L OX 2 M Toal Harmonic Disorion: b Recall, if x b sin hen k kωt+ k 2 k THD k0 b 2 k Thus, for his amplifier, as long as M 1 says in he sauraion region μc 2 OXW A R M 2 M A2 M THD 4L A A μc M OXW R( 4(GS- T ) GS- T ) L Disorion will be small for M << Disorion will be much worse (larger and more harmonic erms) if M 1 leaves sauraion region. 1

20 Consider he following MOSFET and BJT Circuis BJT CC R 1 MOSFET DD R 1 IN () 1 IN () M 1 EE SS One of he mos widely used amplifier archiecures Analysis was very ime consuming Issue of operaion of circui was obscured in he deails of he analysis

21 Consider he following MOSFET and BJT Circuis BJT CC R 1 MOSFET DD R 1 IN () 1 IN () M 1 EE SS One of he mos widely used amplifier archiecures

22 Small signal analysis using nonlinear models IN () CC R 1 1 EE IN = IN + M sinω M is small IN M - M By selecing appropriae value of SS, M 1 will operae in he forward acive region Assume 1 operaing in forward acive region = -I R OUT CC C 1 I J A e C S E IN - EE - IN EE be I J A e J A e C S E S E J A R e OUT CC S E 1 J A R e OUT CC S E 1 be sin + M be

23 IN () Small signal analysis using nonlinear models CC R 1 1 EE IN = IN + M sinω M is small \ J A R e OUT CC S E 1 M I J A e C S E be sin + J A R e e OUT CC S E 1 Recall ha if x is small be sin be sin M J A R e OUT CC S E 1 1+ be J A R e S E 1 J A R e sin OUT CC S E 1 M M be be e 1+ (runcaed Taylor s series)

24 Small signal analysis using nonlinear models IN () CC R 1 1 EE be J A R e S E 1 J A R e sin OUT CC S E 1 M I J A e C S E be be IN = IN + M sinω I R I R sin uiescen Oupu C 1 M is small OUT CC C 1 M ss olage Gain

25 Comparison of Gains for MOSFET and BJT Circuis IN () A B BJT CC 1 R EE C I R If I D R 1 =I C R=2, GS - T = 1, =25m I R 2 C A - =-80 B 25m IN () Observe A B >>A M Due o exponenial-law raher han square-law model A MOSFET M M 1 DD R 1 SS GS 2I R D 1 2I R D 1 A = - 4 M GS T T 4 1

26 Operaion wih Small-Signal Inpus Analysis procedure for hese simple circuis was very edious This approach will be unmanageable for even modesly more complicaed circuis Faser analysis mehod is needed!

27 Small-Signal Analysis Biasing (volage or curren) INSS or I INSS Nonlinear Circui I OUTSS SS Nonlinear Analysis Map Nonlinear circui o linear small-signal circui INSS or I INSS Linear Small Signal Circui I OUTSS SS Linear Analysis Will commi nex several lecures o developing his approach Analysis will be MUCH simpler, faser, and provide significanly more insigh Applicable o many fields of engineering

28 Small-Signal Analysis Simple dc Model Square-Law Model Small Signal Beer Analyical dc Model Sophisicaed Model for Compuer Simulaions BSIM Model Square-Law Model (wih exensions for λ,γ effecs) Shor-Channel α-law Model Frequency Dependen Small Signal Simpler dc Model Swich-Level Models Ideal swiches R SW and C GS

29 Operaion wih Small-Signal Inpus Why was his analysis so edious? Because of he nonlineariy in he device models Wha was he key echnique in he analysis ha was used o obain a simple expression for he oupu (and ha relaed linearly o he inpu)? J A R e e OUT CC S E 1 sin C 1 I R sin OUT CC C 1 M be I R Linearizaion of he nonlinear oupu expression a he operaing poin M

30 Operaion wih Small-Signal Inpus I J A e C S E be I R C 1 I R sin OUT CC C 1 M uiescen Oupu ss olage Gain Small-signal analysis sraegy 1. Obain uiescen Oupu (-poin) 2. Linearize circui a -poin insead of linearize he nonlinear soluion 3. Analyze linear small-signal circui 4. Add quiescen and small-signal oupus o obain good approximaion o acual oupu

31 Small-Signal Principle y Nonlinear funcion y=f(x) Y -poin X x

32 Small-Signal Principle y Region around -Poin y=f(x) Y -poin X x

33 Small-Signal Principle y Region around -Poin y=f(x) Y -poin X x Relaionship is nearly linear in a small enough region around -poin Region of lineariy is ofen quie large Linear relaionship may be differen for differen -poins

34 Small-Signal Principle y y=f(x) Region around -Poin Y -poin x X Relaionship is nearly linear in a small enough region around -poin Region of lineariy is ofen quie large Linear relaionship may be differen for differen -poins

35 Small-Signal Principle y -poin y ss y=f(x) Y x ss X x Device Behaves Linearly in Neighborhood of -Poin Can be characerized in erms of a small-signal coordinae sysem

36 Small-Signal Principle y (x,y) y=mx+b y=f(x) Y - poin (x,y ) or (x,y ) y y-y x-x f = x X= X X X INT f f x y x x x x=x x=x x=x f y- y x- x x x x=x f m f b x y x x x= x

37 Small-Signal Principle y -poin y SS y=f(x) y x SS x Changing coordinae sysems: f y SS =y-y y - y x - x x=x x SS =x-x x x y SS f x x=x x SS

38 Small-Signal Principle y -poin y SS y=f(x) y x SS x Small-Signal Model: y SS f x Linearized model for he nonlinear funcion y=f(x) alid in he region of he -poin Will show he small signal model is simply Taylor s series expansion of f(x) a he -poin runcaed afer firs-order erms x=x x x SS

39 Small-Signal Principle Observe: y - y f x x=x x - x y f x y SS f x x=x x SS y y -poin y SS y=f(x) x SS f y f x x - x x x=x x x Recall Taylors Series Expansion of nonlinear funcion f a expansion poin x 0 1 df k y=f x 0+ x-x0 k=1k! dx x=x Truncaing afer firs-order erms (and defining o as ): f y f x x- x x x=x 0 Small-Signal Model: y SS f x Mahemaically, linearized model is simply Taylor s series expansion of he nonlinear funcion f a he -poin runcaed afer firs-order erms wih noaion x =x 0 x=x x SS

40 End of Lecure 23

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