4 5 6 EE 0 Class Seating 4 5 6 7 8 Zechariah Daniel Liuchang Andrew Brian Dieng Aimee Julien Di Pettit Borgerding Li Mun Crist Liu Salt Tria Erik Nick Bijan Wing Yi Pangzhou Travis Wentai Hisham Lee Robbins Choobineh Lwe Li Cook Wang Abbas Ran Jiau Jean-Francois Morgan Bodhisatta Nagulapall Alonso Core Wade Hong Burnier Hard Pramanik Spurthi Ramundo Wright Mohamad Aqila-Sarah Honghao Claton Christopher Antonio Jaehuk Logan Samusdin Zulkili Liu Hawken Little Montoa Han Heinen Nicholas Satvik Ale Wei Shen Minh Trevor Zhong Mingda Riesen Shah McCullough Theh Nguen Brown Zhang Yang Abdussamad Brenda Benjamin Blake Mark Daniel la Brce Hisham Lopez Engh Burns Rusciano Mallek Simirov Roone
EE 0 Lecture Small Signal Analsis Small Signal Analsis o BJT Ampliier
Review rom Last Lecture Comparison o Gains or MOSFET and BJT Circuits N (t) A B BJT CC R EE OUT R C t D R = C R =, SS + T = -, t =5m R C A - =-80 B 5m t N (t) A MOSFET M M DD R SS SS OUT R D R 4 A = - 4 - D M SS T Observe A B >>A M Due to eponential-law rather than square-law model T
Review rom Last Lecture Operation with Small-Signal nputs Analsis procedure or these simple circuits was ver tedious This approach will be unmanageable or even modestl more complicated circuits Faster analsis method is needed!
Small-Signal Analsis Biasing (voltage or current) NSS or NSS Nonlinear Circuit OUTSS OUTSS NSS or NSS Linear Small Signal Circuit OUTSS OUTSS Will commit net several lectures to developing this approach Analsis will be MUCH simpler, aster, and provide signiicantl more insight Applicable to man ields o engineering
Small-Signal Analsis Simple dc Model Square-Law Model Small Signal Better Analtical dc Model Sophisticated Model or Computer Simulations BSM Model Square-Law Model (with etensions or λ,γ eects) Short-Channel α-law Model Frequenc Dependent Small Signal Simpler dc Model Switch-Level Models deal switches R SW and C GS
Operation with Small-Signal nputs Wh was this analsis so tedious? Because o the nonlinearit in the device models What was the ke technique in the analsis that was used to obtain a simple epression or the output (and that related linearl to the input)? t J A R e e O U T C C S E - EE sin t M t R t C R sin t OUT CC C M Linearization o the nonlinear output epression at the operating point
Operation with Small-Signal nputs J A e C S E - EE t R t C R sin t OUT CC C M uiescent Output ss oltage Gain Small-signal analsis strateg. Obtain uiescent Output (-point). Linearize circuit at -point instead o linearize the nonlinear solution. Analze linear small-signal circuit 4. Add quiescent and small-signal outputs to obtain good approimation to actual output
Small-Signal Principle Nonlinear unction =() Y -point X
Small-Signal Principle Region around -Point =() Y -point X
Small-Signal Principle Region around -Point =() Y -point X Relationship is nearl linear in a small enough region around -point Region o linearit is oten quite large Linear relationship ma be dierent or dierent -points
Small-Signal Principle =() Region around -Point Y -point X Relationship is nearl linear in a small enough region around -point Region o linearit is oten quite large Linear relationship ma be dierent or dierent -points
Small-Signal Principle -point ss =() Y ss X Device Behaves Linearl in Neighborhood o -Point Can be characterized in terms o a small-signal coordinate sstem
Small-Signal Principle (,) =m+b =() Y - point (, ) or (, ) - - = X= X X X NT - m = = = = - - = -
Small-Signal Principle -point SS =() SS Changing coordinate sstems: SS =- - - = SS =- SS = SS
Small-Signal Principle -point SS =() SS Small-Signal Model: SS Linearized model or the nonlinear unction =() alid in the region o the -point Will show the small signal model is simpl Talor s series epansion o () at the -point truncated ater irst-order terms = SS
Small-Signal Principle Observe: - = - SS = SS -point SS =() SS - = Recall Talors Series Epansion o nonlinear unction at epansion point 0 d k = 0+ -0 k=k! d = Truncating ater irst-order terms (and deining o as ): - = 0 Small-Signal Model: SS Mathematicall, linearized model is simpl Talor s series epansion o the nonlinear unction at the -point truncated ater irst-order terms with notation = 0 = SS
Small-Signal Principle -point SS =() = SS SS uiescent Output ss Gain How can a circuit be linearized at an operating point as an alternative to linearizing a nonlinear unction at an operating point? Consider arbitrar nonlinear one-port network Nonlinear One-Port
Arbitrar Nonlinear One-Port Nonlinear One-Port i SS = v SS i SS i SS de = = i v SS -point v SS de = den = Linear model o the nonlinear device at the -point i
Arbitrar Nonlinear One-Port Nonlinear One-Port -Terminal Nonlinear Device (v) = Linear small-signal model: i A Small Signal Equivalent Circuit: i The small-signal model o this -terminal electrical network is a resistor o value / or a conductor o value One small-signal parameter characterizes this one-port but it is dependent on - point This applies to ANY nonlinear one-port that is dierentiable at a -point (e.g. a diode)
Small-Signal Principle Goal with small signal model is to predict perormance o circuit or device in the vicinit o an operating point (-point) Will be etended to unctions o two and three variables (e.g. BJTs and MOSFETs)
Solution or the eample o the previous lecture was based upon solving the nonlinear circuit or OUT and then linearizing the solution b doing a Talor s series epansion Solution o nonlinear equations ver involved with two or more nonlinear devices Talor s series linearization can get ver tedious i multiple nonlinear devices are present Standard Approach to small-signal analsis o nonlinear networks. Solve nonlinear network. Linearize solution Alternative Approach to small-signal analsis o nonlinear networks.linearize nonlinear devices (all). Obtain small-signal model rom linearized device models. Replace all devices with small-signal equivalent 4.Solve linear small-signal network
Alternative Approach to small-signal analsis o nonlinear networks.linearize nonlinear devices. Obtain small-signal model rom linearized device models. Replace all devices with small-signal equivalent 4.Solve linear small-signal network Must onl develop linearized model once or an nonlinear device (steps. and.) e.g. once or a MOSFET, once or a JFET, and once or a BJT Linearized model or nonlinear device termed small-signal model derivation o small-signal model or most nonlinear devices is less complicated than solving even one simple nonlinear circuit Solution o linear network much easier than solution o nonlinear network
Alternative Approach to small-signal analsis o nonlinear networks.linearize nonlinear devices. Obtain small-signal model rom linearized device models. Replace all devices with small-signal equivalent 4.Solve linear small-signal network The Alternative approach is used almost eclusivel or the small-signal analsis o nonlinear networks
Alternative Approach to small-signal analsis o nonlinear networks Nonlinear Network dc Equivalent Network -point alues or small-signal parameters Small-signal (linear) equivalent network Small-signal output Total output (good approimation)
Linearized nonlinear devices Nonlinear Device Linearized Small-signal Device This terminolog will be used in THS course to emphasize dierence between nonlinear model and linearized small signal model
Eample: t will be shown that the nonlinear circuit has the linearized small-signal network given R DD OUT M OUT N M R N SS Nonlinear network Linearized smallsignal network
Linearized Circuit Elements Must obtain the linearized circuit element or ALL linear and nonlinear circuit elements DC DC AC C Large R L Small L Large C Small AC (Will also give models that are usuall used or -point calculations : Simpliied dc models)
Small-signal and simpliied dc equivalent elements Element ss equivalent Simpliied dc equivalent dc oltage Source DC DC ac oltage Source AC AC dc Current Source DC DC ac Current Source AC AC Resistor R R R
Small-signal and simpliied dc equivalent elements Element ss equivalent Simpliied dc equivalent C Large Capacitors C Small C L Large nductors L Small L Diodes Simpliied MOS transistors (MOSFET (enhancement or depletion), JFET) Simpliied Simpliied
Small-signal and simpliied dc equivalent elements Element ss equivalent Simpliied dc equivalent Bipolar Transistors Simpliied Simpliied Dependent Sources (Linear) O =A N O =R T N O =A N O =G T N
Eample: Obtain the small-signal equivalent circuit DD R C R OUT NSS R C is large R R OUT R OUT N R //R N R
Eample: Obtain the small-signal equivalent circuit R DD OUT M N SS R OUT N M OUT N M R
Eample: Obtain the small-signal equivalent circuit DD C R R 4 R5 R 7 C DD OUT 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 M N R R R 6 C 4 OUT R L N R //R R 4 R 5 C 4 R 6 M R 7 OUT R L
How is the small-signal equivalent circuit obtained rom the nonlinear circuit? What is the small-signal equivalent o the MOSFET, BJT, and diode?
Small-Signal Diode Model -Terminal Nonlinearl Device () = i A Small Signal Equivalent Circuit i Thus, or the diode Rd - D D
Small-Signal Diode Model For the diode Rd - D D = e D S D t D = Se D t D t D t R t d= D
Eample o diode circuit where small-signal diode model is useul REF REF R R R R X X R 0 R 0 D D D D D D R D R D oltage Reerence Small-signal model o oltage Reerence (useul or compensation when parasitic Cs included)
Small-Signal Model o BJT and MOSFET Consider 4-terminal network 4-Terminal Device,,,,,, Deine i i i v v v Small signal model is that which represents the relationship between the small signal voltages and the small signal currents
Small-Signal Model o 4-Terminal Network,,,,,, i i i g g g 4-Terminal Device Small signal model is that which represents the relationship between the small signal voltages and the small signal currents For small signals, this relationship should be linear Can be thought o as a change in coordinate sstems rom the large signal coordinate sstem to the small-signal coordinate sstem
Recall or a unction o one variable Talor s Series Epansion about the point 0...! ) ( 0 0 0 0 0 () - 0 is small 0 0 0 0 0 0
Recall or a unction o one variable () - 0 is small 0 0 0 0 0 0 we deine the small signal variables as 0 0
Recall or a unction o one variable () - 0 is small 0 0 0 we deine the small signal variables as 0 Then 0 This relationship is linear! 0
Consider now a unction o n variables (,... ) ( ) n we deine the small signal variables as X {,,... n } 0 0 0 0 The multivariate Talor s series epansion around the point ( ) - 0 n k k 0 k k0 X 0 is given b nn, k j j - j0 k - k0.. (H.O.T.)! j k 0 Truncating ater irst-order terms, we obtain the approimation where 0 0 0 n k -k0 k k 0
Multivariate Talors Series Epansion (,... ) ( ) Linearized approimation This can be epressed as n - - 0 0 n k k0 k k where ss ss 0 n k a k ss k n k ak 0 k - a ss k k k k k0 0 k - k k0
n the more general orm, the multivariate Talor s series epansion can be epressed as http://www.chem.mtu.edu/~tbco/cm46/talor.html
Consider 4-terminal network 4-Terminal Device,,,,,, Nonlinear network characterized b unctions each unctions o variables
Consider now unctions each unctions o variables,,,,,, Deine n what ollows, we will use series epansion. as an epansion point in a Talor s
,,,,,, Consider now unctions each unctions o variables Deine,,,,,,,,,,,,,,,, The multivariate Talors Series epansion o, around the operating point. when truncated ater irst-order terms, can be epressed as: Consider irst the unction or equivalentl as:
Make the ollowing deinitions,,,,,,,,,,,, i i i v v v repeating rom previous slide: t thus ollows that v v v i This is a linear relationship between the small signal electrical variables
,, i v v v Small Signal Model Development Nonlinear Model,,,, Linear Model at (alt. small signal model) Etending this approach to the two nonlinear unctions and where i v v v i v v v ij i, j,
Small Signal Model Development Nonlinear Model,, i v v v,,,, Linear Model at (alt. small signal model) i v v v i v v v where ij i, j,
Small Signal Model i v v v i v v v i v v v where ij i, j, This is a small-signal model o a 4-terminal network and it is linear 9 small-signal parameters characterize the linear 4-terminal network Small-signal model parameters dependent upon -point!
End o Lecture