EE 330 Lecture 21. Small Signal Analysis Small Signal Analysis of BJT Amplifier

Transcription

1 EE 330 Lecture 21 Small Signal Analsis Small Signal Analsis of BJT Amplifier

2 Review from Last Lecture Comparison of Gains for MOSFET and BJT Circuits IN (t) A B BJT CC 1 R EE OUT I R C 1 t If I D R =I C R 1 =2, SS + T = -1, t =25m I R 2 C 1 A - =-80 B 25m t IN (t) A MOSFET M M 1 DD R 1 SS SS OUT 2I R D 2I R 4 A = D M SS T Observe A B >>A M Due to exponential-law rather than square-law model T

3 Review from Last Lecture Operation with Small-Signal Inputs Analsis procedure for these simple circuits was ver tedious This approach will be unmanageable for even modestl more complicated circuits Faster analsis method is needed!

4 Small-Signal Analsis Biasing (voltage or current) INSS or I INSS Nonlinear Circuit I OUTSS OUTSS INSS or I INSS Linear Small Signal Circuit I OUTSS OUTSS Will commit next several lectures to developing this approach Analsis will be MUCH simpler, faster, and provide significantl more insight Applicable to man fields of engineering

5 Small-Signal Analsis Simple dc Model Square-Law Model Small Signal Better Analtical dc Model Sophisticated Model for Computer Simulations BSIM Model Square-Law Model (with extensions for λ,γ effects) Short-Channel α-law Model Frequenc Dependent Small Signal Simpler dc Model Switch-Level Models Ideal switches R SW and C GS

6 Operation with Small-Signal Inputs Wh was this analsis so tedious? Because of the nonlinearit in the device models What was the ke technique in the analsis that was used to obtain a simple expression for the output (and that related linearl to the input)? t J A R e e O U T C C S E 1 - EE sin t M t I R t C 1 I R sin t OUT CC C 1 M Linearization of the nonlinear output expression at the operating point

7 Operation with Small-Signal Inputs I J A e C S E - EE t I R t C 1 I R sin t OUT CC C 1 M uiescent Output ss oltage Gain Small-signal analsis strateg 1. Obtain uiescent Output (-point) 2. Linearize circuit at -point instead of linearize the nonlinear solution 3. Analze linear small-signal circuit 4. Add quiescent and small-signal outputs to obtain good approximation to actual output

8 Small-Signal Principle Nonlinear function =f(x) Y -point X x

9 Small-Signal Principle Region around -Point =f(x) Y -point X x

10 Small-Signal Principle Region around -Point =f(x) Y -point X x Relationship is nearl linear in a small enough region around -point Region of linearit is often quite large Linear relationship ma be different for different -points

11 Small-Signal Principle =f(x) Region around -Point Y -point X x Relationship is nearl linear in a small enough region around -point Region of linearit is often quite large Linear relationship ma be different for different -points

12 Small-Signal Principle -point ss =f(x) Y x ss X x Device Behaves Linearl in Neighborhood of -Point Can be characterized in terms of a small-signal coordinate sstem

13 Small-Signal Principle (x,) =mx+b =f(x) Y - point (x, ) or (x, ) - x-x f = x X= X X X INT f f x x x x - x=x x=x x=x f m x f x x x=x - x - x f x x=x x - x

14 Small-Signal Principle -point SS =f(x) x SS x Changing coordinate sstems: f SS =- - x - x x=x x SS =x-x x x SS f x x=x x SS

15 Small-Signal Principle -point SS =f(x) x SS x Small-Signal Model: SS f x Linearized model for the nonlinear function =f(x) alid in the region of the -point Will show the small signal model is simpl Talor s series expansion of f(x) at the -point truncated after first-order terms x=x x x SS

16 Small-Signal Principle Observe: - f x x=x x - x f x f f x x - x x x=x SS f x x=x x SS -point x SS =f(x) x SS x Recall Talors Series Expansion of nonlinear function f at expansion point x 0 1 df 0 k=1k! dx x=x 0 =f x + x-x 0 Truncating after first-order terms (and defining o as ): f f x x - x x x=x k Small-Signal Model: SS f x Mathematicall, linearized model is simpl Talor s series expansion of the nonlinear function f at the -point truncated after first-order terms with notation x =x 0 x=x x SS

17 Small-Signal Principle -point SS =f(x) f f x x x x=x SS x x SS x uiescent Output ss Gain How can a circuit be linearized at an operating point as an alternative to linearizing a nonlinear function at an operating point? Consider arbitrar nonlinear one-port network I Nonlinear One-Port

18 Arbitrar Nonlinear One-Port I I I Nonlinear One-Port i SS I = f v SS -point i SS i SS v I SS defn def = def = = I i v = SS Linear model of the nonlinear device at the -point i

19 Arbitrar Nonlinear One-Port I Nonlinear One-Port 2-Terminal Nonlinear Device f(v) I I = Linear small-signal model: i A Small Signal Equivalent Circuit: i The small-signal model of this 2-terminal electrical network is a resistor of value 1/ or a conductor of value One small-signal parameter characterizes this one-port but it is dependent on - point This applies to ANY nonlinear one-port that is differentiable at a -point (e.g. a diode)

20 Small-Signal Principle Goal with small signal model is to predict performance of circuit or device in the vicinit of an operating point (-point) Will be extended to functions of two and three variables (e.g. BJTs and MOSFETs)

21 Solution for the example of the previous lecture was based upon solving the nonlinear circuit for OUT and then linearizing the solution b doing a Talor s series expansion Solution of nonlinear equations ver involved with two or more nonlinear devices Talor s series linearization can get ver tedious if multiple nonlinear devices are present Standard Approach to small-signal analsis of nonlinear networks 1. Solve nonlinear network 2. Linearize solution Alternative Approach to small-signal analsis of nonlinear networks 1.Linearize nonlinear devices (all) 2. Obtain small-signal model from linearized device models 3. Replace all devices with small-signal equivalent 4.Solve linear small-signal network

22 Alternative Approach to small-signal analsis of nonlinear networks 1.Linearize nonlinear devices 2. Obtain small-signal model from linearized device models 3. Replace all devices with small-signal equivalent 4.Solve linear small-signal network Must onl develop linearized model once for an nonlinear device (steps 1. and 2.) e.g. once for a MOSFET, once for a JFET, and once for a BJT Linearized model for nonlinear device termed small-signal model derivation of small-signal model for most nonlinear devices is less complicated than solving even one simple nonlinear circuit Solution of linear network much easier than solution of nonlinear network

23 Alternative Approach to small-signal analsis of nonlinear networks 1.Linearize nonlinear devices 2. Obtain small-signal model from linearized device models 3. Replace all devices with small-signal equivalent 4.Solve linear small-signal network The Alternative approach is used almost exclusivel for the small-signal analsis of nonlinear networks

24 Alternative Approach to small-signal analsis of nonlinear networks Nonlinear Network dc Equivalent Network -point alues for small-signal parameters Small-signal (linear) equivalent network Small-signal output Total output (good approximation)

25 Linearized nonlinear devices Nonlinear Device Linearized Small-signal Device This terminolog will be used in THIS course to emphasize difference between nonlinear model and linearized small signal model

26 Example: It will be shown that the nonlinear circuit has the linearized small-signal network given R DD OUT M 1 OUT IN M 1 R IN SS Nonlinear network Linearized smallsignal network

27 Linearized Circuit Elements Must obtain the linearized circuit element for ALL linear and nonlinear circuit elements DC I DC I AC C Large R L Small L Large C Small AC (Will also give models that are usuall used for -point calculations : Simplified dc models)

28 Small-signal and simplified dc equivalent elements Element ss equivalent Simplified dc equivalent dc oltage Source DC DC ac oltage Source AC AC dc Current Source I DC I DC ac Current Source I AC I AC Resistor R R R

29 Small-signal and simplified dc equivalent elements Element ss equivalent Simplified dc equivalent C Large Capacitors C Small C L Large Inductors L Small L Diodes Simplified MOS transistors (MOSFET (enhancement or depletion), JFET) Simplified Simplified

30 Small-signal and simplified dc equivalent elements Element ss equivalent Simplified dc equivalent Bipolar Transistors Simplified Simplified Dependent Sources (Linear) O =A IN O =R T I IN I O =A I I IN I O =G T IN

31 Example: Obtain the small-signal equivalent circuit DD R 3 C R 1 OUT INSS R 2 C is large R 3 R 1 OUT R 1 OUT IN R 2 //R 3 IN R 2

32 Example: Obtain the small-signal equivalent circuit R DD OUT M 1 IN SS R OUT IN M 1 OUT IN M 1 R

33 Example: Obtain the small-signal equivalent circuit DD C 1 R 1 R 4 R5 R 7 C 3 I DD OUT 1 M 1 R L INSS R 2 R 3 C 2 R 6 C 4 SS C 1,C 2, C 3 large C 4 small R 4 R R5 1 R 7 1 M 1 IN R 2 R 3 R 6 C 4 OUT R L IN R 1 //R 2 1 R 4 R 5 C 4 R 6 M 1 R 7 OUT R L

34 How is the small-signal equivalent circuit obtained from the nonlinear circuit? What is the small-signal equivalent of the MOSFET, BJT, and diode?

35 Small-Signal Diode Model 2-Terminal Nonlinearl Device f(x) I I = i A Small Signal Equivalent Circuit i Thus, for the diode Rd -1 ID D

36 Small-Signal Diode Model For the diode Rd -1 ID D I =I e D S D t ID 1 = ISe D t D t I D t R t d= ID

37 Example of diode circuit where small-signal diode model is useful REF REF R 1 R 2 R 1 R 2 I 1 I 2 X X R 0 R 0 I D1 D1 I D2 D 1 D 2 D2 R D1 R D2 oltage Reference Small-signal model of oltage Reference (useful for compensation when parasitic Cs included)

38 End of Lecture 21