EE 330 Lecture 25. Small Signal Modeling

Transcription

1 EE 330 Lecture 25 Small Signal Modeling

2 Review from Last Lecture Amplification with Transistors From Wikipedia: Generall, an amplifier or simpl amp, is an device that changes, usuall increases, the amplitude of a signal. The "signal" is usuall voltage or current. It is difficult to increase the voltage or current ver much with passive RC circuits oltage and current levels can be increased a lot with transformers but not practical in integrated circuits Power levels can not be increased with passive elements (R, L, C, and Transformers) Often an amplifier is defined to be a circuit that can increase power levels (be careful with Wikipedia and WWW even when some of the most basic concepts are discussed) Transistors can be used to increase not onl signal levels but power levels to a load In transistor circuits, power that is delivered in the signal path is supplied b a biasing network

3 Review from Last Lecture Small signal analsis example R DD Assume M 1 operating in saturation region IN = M sinωt OUT M 1 IN SS μc W 2 μc W OX 2L L OX R R sin t OUT DD SS T SS T M uiescent Output ss oltage Gain μ C W OX A R v S S T L μc W 2L 2 OX R OUT DD SS T A sin t OUT OUT M Note the ss voltage gain is negative since SS + T <0!

4 Review from Last Lecture Small signal analsis example IN = M sinωt R DD M 1 OUT A v 2 I R S S D T IN SS Observe the small signal voltage gain is twice the uiescent voltage across R divided b SS + T Can make A large b making SS + T small This analsis which required linearization of a nonlinear output voltage is quite tedious. This approach becomes unwield for even slightl more complicated circuits A much easier approach based upon the development of small signal models will provide the same results, provide more insight into both analsis and design, and result in a dramatic reduction in computational requirements

5 Small signal analsis example (Consider what was neglected in the previous analsis) IN = M sinωt IN R DD M 1 OUT A sin t OUT OUT M However, there are invariabl small errors in this analsis A sin t + ε t OUT OUT M SS To see the effects of the approximations consider again μcox W 2 OUT DD M sin t SS T R 2L μc sin 2 sin OXRW OUT DD M t SS T M t+ SS T 2L μc 1 cos 2 OXRW 2 t 2 sin 2 OUT DD M SS T M t+ SS T 2L 2 2 μcoxrw 2 M μcoxw μcoxrw 2 OUT DD + SS T SS T R M sin t M cos 2 t 2L 2 L 4L Note presence of second harmonic distortion term!

6 IN R IN = M sinωt DD M 1 SS OUT Small signal analsis example Nonlinear distortion term A sin t OUT OUT M A sin t + ε t OUT OUT M 2 μcoxrw 2 M μcoxw μcoxrw 2 OUT DD + SS T SS T R M sin t M cos 2 t 2L 2 L 4L 2 μcoxrw M + 2L 2 2 OUT DD SS T μcoxw A SS T R L A μc RW 4L OX 2 M sin cos 2 A t A t O UT O UT M 2 M

7 IN = M sinωt IN R DD M 1 SS OUT Small signal analsis example Nonlinear distortion term sin cos 2 A t A t O UT O UT M 2 M 2 μcoxrw M + 2L 2 2 OUT DD SS T μcoxw A SS T R L Total Harmonic Distortion: Recall, if b s in kω T + A μc RW 4L OX 2 M then x t k k T H D k 2 k 0 b Thus, for this amplifier, as long as M 1 stas in the saturation region T H D μ C W 2 OX A R M 2 M A 2 4L M A A μ C W 4 + R + L M OX S S T S S T Distortion will be small for M << SS + T Distortion will be much worse (larger and more harmonic terms) 1 2 b k if M 1 leaves saturation region.

8 Consider the following MOSFET and BJT Circuits BJT CC R 1 MOSFET DD R 1 OUT OUT IN (t) 1 IN (t) M 1 EE SS One of the most widel used amplifier architectures Analsis was ver time consuming Issue of operation of circuit was obscured in the details of the analsis

9 Consider the following MOSFET and BJT Circuits BJT CC R 1 MOSFET DD R 1 OUT OUT IN (t) 1 IN (t) M 1 EE SS One of the most widel used amplifier architectures

10 Small signal analsis using nonlinear models CC R 1 B selecting appropriate value of SS, M 1 will operate in the forward active region OUT IN Assume 1 operating in forward active region 1 M t IN (t) - M EE = -I R O U T C C C 1 IN = M sinωt M is small I J A e C S E - IN E E t I J A e C S E - EE t J A R e OUT CC S E 1 J A R e OUT CC S E 1 IN - t EE sin t - M t EE

11 IN (t) Small signal analsis using nonlinear models CC R 1 OUT 1 EE J A R e OUT CC S E 1 I J A e C S E sin t - t J A R e e OUT CC S E 1 M - EE t EE sin t M t - EE t IN = M sinωt M is small \ Recall that if x is small -EE sin t t M J A R e OUT CC S E 1 1+ t -EE -EE sin t t t M J A R e OUT CC S E 1 J A R e S E 1 t e x 1 + x (truncated Talor s series)

12 Small signal analsis using nonlinear models IN (t) CC R 1 OUT 1 EE - - sin t EE EE t t M J A e R J A e R OUT CC S E 1 S E 1 t I J A e C S E - EE t IN = M sinωt M is small I R uiescent Output t C 1 I R sin t OUT CC C 1 M ss oltage Gain

13 Comparison of Gains for MOSFET and BJT Circuits IN (t) BJT CC R OUT 1 IN (t) MOSFET DD R 1 OUT M 1 A B EE 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 - = -8 0 B 2 5 m t A M SS 2 I R S S D D A = - 4 M S S T T 2 I R 4-1 Observe A B >>A M Due to exponential-law rather than square-law model

14 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!

15 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

16 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

17 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

18 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

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

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

21 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

22 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

23 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

24 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 m - x = x x = x x = x f x f x x x=x - x - x f x x=x x - x

25 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

26 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

27 Small-Signal Principle Observe: - f x x=x x - x f x SS f x x=x x SS -point SS =f(x) x SS f f x x - x x x=x x x Recall Talors Series Expansion of nonlinear function f at expansion point x 0 1 d f k = f x 0 + x -x 0 k=1 k! d x x = x 0 Truncating after first-order terms (and defining o as ): f f x x - x x x=x 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

28 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

29 Arbitrar Nonlinear One-Port I I Nonlinear One-Port i SS I = f i S S I = v S S v SS i SS de f = i I -point v SS d ef = d e fn I = Linear model of the nonlinear device at the -point i

30 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)

31 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)

32 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

33 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

34 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

35 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)

36 End of Lecture 25