hisham lopez engh burns rusciano mallek simirov rooney ri

1 2 3 4 5 6 7 8

1 Zechariah

Pettit Daniel

Borgerding Liuchang

Li Andrew

Mun Brian

Crist Difeng

Liu Aimee

Salt Julien Di

2 Erik Lee

Nick Robbins

Bijan Choobineh

Wing Yi Lwe

Pangzhou Li

Travis Cook

Wentai Wang

Hisham Abbas

3 Ryan Wade

Jiayu Hong

Jean-Francois Burnier

Morgan Hardy

Bodhisatta Pramanik

Nagulapally Spurthi

Alfonso Raymundo

Corey Wright

4 Mohamad Samusdin

Aqila-Sarah Zulkifli

Honghao Liu

Clayton Hawken

Christopher Little

Antonio Montoya

Jaehyuk Han

Logan Heinen

5 Nicholas Riesen

Satvik Shah

Alex McCullough

Wei Shen Theh

Minh Nguyen

Trevor Brown

Zhong Zhang

Mingda Yang

6 Abdussamad

Hisham Brenda Lopez

Benjamin Engh

Blake Burns

Mark Rusciano

Daniel Mallek

Ilya Simirov

Bryce Rooney

EE 330 Class Seating

EE 330

Lecture 21

• Small Signal Analysis

• Small Signal Modelling

Small-Signal Operation

Q-point

Output

• If slope is steep, output range can be much larger than input range

• This can be viewed as voltage gain in the circuit

Review from Last Lecture

Small signal operation of nonlinear circuits

VIN=VMsinωt

VM is small

Nonlinear CircuitVIN VOUT = ?

• Small signal concepts often apply when building amplifiers

• If small signal concepts do not apply, usually the amplifier will not perform

• Small signal operation is usually synonymous with “locally linear”

• Small signal operation is relative to an “operating point”

Amplification with Transistors

An amplifier, electronic amplifier or (informally) amp is an

electronic device that increases the power of a signal.

From Wikipedia: (Oct. 2014)

•It is difficult to increase the voltage or current very much with passive RC circuits

•Voltage 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 only signal levels but power levels to a load

• In transistor circuits, power that is delivered in the signal path is supplied by a

biasing network

Consider the following MOSFET and BJT Circuits

VIN(t)

BJT MOSFET

VIN(t)

• MOS and BJT Architectures often Identical

• Circuit are Highly Nonlinear

• Nonlinear Analysis Methods Must be used to analyze these and almost any

other nonlinear circuit

Small signal analysis using nonlinear models

RVVtV2L

WμCVV TSSM

OXDDOUT

Assume M1 operating in saturation region

By selecting appropriate value of VSS, M1

will operate in the saturation region

VIN=VMsinωt

VM is small

D IN SS T

μC WI V -V -V

OUT DD DV =V -I R

OUT DD IN SS T

μC WV V V -V -V R

2TSSOX

DQ VV2L

WC μI

Termed Load Line

Small signal analysis example

RVVtV2L

WμCVV TSSM

OXDDOUT

VIN=VMsinωt

VM is small

tV1-VV

WμCVV

OXDDOUT

Recall that if x is small 2

1+x 1+2x

2 sinOX MOUT DD SS T

μC W 2V tV V V V 1- R

2L V V

2 2OX OX M

OUT DD SS T SS T

μC W μC W 2V sin tV V V V R V V R

2L 2L V V

OUT DD SS T SS T M

μC W μC WV V V V R V V R V sin t

Small signal analysis exampleVDD

VIN=VMsinωt

OUT DD SS T SS T M

will operate in the saturation region

VIN=VMsinωt

OUT DD SS T SS T M

Quiescent Output ss Voltage Gain

v SS T

μC WA V V R

OUTQ DD SS T

μC WV V V V R

OUT OUTQ V MV V A V sin t

Note the ss voltage gain is negative since VSS+VT<0!

VIN=VMsinωt

v SS T

μC WA V V R

But – this expression gives little insight into how large the gain is !

And the analysis for even this very simple circuit was messy!

OUTQ DD SS T

μC WV V V V R

VIN=VMsinωtVIN

VIN=VMsinωt OX

v SS T

μC WA V V R

VIN=VMsinωt OX

v SS T

μC WA V V R

VIN=VMsinωt

v SS T

μC WA V V R

Serious Distortion occurs if signal is too large or Q-point non-optimal

Here “clipping” occurs for high VOUT

VIN=VMsinωt OX

v SS T

μC WA V V R

Serious Distortion occurs if signal is too large or Q-point non-optimal

Here “clipping” occurs for low VOUT

VIN=VMsinωt

v SS T

μC WA V V R

Thus, substituting from the expression for IDQ we obtain

2TSSOX

DQ VV2L

WC μI But recall:

Note this is negative since VSS+VT < 0

VIN=VMsinωt

Observe the small signal voltage gain is twice the

Quiescent voltage across R divided by VSS+VT

Can make |AV |large by making |VSS+VT |small

• This analysis which required linearization of a nonlinear output voltage is quite

tedious.

• This approach becomes unwieldy for even slightly 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 analysis and

design, and result in a dramatic reduction in computational requirements

VIN=VMsinωt

However, there are invariably small errors in this analsis

OUT OUTQ V M

V V A V sin t + ε t

RVVtV2L

WμCVV TSSM

OXDDOUT

To see the effects of the approximations consider again

222 sin 2 sinOXOUT DD SS T M SS T

μC RWV V t V V V t+ V V

21 cos 2

2 sin2

2OXOUT DD M SS T M SS T

μC RW tV V V V V V t+ V V

sin cos 22

22OX OX OXM

OUT DD SS T SS T M M

μC RW μC W μC RWVV V + V V V V R V t V t

2L L 4L

Note presence of second harmonic distortion term !

(Consider what was neglected in the previous analysis)

recall

VIN=VMsinωtOUT OUTQ V M

V V A V sin t

OUT OUTQ V M

V V A V sin t + ε t

sin cos 22

22OX OX OXM

OUT DD SS T SS T M M

μC RW μC W μC RWVV V + V V V V R V t V t

2L L 4L

OX MOUTQ DD SS T

μC RW VV V + V V

OXV SS T

μC WA V V R

μC RWA V

Nonlinear distortion term

sin cos 2OUT OUTQ V M 2 MV V A V t A V t

VIN=VMsinωt

OX MOUTQ DD SS T

μC RW VV V + V V

OXV SS T

μC WA V V R

μC RWA V

Nonlinear distortion term

sin cos 2OUT OUTQ V M 2 MV V A V t A V t

Total Harmonic Distortion:

Recall, if then 0

sink kb kωT+k

Thus, for this amplifier, as long as M1 stays in the saturation region

M2 M 2 M

OXV M V SS TSS T

μC WRVA V A V4LTHD

μC WA V A 4 V +VR V +V

Distortion will be much worse (larger and more harmonic terms) if M1 leaves saturation region.

Distortion will be small for VM<<|VSS+VT|

VIN(t)

BJT MOSFET

VIN(t)

One of the most widely used amplifier architectures

• Analysis was very time consuming

• Issue of operation of circuit was

obscured in the details of the analysis

VIN(t)

BJT MOSFET

VIN(t)

One of the most widely used amplifier architectures

VIN(t)

CQ S EI J A e

Assume Q1 operating in forward active region

will operate in the forward active region

VIN=VMsinωt

VM is small

C S EI J A e

OUT CC C 1V =V -I R

OUT CC S E 1V V J A R e

V sin t -V

VIN(t)

CQ S EI J A e

VIN=VMsinωt

VM is small

V sin t -V

V sin t-V

OUT CC S E 1V V J A R e e

OUT CC S E 1

V sin tV V J A R e 1+

OUT CC S E 1 S E 1

V sin tV V J A R e J A R e

Recall that if x is small xe 1+x (truncated Taylor’s series)

VIN(t)EE

CQ S EI J A e

VIN=VMsinωt

VM is small

OUT CC S E 1 S E 1

V sin tV V J A e R J A e R

OUT CC 1 M

IV V R V sin t

Quiescent Output

ss Voltage Gain

Comparison of Gains for MOSFET and BJT Circuits

VIN(t)

BJT MOSFET

VIN(t)

If IDQR =ICQR1=2V, VSS+VT= -1V, Vt=25mV

I R 2VA - =-8

V 25mV

2I R 4VA = - 4

V V -1V

Observe AVB>>AVM

Due to exponential-law rather than square-law model

Operation with Small-Signal Inputs

• Analysis procedure for these simple circuits was very tedious

• This approach will be unmanageable for even modestly more complicated

circuits

• Faster analysis method is needed !

End of Lecture 21

Small-Signal Analysis

IOUTSS

VOUTSS

VINSS or IINSS

Biasing (voltage or current)

Nonlinear

Circuit

IOUTSS

VOUTSS

VINSS or IINSSLinear Small

Signal Circuit

• Will commit next several lectures to developing this approach

• Analysis will be MUCH simpler, faster, and provide significantly more insight

• Applicable to many fields of engineering

Simple dc Model

Signal

Frequency

Dependent Small

Signal

Better Analytical

dc Model

Sophisticated Model

for Computer

Simulations

Simpler dc Model

Square-Law Model

Square-Law Model (with extensions for λ,γ effects)

Short-Channel α-law Model

BSIM Model

Switch-Level Models

• Ideal switches

• RSW and CGS

Small-Signal Analysis

Why was this analysis so tedious?

What was the key technique in the analysis that was used to obtain a simple

expression for the output (and that related linearly to the input)?

Because of the nonlinearity in the device models

V sin t-V

O U T C C S E 1V V J A R e e

OUT CC 1 M

IV V R V sin t

Linearization of the nonlinear output expression at the operating point

OUT CC 1 M

IV V R V sin t

Quiescent Output

ss Voltage Gain

C Q S EI J A e

Small-signal analysis strategy

1. Obtain Quiescent Output (Q-point)

2. Linearize circuit at Q-point instead of linearize the nonlinear solution

3. Analyze linear “small-signal” circuit

4. Add quiescent and small-signal outputs to obtain good approximation

to actual output

Small-Signal Principley

Q-point

Nonlinear function

y=f(x)

Q-point

Region around

Q-Point

y=f(x)

Q-point

Region around

Q-Point

Relationship is nearly linear in a small enough region around Q-point

Region of linearity is often quite large

Linear relationship may be different for different Q-points

y=f(x)

Q-point

Region around

Q-Point

Relationship is nearly linear in a small enough region around Q-point

Region of linearity is often quite large

Linear relationship may be different for different Q-points

y=f(x)

Small-Signal Principle y

Q-point

Device Behaves Linearly in Neighborhood of Q-Point

Can be characterized in terms of a small-signal coordinate system

y=f(x)

Small-Signal Principle

xy - y x - x

Q-point

y=f(x)

(xQ,yQ) or (xQ,yQ)

y=mx+b

x=x x=x

y yf f

xy - y x - x

Q-point y=f(x)

Changing coordinate systems:

ySS=y-yQ

xSS=x-xQ

fy - y x - x

Q-point y=f(x)

• Linearized model for the nonlinear function y=f(x)

• Valid in the region of the Q-point

• Will show the small signal model is simply Taylor’s series expansion

of f(x) at the Q-point truncated after first-order terms

Small-Signal Model:

Q-point y=f(x)

Mathematically, linearized model is simply Taylor’s series expansion of the nonlinear

function f at the Q-point truncated after first-order terms with notation xQ=x0

Small-Signal Model:

fy - y x - x

fy f x x - x

Observe:

Recall Taylors Series Expansion of nonlinear function f at expansion point x0

x=xk=1

1 dfy=f x + x-x

Truncating after first-order terms (and defining “o” as “Q”):

Q-point y=f(x)

fy f x x

Quiescent 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?

V Nonlinear

One-Port

IConsider arbitrary nonlinear one-port network

Arbitrary Nonlinear One-Port

yi VQV=V

defn I

Q-point

Linear model of the nonlinear

device at the Q-point

V Nonlinear

One-Port

I = f V

iA Small Signal Equivalent Circuit:

2-Terminal

Nonlinear

Device

Arbitrary Nonlinear One-Port

• The small-signal model of this 2-terminal electrical network is a resistor of value 1/y

or a conductor of value y

• One small-signal parameter characterizes this one-port but it is dependent on Q-

• This applies to ANY nonlinear one-port that is differentiable at a Q-point (e.g. a diode)

V Nonlinear

One-Port

Linear small-signal model:

Goal with small signal model is to predict

performance of circuit or device in the

vicinity of an operating point (Q-point)

Will be extended to functions of two and

three variables (e.g. BJTs and MOSFETs)

Solution for the example of the previous lecture was based upon solving the

nonlinear circuit for VOUT and then linearizing the solution by doing a Taylor’s

series expansion

• Solution of nonlinear equations very involved with two or more

nonlinear devices

• Taylor’s series linearization can get very tedious if multiple nonlinear

devices are present

Standard Approach to small-signal

analysis of nonlinear networks

1. Solve nonlinear network

2. Linearize solution

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

Alternative Approach to small-signal

• Must only develop linearized model once for any

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

1.Linearize nonlinear devices

equivalent

Alternative Approach to small-signal analysis of

nonlinear networks

1.Linearize nonlinear devices

equivalent

The “Alternative” approach is used almost exclusively

for the small-signal analysis of nonlinear networks

“Alternative” Approach to small-signal

Nonlinear

Network

dc Equivalent

Network

Q-point

Values for small-signal parameters

Small-signal (linear)

equivalent network

Small-signal output

Total output

(good approximation)

Nonlinear

Device

Linearized

Small-signal

Device

Linearized nonlinear devices

This terminology will be used in THIS course to emphasize difference

between nonlinear model and linearized small signal model

Nonlinear network

Linearized small-

signal network

Example:

It will be shown that the nonlinear circuit has the linearized small-signal network given

Linearized Circuit Elements

Must obtain the linearized circuit element for ALL linear and nonlinear

circuit elements

(Will also give models that are usually used for Q-point calculations : Simplified dc models)

SmallL

LargeL

Small-signal and simplified dc equivalent elements

Element ss equivalentSimplified dc

equivalent

VDCVDC

IDC IDC

dc Voltage Source

dc Current Source

Resistor

VACVACac Voltage Source

ac Current Source IAC IAC

Simplified

Capacitors

Inductors

transistors

Diodes

Simplified

Element ss equivalentSimplified dc

equivalent

(MOSFET (enhancement or

depletion), JFET)

Element ss equivalent

Dependent

Sources

(Linear)

Simplified

Bipolar

Transistors

Simplified dc

equivalent

VO=AVVIN IO=AIIIN VO=RTIIN IO=GTVIN

Example: Obtain the small-signal equivalent circuit

C is large

VOUTVIN

R2//R3

VOUT RM1

C1,C2, C3 large

C4 small

R1//R2

How is the small-signal equivalent circuit

obtained from the nonlinear circuit?

What is the small-signal equivalent of the

MOSFET, BJT, and diode ?

A Small Signal Equivalent Circuit

2-Terminal

Nonlinearl

Device

Small-Signal Diode Model

Thus, for the diode

Small-Signal Diode Model

For the diode

D SI =I e

VV DQD

II 1= I e

Example of diode circuit where small-signal

diode model is useful

Voltage Reference

ID1 ID2VD1

RD1 RD2

Small-signal model of

Voltage Reference (useful for compensation

when parasitic Cs included)

End of Lecture 21

hisham lopez engh burns rusciano mallek simirov rooney ri

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