6.002 CIRCUITS ANDELECTRONICS

Incremental Analysis

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Nonlinear AnalysisAnalytical methodGraphical method

TodayIncremental analysis

Reading: Section 4.5

Review

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Method 3: Incremental Analysis Motivation: music over a light beam

Can we pull this off?

LED: Light Emitting expoDweep ☺

Dv+-

)(tvI+–

Di

LEDRi

AMP

light intensity IRin photoreceiver

RR Ii ∝

lightintensity

DD iI ∝

Iv

t

music signal

)(tvI light sound)(tiR)(tiD

nonlinearlinear

problem! will result in distortion

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Problem:The LED is nonlinear distortion

ID vv =Dv

Div

D

t

tDi vD

Di

t

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Insight:

Dv

Di

DI

DV

DC offsetor DC bias

Trick:

dDD iIi +=

IVDv

+-

)(tvi+–

LED+–

Iv

dDD vVv +=

IV iv

small regionlooks linear(about VD , ID)

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Result

vdvery small

Di

Dv

di

DI

DV

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Result

t

DvDV

ID vv =

tDI

Di~linear!

Demo

dv

diDi

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totalvariable

DCoffset

smallsuperimposed

signal

The incremental method:(or small signal method)

1. Operate at some DC offsetor bias point VD, ID .

2. Superimpose small signal vd(music) on top of VD .

3. Response id to small signal vdis approximately linear.

Notation:dDD iIi +=

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( )DD vfi =

What does this meanmathematically?Or, why is the small signal responselinear?

We replacedDDD vVv Δ+=

using Taylor’s Expansion to expandf(vD) near vD=VD :

( ) DVvD

DDD v

dvvdfVfi

DD

Δ⋅+==

)(

+Δ⋅+=

22

2 )(!21

DVvD

D vdv

vfd

DD

large DC

incrementabout VD

nonlinear

dv

neglect higher order termsbecause is smallDvΔ

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( ) DVvD

DDD v

vdvfdVfi

DD

Δ⋅+≈=

)(

equating DC and time-varying parts,

DVvD

DD v

vdvfdi

DD

Δ⋅=Δ=

)(

constantw.r.t. ΔvD

constant w.r.t. ΔvDslope at VD, ID

( )DD VfI = operating point

constant w.r.t. ΔvD

X :

We can write

( ) DVvD

DDDD v

vdvfdVfiI

DD

Δ⋅+≈Δ+=

)(

so, DD vi Δ∝Δ By notation,dD ii =ΔdD vv =Δ

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Equate DC and incremental terms,

DbvD eai =

In our example,

From X : dbVbV

dD vbeaeaiI DD ⋅⋅

constant

+≈+

DbVD eaI =

dbV

d vbeai D ⋅⋅=

operating point

dDd vbIi ⋅⋅= small signalbehaviorlinear!

aka bias pt.aka DC offset

DbVD eaI = operating point

dDd vbIi ⋅⋅=

Dv

Di

DI

DV

slope atVD, ID

operatingpoint

we areapproximatingA with B

A

B

dv

di

Graphical interpretation

We saw the small signal

DI

IVDV

+-

+– LED DbVD eaI =

Large signal circuit:

Small signal reponse: dDd vbIi =

graphicallymathematicallynow, circuit

small signal circuit:

Linear!

di

iv dv+- bID

1+–

behaves like:dv+ -

di

bI1RD

=