Tài liệu Circuits & Electronics P7 pdf

6.002 CIRCUITS ANDELECTRONICS Incremental Analysis... Nonlinear AnalysisX Analytical method X Graphical method Today X Incremental analysis Reading: Section 4.5 Review... total variable

6.002 CIRCUITS AND

ELECTRONICS

Incremental Analysis

Nonlinear Analysis

X Analytical method

X Graphical method

Today

X Incremental analysis

Reading: Section 4.5

Review

Method 3: Incremental Analysis

Motivation: music over a light beam

Can we pull this off?

LED: Light Emitting expoDweep ☺

D

v

+

-)

(t

v I +–

D

i

LED

R

i

AMP

light intensity I R

in photoreceiver

R

R I

light intensity

D

D i

I

v

t

music signal

)

(t

v I i D (t) light i R (t) sound

nonlinear

linear

The LED is nonlinear distortion

I

D v

v = v D

D

i

D

D

i

t

D

v

D

i

D

I

D

V

DC offset

or DC bias

Trick:

d D

D I i

i = +

I

V

D

v

+

-) (t

v i +–

LED

+ –

I

v

d D

v = +

I

small region looks linear

(about V D , I D)

very small

D

i

D

v

d

i

D

I

D

V

t

D

v

D

V

I

D v

t

D

I

D

i

~linear!

Demo

d v

d

i

D

i

total variable offsetDC superimposedsmall

signal

The incremental method:

(or small signal method)

1 Operate at some DC offset

or bias point V D , I D

2 Superimpose small signal v d

(music) on top of V D

3 Response i d to small signal v d

is approximately linear

Notation:

d

D

D I i

i = +

( )D

D f v

What does this mean

mathematically?

Or, why is the small signal response linear?

We replaced

D D

using Taylor’s Expansion to expand

f(v D ) near v D =V D :

V v D

D D

dv

v

df V

f i

D D

∆

⋅ +

=

=

) (

"

+

∆

⋅

+

=

2 2

2 ( )

! 2

1

D V

v D

dv

v f d

D D

large DC

increment

about V D

nonlinear

d v

neglect higher order terms because is small∆v D

( ) D

V v D

D D

v d

v f

d V

f i

D D

∆

⋅ +

≈

=

) (

equating DC and time-varying parts,

D V

v D

D

v d

v f

d i

D D

∆

⋅

=

∆

=

) (

constant

w.r.t ∆v D constant w.r.t ∆v slope at V D , I D D

( )D

D f V

constant w.r.t ∆v D

X :

We can write

V v D

D D

D

v d

v f

d V

f i

I

D D

∆

⋅ +

≈

∆

+

=

) (

Equate DC and incremental terms,

D

bv

D a e

i =

From X :

constant

In our example,

d

bV

bV d

I + ≈ D + D ⋅ ⋅

D

bV

I =

d

bV

d a e b v

i = D ⋅ ⋅

operating point

d D

d I b v

behavior linear!

aka bias pt

aka DC offset

bV

D a e

d D

d I b v

i = ⋅ ⋅

D

v

D

i

D

I

D

V

slope at

V D , I D

operating point

we are approximating

A

B

d v

d i

Graphical interpretation

We saw the small signal

D

I

I

V

D

V

+

-+

D a e

Large signal circuit:

Small signal response: i d = I D b v d

graphically mathematically

now, circuit

small signal circuit:

Linear!

d i i

- I D b

1

+ –

behaves like: + v d

-d

i

b I

1 R

D

=