Tài liệu Circuits & Electronics P16 pptx

Sinusoids important because signals can be represented as a sum of sinusoids.. Response to sinusoids of various frequencies -- aka frequency response -- tells us a lot about the system

6.002 CIRCUITS

6.002 Fall 2000 Lecture 16

Review

5V

C

R

L

v

to sinusoidal drive

Sinusoids important because signals can be

represented as a sum of sinusoids Response to sinusoids of various frequencies aka frequency response tells us a lot about the system

6.002 Fall 2000 Lecture 16

For motivation, consider our old friend,

the amplifier:

S

V

v O

v i

C

v

+ – + –

GS

C

R

V BIAS

Observe v o amplitude as the frequency of the

input v i changes Notice it decreases with

frequency

Also observe v o shift as frequency changes

(phase)

Need to study behavior of networks for

sinusoidal drive

Demo

6.002 Fall 2000 Lecture 16

Example:

+ –

+

–

v I (t) = V i cosωt for t ≥ 0 (V i real)

I

v

6.002 Fall 2000 Lecture 16

11 11

ectu r

+ –

R

Our Approach

i

lectu

re

sneaky approach

very sneaky

Usual approach

2 s

hi

6.002 Fall 2000 Lecture 16

1 Set up DE

2 Find v p

3 Find v H

4 v C = v P + v H, solve for unknowns

using initial conditions

6.002 Fall 2000 Lecture 16

Usual approach…

1 Set up DE

RC dv C + v C = v I

dt

= V i cosωt

That was easy!

6.002 Fall 2000 Lecture 16

2 Find v p

RC dv P +

dt v P = V i cosωt

First try: v P = A Æ nope

Second try: v P = A cosωt Æ nope

Third try: v P = A cos(

amplitude

φ

ωt + ) frequency

phase

− RCAω sin(ωt +φ) + A cos(ωt +φ) = Vi cosωt

− RCAω sin ωt cosφ − RCAω cosωt sin φ +

A cosωt cosφ − Asinωt sinφ = V i cosωt

6.002 Fall 2000 Lecture 16 8

6.002 ll 2000 Lecture 16 9

Let’s get sneaky!

p

PS V e

v =

st

i

st

p

st

p

e V e

V

dt

e

dV

st

i

st

p

st

p e V e V e

i

p V V

) 1 sRC

sRC

1

V

p = +

Nice property

of exponentials

IS

PS

PS v v

dt

dv

st

i e V

=

Find particular solution to another input…

p

sRC

1

V

+

=

st

i e V

where we replace s = jω

i e

V ω

solution for

t j

i e RC

j

V ω

ω ⋅

+

1

Fa

2 Fourth try to find v P…

using the sneaky approach

V i e jωt

was easy

= real[V i e jωt ]= real[v IS ]

from Euler relation,

j

I

IS

real part

real

part

e jωt = cosωt + sin ωt

an inverse superposition argument,

assuming system is real, linear

6.002 Fall 2000 Lecture 16 10



2 Fourth try to find v P…

P

v = Re[v PS ]= Re[V p e jωt ]

V i

= Re 1+ jωRC ⋅ e jωt 



= ReV i (1− jωRC) ⋅ e jωt 

= Re 1 + ω2 R 2 C 2

V i ⋅ e jφe jωt 

 ,tanφ = −ωRC

= Re + 1 ω2 R 2 C 2

V i ⋅ e j( ωt + φ ) 



v P =

C R

1 + ω2 2 2

6.002 Fall 2000 Lecture 16 11

3 Find v

−t

6.002 Fall 2000 Lecture 16

4 Find total solution

v C = v P + v H

t

−

v C = 2 2 2

C R

1 + ω

V i

cos(ωt +φ ) + Ae RC

where φ = tan−1( −ωRC )

so,

A = −

1+ ω2 R2C 2

V i

cos(φ)

6.002 Fall 2000 Lecture 16

We are usually interested only in the

particular solution for sinusoids,

t

−

2 2

2

i

C R

1

V

+

=

ω

tan where φ =

A = −

p

V

RC

t

Ae )

t +φ + −

ω

)

RC (

1 −ω

−

cos(

1 2 2 2 φ

ω R C

V i

+

0

v

)

Described as

SSS: Sinusoidal Steady State

p

V

∠

6.002 Fall 2000 Lecture 16

All information about SSS is contained

Recall

RC j

1

V

p = + ω

Steps 3 , were a waste of time!

4

=

V i 1+ jωRC

V p

ω2 2 2

i 1 R C

1

e jφ where

φ = tan −1 −ωRC

2 2 2

1

1

C R

V

V

i

p

ω

+

=

RC

V

V

i

magnitude

6.002 Fall 2000 Lecture 16

Visualizing the process of finding the

sneak

in

V i e jωt

drive

algebraic equation

+ complex algebra

take real part

t

j

p e

particular solution

t

+ nightmare trig

the sneaky path!

6.002 Fall 2000 Lecture 16

transfer function

V

H ( jω) =

V

p

i

2 2

2

1

1

C R V

V

i

p

ω

+

=

V p

1

V i

log scale

ω

for high frequencies!

6.002 Fall 2000 Lecture 16

φ = tan −1 −ωRC

V

φ = ∠ p

V i

0

π

−

4

π

−

2

ω

RC

1

=

ω

log scale

6.002 Fall 2000 Lecture 16