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6.002x CIRCUITS AND ELECTRONICS

Sinusoidal Steady State

!

Reading Section 13.1 of the textbook

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n We now understand the why of:

Review

5V

C

R

L

v

t

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n Today, look at response of networks 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

Today

0   t 0   t

0   t 0   t

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Observe vO amplitude as the frequency of the input vI changes. Notice it decreases with frequency.

Also observe vO shift as frequency changes (phase).

For motivation, consider our old friend, the amplifier: Motivation

Demo

CvOv

BIASV

+"–"+"–"

GSC

R

Iv

SV

Need to study behavior of networks for sinusoidal drive.

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Sinusoidal Response of RC Network

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+"

–"vIv +"–"

Ri

Determine v(t)

Our Approach

t

Indulge me!

agony sneaky approach

Effo

rt

This sequence

easy

Usual diff eqn. approach

super sneaky

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Let’s try the usual approach…

Effo

rt Usual

diff eqn. approach

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Usual approach… +"

–"vIv +"

–"R

i

Determine v(t)

Effo

rt Usual

diff eqn. approach

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2 Find vP tVvdtdv

RC i ωcos=+

Effo

rt Usual

diff eqn. approach

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Detour: Let’s get sneaky! sneaky approach

Effo

rt Usual

diff eqn. approach

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Detour: Let’s get sneaky!

Try solution stpPS eVv =

ISPSPS vvdtdvRC =+ (S: sneaky :-))

stieV=

Find particular solution to another input… sneaky approach

Effo

rt Usual

diff eqn. approach

Find VP

Sneaky detour: find particular solution to another input sRC1

VV ip +=

sneaky approach

Effo

rt Usual

diff eqn. approach

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Fourth try to find vP… using the sneaky way 2

sneaky approach

Effo

rt Usual

diff eqn. approach

sRC1VV i

p +=

s = jω

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vP particular response to Vi cos ωt Fourth try to find vP… 2 sRC1VV i

p +=

s = jω

sneaky approach

Effo

rt Usual

diff eqn. approach

vPS particular response to Vi ejωt

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vP particular response to Vi cos ωt

so, vP = Re vPS[ ] = Re Vpe

jωt!" #$ ⎥⎦

⎤⎢⎣

⎡ ⋅+

= tji eRCj

V ω

ω1Re

complex Fourth try to find vP… 2 sRC1

VV ip +=

s = jω

sneaky approach

Effo

rt Usual

diff eqn. approach

vPS particular response to Vi ejωt

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Find vH 3 0=+ HH vdtdvRC

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Find total solution 4

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Sinusoidal Steady State sRC1VV i

p +=

s = jω

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Sinusoidal Steady State – VP says it all sRC1VV i

p +=

s = jω

sneaky approach

Effo

rt Usual

diff eqn. approach

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Visualizing Sinusoidal Steady State

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Summary of SSS approach – fits on one slide

sneaky approach

Effo

rt Usual

diff eqn. approach

+"

–"vIv +"

–"R

i

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Magnitude Plot sRC1VV i

p +=

s = jω

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Phase Plot sRC1VV i

p +=

s = jω

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Preview of upcoming attractions: The next Aha moment!

sti

stp

stp eVeVdtedV

RC =+

sRC1VV i

p +=

tVvdtdv

RC i ωcos=+1. Set up DE

sneaky approach

Effo

rt Usual

diff eqn. approach

+"

–"vIv+"

–"R

i

2. Apply sneaky input Viest,    then  find particular solution VPest

stieVv

dtdv

RC =+

)cos( PPP VtVv ∠+= ω

s = jω

3. Find VPest

4. Find vP , which is steady state solution for v