6.002x circuits and electronics€¦ · 2 n sinusoidal steady state (sss) reading 13.1, 13.2 + –...

1

6.002x CIRCUITS AND ELECTRONICS

The Impedance Model !!!

Reading: Section 13.3 from text

2

n Sinusoidal Steady State (SSS) Reading 13.1, 13.2

+

–

vtVv iI cos+ –

RC

Review

n Focus on sinusoids. n Focus on steady state, only care about vP as vH dies away.

SSS

3

3

4

Hv

total

Review

Vp contains all the information we need:

p

p

V

V

Amplitude of output cosine

Phase

sneak in Vi e jωt

drive

complex algebra

take real part

The Sneaky Path

Vp

tVi cos Vp cos wt +ÐVpéë ùû

set up DE

usual circuit model

nightmare trig.

1

tj

p eV

RCj

Vi

1

2

Sinusoidal Steady State SSS Approach

4

transfer function

jHRCjV

V

i

p

1

1

Review Frequency response - magnitude RCj

VV i

p

1

pp VtVv cos

The Frequency View

5

RC

1

4

2

0

i

p

V

V

1

RCtan 1

The Frequency View

Review

CvOv

BIASV

+ –

+ –

GSC

R

SVFrequency response - phase

6

Is there an even simpler way to get Vp ?

RCj

VV i

p

1

t

agony

sneaky

approach

Eff

ort

easy

Usual

diff eqn.

approach

? Is there

any even

simpler

way

without

DEs?

RCj

VV i

p

1

7

The Impedance Model

Is there an even simpler way to get Vp ?

Consider resistor:

R

Ri+

– RvResistor

8

The Impedance Model -- Capacitor Consider capacitor:

C

Ci+

– CvCapacitor

9

The Impedance Model -- Inductor Consider Inductor:

L

Li+

– LvInductor

10

The Impedance Model -- Inductor Consider Inductor:

L

Li+

– Lv

ts

lL eIi

ts

lL eVv

dt

diLv L

L

Inductor

11

inductor resistor capacitor

The Impedance Model -- Summary

12

Back to RC example…

+

– Cv

Iv + –

R

C

capacitor

sCZC

1

cCc IZV

impedance

cI

+

– cV CZ

Is there an even simpler way to get Vp ?

13

Impedance model:

Back to RC example…

+

– Cv

Iv + –

R

C

+

– cViV +

–

RZR

sCZC

1

cI

capacitor

sCZC

1

cCc IZV

impedance

cI

+

– cV CZ

Is there an even simpler way to get Vp ?

14

t

sneaky

approach

Eff

ort

agony

easy

Usual

diff eqn.

approach

There you have it

super

sneaky… no DEs!

Impedance method Today

15

Signal Notation

16

Impedance Method Summary

17

Another example, recall series RLC Remember, we want only the steady-state response to sinusoid (SSS)

18

Another example, recall series RLC Remember, we want only the steady-state response to sinusoid (SSS)

19

The Big Picture… Vi coswt Vp cos wt +ÐVpéë ùû

set up DE

usual circuit model

nightmare trig.

Vi e st

drive complex algebra

take real part

complex algebra

impedance-based circuit model

No D.E.s, No trig!

t

agony

sneaky

approach

Eff

ort

easy

Usual

diff eqn.

approach

Super

sneaky…

no DEs

20

Back to

21

L

Rj

LC

L

Rj

V

V

i

r

+ –

L

rI

C +

– rV

iV RLet’s study this transfer function

Review complex algebra in Appendix C of textbook

21

Graphically

2221 RCLC

RC

V

V

i

r

:Low

:High

:1

LC

Find limit

+ –

L

rI

C +

– rV

iV R

22

Graphically

2221 RCLC

RC

V

V

i

r

:Low

:High

:1

LC

Observe

+ –

L

rI

C +

– rV

iV R

1

Is there an even simpler way to get Vp ?

RCj

VV i

p

1

t

agony

sneaky

approach

Eff

ort

easy

Usual

diff eqn.

approach

? Is there

any even

simpler

way

without

DEs?

RCj

VV i

p

1

1

The Impedance Model

Is there an even simpler way to get Vp ?

Consider resistor:

R

Ri+

– RvResistor

2

The Impedance Model -- Capacitor Consider capacitor:

C

Ci+

– CvCapacitor

3

The Impedance Model -- Inductor Consider Inductor:

L

Li+

– LvInductor

1

The Impedance Model -- Inductor Consider Inductor:

L

Li+

– Lv

ts

lL eIi

ts

lL eVv

dt

diLv L

L

Inductor

2

inductor resistor capacitor

The Impedance Model -- Summary

1

Back to RC example…

+

– Cv

Iv + –

R

C

capacitor

sCZC

1

cCc IZV

impedance

cI

+

– cV CZ

Is there an even simpler way to get Vp ?

1

Impedance model:

Back to RC example…

+

– Cv

Iv + –

R

C

+

– cViV +

–

RZR

sCZC

1

cI

capacitor

sCZC

1

cCc IZV

impedance

cI

+

– cV CZ

Is there an even simpler way to get Vp ?

2

t

sneaky

approach

Eff

ort

agony

easy

Usual

diff eqn.

approach

There you have it

super

sneaky… no DEs!

Impedance method Today

1

Signal Notation

2

Impedance Method Summary

1

Another example, recall series RLC Remember, we want only the steady-state response to sinusoid (SSS)

1

Another example, recall series RLC Remember, we want only the steady-state response to sinusoid (SSS)

1

The Big Picture… Vi coswt Vp cos wt +ÐVpéë ùû

set up DE

usual circuit model

nightmare trig.

Vi e st

drive complex algebra

take real part

complex algebra

impedance-based circuit model

No D.E.s, No trig!

t

agony

sneaky

approach

Eff

ort

easy

Usual

diff eqn.

approach

Super

sneaky…

no DEs

1

Back to

21

L

Rj

LC

L

Rj

V

V

i

r

+ –

L

rI

C +

– rV

iV RLet’s study this transfer function

Review complex algebra in Appendix C of textbook

2

Graphically

2221 RCLC

RC

V

V

i

r

:Low

:High

:1

LC

Find limit

+ –

L

rI

C +

– rV

iV R

1

Graphically

2221 RCLC

RC

V

V

i

r

:Low

:High

:1

LC

Observe

+ –

L

rI

C +

– rV

iV R