bode plots & frequency response

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Bode Plots & Frequency Response Read Chapter 11 of Razavi 1

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Page 1: Bode Plots & Frequency Response

Bode Plots & Frequency Response

Read Chapter 11 of Razavi

1

Page 2: Bode Plots & Frequency Response

2

1 2

1 2

( )

1 1 1

( )

1 1 1

We must factor polynomials in variable

where

z z znn

m

p p pm

Transfer Function H s

s s s

bH s

a s s s

s

s j

General Form of H(s)

Page 3: Bode Plots & Frequency Response

3

( )( ) ( ); Let ( ) ( )

,

( )

( ) 1( )

( )

, ( )

st st

st st st

st st

di tRi t L v t i t Ie and v t Ve

dtTherefore

Re sLe I Ve

i t IH s

v t V R sL

So Ie H s Ve

i(t) v(t)

R

L

Example: R-L Circuit

Page 4: Bode Plots & Frequency Response

4

1( ) 1

( )( ) 1

where 2 p p

i t I RH ssv t V R sLRL

Rf

L

Example: R-L Circuit

|H(f)|

f fp

0 dB

Slope = + 6 dB/octave & 20 dB/decade

+ -3 dB

f fp

Slope = - 45/decade +

90

45

0

0.1 fp 10 fp

These are log-log plots – frequency is logarithmic

Page 5: Bode Plots & Frequency Response

5

R-L and R-C Circuits

Low-pass

High-pass

Page 6: Bode Plots & Frequency Response

6

Bode Plots for R-L and R-C Circuits Summary

Page 7: Bode Plots & Frequency Response

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1 / 2 1 / 2

1 12 2

1 14 4

1 15 5

1 110 10

( ) ( )

1 20 log(1) 0

2 20 log( 2 ) 3

2 20 log(2) 6

4 20 log(4) 12

5 20 log(5) 14

10 20 log(10) 20

20 log( ) 3

20 log( ) 6

20 log( ) 12

20 log( ) 14

20 log( ) 20

dBH s H s

dB

dB

dB

dB

dB

dB

dB

dB

dB

dB

dB

Some Logarithm values convenient for Circuit Analysis

Page 8: Bode Plots & Frequency Response

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Low-pass Filter R-C Network’s Bode Plot

Page 9: Bode Plots & Frequency Response

9

1

1

1 1

( )Suppose we have ( ) , then

( )

( ) 20 log 20 log 20 log

10( 100): ( )

( 1)

dB

A S zH s

S p

H s A S z S p

sExample H s

s

A = 10 Pole at s = -1 Zero at s = -100

100 1

20 dB 40 dB 0 dB

( ) 10H s 1( )

( 1)H s

s

( ) ( 100)H s s

frequency

Now add all three together

Example

Page 10: Bode Plots & Frequency Response

10

10( 100)( )

( 1)

sH s

s

frequency

0.01 0.1 1 10 100 1000 104 105

60 dB

40 dB

20 dB

0 dB

80 dB

lim ( ) 20s

H s dB

0lim ( ) 60s

H s dB

|H(s

)|

All together now

Page 11: Bode Plots & Frequency Response

11

Phase in Bode Plots

The transfer function H(s) is a phasor.

For a rational function H(s) we add the phases from the

numerator and subtract the phases from the denominator:

( )( ) ( ) ( ) ( ),

( )

Example:

( )

N jH j H j N j D j

D j

H j

1 1( )( ) tan tan

( )

j zH j

j p z p

Page 12: Bode Plots & Frequency Response

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• The capacitive load, CL, is the culprit for gain roll-off since at high frequency, it will “steal” away some signal current and shunt it to ground.

1||out m in D

L

V g V RC s

Poles are Associated with Nodes in Circuits

Page 13: Bode Plots & Frequency Response

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• The circuit only has one pole (no zero) at 1/(RDCL), so the slope drops from 0 to -20dB/dec as we pass ωp1.

LD

pCR

11

Corresponding Bode Plot for Previous Circuit

Page 14: Bode Plots & Frequency Response

14

inS

pCR

11

LD

pCR

12

2

2

22

1

2 11 pp

Dm

in

out Rg

V

V

Example of a Circuit with Two Poles

Page 15: Bode Plots & Frequency Response

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• For a MOS, there exist oxide capacitance from gate to channel, junction capacitances from source/drain to substrate, and overlap capacitance from gate to source/drain.

Origin of Capacitances in MOSFET – I

Page 16: Bode Plots & Frequency Response

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• The gate oxide capacitance is often partitioned between source and drain. In saturation, C2 ~ Cgate, and C1 ~ 0. They are in parallel with the overlap capacitance to form CGS and CGD.

Origin of Capacitances in MOSFET – II

Page 17: Bode Plots & Frequency Response

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• At high frequency, capacitive effects come into play. Cb represents the base charge, whereas C and Cje are the junction capacitances.

b jeC C C

Origin of Capacitances in BJT – I

Page 18: Bode Plots & Frequency Response

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Origin of Capacitances in BJT – II

• Since an integrated bipolar circuit is fabricated on top of a substrate, another junction capacitance exists between the collector and substrate, namely CCS.

Page 19: Bode Plots & Frequency Response

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Example of Capacitors in BJT Circuit

Page 20: Bode Plots & Frequency Response

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• The frequency response refers to the magnitude of the transfer function.

• Bode’s approximation simplifies the plotting of the frequency response if poles and zeros are known.

• In general, it is possible to associate a pole with each node in the signal path.

• Miller’s theorem helps to decompose floating capacitors into grounded elements.

• Bipolar and MOS devices exhibit various capacitances that limit the speed of circuits.

Summary

But . . . We will soon deal with Miller’s Theorem to simplify circuits analysis.