L02 Lecture

7/28/2019 L02 Lecture

1/13

EECS 247 Lecture 2: Introduction to Filters 2002 B. Boser 1A/DDSP

Introduction to Filters

Filtering = frequency-selective signal processing

Its the most common type of signal processing

Examples:

Extract desired signal from many (radio)

Separating signal and noise

Amplifier bandwidth limitations

Where to start

Perfectionist: ideal (low-pass) filter

Engineer: continuous time, first-order low-pass filter


First-Order RC Filter (LPF1)

Steady-state frequency response:

kHzRC

ssV

sVsH

o

o

in

out

10021

with

1

1

)(

)()(

==

+==

7/28/2019 L02 Lecture

2/13


Poles and Zeros

s-plane (pzmap):

=

+=

z

p

ssH

o

o

:Zero

:Pole

1

1)(

j

p=-o


Magnitude Response

-5

0

5

x 105

-5

0

5

x 105

0

0.5

1

1.5

2

2.5

3

Frequency [Hz]

Magnitude Response (s-plane)

Sigma [Hz]

Magnitude[linear]

L02_bode3_lpf1.m

7/28/2019 L02 Lecture

3/13


Frequency Response

Asymptotes:

- 20 dB/dec rolloff

- 90 degrees phase shift per 2 decades

Matlab code (L02_bode_lpf1.m):wo = 2*pi*100e3;

s = tf('s');

h = 1 / (1+s/wo);

bodehz(h, logspace(1, 10, 100));

Note: bodehz is same as bode, but frequency axisis in Hz, rather than rad/s.

-120

-100

-80

-60

-40

-20

0Bode Diagram

Frequency [Hz]

Phase(deg)

Magnitude(dB)

10

1

10

2

10

3

10

4

10

5

10

6

10

7

10

8

10

9

10

10-90

-60

-30

0

0)(

1)(0

==

==

=

jsH

jsH


Parasitics

Can we really get 100dB attenuation at 10GHz?

Probably not

Parasitics limit the performance of analog

components

E.g.

Shunt capacitance

Feed-through capacitance

Finite inductor, capacitor Q

7/28/2019 L02 Lecture

4/13


LPF2

( )P

P

CCsR

sRC

sH +++

= 11

)(( )

P

P

RCz

RCCCRp

1:Zero

11:Pole

=

+

=


Frequency Response

dB

C

C

CC

CjH

jH

P

P

P

6010

)(

1)(

3

0

==

+=

=

=

LPF2

Frequency [Hz]

Pha

se(deg)

Magnitude(dB)

-80

-70

-60

-50

-40

-30

-20

-10

0

102

103

104

105

106

107

108

109

1010

-90

-45

0

Why not just make C larger?

Beware of other parasitics not included in this model

7/28/2019 L02 Lecture

5/13


Continuous Time Analog

Analog passive components arent ideal

Extra real poles/zeroes result from parasitics

Parasitic effects begin to appear 50dB beyond desired

component characteristics

Common sense helps you anticipate them

Digital filters do not suffer from these effects


Second-Order LPF

Improved attenuation (compared to 1st order)

Complex poles (rather than multiple real ones)

Why?

Visualize 3D s-plane plot!

Biquadratic (2nd order) transfer function:

2

2

1

1)(

PPP

s

Q

ssH

++=

PQjH

jH

jH

P

=

=

=

=

=

)(

0)(

1)(0

7/28/2019 L02 Lecture

6/13


Biquad Poles

( )

otherwisecomplexreal,arepolesfor

4112

satpoleshas

1

1)(

21

2

2

2

=

++=

P

P

P

P

PPP

Q

QQ

s

Q

ssH


Complex Poles

( )1412

s 2

21

=

>

P

P

P

P

QjQ

Q

Distance from origin in s-plane:

( )

2

2

2

2 1412

P

P

P

P QQ

d

=

+

=

7/28/2019 L02 Lecture

7/13


s-Plane

poles

j

Pradius =

P2Q-partreal P

=


LPF3

10

1002

==

P

P

Q

kHz

-2

-1

0

1

2

x 105

-2

-1

0

1

2

x 105

0

0.5

1

1.5

2

2.5

3

Frequency [Hz]

Magnitude Response (s-plane)

Sigma [Hz]

Magnitude[linear]

7/28/2019 L02 Lecture

8/13


Bode Diagram

Frequency (rad/sec)

Phase(deg)

Magnitude(dB)

-200

-150

-100

-50

0

50

102

103

104

105

106

107

108

109

1010

-180

-135

-90

-45

0

Frequency Response

-40 dB/dec

-180o


Varying Q Magnitude

Gain at p:

20 log Q [dB]

104

105

106

-50

-40

-30

-20

-10

0

10

20

30

40LPF3 Magnitude Response

Frequency [Hz]

Magnitude

[dB]

Q = 0.5Q = 10.0Q = 100.0

7/28/2019 L02 Lecture

9/13


Phase

Slope at p :

-45 Q deg/decade

104

105

106

-180

-160

-140

-120

-100

-80

-60

-40

-20

0LPF3 Phase Response

Frequency [Hz]

Phase

[degrees]

Q = 0.5Q = 10.0Q = 100.0


Implementation of Biquads

Passive RC: only real poles

Terminated LC lowest power (well its passive!)

No noise (except load and source)

Active Biquad Filter texts give you dozens of topologies.

Who needs or wants that many choices?

Single-opamp biquad: Sallen-Key

Two-opamp biquad: Tow-Thomas

7/28/2019 L02 Lecture

10/13


Sallen-Key LPF

Single gain element

Parasitic sensitive

Versions for LPF, HPF, BP,

Ref: K. L. Su, Analog Filters, Chapman & Hall, 1996, pp. 215.

221211

2211

2

2

111

1

1

)(

CR

G

CRCR

Q

CRCR

s

Q

s

GsH

PP

P

PPP

++

=

=

++=


Component Sizing Choice 1

4 unknowns: R1, R2, C1, C22 knowns: P, QP problem is underdetermined

Choice 1: minimum component spread

9.21

3

6.11

1

1

21

21

==

===

==

P

P

QG

kC

RR

nFCC

7/28/2019 L02 Lecture

11/13


SK Magnitude Response 1

10% increase of R1more than doubles QP!

The circuit is very sensitive

to component variations.

104

105

106

-50

-40

-30

-20

-10

0

10

20

30

40Sallen-Key Choice 1 Magnitude Response

Frequency [Hz]

Magnitude

[dB]

nominal R1R1 10% large


Component Sizing Choice 2

Choice 2: minimum sensitivity

pFRQ

C

nFR

Q

C

kRR

G

PP

P

P

402

1

16

2

2

1

1

2

11

21

==

==

===

4004

2

2

1

== PQCC

Note also:

Huge element spread

This topology is suitable only for

low-Q filter implementations.

7/28/2019 L02 Lecture

12/13


SK Magnitude Response 2

10% increase of R1 has only small

effect on response!

The circuit is NOT very sensitive

to component variations.

104

105

106

-50

-40

-30

-20

-10

0

10

20

30Sallen-Key Choice 2 Magnitude Response

Frequency [Hz]

Magnitude

[dB]

nominal R1R1 10% large


Sensitivity

Implementation and component

sizing have huge impact on

sensitivity

High-sensitivity circuits are

problems in practice

No theory for finding a low-

sensitivity architecture

Ladder filters are usually low

sensitivity

Use proven circuits & check!02Choice

%955.9

5.95.01Choice

Example

with

Definition

1

1

1

1

1

1

1

=

=

==

=

=

=

P

P

P

Q

R

P

P

P

Q

R

Q

R

P

P

y

x

y

x

S

R

R

Q

Q

QS

R

RS

Q

Q

dx

dy

y

xS

x

xS

y

y

Common sense: Sensitivity is a first order approximation,correct only for infinitesimally small errors

7/28/2019 L02 Lecture

13/13


Summary

Frequency Response

Poles and zeros are like tent poles and pegs

Frequency response is evaluated on j axis Poles and zeros close to j axis dominate resonse

Practical Implementation Constraints

Components are not ideal

Avoid solutions requiring large element spread

Beware of high-sensitivity architectures

Documents

L02 Lecture