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EECS 247 Lecture 3: Second Order Transfer Functions 2002 B. Boser 1A/DDSP

2nd Order Transfer Functions

Imaginary axis zeroes

Tow-Thomas Biquad

Example


Imaginary Axis Zeros

Sharpen transition band

notch out interference

High-pass filter (HPF)

Band-reject filter2

2

2

)(

1

1

)(

=

++

+

=

Z

P

PPP

Z

KjH

s

Q

s

s

KsH

Note: Always represent transfer functions as a product of a gainterm, poles, and zeros (pairs if complex). Then allcoefficients have a physical meaning, reasonablemagnitude, and easily checkable unit.

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Imaginary Axis Zeros

-5

0

5x 10

5

-5

0

5

x 105

0

0.5

1

1.5

2

Sigma [Hz]

Magnitude Response (s-plane)

Frequency [Hz]

Magnitude[linear]

-5

0

5x 105

-5

0

5

x105

0

0.5

1

1.5

2

Sigma [Hz]

Magnitude Response (s-plane)

Frequency [Hz]

Magnitude[linear]

No finite zeros With finite zeros


Imaginary Zeros Zeros substantially sharpen

transition band

At the expense of reduced stop-band attenuation at high frequency

PZ

P

P

ff

Q

kHzf

3

2

100

===

Pole-ZeroMap

Real Axis

ImagAxis

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x 106

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x106

104

105

106

107

-50

-40

-30

-20

-10

0

10

Frequency [Hz]

Magnitude

[dB]

With zerosNo zeros

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Moving the Zeros

PZ

P

P

ff

Q

kHzf

===

2

100

Pole-ZeroMap

RealAxis

ImagAxis

-6 -4 -2 0 2 4 6

x105

-6

-4

-2

0

2

4

6

x105

104

105

106

107

-50

-40

-30

-20

-10

0

10

20

Frequency [Hz]

Magnitude

[dB]


Tow-Thomas Biquad

Ref: P. E. Fleischer and J. Tow, Design Formulas for biquad active filters usingthree operational amplifiers, Proc. IEEE, vol. 61, pp. 662-3, May 1973.

Parasitic insensitive

Multiple outputs

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Frequency Response

( ) ( )

( ) ( )

01

2

1001020

01

3

01

2

01

2

22

01

2

0021122

1

1

asas

babasabb

akV

V

asas

bsbsb

V

V

asas

babsbabk

V

V

in

o

in

o

in

o

+++

=

++++

=

+++

=

Vo2 implements a general biquad section with arbitrary poles and zeros Vo1 and Vo3 realize the same poles but are limited to at most one finite zero


Component Values

8

72

173

2821

11

1

21732

80

6

82

74

81

6

8

11

1

21753

80

1

1

R

Rk

CRR

CRRk

CRa

CCRRR

Ra

R

Rb

RR

RR

R

R

CRb

CCRRR

Rb

=

=

=

=

=

=

=

827

2

86

20

01

5

112124

1021

3

20

12

11

1

111

11

1

RkR

b

RR

Cb

akR

CbbakR

CakkR

Ca

kR

CaR

=

=

=

=

=

=

=

821 and,,,,given RCCkba iii

thatfollowsit

11

21732

8

CRQ

CCRRR

R

PP

P

=

=

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Filter Design Example

Application: testing of ultra-linear ADC

Problem: sinusoidal source has higher distortion thanthe ADC!

Solution Filter source with bandpass before converting

Check resulting source with spectral analyzerTwist: the analyzer is not sufficiently linear either notch out sinusoid and look just at harmonics

Implementation

Bandpass & Notch at 1kHz Use Vo2 for bandpass (only possibility),

Vo1 for notch



1kHz

Generator

1kHz

BPF

1kHz

Notch

spectrum

analyzer

ADC undertest

Principle: IC test circuits are useless if you cant

verify their performance!

Our filter

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Filter Coefficients

( ) ( )

( ) ( )

01

2

1001020

01

3

01

2

01

2

22

01

2

0021122

1

1

asas

babasabb

akV

V

asas

bsbsb

V

V

asas

babsbabk

V

V

in

o

in

o

in

o

+++

=

++++=

+++=

( )

1k

outputs)otherbelowslightlyVunusedkeep(to05.1k

:levelssignalreasonableChoose

bandpass)ainwantweas(just0

:free""forssGet Bandpa

1

0

12

:NotchDesign

2

o31

002

1112

2

1

22

00

=

=

==

==

===

bab

abab

b

b

kHzabP


Final Filter

01

2

013

01

2

0

2

2

01

2

11

05.1

1

asas

aa

V

V

asas

as

V

V

asas

sa

V

V

in

o

in

o

in

o

++=

+++=

++=

Choose:

C1=C2=112nF (large to minimize noise)R8=1kfP=1kHz, QP=30 (check sensitivity!)

Solve equations R1=42.631kR2=1.4921kR3=1.3534kR4=42.631kR5=1.4921kR6=R7=R8

Lets order the parts

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Capacitors

C0G capacitors Vishay Vitramon, C0G Dielectric Capacitor datasheet, 2000.

http://www.vishay.com/document/45002/45002.pdf

Negligible voltage coefficient (for linearity)

Excellent tempco (30ppm/C)

2% initial accuracy is easy to get

No high-value capacitors are trimmable

Resistors will be trimmed to compensate for capacitorvariations


Resistors

Trimmed resistors combine fixed metal film resistors andprecision trim potentiometers in series 1%-accurate, 5ppm/C, lab grade metal film resistors provide 90%

of the nominal resistance

Ref: Caddock Electronics, Type TN Lab Grade Low TC Precision Film Resistordatasheet, 1999.

50ppm/C trim pots provide between 0% and 20% of the nominal

resistanceRef: Vishay Foil Resistors, Model 1268 Precision Trimming Potentiometers datasheet

Use two fixed resistors in series with the trimpot to minimize trimpotvalue and optimize overall tempco

R6-R8 are 0.1%-accurate, 5ppm/C metal film

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Opamps

For opamps, well use the Burr-Brown OPA627

Ref: Texas Instruments / Burr-Brown, OPA627 and OPA604datasheets, 1989.

The finest audio opamp in the world, and, at $15/each,priced accordingly!

But money is no object when designing IC test fixtures (onlya few are ever built)

Adequate speed for this application


Bandpass/Bandstop Responses

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Filter Design Example (cont.)

Note that the bandpass output H1 provides>30dB attenuation to all harmonics present inthe 1kHz generator output

Opamp outputs have 0.00.5dB peak gain

This maximizes each opamps output swing forbest dynamic range

Lets magnify the frequency axis for the two

responses of interest


Bandpass/Bandstop Responses

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Temperature changes wont change theseresponses too much

Lab temperatures are stable to 253C Our lab-grade RC products move

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Obviously, weve got to tune the filterback to its original specification

How is that tuning done?

Do you tell your technician to twiddle potsrandomly until it works?

Or do you document a robust tuning

procedure?


RC Filter Tuning Strategy

Famous biquads like the Tow-Thomas comecomplete with their own tuning strategies The circuit topologies allow 1 trim operation to adjust 1

design parameter (such as fP, fZ, QP, QZ, gain) withoutchanging the others

Rationale for a biquads tuning strategy becomesapparent when studying design equations such asthe Tow-Thomas equations on slide 6

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Tow-Thomas Tuning Strategy

R3 will be set to a fixed value to keep theunused OPAMP3 output below 0dB

Tuning involves the following steps performedin the specified sequence: Adjust R2 to center the bandpass at 1kHz

Adjust R5 to center the notch at 1kHz

Adjust R1 to set the bandpass Q to 30

Adjust R4 to deepen the notch


Tow-Thomas Tuning Strategy

The design equations also provide the range ofadjustment required for a given resistor Remember that an excessively large adjustment range

translates into excessively large tempco

R1 tuning range (from slide 7):

a11

R1C1

1

a1C1MAX

1

a1C1MIN< R1

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