1. Cascade of Biquadratic (Biquad) and Bilinear cells 2...

Progettazione Mixed SignalPaolo Bruschi 1

Analog Active Filters: Approaches

1. Cascade of Biquadratic (Biquad) and Bilinear cells

2. State Variable Filters

3. Simulation of LC filters with active RC networks


fc synonyms:

Corner frequency

Cut-off frequency

fCL Corner Freq – Low

fCH Corner Freq – High

Common Filter Responses


Parameters of a real frequency response

The Low Pass filter is the

reference for other types

Frequency are general given as

angular frequencies (w)

A Transition Band (TB) is

introduced

Note: wp is generally different from wc (-3 dB frequency)


Analog Filters: State Variable Filter

01

1

1

01

1

1

....

....

)(

)(

bsbsbs

asasasa

sv

svN

N

N

N

N

N

N

in

out

000222111

021021211 ...

bcabcabcaca

bbb

NNNNNNNN

NNNNNNN

wwwwww

It is an Integrator-based Architecture


Biquad with State Variable Archirecture

2

002

2

000

1

2

2

)(

)(

ww

ww

sQ

s

bsQ

bsb

sV

sV

in

out Function b0 b1 b2

Low Pass 1 0 0

High Pass 0 0 1

Band Pass 0 1 0

Notch 1 0 1

02

01

0201

w

w

wwww

zp

zp

QQ


OTA: definitions and basic circuits

Typical non-idealities:

Finite Rout

Input Capacitance

Frequency dependence of Gm

Input/Output ranges

21 vvGi mout

Ideal operation

OTA (Transconductor)

21 vvsC

G

sC

iv mout

out

OTA-C (Gm-C) Integrator


Ota-Based summing circuits

Summing amplifier

(inverting / non-inverting)

Summing Integrator

(inverting / non-inverting)

43

1

2

21

1 vvG

Gvv

sC

Gv

m

mm

out 43

3

2

21

3

1 vvG

Gvv

G

Gv

m

m

m

m

out


Gm-C integrator with feed-forward input

321 vvvsC

Gv m

out C

Gmw1


Gm-C Universal Biquad

2

2

02

1

1

01C

G

C

G mm ww


Self-Tuning of OTA Filters

Master

Slaves

Phase detector

Characteristic

Tuning law of

transconductors


Switched Capacitors Integrator Parasitic Insensitive SC resistors

Positive SC resistorNegative SC resistor

Virtual ground

SC Integrator

Phase 1: Hold

Phase 2:

2

)1()(C

Qnvnv outout


SC resistors: charge transfer

Phase 1

Phase 2

1

11

1

CfRVCfIVCQ

ck

EQSckEQS

Negative Resistor Positive Resistor

Q<0 Q>0


SC Summing Integrator

1)1()1()1()(2

1

2

12

2

11 nvnvC

Cnv

C

Cnv

C

Cnvnv finfin

Fninpinoutout

ck

ck

zVC

CzV

z

z

C

CzV

z

z

C

CzV fin

Fninpinout

2

1

1

1

2

12

1

1

2

11 )(1

)(1

)(

We assume that

inputs vin-p and vin-n

are sampled at the

end of phase 2


zVC

CzV

z

z

C

CzV

z

z

C

CzV fin

Fninpinout

2

1

1

1

2

12

1

1

2

11 )(1

)(1

)(

Discrete time integral

1

1

1)(

z

zzH IZ

wwww

w

w

ww

2221

)()(TjTjTj

Tj

Tj

TjTj

IZIF

eee

e

e

eeHjH

ww

w

2sin2

)(2

Tj

ejH

Tj

IF

TjjH

ff

TIF

ck

ww

w 1)(1

2

ckfT

1

Forward Euler Integration

Approximation for f<<fckExact response


bilinear

Continuous Time

Euler

0.5

Integrator comparison

1

1

1

z

z

1

1

1

1

2

1

z

z


SC-integrator equivalent block diagram

2

11

12

111

2

121

C

Ck

C

Ck

C

Cf F

F w

1. Cascade of Biquadratic (Biquad) and Bilinear cells 2...

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