Analog Filter Design - unipi.itdocenti.ing.unipi.it/~a008309/mat_stud/AIF/2018/lecture... · 2018. 4. 23. · SAB (Single Opamp Biquad) Finite Gain SABs ... Inductance simulation

P. Bruschi - Analog Filter Design 1

Analog Filter Design

Part. 3: Time Continuous (TC) Filter Implementation

Sect. 3-a: Active Filters

Motivations

• Inductors are generally difficult to miniaturizeL ~ (coil area) x (number of coils)2 x (magnetic permeability)Integrated inductors limited to a few nH (max)Stray magnetic field cause unwanted coupling

• Resistors and capacitors can be easily integrated: feasible ranges are much wider than for inductors

• Active Filters Target: Synthesis of arbitrary transfer functions using only resistors, capacitors and active elements.


Design approaches for active TC filters

• Cascade of Biquadratic (Biquad) and Bilinear cells

• State Variable Filters (MLF: Multiple Loop Feedback circuits)

• Simulation of LC filters with active RC networks


System-level architectures

Circuit-level architectures

Op-amp based

OTA (Operational transconductance amplifier) – based

Cascade of Biquad (Bilinear) functions


Biquad Transfer Function

22

2

01

2

2

0

01

2

01

2

2)(

p

p

p

z

z

z

BQ

sQ

s

bsQ

bsb

Hdsds

cscscfH

p

zBL

s

bsbHfH

010)(

b2,b0 : 0, 1

b1: 0, ±1

Bilinear Transfer Function

b1 : 0, 1

b0: 0, ±1

“Bits” b2,b1,b0 determine

which terms are present in

the numerator

Poles vs. Biquad coefficients


2 Rep

p p P

p

ss Q

s

2

* 2 * * 22 Re

p p p p p p p ps s s s s s s s s s s s s s

For a 2nd order polynomial with complex roots:

Biquads can be easily

extracted from "zpk" output

of python or Matlab filter

synthesis functions

For the zeroes, identical rules apply, with the exception of :

Zeros in the origin, s2 or s term only (b0=0)

Zeros to infinity, only constant term is present (b1,b2=0)

Notable cases


22

p

p

p

p

p

sQ

s

sQ

22

2

p

p

ps

Qs

s

22

22

p

p

p

p

sQ

s

s

22

22

p

p

p

p

p

p

sQ

s

sQ

s

22

2

p

p

p

p

sQ

s

Low pass High pass Band pass Band Stop

All pass

(phase equalizer)

All these biquads have

unity gain in their

respective pass-bands

Sequencing criteria for biquad cascades


Requirement: Zout


Sequencing criteria: Targets and Rules of Thumb

Pairing: couple together poles and zeroes which are closer in the s-plane

(flatter response, less component spread)

Position: Place the biquads with lower Q closer to the inputs

Keep biquads with similar frequency of maximum as far away as

possible

If possible, place LP Biquads first and HP or BP Biquads last

Gain distribution: balance the signal amplitude over the various biquads

Targets

Maximize the Dynamic Range (DR)

Minimize the transmission sensitivity (to component variations)

Minimize the pass-band attenuation

Rules

Biquad implementations


Op-amp Based:

SAB (Single Opamp Biquad)

Finite Gain SABs – positive feedback

Finite Gain SABs – negative feedback

Infinite Gain SABs

Multiple op-amp Biquads (e.g. MFL )

OTA based (Gm-C filters)

SABs


kVV

VyVyVyI

/

0

23

3332231133

kyy

y

V

V

V

V

in

out

/3323

13

1

2

02231133 VyVyI

23

13

1

2

y

y

V

V

V

V

in

out

Finite gain Infinite gain

Example: Sallen-Key Biquads


SK General configuration SK- Low pass filter

in

out

V

V

(R.P. Sallen, E.L. Key – MIT Labs, 1955)

Example II: SAB with infinite amplifier gainand "bridged-T" network


2 1 3 4 1 43

23

2 1 3 4

Y Y Y Y Y Yiy

v Y Y Y

3 41

13

2 1 3 4

Y Yiy

v Y Y Y

13 3 4

23 2 1 3 4 1 4

y Y YH

y Y Y Y Y Y Y

Bridged –T network

For infinite (negarive) amplifier gain:

Band-pass Delyannis-Friend Biquad


1 2

2

2 1 2 1 2

( )1 1

s

s

R CH s

s sR C R R C C

1 1

2 2

3 1

4 2

1 /

1 /

Y sC

Y R

Y R

Y sC

Band-pass

Biquad

Multiple Feedback Loop Filters


Cascaded Biquads: Feedback exist only inside blocks

MFL Filters: Feedback involve all stages together

More Interaction: less sensitivity to component variations

“Follow the Leader Filter” (FLF) architecture”

Ti(s) can be:

• Integrators

• Lossy Integrators

• Biquads

coefficients

Integral representation of transfer functions


01

1

1

01

1

1

1

2

....

....

bsbsbs

asasasa

V

Vn

n

n

n

n

n

n

1 11 1 0 2 1 1 0 1.... ....n n n n

n n ns b s b s b V a s a s a s a V

1 1 0 2 1 1 0 11 1

1 1 1 1 1 11 ,..., ,...,

n n nn n n nb b b V a a a a V

s s s s s s

2 1 1 0 2 1 1 0 11 1

1 1 1 1 1 1,..., ,...,

n n nn n n nV b b b V a a a a V

s s s s s s

divide

by sn

generic rational

transfer function

State variable filters(MFL filters based on Integrators)


1

2 1 0 2 1 2 1 2

2

1

1 1 1 0

1 1 1 1.....

....

n n n

n

n n

n

V V b V b V b Vs s s s

V K

V s b s b s b

Low pass, all poles filters

Multi Feedback – Multi Feed Forward


01

1

1

01

1

1

1

2

....

....

bsbsbs

asasasa

V

Vn

n

n

n

n

n

n

Arbitrary Functions

(poles and zeros)

'

2

1

1 1 1 0

1

....n n

n

V

V s b s b s b

-b0

Integrators: Op-amp based solution


sRCv

v

in

out 11

CRs

CR

R

R

v

v

in

out

1

11

1

1

Integrator (inverting) Lossy Integrator (inverting)

Example: State variable Filter with Op-amp Integrators


Inverting amplifiers

are required to cope

with the inverting function

of the OA based Integrators

Opamp integrators

(inverting)

Resistor values

(inverse of

summing

coefficients)

MLF: Example Universal 2nd order Filter


Kerwin-Huelsman-Newcomb (KHN) filter(Produces LP, BP and HP outputs: Single Input – Multiple Output)

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 moutout

OTA-C (Gm-C) Integrator

OTA-C (Gm-C) Eq. Resistor

(0 )P out m P

P m P

I I G V

I G V

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 433

2

21

3

1 vvG

Gvv

G

Gv

m

m

m

m

out

Equivalent resistor:

R=1/Gm3

Gm-C integrator with feed-forward input


321 vvvsC

Gv mout C

Gm1

Iout

Gm-C integrator cascade


Two signals with opposite signs

can be added at each internal

node of the cascade

State variable filters – alternative solution


1

1 1 0

1

1 1 0

....

....

n n

out n

n n

in n

V s a s a s a

V s b s b s b

Differently from the FL filter (Follow the Leader), where all the integrator outputs are fed back to the first

integrator and fed forward to the summing node, here the output voltage is fed back to the input of each

integrator where also the input signal is fed forward. This architecture is more suitable for Gm-C

implementations, where summing several inputs would require several OTAs

Target f.d.t.

Gm-C state variable filter: coefficients


1 1 0 1 1 01 1

1 1 1 1 1 1,..., ,...,

out n out n n inn n n nV b b b V a a a a V

s s s s s s

1 1

2 1 2

0 1 2 0...

n n

n n n

n n

b

b

b

1 1 1

2 2 2

0 0 0

n n

n n n

n n n

a c

a c b

a c b

a c b

bn-1bn-2b0

Example: First order high pass / low pass filters


( )p

p

H ss

( )p

sH s

s

low pass

high pass

m

p

G

C

Example: State variable Gm-C biquad


2 2

2 1 0

2 2

( )

p

p

p

p

p

p

B s B s BQ

H s

s sQ

01 02

01

02

p

pQ

02

in

in

in

vBv

vBv

vBv

23

12

01

}1,0{,, 210 BBB

Flexible Biquad

State variable Flexible Biquad – OTA implementation


2

2

02

1

1

01C

G

C

G mm

Function B0 B1 B2

Low pass 1 0 0

High pass 0 0 1

Band-Pass 0 1 0

Notch 1 0 1

1 2 1 2

1 2 2 1

m m m

p P

m

G G G CQ

C C G C

Simulation of Ladder Filters with OTAs


Simulation of the inductor: application of the OTA

based Gyrator

Simulation of the nodal equations by means of OTAs

(signal flow path) May require inductor simulation,

depending on the transfer function to synthesize and/or

architecture

Gyrator


0

0

1

2

G

GY

2 1 1

1 2 2

i G v

i G v

Y-parameters equivalent circuit

Inductance simulation by means of a gyrator and a capacitor.


Generic Impedance Inversion

21GG

CLEQ

2 2

P P

V

P

v vZ

i G v

Inductor Synthesis

21

1

GG

CsZ

CsZ V

2 1 1 2

1P P

V

P P

v vZ

i G G v Z G G Z

2 2 1 pv i Z G v Z

OTA Based Gyrators


Grounded gyrator

2 1 1

1 2 2

i G v

i G v

Inductor simulation with OTAs


21 mm

EQGG

CL

Grounded Inductor

Floating Inductor

IA IBIOA IOB

VL+

-

VL

+

-

VC

+-

sL

VII LBA

IB

IA

Lmm

BA

CmOBBCmOAA

LmC

VsC

GGII

VGIIVGII

VGsC

V

12

22

1

;

;1

21 mm

EQGG

CL


Inductor simulation with OTAs - Example

Initial passive filter

2

1 1

2

2 2

2

3 3

L m

L m

L m

C L G

C L G

C L G

L1

L2

L3

L1,L3: floating inductors

L2: grounded inductor

Same filter, with simulated inductors

Signal flow simulation of ladder (LC) networks with OTAs


566

6455

5344

4233

3122

211

IZV

VVYI

IIZV

VVYI

IIZV

VVYI in

Network Equations

Variable transformations


Target: Transform current variables (I1,I3,I5)

into voltage variables

553311

111I

gVI

gVI

gV

1

1 2

2 2 1 3

3

3 2 4

4 4 3 5

5

5 4 6

6 6 5

in

YV V V

g

V gZ V V

YV V V

g

V gZ V V

YV V V

g

V gZ V

Homogeneous

equivalent

equations

Leap-Frog architecture


423

3

3122

21

1

VVg

YV

VVgZV

VVg

YV in

4 4 3 5

5

5 4 6

6 6 5

V gZ V V

YV V V

g

V gZ V

Homogeneous equivalent equations

OTA implementation of the Leap-Frog structure


3122

21

1

VVgZV

VVg

YV in

31222

2111

VVGZV

VVGZV

mL

inmL

2

22

1

11

m

L

m

L

G

gZZ

gG

YZ

same for all

odd indexes

same for all

even indexes

Example: 5th Order Chebyshev Filter


C1=33.9 mF

L1=17.35 mH

C2=47.7 mF

L2=17.35 mH

C3=33.9 mF

fp=10 kHz

Y1

Z2 Z4

Y3 Y5

Z6

11

1

11

m

LgGR

Z

1R

S1

1

g

1

1

1

m

LG

Z mS1k1 11 mL GZ

L

L

mm

LsC

ZGsC

g

G

gZZ

2

2

212

22

1 1

2

2 Cg

GC mL

2

1 S

10 Sm

g

G m

2339 pF

LC


Example: 5th Order Chebyshev Filter

1

3

1

3

3

1

sLY

gG

YZ

m

L Lm

LsCgGsL

Z311

3

11 113 mL gGLC

3

1 S

10 Sm

g

G m

3

173.5 pFL

C 1

2

3

4

5

6L

1

2 6

1 k

339 pF

173.5 pF

47.7 pF

173.5 pF

Z 100 k 339 pF

1mS

G 10 S

L

L

L

L

L

m

m

R

C

C

C

C

G

m

36

2

63

2

66

6

6

1

1

1

CsGR

Gg

sCR

G

g

G

gZZ

mmmm

L

2

6m

R

gG

6 3mG C

g4 5,

L LZ Z

Same procedure 6LZ

Frequent choices in active ladder filters

• Inductor synthesis :

HP filters (ideal for all-grounded inductors)BP filters LP – filters with zeroes

• Leapfrog architectures:

LP all – pole filtersBP filters (resonant groups simulated by biquads)



Self-Tuning of OTA FiltersIn integrated circuits, Gm's and capacitances are strongly affected by PVT variations (up to ± 30 %

variations). For these reasons, in most OTAs the Gm can be controlled by means of a voltage

applied to a proper terminal (Vtune). In this way self-tuning of the filter can be accomplished.

To understand the self-tuning

principle, consider the effect of tuning

on the phase response of a 1st order

LP filter

Self-Tuning of OTA Filters: master slave approach


Slaves

Master

The loop varies Vtune in

such a way that wC of the

two master LP filters

equals the reference

frequency. Since the same

Vtune is fed to the slaves,

their Gm/C ratios are also

proportional to the ref.

frequency.

Reference

frequency

source

Documents

Analog Filter Design - unipi.itdocenti.ing.unipi.it/~a008309/mat_stud/AIF/2018/lecture... · 2018. 4. 23. · SAB (Single Opamp Biquad) Finite Gain SABs ... Inductance simulation