– 1 – Advanced Analog IC DesignProfessor Y. Chiu EECT 7326Fall 2013 Active Filters.

– 1 –

Advanced Analog IC DesignProfessor Y. Chiu

EECT 7326Fall 2013

Active Filters

Introduction

– 2 –

EECT 7326Fall 2013

Filtering: most common linear time-invariant (LTI) signal processing function – selecting the signal bandwidth of interest (in reality neither linear nor time-invariant)…

Categories: continuous time (CT), discrete time (DT)

analog filter, digital filter

(will focus on CT analog filters for this course)

Frequency domain: low-pass (LP), high-pass (HP), band-pass (BP)…

Time domain: impulse response, FIR, IIR…

Simple RC Filter

– 3 –

EECT 7326Fall 2013

1V 1sCH(s)= s = =

1V 1+sRCR +sC

CVi Vo

1s =-ω =-

Continuous-time, 1st-order, one real pole, low-pass

s-plane

0sp=-ω0

Simple RC Filter

– 4 –

EECT 7326Fall 2013

1/τe-t/τ

1h(t)= e for t≥0,

τ=RC, ω =RC

|H(jω)|

-20dB/dec-6dB/oct

ÐH(jω)

Frequency response Impulse response

General Filter Specs

– 5 –

EECT 7326Fall 2013

More realistic:• Magnitude response

ωp, ωs, αp, and αs

• Phase response

Ideal LPF:• Non-causal• Infinite complexity

|H(jω)|

ω0 ωp

αp - passband ripple (< 1dB)

αs - stopband attenuation

Transition band

Passband

Stopband

Example: 2nd-order VTF

– 6 –

EECT 7326Fall 2013

V bH(s)= s =

V s +as+b

• Continuous-time

• Where are the poles? (complex conjugate poles for maximum flatness)

• Low-pass, high-pass, or band-pass?

Passive RLC Filter

– 7 –

EECT 7326Fall 2013

CVi Vo

1LCH(s)=R 1

s +s +L LC

• No active component (low power)

• Inductors are bulky and expensive to realize in IC’s

• Values of R, L, and C will not track each other

Active OP-RC Filter

– 8 –

EECT 7326Fall 2013

• It’s active• Inductor-less• Area efficient• Values of R’s and

C’s and their time co.’s won’t track each other – may lead to RC time constant variations of as high as 20%

• RC time constant enters the VTF in product form – can be tuned for accuracy

1 3 A B

4 B 2 3 A B

1R R C C

H(s)=-1 1

s +s +R C R R C C

Assuming ideal op amps,

Continuous-Time Integrator

– 9 –

EECT 7326Fall 2013

o in1 2 -∞

1v t =- v τ dτ

V 1 1H s = s =-

V s R C

Assuming ideal op amp,

– 10 –

EECT 7326Fall 2013

Cascade Filter Design‒ Biquads

Cascade Filter Design

– 11 –

EECT 7326Fall 2013

bq1 bq2 sH s =H s H s H s

s-plane

For a real-coefficient H(s):

2nd-order 1st-order

22 1 0

bq2 20

K s +K s+KH s =-

ωs + s+ω

Biquad:

The leading minus sign in Hbq(s) is only for convenience

Special Cases of Biquad

– 12 –

EECT 7326Fall 2013

2 11) If K =K =0 ⇒ LPF

22 1 0

bq2 20

K s +K s+KH s =-

ωs + s+ω

0 12) If K =K =0 ⇒ HPF

0 23) If K =K =0 ⇒ BPF

14) If K =0 ⇒ BSF

Q Factor of Poles

– 13 –

EECT 7326Fall 2013

2 20 p p pω =s = σ +ω

s-plane

s ωQ= =

2 σ 2 σ

ω1= 1+

Signal Flow Graph (SFG) for Biquad

– 14 –

EECT 7326Fall 2013

2o 2 1 0

bq2 20i

V K s +K s+KH s = =-

ωV s + s+ωQ

0o 1 2 i o 0 m

0m i 0 o

ω1V =- K +K s V + V -ω V

K1V =- V +ω V

Note: the partition of the biquadratic VTF is not unique

-1/s -1/sVi Vo1 1 1 1 1

1 -ω0K0/ω0

K1+K2s

OP-RC Implementation

– 15 –

EECT 7326Fall 2013

ω0/K0

-1/ω0

22 1 0

bq2 20

K s +K s+KH s =-

ωs + s+ω

Alternative SFG for Biquad

– 16 –

EECT 7326Fall 2013

2o 2 1 0

bq2 20i

V K s +K s+KH s = =-

ωV s + s+ωQ

o 2 i 0 m

0 1m i 0 o

1V =- K sV -ω V

K K1 sV =- + s V + ω + V

s ω ω Q

-1/s -1/sVi Vo1 1 1 1 1

1 -ω0K0/ω0+sK1/ω0

ω0+s/Q

Recast Hbq(s):

Alternative OP-R-C Prototype

– 17 –

EECT 7326Fall 2013

22 1 0

bq2 20

K s +K s+KH s =-

ωs + s+ω

ω0/K0

-1/ω0

K1/ω0

Cascade Filter Design

– 18 –

EECT 7326Fall 2013

bq1 bq2 sH s =H s H s H s

s-plane

For a real-coefficient H(s):

2nd-order 1st-order

• Order of cascade determines the signal dynamic range

• Optimized using engineering rule of thumb or thru simulation

1st-orderSection

Biquad 1(ω01, Q1)

Biquad 2(ω02, Q2)

Biquad Cascade Filter Design

– 19 –

EECT 7326Fall 2013

• Most flexible arrangement of cascade filter design

• Allow independent, non-interacting control of (ω0, Q) for

pole pairs

• Easy design

• Components need to be scaled for maximum DR and minimum component spread

• Pass-band sensitivity to capacitance variation is finite

→ Ladder filter can achieve zero sensitivity

– 20 –

EECT 7326Fall 2013

Scaling of Active Filter

Typical Active Multi-Stage Filters

– 21 –

EECT 7326Fall 2013

Initial component values may not be optimal...

1st-orderSection

Biquad 1(ω01, Q1)

Biquad 2(ω02, Q2)

Vin Vout

Freq. Response of Internal Nodes

– 22 –

EECT 7326Fall 2013

• Internal signal swings need to be large to max SNR

• But not too large such that op amps saturate (producing distortion)

0 ωi ωj

Vo,max of op amps

|H(jω)|

DR Scaling of ith Integrator (Vi)

– 23 –

EECT 7326Fall 2013

• First find out the peak value of Vi(ω), mostly done with simulation

• Then find out the ratio ki = Vi,peak/Vo,max

• Multiply all capacitors connecting at Vi by ki: Fi → Fi*ki, Sij → Sij*ki, …

• Divide all resistors connecting at Vi by ki: Fi → Fi*ki, Sij → Sij*ki, …

• Repeat for all internal nodes…

Vi AjSik

Vin Vout

After DR Scaling

– 24 –

EECT 7326Fall 2013

Max internal signal swings all line up to Vo,max

0 ωi ωj

V1 Vi VjVo,max of op amps

|H(jω)|

Scaling for Min. Component Spread

– 25 –

EECT 7326Fall 2013

• Find out the smallest cap/res connected to Xi – the summing node of Ai

• determine the optimum scaling factor mi to minimize spread

• Multiply all capacitors connected to Xi by mi: Fi → Fi*mi, Sji → Sji*mi, …

• Divide all resistors connected to Xi by mi: Fi → Fi*mi, Sji → Sji*mi, …• Repeat for all integrators…

Vi AjSik

Vin Vout

Scaling of Active Filter

– 26 –

EECT 7326Fall 2013

• DR and min spread scaling do not take op-amp loading into

account – lots of work if individual op amps are sized to meet

the settling time constraint

• Upon the completion of scaling, simulation needs to be performed on the resulting filter to find out the overall SNR

• If SNR is lower than the spec, capacitors and op amps need to be scaled up and resistors scaled down to meet the SNR spec (think about how integrated output noise behaves)

– 27 –

EECT 7326Fall 2013

Ladder Filter Design

Motivation

– 28 –

EECT 7326Fall 2013

• Cascade filter design– Sensitive to component variations, especially high-Q poles

• Ladder filter design– Achieves zero sensitivity to component variations– Discrete CT LC filters with very high-Q poles are built with ladder

structures over the years

Ladder Filter

– 29 –

EECT 7326Fall 2013

• Doubly terminated reactance two-port network

• Delivers the optimum power matching in the passband

• ∂|Vout|/∂Zi = 0 for all L’s and C’s → low sensitivity

V1 V3 V5

Reactance two-port

State Space of Ladder Filter

– 30 –

EECT 7326Fall 2013

Pick –V1, -I2, V3 as

the state variables

for synthesis

in 11 2

2 1 32

V -V1-V =- -I (1)

1-I =- V -V (2)

V1V =- -I (3)

Signal Flow Graph (SFG)

– 31 –

EECT 7326Fall 2013

1/RS 1/RL

-V1 V3

-I21/RS 1

in 11 2

2 1 32

V -V1-V =- -I (1)

1-I =- V -V (2)

V1V =- -I (3)

CT OP-RC Ladder Filter

– 32 –

EECT 7326Fall 2013

C1RS C3 RLL2

-V1 V3

-I2Vin

• Three free state variables → three op amps

• A.k.a the leapfrog ladder structure

Transmission Zeros

– 33 –

EECT 7326Fall 2013

Elliptic LPF

Transmission Zeros

– 34 –

EECT 7326Fall 2013

C 2 3 1 2 3 2 1

C 2 1 3 2 1 2 3

I into node 1 =sC V -V =sC V +sC -V

I into node 3 =sC V -V =sC V +sC -V

VoutC2 C2

sC2·V3 sC2·V1

IC2 IC2'

Modified SFG with Derivatives

– 35 –

EECT 7326Fall 2013

-V1 V3

-1s(C1+C2)

-1s(C2+C3)

1/RS 1/RL

Vin1/RS

OP-RC Ladder Filter w/ Derivatives

– 36 –

EECT 7326Fall 2013

• Derivative input paths implemented with capacitors

C1+C2RS C2+C3 RLL2

-V1 V3

-I2Vin

– 37 –

EECT 7326Fall 2013

Other Active Filters

Tow-Thomas Biquad

– 38 –

EECT 7326Fall 2013

[1] P. E. Fleischer and J. Tow, "Design formulas for biquad active filters using three operational amplifiers,“ Proceedings of the IEEE, vol. 61, pp. 662-3, issue 5, 1973.

• Low sensitivity• Non-interactive

tuning property

Design Equations for Tow-Thomas

– 39 –

EECT 7326Fall 2013

ms +cs+dH s =

s +as+b

28 8 1 8 8

6 1 1 6 4 7 3 5 7 1 2out

1 1 2 3 1 2 7

R R R R R1s + - s+

R R C R R R R R R C CVs =-

R1 1V s + s+R C R R C C R

Note: C1, C2, k1, k2, R8 are free parameters

1 2 3 41 1 2 2 12 1

15 6 8 7 2 8

k1 1 1 1 1R = ,R = ,R = ,R = ,

aC k k k ma-c CbC bC

k b 1R = ,R = R ,R =k R .

Sallen-Key LPF

– 40 –

EECT 7326Fall 2013

• OP-RC active filters are ideally insensitive to bottom-plate stray caps

• Sallen-Key is sensitive to bottom-plate parasitics at node A and B

Vin Vout

Design Equations for SK LPF

– 41 –

EECT 7326Fall 2013

1 1 2 2

1 1 2 1 2 2

1ω = ,

R C R C

ωQ = ,

1 1 1-K+ +

R C R C R C

G ωH s =

ωs + s+ω

Vin Vout

Sallen-Key BPF

– 42 –

EECT 7326Fall 2013

Still sensitive to parasitic capacitance at node A and B

Vin Vout

Design Equations for SK BPF

– 43 –

EECT 7326Fall 2013

03 1 1 2 2

1 1 3 1 3 2 2 1

R +Rω = ,

R R C R C

ωQ = ,

1 1 1 1-K+ + +

R C R C R C R C

G = .1 1 1 1-K

+ + +R C R C R C R C

QH s =ω

s + s+ωQ

Vin Vout

– 44 –

EECT 7326Fall 2013

MOSFET-C Active Filter

MOSFET Resistor

– 45 –

EECT 7326Fall 2013

• MOSFET in triode region is a variable resistor

• Compact, low parasitics compared to large-value resistors

VDS=VGS-Vth

MOSFET Resistor

– 46 –

EECT 7326Fall 2013

But the large-signal response is quite nonlinear

2DS ox GS th DS DS

W 1Intrioderegion, I =μC V -V V - V

DSox GS th DS

ox GS th DS

∂I1 WSmallsignal, = =μC V -V -V

R ∂V L

W=μC V -V for V =0

A Linear (Diff.) MOSFET Resistor

– 47 –

EECT 7326Fall 2013

MOSFET resistor is linear when driven by balanced differential signals!

2DS ox GS th DS DS

ox G th 2 1 2 1 2

2iox G th ic i i

ox G th ic i

W 1Intrioderegion, I =μC V -V V - V

W 1=μC V -V -V V -V - V -V

VW 1=μC V -V -V + V - V

W=μC V -V -V V

i i1 ic 2 ic

V VV =V + , V =V -

Rudell VGA + Mixer

– 48 –

EECT 7326Fall 2013

[2] J. C. Rudell et al., “A 1.9-GHz wide-band IF double conversion CMOS receiver for cordless telephone applications,” IEEE Journal of Solid-State Circuits, vol. 32, pp. 2071-88, issue 12, 1997.

• M9-M15 comprisethe CMFB circuit

• Gain adjustmentby varying IGain

MOSFET-C Integrator

– 49 –

EECT 7326Fall 2013

• Sources of M1 and M2 are ideally always equal-potential

• Fully differential circuit rejects the 2nd-order harmonic (and all even-order distortions)

• Triode resistance significantly depends on process (threshold, mobility, etc.), temperature, and VDD → Filter response needs tuning

Vo-Vi-

– 50 –

EECT 7326Fall 2013

Frequency Tuning

Master-Slave Tuning

– 51 –

EECT 7326Fall 2013

• M1-M1' and C-C' are

matched devices• Master sets M1-C time

constant to an off-chipreference thru f/b

• Slave integrator timeconstant follows that ofthe master

• Subject to device mismatch between the master and slave

Active filter (slave)

Tuning circuit (master)

… …

f/bcontrol

Req Matching

– 52 –

EECT 7326Fall 2013

[3] R. Geiger, P. Allen, and N. Dinh, “Switched-resistor filters - A continuous time approach to monolithic MOS filter design,” IEEE Transactions on Circuits and Systems, vol. 29, pp. 306-315, issue 5, 1982.

VΔQ =VC = T ⇒ RC =T

Req Matching

– 53 –

EECT 7326Fall 2013

Ф1 Ф1Ф2 Ф2

• Charge from R is continuous but discrete from Cr

• VC(t) should be sampled at the end of Ф2 and held before being applied to the MOSFET gate

• LPF can be used instead of ZOH, but error will be introduced

VΔQ =VC = T ⇒ RC =T

Phase Locking

– 54 –

EECT 7326Fall 2013

[4] K. R. Rao, et al., “A novel ‘follow the master’ filter,” Proceedings of the IEEE, vol. 65, pp. 1725-6, issue 12, 1977.

[5] R. Geiger, P. Allen, and N. Dinh, “Switched-resistor filters - A continuous time approach to monolithic MOS filter design,” IEEE Transactions on Circuits and Systems, vol. 29, pp. 306-315, issue 5, 1982.

Phaseshift

Phase Locking

– 55 –

EECT 7326Fall 2013

• Phase detector converts the phase difference b/t V1 and V2 into a pulse with bipolar average <Vp>

• The trailing integrator keeps integrating when <Vp> ≠ 0 holds

Vo1,Vo2

Phase Locking

– 56 –

EECT 7326Fall 2013

• <Vp> = 0 holds for Δφ = 90º, -90º, … In steady state, V2 leads V1 by 90º

• Negative f/b sets the pole freq. of the HPF formed by C & R to ωref

Δφ90o-90o

V jωRC=

V 1+jωRC

Δφ=90o

Δφ=45o

Δφ=135o

1⇒ RC =

R Δφ = 90º

CT Ladder Filter Tuning

– 57 –

EECT 7326Fall 2013

• 5th-order Chebyshev-I all-pole continuous-time filter• Integrators are realized by Gm-C active structures

RLVin Vout∫

∫∫ ∫

Gm-C Active Integrator

– 58 –

EECT 7326Fall 2013

VC IC Cm

IiG = =

V G 1=-

• Differential input with programmable gain constant Gm/C

Frequency Locking

– 59 –

EECT 7326Fall 2013

[6] K.-S. Tan and P. R. Gray, "Fully integrated analog filters using bipolar-JFET technology," IEEE Journal of Solid-State Circuits, vol. SC13, pp. 814-821, issue 6, 1978.

osc 1 2ω = ω ω

PD ∫∫

-1fref VC

Active filter (slave)

∫ ∫

– 60 –

EECT 7326Fall 2013

1 1 2 11 0 2 1 2

ω ω R -RV =- V -V =- V -V

s s R +R

ωV =- V

V0 V1 V2

1 2 1 1

1 21 2 2 11 1 2

2 1 22

ω R -R ω1+ -

Vs R +R s R -R=0 ⇒ Δ =s + ω s+ω ω =0

V R +Rω1

1 2 osc 1 2

1 21 2 1 1 2

s =ω ω ⇒ ω = ω ω ,

R -Rs +s = ω ≥0 ⇒ R ≥R

– 1 – Advanced Analog IC DesignProfessor Y. Chiu EECT 7326Fall 2013 Active Filters.

chiu eect

f i f i

max slide

sensitivity slide

simulation slide

scaling of active filter

convenience slide

iir slide

Documents

Chiu Cervical

Circuit Analysis Final Project KEITH PARKER EECT 111-51C.

Carol Chiu Resume

Bobby chiu

7 Things You’ll Discover About STRESS · to truly...

– 1 – Data ConvertersOversampling ADCProfessor Y. Chiu.....

EECT-111 Presentation By:

Chiu Evolving

Advanced Digital Calibration Techniques for Data...

Queenie Chiu Portfolio

Using Technology: Beliefs and Suggestions English Teacher,.....

By: Shannon Matuszny EECT 111. Thomas Edison and his family.

Mickey Chiu

Maquetas incaicas en Chiu-Chiu: paisaje y ritualidad agraria...

– 1 – Data ConvertersDACProfessor Y. Chiu EECT 7327Fall....

Magneto-optical Kerr Eect Study of Magnetic Anisotropy in...