ee 508 lecture 4 - class.ece.iastate.educlass.ece.iastate.edu/ee508/lectures/ee 508 lect 29 fall...

EE 508

Lecture 29

Integrator Design

Current-Mode Integrators

Metrics for comparing integrators

Anti-aliasing filter often required to limit frequency content at input to SC filters

INV t INFILTV t OUTV kTContinuous-Time

FilterDiscrete-Time

Filter

Anti-aliasing FilterSwitched-

Capacitor Filter

CLKf

Signal Band

Alias Frequencies

f2

CLKf

Why not just make the clock frequency >> signal band edge ?

1POLES CLK

1 Cf f

CRC

Recall in the continuous-time RC-SC counterparts

Since fPOLES will be in the signal band (that is why we are building a filter) large

fCLK will require large capacitor ratios if fCLK>>fPOLES

• Large capacitor ratios not attractive on silicon (area and matching issues)

• High fCLK creates need for high GB in the op amps (area,power, and noise increase)

Often fCLK/fPOLES in the 10:1 range proves useful (20:1 to 5:1 typical)

Review from last time

VIN1

C1φ1

φ2

φ2

φ1

Noninverting Input

VIN2

C1φ1

φ2

φ2

φ1

Noninverting Input

VIN1

C1φ1

φ2

Noninverting Input

VIN2

C1φ1

φ2

φ2

φ1

Noninverting Input

Elimination of Redundant Switches

Redundant Switches

Switched-Capacitor Input

with Redundant Switches

Switched-Capacitor Input with

Redundant Switches Removed

Although developed from the concept of SC-resistor equivalence, SC circuits

often have no Resistor-Capacitor equivalents


Switched Capacitor Amplifiers

VOUTVIN

C1φ1

φ2

C2

φ2

VOUTVIN

C1φ1

φ2

φ2

C2φ1

• Summing, Differencing, Inverting, and Noninverting SC Amplifiers Widely Used

• Significant reduction in switches from what we started with by eliminating C in

SC integrator

• Must be stray insensitive in most applications


Switched-Resistor Voltage Mode Integrators

CX

M1

P F

Filt

er

Pre

ch

arg

e

RF

ET

There are some modest nonlinearities in this MOSFET when operating in the triode region

• Significant improvement in linearity by cross-coupling a pair of triode region resistors

• Perfectly cancels nonlinearities if square law model is valid for M1 and M2

• Only modest additional complexity in the Precharge circuit

CX

M1

P F

CX

M2

Pre

ch

arg

e

Filt

er

RF

ET



Integrator Output with Perfect

Complimentary Clocks

Integrator Output with

Nonoverlapping Clocks

Integrator Output with

Overlapping Clocks

P

F

F

P

• Aberrations are very small, occur very infrequently, and are further filtered

• Play almost no role on performance of integrator or filter



Switched-resistor integrator

• Accurate CRFET products is possible

• Area reduced compared to Active RC structure because RFET small

• Single pretune circuit can be used to “calibrate” large number of resistors

• Clock frequency not fast and not critical (but accuracy of fREF is important)

• Since resistors are memoryless elements, no transients associated with switching

• Since filter is a feedback structure, speed limited by BW of op amp

VIN VOUTRFET

C

RF

ET

CREF

Pretune Circuit

fREF


Voltage Mode Integrators

• Active RC (Feedback-based)

• MOSFET-C (Feedback-based)

• OTA-C

• TA-C

• Switched Capacitor

• Switched Resistor

Sometimes termed “current mode”

Will discuss later

• Other Structures Have introduced a basic voltage-mode integrators in each of these approaches

Performance of all are limited by variability or Op Amp BW

All of these structures have applications where they are useful

How can integrator performance be improved?

• Better op amps

• Better Integrator Architectures

How can the performance of integrator

structures be compared?

Need metric for comparing integrator performance

Are there other integrators in the basic

classes that have been considered?

VIN

VOUTR

C

VIN

R

C

V1

A s = - RCs

V1

A s = - RCs

High-Q Inverting

V1

A s = - RCs

VIN

VOUTR

C

Zero Sensitivity Inverting

V1

A s = - RCs

VIN

VOUTR

C

Cascaded Inverting

Miller Inverting

VIN

VOUTR

C

Zero Second Derivative Inverting

V1

A s = - RCs

VIN

VOUTR

C

RB

RA


V1

A s = - RCs



VIN

R

C

VOUT

R1

R1

V1

A s = RCs

V1

A s = RCs

VIN

R

C

VOUT

R1

R1

Modified Miller Noninverting

VIN

R

C

VOUT

R1

R1 V1

A s = RCs

Phase Lead Integrator

VIN

R

C

VOUT

R1

R1

V1

A s = RCs

High-Q NonInverting

Miller Noninverting

VIN

R

C

VOUT

R1

R1

Modified High-Q Noninverting

V1

A s = RCs

VIN

R

C

VOUT

R1

R1

Zero Sensitivity Noninverting

V1

A s = RCs



VIN

R

C

VOUT

R1

R1

V1

A s = RCsMiller Noninverting

V2

A s = RCs


VIN RC

VOUTR1

R4

R3

R

R2

If R1=R2 and R3=R4

VOUT

VIN

C

C

R

R

V1

A s = RCs

Balanced Time Constant Noninverting

De Boo Integrator

VIN

VX

IOUT

R1

R2

R3

R4

Howland Current Source

GL

2 3OUT IN 1 X 1

4

G GI V G V -G

G

2 31

4

G GG

G

OUT IN 1I V G

If resistors sized so that

If sizing constraints are satisfied, behaves as a

grounded constant-current source

Consider the Howland Current Source

DeBoo Integrator

VIN

VX

IOUT

R1

R2

R1

R2


GLOUT IN 1I V G

CIX

VX

Observe that if a current source drives

a grounded capacitor, then the nodal voltage

on the capacitor is given by

X X1

V =IsC

Thus, if we could make IX proportional to VIN, the

voltage on the capacitor would be a weighted

Integral of VIN

De Boo Integrator

VIN

VX

IOUT

R1

R2

R1

R2


GL

OUT IN 1I V G

INX

1

V 1V =

R sC

VIN

VX

IOUT

R1

R2

R11

R22


C

VOUT

22 IN 22OUT X

11 1 11

R V R1V =V 1+ = 1+

R R sC R

1 22

V1 11

R1A s = 1+

s R C R

If R1=R11 and R2=R22

De Boo Integrator

VIN

VX

IOUT

R1

R2

R1

R2


GLOUT IN 1I V G

1 22

V1 11

R1A s = 1+

s R C R

VIN

VX

IOUT

R1

R2

R1

R2


C

VOUT

De Boo

Integrator

VIN

VX

IOUT

R1

R2

R1

R2


C

VOUT

RA

RB

De Boo

Integrator

1 B 22V

A 1 11

R R1A s = - 1+

s R R C R

Many different integrator architectures that ideally

provide the same gain

Similar observations can be made for other classes of

integrators

How can the performance of an integrator be

characterized and how can integrators be compared?



0V

IA s =

s

Consider Ideal Integrator Gain Function

• Magnitude of the gain at I0=1

• Phase of integrator always 90o

• Gain decreases with 1/ω

0V

IA jω =

jω

Key property of ideal integrator is a phase shift of 90o at frequencies around I0!

Ideal Integrator

Im

Re

Key characteristics of an ideal integrator:

Are any of these properties more critical than others?

Consider a nonideal integrator Gain Function

01V OO

IA s = A s

s+

Im

Re

Nonideal Integrator

In many applications:



Is stability of an integrator of concern?

Ideal Integrator Im

Re

• Ideal integrator is not stable

• Integrator function is inherently ill-conditioned

• Integrator is almost never used open-loop

• Stability of integrator not of concern, stability of filter using integrator is of concern

• Some integrators may cause unstable filters, others may result in stable filters



Express AV(jω) as

where R(ω) and X(ω) are real and represent the real and imaginary parts of

the denominator respectively

V

1A jω =

R ω +jX ω

Ideally R(ω)=0

Definition: The Integrator Q factor is the ratio of the imaginary part of the

denominator to the real part of the denominator

1tan

X ωPhase

R ω

INTX ω

QR ω

Typically most interested in QINT at the nominal unity gain frequency

of the integrator



Express AV(jω) as

V1

I jω =R ω +jX ω

For Phase Lag Integrators, R(ω) is negative

For Phase Lead integrators, R(ω) is positive

Re

Im

1

Phase Lag

Integrator

I(jω0)Ideal Phase

IntegratorPhase Lead

Integrator

Re

Im

1

I(jω0)

Re

Im

1

I(jω0)

Lead/Lag Characteristics for Inverting Integrators



V

1I jω =

R ω +jX ω

For Phase Lag Integrators, R(ω) and X(ω) have opposite signs. For Phase Lead integrators,

R(ω) and X(ω) have the same sign.

Re

Im

1

Phase Lag

Integrator

I(jω0)Ideal Phase

IntegratorPhase Lead

Integrator

Re

Im

1

I(jω0)

Re

Im

1

I(jω0)

Lead/Lag Characteristics for Inverting Integrators

Lead/Lag Characteristics for Noninverting Integrators

ReIm

1

Phase Lag

Integrator

I(jω0)

ReIm

1

Ideal Phase

Integrator

I(jω0)

ReIm

1

Phase Lead

IntegratorI(jω0)

V

-1I jω =

R ω +jX ω

For Phase Lag Integrators, R(ω) and X(ω) have opposite signs. For Phase Lead integrators,

R(ω) and X(ω) have the same sign.

Phase shift ideally 90o

Phase shift ideally 270o

Integrator Q Factor

Consider Miller Inverting Integrator

VIN

VOUTR

C

V-1

A s =RCs+ s 1+RCs

V

-1A s =

RCjω+ jω 1+RCjω

V 2

-1A s =

- ω RC+j ω RC+

V 2n n

-1A s =

- ω +j ω 1+n n

Normalizing by ωn= ωRC and τn = τ/RC = I0n/GB

Observe this integrator has excess phase shift (more than 90o in the denominator) at all

frequencies

1

A ss

Integrator Q Factor

V

1A jω =

R ω +jX ω

VIN

VOUTR

C

V 2

n n

-1A s =

- ω +j ω 1+n n

INTX ω

QR ω

1n nINT 2

n nn n

ω 1+ GBQ = - -A

ω ωω


Since the phase is less than 90o, the Miller Inverting Integrator is a Phase Lag

Integrator and QINT is negative

Re

Im

ω0

Phase Lag

Integrator

I(jω0)

1

A ss

Integrator Pole Locations


VIN

VOUTR

C

V

-1A s =

RCs+ s 1+RCs

Poles at s=0 and s = -I0(1+GB/I0) ; -GB

I0=1/(RC)

Im

Re

-GB

Is the integrator Q factors simply a metric or does it

have some other significance?

VOUT

R0

R1RQ

R4

R3R2

C1 C2

RARB RC

RF

VOLP

VOBP

VIN

Two-Integrator Loop

Summer

INT1 INT2

Tow Thomas Biquad

1P

PN INT1 INT2

Q1 1 1

+ +Q Q Q

It can be shown that the pole Q for the TT Biquad can be approximated by

where QINT1 and QINT2 are evaluated at ω=ωo

Is the integrator Q factors simply a metric or does it

have some other significance?

VOUT

R0

R1RQ

R4

R3R2

C1 C2

RARB RC

RF

VOLP

VOBP

VIN

Two-Integrator Loop

Summer

INT1 INT2

Tow Thomas Biquad

1P

PN INT1 0 INT2 0

Q1 1 1

+ +Q Q ω Q ω

It can be shown that the pole Q for the TT Biquad can be approximated by

Similar expressions for other second-order biquads

Observe that the integrator Q factors adversely affect the pole Q of the filter

Observe that if QINT1 and QINT2 are of opposite signs and equal magnitudes,

nonideal effects of integrator can cancel

What can be done to correct the

phase problems of an integrator?

VIN

VOUTR

C RX

VINVOUTR

CCX

VIN

VOUTR

C

One thing that can help the Miller Integrator is phase-lead compensation

RX and CX will be small components

V

-1A s =

RCs+ s 1+RCs

x

Vx

- 1+R CsA s =

RCs+ s 1+[R+R ]Cs

x

Vx

- 1+RC sA s =

RCs+ s 1+R[C+C ]s

Rx and Cx will add phase-lead by introduction of a zero

Integrator Q Factor

Consider Miller Noninverting Integrator

1 2V

-1 -1A s =

RCs+ s 1+RCs s

V 2

1A jω =

-3 RCω +j ωRC(1+2 ω) VIN

R

C

VOUT

R1

R1

V 2

1A jω

-3 RCω +jωRC

1INT 2

ωRCQ

3 ω-3 RCω

INT-1 GB 1

Q A jω3 ω 3

Observe this integrator has excess phase shift (more than 90o in the denominator) at all

frequencies

Note: The Miller Noninverting Integrator has a modestly poorer QINT than the

Miller Inverting Integrator

Example:

If f0=10KHz, GB=1MHz, QNOM=10, estimate the pole Q for the Tow-Thomas

Biquad if the Miller Integrator and the Miller Noninverting Integrators are used.

Also determine the relative degradation in performance due to each of the integrators.

VOUT

R0

R1RQ

R4

R3R2

C1 C2

RARB RC

RF

VOLP

VOBP

VIN

Two-Integrator Loop

Summer

INT1 INT2

Tow Thomas Biquad

1P

PN INT1 INT2

Q1 1 1

+ +Q Q Q

Example:




1P

PN INT1 INT2

Q1 1 1

+ +Q Q Q

100INT1GB 1MHz

Q -A=ω 10KHz

INT21 -1 GB 1MHz

Q A jω - =-333 3 ω 3•10KHz

1PQ 17.5

1-.01-.033

10

Note that 3 times as much of the shift is due to the noninverting integrator

as is due to the inverting integrator!

Note the nonideal integrators cause about a 75% shift in QP

Similar effects of the integrators will be seen on other filter structures

Example:




1P

PN INT1 INT2

Q1 1 1

+ +Q Q Q

100INT1GB 1MHz

Q -A=ω 10KHz

INT21 -1 GB 1MHz

Q A jω - =-333 3 ω 3•10KHz

1PQ 17.5

1-.01-.033

10

How can the problem be solved?

1. Compensate Integrator

2. Use better integrators

3. Use phase-lead and phase/lag pairs

Example:




1P

PN INT1 INT2

Q1 1 1

+ +Q Q Q

100INT1Q INT2Q -33

How can the problem be solved?

VIN

VOUTR

C RX

Phase Compensation of INT1

Pick Rx so that QINT1=33 at ω=1/(RC)

x

Vx

- 1+R CsA s =

RCs+ s 1+[R+R ]Cs

INT1X

GB-ωQ

1-GB CR

Solving, obtain CRx=4/GB

Useful for hand calibration but not practical for volume production because of

variability in components

What are the integrator Q factors for other

integrators that have been considered?

VIN

VOUTR

C

VIN

R

C

V1

A s = - RCs

V1

A s = - RCs

High-Q Inverting

V1

A s = - RCs

VIN

VOUTR

C

Zero Sensitivity Inverting

V1

A s = - RCs

VIN

VOUTR

C

Cascaded Inverting

Miller Inverting

VIN

VOUTR

C


V1

A s = - RCs

VIN

VOUTR

C

RB

RA


V1

A s = - RCs

QINT= -A

QINT= -A2

QINT= -A2

(stability problems)


integrators that have been considered?

VIN

VOUTR

C

V1

A s = - RCs

Miller Inverting

QINT= -A

VIN

VX

IOUT

R1

R2

R1

R2


CX

VOUT

De Boo

Integrator

V2

A s =RCsIf R1=R2=R

2

1V

2 11 1 1

1 2

R1+

RA s =

R RR Cs+ s 1+ 1+ +sCR

R R

INT2

1

AQ

R1+

R

INTA

Q2


integrators that have been considered? VIN

R

C

VOUT

R1

R1

V1

A s = RCs

V1

A s = RCs

VIN

R

C

VOUT

R1

R1

Modified Miller Noninverting

VIN

R

C

VOUT

R1

R1 V1

A s = RCs

Phase Lead Integrator

VIN

R

C

VOUT

R1

R1

V1

A s = RCs

High-Q NonInverting

Miller Noninverting

VIN

R

C

VOUT

R1

R1

Modified High-Q Noninverting

V1

A s = RCs

VIN

R

C

VOUT

R1

R1


V1

A s = RCs

INT1

Q A jω3

INTQ A jω

INTQ A jω

3

INTQ A jω

Improving Integrator Performance:

1. Compensate Integrator

2. Use better integrators

3. Use phase-lead and phase/lag pairs

• These methods all provide some improvements in integrator performance

• But both magnitude and phase of an integrator are important so focusing only

on integrator Q factor only may only improve performance to a certain level

• In higher-order integrator-based filters, the linearity in 1/ω of the integrator

gain is also important. The integrator magnitude and Q factor at ω0 ignore

the frequency nonlinearity that may occur in the 1/ω dependence

• There is little in the literature on improving the performance of integrated

integrators within a basic class. At high frequencies where the active device

performance degrades, particularly in finer-feature processes, there may be

some benefits that can be derived from architectural modifications along the

line of those discussed in this lecture

ee 508 lecture 4 - class.ece.iastate.educlass.ece.iastate.edu/ee508/lectures/ee 508 lect 29 fall...

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