Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
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
Review from last time
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
Review from last time
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
Review from last time
Switched-Resistor Voltage Mode Integrators
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
Review from last time
Switched-Resistor Voltage Mode Integrators
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
Review from last time
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
Zero Second Derivative Inverting
V1
A s = - RCs
Are there other integrators in the basic
classes 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
Zero Sensitivity Noninverting
V1
A s = RCs
Are there other integrators in the basic
classes that have been considered?
VIN
R
C
VOUT
R1
R1
V1
A s = RCsMiller Noninverting
V2
A s = RCs
Zero Sensitivity Noninverting
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
Howland Current Source
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
Howland Current Source
GL
OUT IN 1I V G
INX
1
V 1V =
R sC
VIN
VX
IOUT
R1
R2
R11
R22
Howland Current Source
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
Howland Current Source
GLOUT IN 1I V G
1 22
V1 11
R1A s = 1+
s R C R
VIN
VX
IOUT
R1
R2
R1
R2
Howland Current Source
C
VOUT
De Boo
Integrator
VIN
VX
IOUT
R1
R2
R1
R2
Howland Current Source
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?
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:
How can the performance of an integrator be
characterized and how can integrators be compared?
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
How can the performance of an integrator be
characterized and how can integrators be compared?
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
How can the performance of an integrator be
characterized and how can integrators be compared?
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
How can the performance of an integrator be
characterized and how can integrators be compared?
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
Consider Miller Inverting Integrator
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
Consider Miller Inverting Integrator
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:
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.
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:
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.
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:
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.
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
Zero Second Derivative Inverting
V1
A s = - RCs
VIN
VOUTR
C
RB
RA
Zero Second Derivative Inverting
V1
A s = - RCs
QINT= -A
QINT= -A2
QINT= -A2
(stability problems)
What are the integrator Q factors for other
integrators that have been considered?
VIN
VOUTR
C
V1
A s = - RCs
Miller Inverting
QINT= -A
VIN
VX
IOUT
R1
R2
R1
R2
Howland Current Source
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
What are the integrator Q factors for other
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
Zero Sensitivity Noninverting
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