summary last lecture - university of california, berkeleyee247/fa05/lectures/l6_2_f05.pdf · eecs...

EECS 247 Lecture 6: Filters © 2005 H.K.

EE247 Lecture 6

• Summary last lecture• Continuous-time filters

– Opamp MOSFET-C filters– Opamp MOSFET-RC filters– Gm-C filters

• Frequency tuning for continuous-time filters– Trimming via fuses– Automatic on-chip filter tuning

• Continuous tuning– Master-slave tuning

• Periodic off-line tuning– Systems where filter is followed by ADC & DSP, existing hardware

can be used to periodically update filter freq. response

EECS 247 Lecture 6: Filters © 2005 H.K. Page 2

Summary Last Lecture

• Continuous-time filters– Effect of integrator non-idealities on continuous-

time filter behavior– Facts about monolithic Rs & Cs and its effect on

integrated filter characteristics– Opamp RC filters– Opamp MOSFET-C filters

• Frequency tuning for continuous-time filters– Frequency adjustment by making provisions to

have variable R or C


Use of MOSFETs as ResistorsSingle-Ended Integrator

Problem: Single-ended MOSFET-C Integratorà Effective R non-linearNote that the non-linearity is mainly 2nd order type

( )

( )

( )

2ds

2i

W VCI V V VD ox gs th dsL 2

W VC V V VI ox gs th iD L 2I WD V V VCG gs th ioxV Li

µ

µ

µ

= − −

= − −

∂ − −= =∂

VG

oV

C

-

+

inV

ID

àTunable by varying VG:


Use of MOSFETs as ResistorsDifferential Integrator

Opamp-MOSFET-C

VGC

• Non-linear term cancelled!• Admittance independent of Vi

Problem: Threshold voltage dependence

+

+-

- outV

Vi/2

-Vi/2

( )( ) ( )

W VdsCI VV VD ox dsgs thL 2VVW iiCI V VD1 ox gs thL 24

VVW iiCI V VD2 ox gs thL 24W V VCI I Vgs thD1 D2 ox iL

I I WD1 D2 V VCG gs thoxV Li

µ

µ

µ

µ

µ

= − −

= − −

= − − +

−− =

∂ −−= =

∂

ID1

M1

M2ID2

C


MOSFET-C Integrator

VGC

à Common-mode voltage sensitivity

+

+-

- outV

Vi/2

-Vi/2

Vx

Vx

+

+-

-

Vi/2

-Vi/2

outV0V

0V

•For the Opamp-RC integrator, opamp input stays at 0V (virtual gnd.)

•For the MOSFET-C integrator, opamp input stays at the voltage Vx which is a function of 2nd order MOSFET non-linearities


Use of MOSFET as Resistor Issues

• Distributed nature of gate capacitance & channel resistance results in infinite no. of high-frequency poles à excess phase

• Filter performance mandates well-matched MOSFETs à long channel devices

• Excess phase increases with L2

àTradeoff between matching and integrator QàThis type of filter limited to low frequencies

MOS xtor operating in triode regionCross section view

Distributed channel resistance & gate capacitance


Example:Opamp MOSFET-C Filter

•Suitable for low frequency applications•Issues with linearity•Linearity achieved ~40-50dB•Needs tuning

5th Order Elliptic MOSFET-C LPF with 4kHz Bandwidth

Ref: Y. Tsividis, M.Banu, and J. Khoury, “Continuous-Time MOSFET-C Filters in VLSI”, IEEE Journal of Solid State Circuits Vol. SC-21, No.1 Feb. 1986, pp. 15-30


Improved MOSFET-C IntegratorVG1

C

No threshold dependenceFirst order Common-mode non-linearity cancelledLinearity achieved in the order of 60-70dB

+

+-

- outV

Vi/2

-Vi/2

VG2

ID1

M1

M2

ID2

M3

M4

I D3

ID4

IX1

IX2

( )( )

W VdsCI VV VD ox dsgs thL 2VW V iiCI V VD1 ox gs1 thL 24

VW V iiCI V VD3 ox gs2 thL 24I I IX 1 D1 D3

VW V iiC V Vox gs1 gs2L 22VW V iiCI V VX 2 ox gs2 gs1L 22

W V VCI I Vgs1 gs2X 1 X 2 ox iLI IX 1 X 2G

µ

µ

µ

µ

µ

µ

= − − = − −

= − − +

= +

= − − = − −

−− =

∂ −= ( )W V VC gs1 gs2oxV Li

µ −=∂

C

Ref: Z. Czarnul, “Modification of the Banu-Tsividis Continuous-Time Integrator Structure,” IEEE Transactions on Circuits and Systems, Vol. CAS-33, No. 7, pp. 714-716, July 1986.


R-MOSFET-C IntegratorVG1

C

•Improvement over MOSFET-C by adding resistor in series with MOSFET•Voltage drop primarily across fixed resistor à small MOSFET Vds àimproved linearity & reduced tuning range•Linearity in the order of 90dB possible•Generally low frequency applications

+

+-

- outV

Vi/2

-Vi/2

VG2

M1

M2

M3

M4

C

Ref: U-K Moon, and B-S Song, “Design of a Low-Distortion 22-kHz Fifth Order Bessel Filter,”IEEE Journal of Solid State Circuits, Vol. 28, No. 12, pp. 1254-1264, Dec. 1993.

R

R


R-MOSFET-C Lossy Integrator

VG1

C

Negative feedback around the non-linear MOSFETs improves linearityCompromises frequency response accuracy

+

+-

- outV

Vi/2

-Vi/2

VG2M1

M2

M3

M4

C

Ref: U-K Moon, and B-S Song, “Design of a Low-Distortion 22-kHz Fifth Order Bessel Filter,” IEEE Journal of Solid State Circuits, Vol. 28, No. 12, pp. 1254-1264, Dec. 1993.

R1

R2

R2

R2


Example:Opamp MOSFET-RC Filter

Ref: U-K Moon, and B-S Song, “Design of a Low-Distortion 22-kHz Fifth Order Bessel Filter,” IEEE Journal of Solid State Circuits, Vol. 28, No. 12, pp. 1254-1264, Dec. 1993.

•Suitable for low frequency applications•Significant improvement in linearity compared to MOSFET-C•Needs tuning

5th Order Bessel MOSFET-RC LPF -22kHz bandwidthTHD à-90dB for 4Vp-p 2kHz input signal


Operational Amplifiers (Opamps) versus Operational Transconductance Amplifiers (OTA)

• Low output impedance• Output in the form of voltage • Can drive R-loads• Good for RC filters,

OK for SC filters

• Extra buffer adds complexity, power dissipation

• High output impedance• In the context of filter design called

gm-cells• Output in the form of current • Cannot drive R-loads• Good for SC & gm-C filters• Typically, less complex compared to

opampà higher freq. potential• Typically lower power

Opamp OTAVoltage controlled Voltage controlled

voltage source current source


Integrator ImplementationTransconductance-C & Opamp-Transconductance-C

inV

oVGm

oV

C

inV

Gm

whereo o mo

in

V G

V s C

ωω

−= =

-

+∫

GmC Intg. GmC-OTA Intg.

-

+


Gm-C FiltersSimplest Form of CMOS Gm-C Integrator

• Transconductance element formed by the source-coupled pair

• All MOSFETs operating in saturation region

• Current in M1& M2 can be varied by changing Vcontrol

à Transconductance of M1& M2 varied through Vcontrol

Ref: H. Khorramabadi and P.R. Gray, “High Frequency CMOS continuous-time filters,” IEEE Journal of Solid-State Circuits, Vol.-SC-19, No. 6, pp.939-948, Dec. 1984.

controlV

oV

inV

-

+

+

-

int gCM1 M2

M10


Simplest Form of CMOS Gm-C IntegratorAc Half Circuit

int gC

oV

inV

-

+

+

-

M1 M2

M10 controlV

oV

inV

-

+

+

-

int g2C

M1 M2

M10

int g2C

controlV

inV intg2CM1

oV

AC half circuit



• Use ac half circuit & small signal model to derive transfer function:

M 1,2o m in int g

M 1,2o m

in int g

o o

inM 1,2m

oint g

V g V 2C s

V gV 2C s

VV s

g

2 C

ω

ω

= − × ×

= −

−=

→ =×

inV

intg2C

oVing Vm

CGS

inV intg2CM1

oV

AC half circuit

Small signal model



• MOSFET in saturation region:

• Gm is given by:

( )

( )

( )

2

M 1&M 2m

1 / 2

C Wox V VI gs thd 2 L

I Wd V VCg gs thoxV LgsId2

V Vgs th

1 WC2 Iox d2 L

µ

µ

µ

−=

∂ −= =∂

=−

=

Id varied via Vcontrol à gm tunable via Vcontrol

controlV

oV

inV

-

+

+

-

int gCM1 M2

M10


Gm-C Filters2nd Order Gm-C Filter

• Use the Gm-cell to build a 2nd order bandpass filter

controlV

oV

inV

-

+

+

-

int gCM1 M2

M10


2nd Order Bandpass Filter1 1

*RR *

1

sCR

'1V

2V

inV1− 1

1V

*R

sL

1−

oV

'3V

inV

1*R R 21

sτ11

sτ

∑

oV

--

* *1 2R C L Rτ τ= × =

oV

R CinI L CV

LI

+

- CILV

+

-

+

-RV

RI


2nd Order Integrator-Based Bandpass Filter

inV

11Q−

1sτ1

sτ

BPV

-

∑

BP 22in 1 2 2

* *1 2

*

0 1 2

1 2

1 2 *0

V sV s s 1

R C L R

R R

11

Q 1

Frommatch ing pointof viewdes i rab le :1 RQ

R

L C

β

β

β

ω

ττ τ τ

ω τ τ

τ τ

τ τ

τ τ

=+ +

= × =

=

= =

= ×

= = → =


2nd Order Integrator-Based Bandpass Filter

∑

inV

11Q−

1sτ

1sτ

BPV

-First implement this partWith transfer function:

0in

0

V 1V s 1

Qω

−=+


Terminated Gm-C Integrator

oV

inV

+

-

+

-

M1 M2

M10controlV

M3 M4

M11

int gC

inV intg2CM1

oV

AC half circuit

M3



inV intg2CM1

oV

AC half circuit

M3

oM 3in int g m

M 1 M 1m m

V 1V s 2C g

g g

−=

×+

inV

intg2C

oVinM 1g Vm

CGS

Small signal model

M 3gm

1



oM 3in int g m

M 1 M 1m m

M 1 M 1m m

0 M 3int g m

V 1V s 2C g

g g

g g& Q

2C gω

−=

×+

= =→

inV

intg2C

oVinM 1g Vm

CGS

Small signal model

M 3gm

1

∑

inV

11Q−

1sτ

1sτ

BPV

-

0in

0

V 1V s 1

Qω

−=+

Question: How to define Q accurately?



int gC

controlV

oV

inV

+

-

+

-

M1 M2

M10

M3 M4

M11

1 / 2M 1 M 1M 1m d

M 11 / 2

M 3 M 3M 3m d

M 3

1 / 2M 1M 1m d M 1M 3 M 3 M 3m d

W1 Cg 2 Iox2 L

W1 Cg 2 Iox2 L

Let us assume equal channel lengths for M1, M3 then:

Ig WWg I

µ

µ

=

=

= ×



int gC

controlV

oV

inV

+

-

+

-

M1 M2

M10

M3 M4

M111 / 2

M 1 M 10d dM 3 M 11d d

M 10d M 10M 11

M 11d

Note that:

I I

I I

Assuming equal channel lengths for M10, M11:

I W

I W

M1 Wg Wm M 10 M 1M 3 Wg W M 3m M 11

=

=

=

→

×


2nd Order Gm-C Filter

M 1,2mM 3,4m

gQg

=

• Simple design• Tunable• Q function of device ratios:


Filter Frequency Tuning Techniques

• Component trimming

• Automatic on-chip filter tuning– Continuous tuning

• Master-slave tuning

– Periodic off-line tuning• Systems where filter is followed by ADC & DSP, existing

hardware can be used to periodically update filter freq. response


Example: Tunable Opamp-RC Filter

oV

C-

+

inV

D0

R1 R2 R3 R4

R1

D1D2

R2 R3 R4

Post manufacturing:•Usually at wafer-sort tuning performed

•Measure -3dB frequency•If frequency too high decrement D to D-1•If frequency too low increment D to D+1•If frequency within 10% of the desired corner freq. stop

Not practical to require end-user to tune the filter à Need to fix the adjustment at the factory


Trimming• Component trimming

– Build fuses on-chip• Based on measurements @

wafer-sort blow fuses selectively by applying high current to the fuse– Expensive– Fuse regrowth problems!– Does not account for temp.

variations & aging– Laser trimming

• Trim components or cut fuses by laser– Even more expensive– Does not account for temp.

variations & aging

Fuse not blown à D1=1Fuse blown à D1=0

Fuse

To switch

D1


Example:Tunable/Trimmable Opamp-RC Filter

oV

C-

+inV

R1 R2 R3 R4

R1 R2 R3 R4

Fuse

D0

Fuse

D1

Fuse

D2

D2 D1 D0 Rnom1 1 1 7.2K1 1 0 8.28K1 0 1 9.37K………………………………………………………………………

0 0 0 14.8K


Automatic Frequency Tuning

• By adding additional circuitry to the main filter circuit– Have the filter critical frequency automatically

tuned à Expensive trimming avoidedà Accounts for critical frequency variations due to temperature, supply voltage, and effect of aging


Master-Slave Automatic Frequency Tuning

• Following facts used in this scheme:– Use a replica (master) of the main filter

(called the slave) in the tuning circuitry– Place the replica in close proximity of the main

filter – Use the tuning signal generated to tune the

replica, to also tune the main filter– In the literature, this scheme is called master-

slave tuning!


Master-Slave Frequency TuningReference Filter (VCF)

• Use a biquad for master filter (VCF)• Utilize the fact that @ the frequency fo the lowpass (or highpass)

outputs are 90 degree out of phase wrt to input

• Apply a sinusoid at the desired fo• Compare the LP output phase to the input• Based on the phase difference

– Increase or decrease filter critical freq.

oLP o2in2o o

V 1 @ 90V s s 1Q

ω ω φ

ω ω

= = = −+ +

inV

11Q

os

ω

BPV

--LPV

HPV

∑

os

ω



rms rmstune LPrefV K V V cosφ≈ − × × ×

Input Signal Frequency

of

of Q

LPV

re fV

11Q

os

ω

--

os

ω

Phase Comparator

Amp.+ Filter

TuneV

∑

Vtu

ne



• By closing the loop, feedback tends to drive the error voltage to zero.à Locks fo, the critical

frequency of the filter to the accurate reference frequency

• Typically the reference frequency is provided by a crystal oscillator with accuracies in the order of few ppm

LPV

re fV

11Q

os

ω

--

os

ω

Phase Comparator

Amp.+ Filter

TuneV



tuneV

inV

1 + -

-+ -+

+ - + -

*R Rs−

*R RL21

sτ 31

sτ 41

sτ 51

sτ11

sτ

LPV

re fV

11Q

--

01

sτ

Phase Comparator

Amp.+ Filter

Main Filter (Slave)

Replica Filter(Master)

oV

Ref: H. Khorramabadi and P.R. Gray, “High Frequency CMOS continuous-time filters,” IEEE Journal of Solid-State Circuits, Vol.-SC-19, No. 6, pp.939-948, Dec. 1984.

01

sτ



• Issues to be aware of:– Input reference tuning signal needs to be sinusoid à Disadvantage since clocks are usually available as square waveform

– Reference signal feed-through to the output of the filter can limit filter dynamic range (reported levels or about 100µVrms)

– Ref. signal feed-through is a function of:• Reference signal frequency wrt filter passband• Filter topology • Care in the layout• Fully differential topologies beneficial


Master-Slave Frequency TuningReference Voltage-Controlled-Oscillator (VCO)

•Instead of VCF a voltage-controlled-oscillator (VCO) is used •VCO made of replica integrator used in main filter •Tuning circuit operates exactly as a conventional phase-locked loop (PLL)•Tuning signal used to tune main filterRef: K.S. Tan and P.R. Gray, “Fully integrated analog filters using bipolar FET technology,” IEEE, J.

Solid-State Circuits, vol. SC-13, no.6, pp. 814-821, December 1978..


Master-Slave Frequency TuningReference Voltage-Controlled-Oscillator (VCO)

• Issues to be aware of:– Design of stable & repeatable oscillator challenging– VCO operation should be limited to the linear region of

the amp or else the operation loses accuracy– Limiting the VCO signal range to the linear region not

a trivial design issue– In the case of VCF based tuning ckt there was only ref.

signal feedthrough. In this case, there is also the feedthrough of the VCO signal!!

– Advantage over VCF based tuning à Reference input signal square wave (not sin.)


Master-Slave Frequency TuningChoice of Ref. Frequency wrt Feedthrough Immunity

Ref: V. Gopinathan, et. al, “Design Considerations for High-Frequency Continuous-Time Filters and Implementation of an Antialiasing Filter for Digital Video,” IEEE JSSC, Vol. SC-25, no. 6 pp. 1368-1378, Dec. 1990.


Master-Slave Frequency TuningReference C/Gm Locked to Ref. Frequency

C1 refV Gm V T C1≈ × ×

tuneV

Gm

C

Vin

•Replica of main filter Gm-C building block used •Utilizes the fact that a DC voltage source connected to the input of the Gm cell generates a constant current

I=Gm.Vref•If the integrating capacitor is fully discharged and at t=0 is connected to the output of the Gm cell then:

•If VC1 is forced to be equal to Vref then:

Replica of main filter Gm

Vout

VC1

T

Vref

I=Gm*Vref

clkC NTGm f= =


Master-Slave Frequency TuningReference C/Gm Locked to Ref. Frequency

S2

S1

S3Gm

C1 C2

Vref

A

•Three phase operation •Feedback loop forces:

Replica of main filter Gm

clkC N

Gm f≈

Ref: A. Durham, J. Hughes, and W. Redman- White, “Circuit Architectures for High Linearity Monolithic Continuous-Time Filtering,” IEEE Transactions on Circuits and Systems, pp. 651-657, Sept. 1992.


Reference C/Gm Locked to Ref. FrequencyP1 highà S1 closed

S2

S1

S3Gm

C1 C2

Vref

Discharge C1

A


Reference C/Gm Locked to Ref. FrequencyP2 high à S2 closed

S2 S3Gm

C1 C2

Vref

A

Charge C1 with I=Gm*Vref

I=Gm*Vref

P2

VC1

C1 refV Gm V T 2 C1≈ × ×

T1 T2


Reference C/Gm Locked to Ref. Frequency P3 high à S3 closed

Charge on C1 shared with C2Feedback forces Gm to assume a value:

S2 S3Gm

C1 C2

Vref

A

C1 C2

C1 ref

ref re f

V V Vref

s ince V Gm V T 2 C1then: V Gm V T2 C1

C1o r : T 2 N / fclkGm

:= =

= × ×

= × ×

= =

T1 T2


SummaryReference C/Gm Locked to Ref. Frequency

Feedback forces Gm to vary so that :

S2 S3Gm

C1 C2

Vref

A

in t g

int g0

C1 N / fclkGmor

Gm fclk / NC1

τ

ω

= =

= =

Integrator time constant locked to an accurate frequencyTuning signal used to adjust the time constant of the main filter integrators

Problems to be aware of:à Tuning error due to Gm-cell DC offset


Issues Reference C/Gm Locked to Ref. Frequency

What is DC offset?

Simple example:

For the gm-cell shown here, difference between the threshold voltage of the input devices ( M1 & M2) would cause DC offset.à A non-zero voltage should be applied to input to have Vo=0Offset is usually models as a small DC voltage source at the input

Example: Gm-cell

controlV

oV

inV

-

+

+

-

int gCM1 M2

M10


Gm-Cell Offset Induced Error

( )

C1 C2

C1 ref

C1 osref

os

ref

V V Vref

Ideal V Gm V T2 C1with of fset : V Gm V V T2 C1

VC1or : T2 1Gm V

:= =

= × ×

= × − ×

= −

Vref

Vos S2 S3Gm

C1 C2

A

I=Gm(Vref-Vos)

•Effect of Gm-cell DC offset:

Voltage sourceRepresenting

DC offset


Reference C/Gm Locked to Ref. FrequencyGm-Cell Offset Induced Error

os

ref

os

ref

VC1 T2 1Gm V

Vfor 1/10V

10% error in tuning!

= −

=

Vref

Vos S2 S3Gm

C1 C2

A

I=Gm(Vref+Vos)

•Example:


Gm-cell à two sets of input pairs Aux. input pair + C3a,b à Offset cancellation Same clock timing

Reference C/Gm Locked to Ref. FrequencyIncorporating Offset Cancellation

+

-

-+

P2

P2B-

+

P3

P1+

-

+

-

P1

P2

P3

P2B

P3

P2 P3

P2

Vcm

+Vref/2

-Vref/2

Vtune

C1 C2C3a

C3b


Reference C/Gm Locked to Ref. FrequencyP3 High (Update & Store Vos)

out osV V= −osV

+

-

-+

-

+

+

-

+

-

Vcm

+Vref/2

-Vref/2

Vtune

C1 C2

C3a

C3b

Gm-cell à Unity gain config.C3a,b à Store Gm-cell offsetC1, C2à Charge sharing


Reference C/Gm Locked to Ref. FrequencyP1 High (Reset)

+

-

-+

-

+

+

-

+

-

Vcm

+Vref/2

-Vref/2

Vtune

C1 C2C3a

C3b

Gm-cell à Reset.C1 à DischargeC2 à Hold ChargeC3a,b à Hold Chargeà Offset stored on C3a,b cancels gm-cell offset

osV

osC3aV V= −


Reference C/Gm Locked to Ref. FrequencyP2 High (Charge)

osV+

-

-+

-

+

+

-

+

-

Vcm

+Vref/2

-Vref/2

Vtune

C1 C2C3a

C3b

osC3aV V= −

Gm-cell à Charging C1C3a,b à Store Gm-cell offsetC2 à Hold charge

Key point: Tuning error due to Gm-cell offset cancelled



osV+

-

-+

-

+

+

-

+

-

Vcm

+Vref/2

-Vref/2

Vtune

C1 C2C3a

C3b

osC3aV V= −

Gm-cell à Charging C1C3a,b à Store Gm-cell offsetC2 à Hold charge

Key point: Tuning error due to Gm-cell offset cancelled



Feedback forces Gm to vary so that :

S2 S3Gm

C1 C2

Vref

A

in t g

int g0

C1 N / fclkGmor

Gm fclk / NC1

τ

ω

= =

= =

Tuning error due to gm-cell offset voltage resolvedHas the advantage over previous scheme that fclk can be chosen to be at much higher frequencies compared to filter bandwidth (N>1) à Feedthrough of Vref attenuated by filter


DC Tuning of Resistive Timing ElementVtune

Rext used to lock Gm or on-chip R

Feedback forces Gm=1/Rext

Account for Cap. variations in the gm-C implementation by trimming

Rext.

-

+-

+

I

I

Gm

Ref: C. Laber and Gray, “A 20MHz 6th Order BiCMOS Parasitic Insensitive Continuous-time Filter and Second Order Equalizer Optimized for Disk Drive Read Channels,” IEEE Journal of Solid State Circuits, Vol. 28, pp. 462-470, April 1993

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