Opamp Razavi

5/26/2018 Opamp Razavi

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Ching-Yuan Yang

National Chung-Hsing UniversityDepartment of Electrical Engineering

Operational Amplifiers

(3349) - 2004

9-1 Ching-Yuan Yang / EE, NCHUAnalog-Circuit Design

Overview

Reading

B. Razavi Chapter 9.

Introduction

Operational amplifiers (op amps) are an integral part of many analog and

mixed-signal systems. Op amps with vastly different levels of complexity are

used to realize functions ranging from dc bias generation to high-speed

amplification or filtering.This lecture deals with the analysis and design of CMOS op amps.

Following a review of performance parameters, we describe simple op amps

such as telescopic and folded cascode topologies. Next, we study two-stage

and gain-boosting configurations and the problem of common-mode

feedback. Finally, we introduce the concept of slew rate and analyze the

effect of supply rejection and noise in op amps.


2/33


Performance parameters

Gain

Example: the circuit is designed for a nominal of 10, i.e.,

1 + R1/R

2=10.

Discussion

The close-loop gain:

1

2

21

1

2

21

1

21

2

1

1 AR

RR

A

R

RR

ARR

R

A

V

V

in

out

++

+

=

++

=

If A1

>> (R1

+ R2)/R

2, then

+

+

12

21

2

1 111AR

RR

R

R

V

V

in

out

The term (R1

+ R2)/(R

2A

1) = (1 + R

1/R

2)/A

1represents the relative error.

To achieve a gain error less than 1%, we must have A1

> 1000.

Simple CS stage an open-loop implementation:

10== Dmin

out RgV

V

However, it is difficult to guarantee an error less than 1%.The variations in the mobility and gate oxide thickness of the transistor and

the value of the resistor typically yield an error greater than 20%.


Performance parameters (contd)

Small-signal bandwidth

Large-signal bandwidth slew rate

unity-gain

dB

Gain roll-off with frequency


3/33


Performance parameters (contd)

Output swing Most systems employing opamps require large voltage swings to

accommodate a wide range of signal amplitudes.

Linearity Open-loop op amps suffer fromsubstantial nonlinearity. For example, the input

pair M1M

2exhibits a nonilinear relationship

between its differential drain current and input

voltage. In many feedback circuits, the linearity

requirement, rather than the gain error

requirement, governs the choice of the open-loop

gain.

Noise and offset The input noise and offset ofop amps determine the minimum signal level that

can be processed with reasonable quality.

Supply rejection Op amps are often employedin mixed-signal systems and sometimes

connected to noise digital supply lines. Thus, the

performance of op amps in the presence of supply

noise is quite important. For this reason, fullydifferential topologies are preferred.


One-stage op amps

Simple op amp topologies

Differentialinput & single-endedoutput Differentialinput & differentialoutput

For small-signal:

Low frequency gain = gmN(roN||roP). In general, this value hardlyexceeds 20 in submicron devices with typical current levels.

The bandwidth is usually determined by the load capacitance, CL.

The circuits suffer from noise contributions of M1-M4. In all op amp

topologies, at least four devices contribute to the input noise: two

input transistors and two load transistors.


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Unit-gain buffer

VCSS

Input common-mode voltage range

Vin,min= VCSS+ VGS1Vin,max= VDD |VGS3| + VTH1

If each device has a threshold voltage of 0.7V and an overdrive

of 0.3V, then Vin,min= 1.3V, and Vin,max= 2.7V. Thus, the inputCM range equals 1.4V with a 3-V supply.

Output impedance

( ) mNoNoPmNoNoP

openv

openoutout

grrg

rr

A

RR

1

11 ,

, +

=+

=

The close-loop output impedance is relatively

independentof the open-loop output impedance.

Allowing us to design high-gain op amps by increasing

the open-loop output impedance while still achieving a

relatively low close-loop output impedance.


Telescope cascode op amps

In order to achieve a high gain, the differential cascode topologies can be used.

Low-frequency gain Av= gmN[(gmNroN2) || (gmProP2)], but at the cost of output

swing and adding poles.


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(a): The circuit providing a single-ended output suffers from a mirror pole at

nodeX, creating stability issues.

(b): Fully differential topology, the output swing is given by

2[VDD (VOD1 + VOD3 + VCSS+ |VOD5| + |VOD7|)]

where VODjdenotes the overdrive voltage ofMj.

Another drawback of telescopic cascodes is the difficult in shorting their inputs

and outputs, e.g., to implement a unity-gain buffer.


Telescope cascode op amps (contd)

Cascode op amp with input and output shorted

unit gain feedback topology

Output swing: M2 and M4 in saturation:

244

4

2

THGSboutTHb

THbout

THXoutVVVVVV

VVV

VVV+

+

the voltage range VmaxVmin= VTH4 (VGS4VTH2)

Since the op amp attempts to force Voutto be equal to

Vin, for Vin< VbVTH4, we have VoutVinand M4 is in

triode region while others are saturated. Under this

condition, the open-loop gain of the op amp is reduced.

As Vinand Vouthence exceed VbVTH4, M4 enters

saturation and the open-loop gain reaches a maximum.

For VbVTH4


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Design of fully differential telescope op amp

Specifications:

VDD= 3V, differential output swing = 3V,

power dissipation = 10mW, voltage gain = 2000.

Assume nCox= 60 A/V2, pCox= 30 A/V2, n= 0.1V1, p= 0.2V1 (for an

effective channel length of 0.5 m), = 0, VTHN

= |VTHP

| = 0.7V.

Power budget:

IM9 = 3mA, IMb1 + IMb2 = 330A Output swing:

Node X(Y) swing = 1.5V, M3-M6 in saturation.

For M9,

|VOD7| + |VOD5| +VOD3 +VOD1 +VOD9 = 1.5VSinceM9 carrying largest current,

VOD9 0.5V is chosen.|VOD5| = |VOD7| 0.3V,VOD1 =VOD3 0.2V.

W/L

By ID= (1/2)Cox(W/L)(VGSVTH )2, we have(W/L)14 = 1250, (W/L)58 = 1111, (W/L)9 = 400.


Gain:

Avgm1[(gm3ro3ro1)|| (gm5ro5ro7)]. In order to Increase the gain,

we recognize

where 1/L. We can therefore increase the width or length.Choose (W/L)58 = 1111m/1m,

thenAv 4000.

CM level & bias:

Min. allowable input CM level= VGS1 +VOD9 = 1.4V.

Vb1,min = VGS3 +VOD1 +VOD9 = 1.6V.

Vb2,max = VDD (|VGS5 |+|VOD7|) = 1.7V.

DDDoxom IWLIILWCrg /)/()/(2 =


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Folded cascode op amps

In order to alleviate the drawbacks

of telescopic cascode op amps. The

primary advantage of the folded

structure lies in the choice of the

voltage levels because it does not

stack the cascode transistor on the

top of the input device.


Folded cascode op amps (contd)

Two important differences between the two circuits:

In Fig.(a), one bias current, ISS, provides the drain current of both the input

transistors and the cascode devices.

In Fig.(b), the input pair requires an additional bias current, ISS1 = ISS/2 + ID3.

In Fig.(a), the input CM level cannot exceed Vb1 VGS3 + VTH1,whereas inFig.(b), it cannot be less than Vb1 VGS3 + |VTH1|.

In Fig.(b), it is possible to tie the n-well of M1 and M2 to their common source point.


8/33



Folded cascode op amp with cascode PMOS loads

Max. output voltage swing: With proper choice of Vb1 and Vb2,

Peak-peak swing = [VDD (|VOD7| + |VOD9|)] (VOD3 + VOD5 ) for one side.

The swing is lager by the overdrive of the tail current source in the telescopic

cascode.M5 and M6 may require a high overdrive voltage if their capacitance

contribution to nodesXand Yis to be minimized.



Small-signal voltage gain

Half circuit

|Av| = Gm Rout

Equivalent circuit with

output shorted to ground

Since (gm3+gmb3)1||ro3


9/33



Effect of device capacitance on the nondominant pole in telescopic and folded cascode

op amps

Ctot= CGS3 + CSB3 + CDB1 + CGD1 Ctot= CGS3 + CSB3 + CDB1 + CGD1 + CGD5 + CDB5

The pole at the folding point, i.e., the sources of M3 and M4, is quite closer to the

origin than that associated with the source of cascode devices in a telescopic

topology.


A high-gain folded cascode op amp

The circuit provides a higher gain because of the greater mobility of NMOS

devices, but at the cost of lowering the pole at the folding point,

p,X (gm3 + gmb3) / Ctot,X.


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Telescopic- & folded-cascode op amps: Discussion

The overall voltage swing of a folded-cascode op amp is only slightly higher than that of a

telescopic configuration. This advantage comes at the cost of higher power dissipation,

lower voltage gain, lower pole frequencies, and higher noise.

Folded-cascode op amps are used quite widely, even more than telescopic topologies,

because the input and outputs can be shorted together and the choice of the input

common-mode level is easier. In a telescopic op amp, three voltages must be defined carefully: the input CM level

and the gate bias voltages of the PMOS and NMOS cascode transistors, whereas in

folded-cascode configurations only the latter two are critical.

In folded-cascode op amps, the capability of handling input CM levels are close to

one of the supply rails.


Design of folded-cascode op amp

Specifications:

VDD= 3V, differential output swing = 3V,

power dissipation = 10mW, voltage gain = 2000.

Assume nCox= 60 A/V2, pCox= 30 A/V2, n= 0.1V1, p= 0.2V

1 (for an

effective channel length of 0.5 m), = 0, VTHN= |VTHP| = 0.7V.


11/33


Power budget: IM11 = 1.5mA, IM9 +IM10 = 1.5mA, IMb1 +IMb2 +IMb3 = 330A.

Output swing: one side o/p swing = 1.5V, M3-M10 in saturation.

Choose |VOD5,6| 0.5V, |VOD3,4| 0.4V, VOD7,8 =VOD9,10 0.3V.

W/LBy ID= (1/2)Cox(W/L)(VGSVTH )2, we have

(W/L)5,6 = 400, (W/L)3,4 = 313, (W/L)710 = 555. O/p CM level: CMmin = 0.6V, CMmax= 2.1V, thus CMopt= 1.35V.


Determine (W/L)1,2: min. input CM level = VGS1 +VOD11.

If input and output are shorted, then VGS2 +VOD11 = 1.35V,

and VGS1 = 0.95V VOD1,2 = 0.25V (W/L)1,2 = 400.

The maximum dimensions of M1,2 are determined by the tolerable input

capacitance at nodesXand Y.

Gain: gm= 2ID/(VGSVTH), we havegm1,2 = 0.006 A/V,gm3,4 = 0.0038 A/V,gm7,8 = 0.05 A/V.

For L= x m, findro.

Note |Av|gm1{[(gm3 + gmb3)ro3(ro1|| ro5)] || [(gm7 + gmb7) ro7ro9]}


12/33


Cascode op amps with single-ended output

Fig(a): VX= VDD |VGS5| |VGS7|,

limiting the maximum value of

VouttoVDD |VGS5| |VGS7|

|VTH6| and wasting one PMOS

threshold voltage in the swing.

Fig(b): To solve above issue, M7 and

M8 are biased at the edge of

the triode region.

Disadvantages: (1) it provides only half the output voltage swing.

(2) it contains a mirror pole at nodeX, thus limiting the speed

of feedback systems employing such an amplifier.

It is preferable to use the differential topology, although it requires a feedbackloop to define the output CM level.


Triple-cascode op amp

The triple cascode topology provides

a gain on the order of (gmro)3/2 but

further limits the output swings. With

six overdrive voltages subtracted fromVDDin this circuit, it is difficult to

operate the amplifier from a supply

voltage or lower while obtaining

reasonable output swings.


13/33


Two-stage op amps

The gain of one-stage topologies is limited to the input pair transconductance

and the output impedance.

Two-stage op amps consist of first stage providing a high gain and the second

providing large swing. The first stage incorporates various amplifier topologies,

but the second stage is typically configured as a simple common- source

stage to allow maximum output swings.

Can we cascade more than two stages to achieve a higher gain?

Each gain stage introduces at least one pole in the open-loop transfer function,

making it difficult to guarantee stability in a feedback system using such an op

amp. For this reason, op amps having more than two stages are rarely used.


Simple implementation of a two-stage op amp

Gain:Av,1st stage = gm1,2(ro1,2||ro3,4)

Av,2nd stage = gm5,6(ro5,6||ro7,8)

Overall gain Av= Av,1st stageAv,2nd stage

Output swing = VDD |VOD5,6| VOD7,8


14/33


Two-stage op amp employing cascoding

To obtain a higher, the first stage incorporate cascode devices. The overall voltage gain is

Av

{gm1,2

[(gm3,4

+ gmb3,4

)ro3,4

ro1,2

] || (gm5,6

+ gmb5,6

)ro5,6

ro7,8

]} [gm9,10

(ro9,10

||ro11,12

)]


Two-stage op amp with single-ended output

Note that if the gate of M1 is shorted to Voutto form a unity-gain buffer,

then the minimum allowable output level is equal to VGS1 + VISS, severely

limit the output swing.


15/33


Gain boosting

Increasing the output impedance by feedback

Rout= gm2ro2ro1

M1 operates as a degeneration resistor.

The voltage variations at the drain of M2 effect VX to a

lesser extent because A1 regulates this voltage. (VX= Vb)

With smaller variations at X, the current throughro1 and

hence the output current remains more constant,

yielding a higher output impedance.

RoutA1gm2ro2ro1,

Rout is booted substantially without stacking morecascode devices on top of M2.


Gain boosting in cascode stage

For small-signal operation, Vb is set to zero.

regulated cascode

Gain:

|Av| gm1 (gm2ro2ro1) (gm3 ro3)

Min. output swing:

Since VX= VGS3, the min.value ofVoutis VOD2 + VGS3. The auxiliary

amplifier limits the output swing.

Note: Min. output swing is VOD2 +

VOD1 in a simple cascode.


16/33


Boosting output impedance of a differential cascode stage

The minimum level at the drain of M3 is

equal to VOD3 + VGS5 + VISS2.

The voltage swing limitation results the fact

that the gain-boosting amplifier incorporates

an NMOS differential pair.


Folded-cascode circuit used as auxiliary amplifierHalf circuit

If nodesXand Yare sensed by a PMOS pair, the minimum value of VXand

VY is not dictated by the gain-boosting amplifier.

The minimum allowable level of VXand VY is given by VOD1,2 + VISS1.

Output impedance: Since 15 outmX

P RgV

V=

where Rout1 [gm7ro7(ro9||ro5)] || (gm11ro11ro13), Routgm3ro3ro1gm5Rout1.


17/33


Gain boosting applied to both signal path and load devices

Regulated cascodes can also be utilized in the load current sources of a

cascode op amp.


Comparison of performance of various op amp topologies

Gain

Medium

Medium

High

High

Output

Swing

Medium

Medium

Highest

Medium

Speed

Low

Medium

Highest

High

Low

Medium

Medium

High

Power

DissipationNoise

Low

Medium

Low

Medium

Telescopic

Folded-Cascode

Two-stage

Gain-Boosted


18/33


Common-mode feedback (CMFB)

Full differential circuits have many advantages over their single-ended

counterparts such as greater output swings, avoiding mirror poles, higher

closed-loop speed. However, high-gain differential circuits require common-

mode feedback.

Simple differential pair

Input & output common-mode

level is equal to VDDISS RD /2


High-gain differential pair with inputs shorted to outputs

What is the common-mode level at nodesXand Y?

Since each of the input transistors carries a current ISS/2, the CM level depends on

how close ID3 and ID4 are to this value.

Effect of current mismatches: Mismatches in the PMOS and NMOS current mirrors

defining ISSand ID3,4 create a finite error betweenID3,4 and ISS/2.

IfID3,4 >ISS/2, then both M3 and M4 must enter the triode region so that their drain

currents fall toISS/2. Conversely, IfID3,4


19/33


Simplified model of high-gain amplifier

In high-gain amplifiers, we wish ap-type current source to balance an n-type current

source.

( )( )NPNP RRIIV =

Since the current error depends on mismatches and RP||RN is quite high, the voltageerror may be large, thus driving thep-type or n-type current source into triode region.

As a general rule, if the output CM level cannot be determined by visual inspection

and requires calculations based on device properties, then it is poorly defined.

In high-gain amplifiers, the output CM level is quite sensitive to device properties and

mismatches and it cannot be stabilized by means of differential feedback. Thus a

CMFB network must be added to sense the CM level of the two outputs and

accordingly adjust one of the bias currents in the amplifier.


Conceptual topology for CMFB

In high-gain amplifiers, the output CM level is quite sensitive to device properties and

mismatches and it cannot be stabilized by means of differential feedback. Thus a CMFB

network must be added to sense the CM level of the two outputs and accordingly adjust

one of the bias currents in the amplifier.


20/33


CMFB with resistive sensing

Output CM level: Vout,CM= (Vout1 + Vout2)/2

Resistive divider level: Vout,CM= (R1Vout2 + R2Vout1)/(R1 + R2)

= (Vout1 + Vout2)/2, ifR1 = R2.

R1 and R2 must be much larger than the output impedance of the op ampso as to avoid lowering the open-loop gain.


CMFB using source followers

Current starvation of source followers for large swings

This technique produces a CM level

that is lower than the output CM

level by VGS7,8, but this shift can be

taken into account in the

comparison operation.

R1 and R2 or I1 and I2 must be large

enough to ensure that M7 or M8 is

not starved when a large differential

swing appears at the output.

If Vout2 is quite higher than Vout1, then I1must sink both IX (Vout2Vout1)/(R1 + R2)and ID7. Consequently, if (R1 + R2) or I1 is

not sufficiently large, ID7 drops to zero

and Vout,CMno longer represents the true

output CM level.

This sensing method limits the differential

output swings. The swing at each output is

reduced by approximately VTH, a significant

value in low-voltage design.


21/33


CMFB using MOSFET operating in deep triode region

( ) ( )

( )THoutoutoxn

THoutoxnTHoutoxn

ononP

VVVL

WC

VVL

W

CVVL

W

C

RRR

2

1

11

21

21

87

+=

=

=

RP is a function of Vout1 + Vout2 but independent of

Vout2Vout1.

The use of M7 and M8 limits the output voltage swings, Vout,min= VTH7,8, which is

relatively close to two overdrive voltages, but the difficulty arises from the assumption

above that both M7 and M8 operate in deep triode region. If Vout1drops from the

equilibrium CM level to one threshold voltage above ground and Vout2 rises by the same

amount, then M7 enters the saturation region, thus exhibiting a variation in its on-resistance that is not counterbalanced by that of M8.

Identical transistors M7 and M8 operate in deep triode region,


Sensing and controlling output CM level

We employ a simple amplifier to detect the difference between Vout,CMand a reference

voltage, VREF, applying the result to the NMOS current sources with negative feedback.

If the loop gain is large, the feedback network forces the CM level of Vout1 and Vout2 to

approach VREF.

Also, the feedback may control only a fraction of the current to allow optimization of the

settling behavior. For example, each M3 and M4 can be decomposed into two parallel

devices, one biased at a constant current and the other driven by the error amplifier.


22/33


Alternative method of controlling output CM level

In a folded-cascode op amp, the CM feedback may control the tail current

of the input differential pair. This method increases the tail current if Vout1and Vout2 rise, lowering the drain currents of M5M6 and restoring the output

CM level.


CMFB using triode devices

The output CM level sets Ron7||Ron8 such thatID5 and ID6 exactly balance ID9 and ID10, respectively.

Assuming ID9 = ID10 = ID,

RP= Ron7||Ron8 = (VbVGS5)/(2ID ), and also

( )THoutoutoxnP

VVVL

WC

R

2

1

12

8,7

+

=

where( ) 55

5/

2TH

oxn

DGS V

LWC

IV +=

Drawbacks:

1.The value of the output CM level is a function of device

parameters.

2.The voltage drop across Ron7||Ron8 limits the output voltage swing.

3.To minimize this drop, M7 and M8 are usually quite wide devices, introducing

substantial capacitance at the output.


23/33


Alternative method of controlling output CM level

If Vb is higher than expected, the tail current of M1 and

M2 increases and the output CM level falls. Since the

feedback through M7 and M8 attempts to correct this

error, the overall change in Vout,CMdepends on the loop

gain in the CMFB network.

Determine the sensitivity dVout,CM/dVb:M7,8 in triode region: gm7,8 = nCox(W/L)7,8VDS7,8Feedback factor:

( )( )THGS

DSononmm

I VV

VRRgg

V

V

=+==

= 8,7

8,78787

01

2

2

Thus,8,7

8,7, 1

DS

THGS

b

CMout

V

VV

dV

dV =

Since VGS7,8 (i.e., the output CM level) is typically

in the vicinity of VDD

/2, the above equation

suggests that VDS7,8 must be maximized.


Modification of CMFB for more accurate definition of output MC level

The idea is to define Vb by a current mirror arrangement such that ID9 tracks I1 and IREF.

Suppose (W/L)15 = (W/L)9 and (W/L)16 = (W/L)7 + (W/L)8.

Thus, ID9 = I1 only if Vout,CM= VREF.

The circuit produces an output CM level equal to a reference but it requires no resistors in

sensing Vout,CM.

In practice, since VDS15VDS9, channel-length modulation results in a finite error.


24/33


Modification to suppress error due to channel-length modulation

Transistors M17 and M18 reproduce at the drain of M15 a voltage equal to the

source voltage M1 and M2, ensuring that VDS15 = VDS9.


Another CM feedback topologies

Differential pair using diode-connected loads

The input CM level, VDDVGS3,4, is relativelywell-defined, but the voltage gain is quite low.

Resistive CMFB

To increase the differential gain, the PMOS

device must operate as current sources for

differential signals.

For differential change at Vout1 and Vout2, node

Pis a virtual ground and the gain can be

expresses as

Av= gm1,2(ro1,2||ro3,4||RF)

For CM levels, M3 and M4 operate as diode-

connected devices.


25/33


Input range limitations

Limitation: While the differential input swings are usually much smaller, the input

common-mode level may need to vary over a wide range in some applications.

Unity-gain buffer

The voltage swings are limited by the input differential pair rather than the

output cascode branch. Specifically, Vin,minVout,min = VGS1,2 + VISS, approximatelyone threshold voltage higher than the allowable minimum provided by M5-M8.

If Vin < Vin,min: The MOS transistor operating as ISS enters the triode region,

decreasing the bias current of the differential pair and hence lowering thetransconductance.


Extension of input CM range

Variation of equivalent transconductance withthe input CM level.

A simple approach to extending the

input CM range is to incorporate

both NMOS and PMOS differential

pairs such that when one is dead,

the other is alive. This idea is to

combine two folded-cascode op amps

with NMOS and PMOS input

differential pairs.

As the input CM level approaches the

ground potential, the NMOS pairs

transconductance drops, eventually

falling to zero. Nonetheless, the PMOS

pair remains active, allowing normal

operation. Conversely, if the input CM

level approaches VDD, M1Pand M2Pbegin to turn off but M1 and M2 function

properly.


26/33


Slew rate

Response of a linear circuit to input

step

dVout/dt: Since Vout= V0[1 exp(t/)], where = RC, we have

tV

dt

dVout = exp0

dVout/dtV0; if we apply a larger input step, the output rises more rapidly.


Slew rate (contd)

Response of a linear op amp to step response

Assume op amp is linear,

sCVRR

V

RVA

RR

RVV Lout

out

out

outoutin ++=

+

2121

2 1

Assume R1 + R2 >> Rout, we have

( )

( )

+++

++

sRRAR

CRRR

RA

As

V

V

Loutin

out

21221

2

111

The step response is given by

( )

( )tu

RRAR

RC

t

RR

RA

AVV

outLout

++

++

=

21221

20

1

exp1

1

indicating that the slope is proportional to the final value.

This type of response is called linear settling.


27/33


Slew rate (contd)

Slewing in an op amp circuit

( )

( )tu

RRAR

RC

t

RR

RA

AVV

outLout

++

++

=

21221

20

1

exp1

1

The response to sufficiently small inputs follows the exponential of Eq.(A), but

with large input steps, the output displays a linear ramp having a constant slope.

Under this condition, we say the op amp experiences slewing and call the slop ofthe ramp the slew rate.

..(A)


Small-signal operation of a simple op amp

Assuming that R1 + R2 is quite large.If Vin experiences a change of V, the total small-signal current providedby the op amp equals gmV. This current begins to change CL, but as Voutrises, so does VX, reducing the difference between VG1 and VG2 and hence

the output current of the op amp.


28/33


Slewing during large signal transition

Slewing during low-to-high transition

M1 absorbs all of ISS and M2 turns off.

So long as M2 remains off, the feedback

loop is broken and the current charging

CL

is constant and independent of the

input level.

Slewing during high-to-low transition

Slope =ISS /CL


Discussion of slew rate

While the small-signal bandwidth of a circuit may suggest a fast time-domainresponse, the large-signal speed may be limited by the slew rate simplybecause the current available to charge and discharge the dominantcapacitor in the circuit is small.

Since the input/output relationship during slewing is nonlinear, the output ofa skewing amplifier exhibits substantial distortion.

For example, if a circuit is to amplify a sinusoid V0sin0t(in the steadystate), then its slew rate must exceed V00.


29/33


Slewing in telescopic op amp

Vout1 and Vout2 appear as a ramps with slopes equal to ISS /(2CL), and

consequently Vout1Vout2 exhibits a slew rate equal to ISS /CL.


Slewing in folded-cascode op amp

If IPISS, the slew rate is equal to ISS /CL, independent of IP.


30/33


Slewing in folded-cascode op amp (contd)

If ISS > IP, then during slewing M3 turns off and VX falls to a low level such that M1and the tail current source enters the triode region. Thus, for the circuit to return to

equilibrium after M2 turns on, VXmust experience a large swing, slow down the

settling.


Slewing in folded-cascode op amp (contd)

Clamp circuit to limit swings atXand Y

The difference between ISS and IPflows through

M11, or M12, requiring only enough drop in VXor

VYto return on one of these transistors.

M11 and M12 clamp the two nodes directly to VDD.

Since the equilibrium value VXand VYis usually

higher than VDDVTHN, M11 and M12 are offduring small signal-signal operation.


31/33


Power supply rejection

If the circuit in the figure is perfectly symmetric, Vout= VX.

Since the diode-connected device clamps nodeXto VDDVX and hence Voutexperience approximately the same

change as does VDD. In other words, the gain from VDD to

Vout is

1

DD

out

V

V

The power supply rejection ratio (PSRR) is defined as

the gain from the input to the output divided by the gain

from the supply to the output. At low frequencies:

( )oNoPmNDDout

inout rrgVV

VVPSRR

=


Noise in a telescopic op amp

Guide: With many transistors in an op amp, it may seem difficult to intuitively identify

the dominant sources of noise. A simple rule for inspection is to change the gate

voltage of each transistor by a small amount and predict the effect at the output.

At relatively low frequency, the

cascode devices contribute negligible

noise, leaving M1-M2 and M7-M8 as the

primary noise sources.

The input-referred noise voltage per unit bandwidth is given by

( ) ( ) 22,1

28,7

8,72,12

2,1

8,7

2,1

222

3

22

3

224

m

m

ox

P

ox

N

m

m

mn

g

g

fCWL

K

fCWL

K

g

g

gkTV ++

+=

where KNand KPdenote the 1/fnoise coefficients of NMOS

and PMOS devices, respectively.


32/33


Noise in a fold-cascode op amp

The noise of the cascode devices is negligible at low frequencies,

leaving M1-M

2, M

7-M

8, and M

9-M

10as potentially significant sources.

Thermal noise:

= 22 8,7

8,7

8,7

2

,3

242 outm

m

Moutn Rgg

kTV

= 22 10,9

10,9

10,92,

3

242 outm

m

Moutn Rgg

kTV

= 22 2,1

2,1

2,12,

3

242 outm

m

Moutn Rgg

kTV

andAv

= gm1,2

Rout

.

Total input-referred thermal noise:

++==2

2,1

10,9

22,1

8,7

2,12

2,,2

,3

2

3

2

3

28

m

m

m

m

mv

totoutn

inng

g

g

g

gkT

A

VV

, (uncorrelated noise)

where the factor 2 accounts for noise of M7 and M8, and

Routdenotes the open-loop output resistance of the op

amp.


Noise in a fold-cascode op amp (contd)

( )

= 22 8,7

8,7

8,7

2

,

12 outm

ox

PMoutn Rg

fWLC

KV

( )

= 22 10,9

10,9

10,92,

12 outm

ox

NMoutn Rg

fWLC

KV

( )

= 22 2,1

2,1

2,12,

12 outm

ox

NMoutn Rg

fWLC

KV

( ) ( ) ( ) 2 2,1

28,7

8,72

2,1

210,9

10,92,1

2

2,,2

,

12112

m

m

ox

P

m

m

ox

N

v

totoutn

inn

g

g

WLfC

K

g

g

WLWLfC

K

A

VV

+

+=

=

andAv= gm1,2Rout.

Total input-referred flicker noise:

Flicker noise:


33/33


Noise in a fold-cascode op amp (contd)

The overall noise:

( ) ( ) ( ) 22,1

2

8,7

8,72

2,1

2

10,9

10,92,1

22,1

10,9

22,1

8,7

2,1

2,

12112

3

2

3

2

3

28

m

m

ox

P

m

m

ox

N

m

m

m

m

m

inn

g

g

WLfCK

g

g

WLWLfCK

g

g

g

g

gkTV

+

++

++=

Discussion:

The noise contribution of the PMOS and NMOS

current sources increases in proportion to their

transconductance. This trend results in a trade-off

between output voltage swings and input-referred

noise: for a given current, as implied by

gm = 2ID /(VGSVTH), if the overdrive voltage ofthe current sources is minimized to allow large

swings, then their transconductance is maximized.


Noise in a two-stage op amp

( )( )( )231

25

21

752

2

75752

3

161

3

242

85

oomm

mm

v

oommMnrrgg

ggkT

ArrggkTV

+=+=

In the 1st stage:2

1

312

3

242

41

m

mmMn

g

ggkTV

+=

Total input-referred thermal noise: ( )

+++= 2

312

5

753121

2,

1

3

16

oom

mmmm

m

totnrrg

ggggg

kTV

Note the noise resulting from the second

stage is usually negligible because it is

divided by the gain of the first stage when

referred to the main input.

Total voltage gain: Av= gm1(ro1||ro3)gm5(ro5||ro7).

In the 2nd stage: The noise current of M5 and M7 flows through ro5||ro7.

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Opamp Razavi