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Exercises for Differential Amplifiers ECE 102, Fall 2012, F. Najmabadi

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Page 1: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

Exercises for Differential Amplifiers

ECE 102, Fall 2012, F. Najmabadi

Page 2: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (2/37)

Exercise 1: Compute VD , VS , VDS and VGS if ID3 = 2 mA, RD = 500 Ω, VOV3 = 0.5 V, and identical Q1 &Q2 with µnCox (W/L ) = 8 mA/V2, Vt = 0.5 V, λ = 0. A) For VG = 0 and B) For VG = 1 V. Repeat the exercise for λ = 0.1 V-1.

This exercise shows that precise biasing of Q1 and Q2 is not necessary as VS adjusts itself automatically.

Inclusion of channel-width modulation does not impact the bias points of Q1 and Q2 (which is set by the current source).

Page 3: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (3/37)

Assume Saturation

V 111 =+= tOVGS VVV

Saturation & V 121)2(

V 5.2)1(5.1 V 110

0 A)

3311

3

11

11

1

⇒>>=+−=−−==−−=−=

−=−=−==

OVDSOVDS

SDS

SDDS

GSGS

G

VVVVVV

VVVVVV

V

V 5.01 =OVV

Ignoring channel-width modulation (λ = 0)

21

3211

3 104 )/(5.0101 OVOVoxnD VVLWCI −− ×===× µ

V 5.15.022 11 =−=−= DDD IRVKVL:

Note that as the bias voltage of Q1, VG1, changes, VS is adjusted automatically to get the necessary VOV1 and ID1

mA 15.0 321 === DDD III

Saturation & V 220)2(

V 5.105.1 011

1 A)

3311

3

11

11

1

⇒>>=+=−−==−=−=

=−=−==

OVDSOVDS

SDS

SDDS

GSGS

G

VVVVVV

VVVVVV

V

Page 4: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (4/37)

Assume Saturation

)( )( )(

111

111

111

11

GtDOV

GtOVD

GGSD

SDDS

VVVVVVVV

VVVVVV

−++=−++=

−+=−=

Including channel-width modulation (λ = 0.1)

25.0) 1.01(

)1( )/(5.0101

12

1

12

113

=+

+==× −

DSOV

DSnOVoxnD

VVVVLWCI λµ

Need to write VDS1 in terms of VOV1:

For a given VG1, we then substitute for VDS1 in ID equation which leads to a cubic equation for VOV1

V 5.15.022 11 =−=−= DDD IRVKVL:

mA 15.0 321 === DDD III

Page 5: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (5/37)

V 948.011 =+= tOVGS VVV

1111 )( GtOVDDS VVVVV −++=

V 448.01 =OVV

Including channel-width modulation (λ = 0.1)

25.0) 1.01( 12

1 =+ DSOV VV

V 5.11 =DV

V 948.0948.0011 −=−=−= GSGS VVV

C) VG1 = 0

0.2)05.05.1( 111 +=−++= OVOVDS VVV

025.02.11.0

25.0)]0.2 (1.01[ 2

13

1

12

1

=−+

=++

OVOV

OVOV

VVVV

V 448.20.2 11 =+= OVDS VV

V 967.011 =+= tOVGS VVV

V 467.01 =OVV

V 033.0967.0111 +=−=−= GSGS VVV

0.1)15.05.1( 111 +=−++= OVOVDS VVV

025.01.11.0

25.0)]0.1 (1.01[ 2

13

1

12

1

=−+

=++

OVOV

OVOV

VVVV

V 467.10.1 11 =+= OVDS VV

D) VG1 = 1 V

V 052.12948.0)2(3 =+−=−−= SDS VV V 033.22033.0)2(3 =+=−−= SDS VV

Page 6: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (6/37)

V 00.1 V 50.2

V 00.1 V 0

V 500.0 mA 0.1

3

1

21

21

21

==−=

======

DS

DS

S

GG

OVOV

DD

VVV

VVVV

II

Include channel-width modulation

Specified parameters

Inclusion of channel-width modulation does not impact the bias points of Q1 and Q2 (which is set by the Q3 current source).

V 2.0V 50.1

0V 0.1

V 500.0mA 0.1

V 1.052V 448.2

V 948.00

V 448.0mA 0.1

V 2.033V 467.1V 033.0

V 0.1V 467.0

mA 0.1

Ignore channel-width modulation

Bias voltage of Q1 and Q2 (VG) does not affect ID1 as VS adjusts itself automatically. VG affects only VDS1 and VDS3 and precise biasing is NOT necessary.

Page 7: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (7/37)

Exercise 2: Find the differential gain and (W/L) of all transistors in the circuit below, Q3 & Q4 are matched, Q1 & Q2 are matched, all transistors have VOV = 0.2 V, µnCox = 400 µA/V2, µpCox = 100 µA/V2, and VAn = |VAp |= 3.6 V. Ignore channel-width modulation in biasing calculations.

For symmetric circuits:

d

do

d

do

d

dod

dodododooo

vv

vv

vv

A

vvvvvv

5.02

2

1,1,,

,1,1,2,12

−=×−==

−=−=→−=

Page 8: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (8/37)

Since transistors are matched and have the same VOV :

21

2111

6

)/(5.12)/( )/(5.010100

LWLWVLWCI OVoxnD

====× − µ

A 1004321 µ==== DDDD IIII

43

2333

6

)/(50)/(

)/(5.010100

LWLWVLWCI OVoxpD

==

==× − µ

A/V 102 3

1

11

−==OV

Dm V

Ig k 361

11 ==

D

Ao I

Vr

Differential Mode Half Circuit

k 363

33 ==

D

Ao I

Vr

18101810k) 36||k 36(10)||( 333o3o11 −=××−=−=−= --

md rrgA

)||( 5.0

311

1,,

oom

d

do

d

dod

rrgv

vvv

A

−=−

==

Page 9: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (9/37)

Exercise 3: The differential amplifier below should achieve a differential gain of 40 with a power consumption of 2 mW. All transistors operate with the same VOV . Find (W/L) of all transistors, VG3, VG4, and VG5. (µnCox = 400 µA/V2, µpCox = 100 µA/V2, λn = 0.1 /V, λp = 0.2 /V, and Vtn = |Vtp |= 0.4 V. Ignore channel-width modulation in biasing.

Power Consumption:

mA 556.05.0mA 11.1 1028.1

54321

53

5

======→×== −

DDDDD

DD

IIIIIIIP

Page 10: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (10/37)

Differential Mode Half Circuit

35.0)5.0(||

5.011 1

1

1

11

1131

11113

3

11

o

oo

oooo

oop

n

Dnp

n

DpDpo

Dno

rrrrrrr

rrIII

r

Ir

=+×

=

=×=×===

=

λλ

λλλ

λλ

λ

V 167.0

40 3.0212

31

31)||( ||

1

111

1

11311

=

==××=

=+=

OV

OVDnOV

D

omoomd

VVIV

I

rgrrgA

λ

)||( 5.0

311

1,,

oom

d

do

d

dod

rrgv

vvv

A

−=−

==

Page 11: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (11/37)

V 167.0mA 556.05.0

54321

54321

==========

OVOVOVOVOV

DDDDD

VVVVVIIIII

100)/()/()167.0()/(104005.010556.0

)/(5.0

12

21

63

2111

==×××=×

=−−

LWLWLW

VLWCI OVoxnD µ

400)/()/()167.0()/(101005.010556.0

)/(5.0

34

23

63

2333

==×××=×

=−−

LWLWLW

VLWCI OVoxpD µ

200)/()167.0()/(104005.01011.1

)/(5.0

5

25

63

2555

=×××=×

=−−

LWLW

VLWCI OVoxnD µ

V 567.00567.0V 567.04.0167.0

555

55

=+=+==+=+=

SGSG

tnOVGS

VVVVVV

V 233.1567.08.1

V 567.04.0167.0||

333

33

=−=−=

=+=+=

SGSG

tpOVSG

VVVVVV

Page 12: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (12/37)

Exercise 4: The circuit below is fabricated with VAn = |VAp| = 3.6 V, µnCox = 100 µA/V2 & µpCox = 25 µA/V2. All transistors operate with VOV = 0.5 V. Find (W/L) of all transistors and the differential gain of the circuit.

Page 13: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (13/37)

k 3610100278.0

11...

mA/V 4.05.0

1010022...

V 5.0...A 1005.0...

61

821

6

1

1821

9821

9821

=××

=====

=××

=====

==========

Dooo

OV

Dmmm

OVOVOVOV

DDDD

Irrr

VIggg

VVVVIIII

λ

µ

/V0.2786.3

11====

Apn V

λλ

NMOS: Q1, Q2, Q3, & Q4:

8)/()/()/()/()5.0()/(101005.010100

)/(5.0

1234

21

66

2111

====×××=×

=−−

LWLWLWLWLW

VLWCI OVoxnD µ

PMOS: Q5, Q6, Q7, & Q8:

32)/()/()/()/()5.0()/(10255.010100

)/(5.0

5678

21

66

2555

====×××=×

=−−

LWLWLWLWLW

VLWCI OVoxpD µ

Page 14: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (14/37)

Differential Mode Half Circuit

7.6340.6k)||k 36(100.4

)||()5.0/(3

31111

−=××−=

−=−=−

iomdvQ RrgvvA

k 6.401

33

33 =

+′+

=om

Loi rg

RrR

8.103 13 −== vQvQd AAA

k 36...mA/V 4.0...

821

821

========

ooo

mmm

rrrggg

k 5901036)364.01(1036

)1(33

7755

=×+×+××=

++=′ oomoL rrgrR

Method 1: Use formula for Cascode Amplifier on Lecture Set 6, slide 14 (which assumes gmro >> 1):

104) 1036100.4(5.0

)( 5.05.0

233

21,

−=×××−=

−=−

=

−d

omd

dod

A

rgv

vA

d

do

d

do

d

dod v

vv

vvv

A5.0

2 1,1,,

−=×−==

Method 2: Use multistage amplifier calculations (similar to Lecture Set 6, slide 14 but not assuming gmro >> 1):

13.6590k)||k 36(100.4

)||(/3

331,13

=××=

′≈=−

LomdovQ RrgvvA

gmro >> 1 is a good approximations

Page 15: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (15/37)

Exercise 5: Assume Q3 and Q4 as well Q1 and Q2 are identical. Compute the differential gain.

This is a practice problem in constructing half-circuit.

Page 16: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (16/37)

Half-circuit for differential Gain Zero voltage at symmetry line Replace Q3 by

Elementary R forms

)||||(

5.0

311

1,,

Poom

d

do

d

dod

Rrrgv

vvv

A

−=−

==

Page 17: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (17/37)

Exercise 6: Compute the differential gain.

This problem has it all, half circuit, constructing resistances from elementary R form, and Cascode amplifier.

Page 18: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (18/37)

Differential-Mode half-circuit

2/|| )]2/||(1[ 775 popomoL RrRrgrR ++=′

)||()5.0/( 31111 iomdvQ RrgvvA =−=33

33 1

om

Loi rg

RrR+

′+=

)||)(||( 3313113 LoiommvQvQd RrRrggAAA ′−==

)||(/ 331,13 LomdovQ RrgvvA ′≈=

Since Rp value is not given, we cannot simplify R’L expression using gmro >> 1 .

v1

Page 19: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (19/37)

Exercise 7: What is the input common-mode range in the circuit below. Q1 and Q2 are Identical and RD = 500. Use µnCox (W/L ) = 8 mA/V2 , Vt = 0.5 V and VG3 = −1 V.

The input common-mode level is the range of DC values that can be applied to the gate of Q1 and Q2 (bias + signal) for which transistors remain in saturation. o Basically we are looking for range of DC

voltages (i.e., bias) that can be applied to Q1 and Q2 while keeping them in saturation.

o Then, for any given bias voltage, we can calculate the range of common-mode signals that can be applied to the circuit.

There are two limits: 1) for Q1 and Q2 remain in saturation, 2) for Q3 to remain in saturation.

It is straight forward to extend this to active loads.

Page 20: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (20/37)

Assume Q1 and Q2 in Saturation

21

3211

3 104 )/(5.0101 OVOVoxnD VVLWCI −− ×===× µ

V 111 =+= tOVGS VVV

V 5.15.022 11 =−=−= DDD IRV

V 5.01 =OVV

115.11

1

≤−≤−−=−=

CM

CMS

SCMGS

VVV

VVV

V 15.1 ≤≤− SVV 5.15.025.0)2(33

33

−=+−≥≥−−=−

S

SSD

OVDS

VVVV

VVFor Q3 in saturation:

V 5.05.0)2(13333 =−−−−=−−=−= tSGtGSOV VVVVVV

For Q1/Q2 in saturation:

V 15.05.15.05.111

11

=−≤≥−=−

S

SSD

OVDS

VVVV

VV

V 25.0 ≤≤− CMV

Page 21: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (21/37)

Exercise 8: Circuit below is designed to operate at zero bias voltage at the gate of Q1 and Q2 (Q1 & Q2 are matched and λ = 0). The practical circuit, however includes a slight mis-match of RD1 = RD − 0.5 ∆RD and RD2 = RD + 0.5 ∆RD (∆RD /RD is small). A) If v1 = v2 = 0, find Vo = vo2 − vo1 (Differential DC voltage at the output). B) For what values of VOS = v2 − v1, the DC output voltage will be zero. Ignore channel-width modulation.

No amplifier chip can be manufactured with perfect symmetry. Mis-matches not only affect CMRR but DC voltages.

Differential DC voltage at the output and the input offset voltage, VOS , are important specs. Chips typically include pins for feedback to zero out these voltages.

Note: v1 and v2 are DC values in this problem, they can be viewed either as mis-matched bias (and no signal) and/or signal (but with a matched “zero” bias).

Page 22: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (22/37)

Since transistors are matched and VGS1 = VGS2 (because v1 = v2 ):

)5.0(5.0)5.0(5.0

2222

1111

DDoDDDDDDDo

DDoDDDDDDDo

RRIVIRVVvRRIVIRVVv∆+−=−==∆−−=−==

oDD III 5.021 ==

Doooo RIvvV ∆−=−= 5.012 Output Offset Voltage

A) If v1 = v2 = 0, find Vo = vo2 − vo1 (Differential DC voltage at the output):

DDDDDD RRRRRR ∆+=∆−= 5.0 & 5.0 21

Page 23: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (23/37)

oDD III 5.021 ==

Method 1: Viewing VOS as the signal.

The bias voltages remain at zero and Vo has the above value. A differential signal vd = VOS is applied to the circuit leading to a differential output , vo,d . We want to find VOS such that vo,d + Vo = 0

Doooo RIvvV ∆−=−= 5.012 Output Offset Voltage

oDOSmdo

DDDDOSmdo

DDOSmdododo

VRVgvRRRRVgvRRVgvvv

−=−=

∆−+∆+−=

+−=−=

,

,

12,1,2,

][5.0)(5.0

DDOSDOV

DDoOSDm RIVR

VIRIVRg ∆−=→∆−= 1

2 5.0 5.0 1D

DOVOS R

RVV ∆×−=

OSOS VvVv 5.0 and 5.0 21 +=−=

Input Offset Voltage

B) For what values of VOS = v2 − v1, the DC output voltage will be zero. Ignore channel-width modulation.

OSDmOSDmdo

OSDmOSDmdo

VRgVRgvVRgVRgv

22,2

11,1

5.0)5.0(5.0)5.0(

−=+−=

+=−−=

Page 24: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (24/37)

221121 DDDDDDDDoo IRVIRVvv −=−→=

Method 2: Viewing VOS as the bias voltage:

0 such that 5.0 and 5.0 1212 =−=−=+= oooosGosG vvVVVVV

)/(5.05.0 0 22121 OVoxnoDDGG VLWCIIIVV µ===⇒==

)/1(5.0 )/1()/(5.0

)( )/(5.0

)50( )/(5.0

2

2

21

OVOSo

OVOSOVoxn

OVOSOVoxn

OSOVoxnD

VVIVVVLWC

VVVLWCV.VLWCI

−=−=

−≈

−=

µ

µ

µ

)/1(5.0)50( )/(5.0 22 OVOSoOSOVoxnD VVIV.VLWCI +≈+= µ

0/)/(2 =∆+ DDOVOS RRVV

)/5.01)(/1(5.0)/5.01)(/1(5.0 DDOVOSDoDDOVOSDo RRVVRIRRVVRI ∆++=∆−−

For:

Find:

5.0 1D

DOVOS R

RVV ∆×−=

2211 DDDD RIRI =

Dropping V2OS terms by

assuming VOS << VOV

Page 25: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (25/37)

Exercise 9: Consider the circuit below with µnCox = 90 µA/V2, µpCox = 30 µA/V2, Vtn = − Vpn = 0.7 V and VAn = − VAp = 20 V. The circuit is to operate such that all transistors operate at VOV = 0.5 V, ID1 = ID2 = ID3 = ID4 = Iref = 0.2 mA, and (W/L )5 = (W/L )6 . a) Design the circuit (i.e., find (W/L) of all transistors). b) Find the differential gain. c) Find the common mode response at vo1 (i.e., vo1/vCM). d) Find the input common-mode range e) Find the allowable range of the output voltage. Ignore channel-width modulation in biasing calculations.

Page 26: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (26/37)

1) Q1 & Q2: PMOS Differential amplifier

2) Q5: Biasing current mirror

5) Q6: Providing reference voltage (or current) for Q5

3) Q3& Q4: current-source/active loads

6) Q7: Providing “Iref” for Q6

4) Qref: The reference leg of current mirror for the circuit

Page 27: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (27/37)

8.17)/(

)5.0()/(10900.5 )/(5.0102.0 2-623

=

××===× −

ref

refOVrefoxnref

LWLWVLWCI µ

mA 4.0 215 =+== DDD IIII

a) Find (W/L of all transistors) ID1 = ID2 = ID3 = ID4 = Iref = 0.2 mA, and (W/L )5 = (W/L )6 .

Step 1: Compute all currents.

NMOS: Qref, Q3, Q4, and Q7

( )( ) mA 4.0

// 5

5

66 =×= DD I

LWLWI

mA 4.067 == DD II

Step 2: Compute (W/L)s (Vov= 0.5 V)

6.35)/(2)/( mA 4.02 77 ==⇒== refrefD LWLWII

8.17)/()/()/( mA 2.0 4343 ===⇒=== refrefDD LWLWLWIII

Page 28: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (28/37)

3.53)/(

)5.0()/(10300.5 )/(5.0102.0

1

21

-6211

3

=

××===× −

LWLWVLWCI OVoxpD µ

PMOS: Q1, Q2, Q5, and Q6

107)/(2)/()/( mA 4.02 165165 ===⇒=== LWLWLWIII DDD

3.53)/()/( mA 2.0 1212 ==⇒== LWLWII DD

A/V 1085.0

102.022 43

1

112

−−

×=××

===OV

Dmm V

Igg

k 001102.0

203

1

112 =

×=== −

D

Aoo I

Vrr

Small signal parameters:

k 001102.0

203

3

334 =

×=== −

D

Aoo I

Vrr

k 50104.0

203

5

55 =

×== −

D

Ao I

Vr

Page 29: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (29/37)

40k)100||k100(108

)||(5.0

4

3111,

−=×−=

−=−

=

−d

oomd

dod

A

rrgv

vA

98.01801

801105010821

10100108

/21

,1,2

34

34,1

1351

31,1

−=++

−==

+××××+×××

−=

++−=

c

co

c

co

c

co

ooom

om

c

co

vv

vvv

vrrrg

rgv

v

b) Find the differential gain: c) Find common mode response, vo1:

Page 30: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (30/37)

d) Find input common mode range:

1) Q5 in saturation:

2.1 V 1.2 || 11 +=⇒=+=− CMStpOVCMS VVVVVV

V 25.05.2 5.2 5155 =−≤=⇒≥−= DSOVDSD VVVVV

2) Q1/Q3 in saturation:

33

11

OVDS

OVSD

VVVV≥≥

5.1 1)5.2(V 1

11

31

−≥⇒≥−−=+≥+

SS

OVOVDSSD

VVVVVV

V 2.0 .51 1 ≤≤− SV

The above equation indicates VS1 changes and tracks VCM as VCM changes. VS1 is limited by two criteria below:

V 2.0 1.2 .51 ≤+≤− CMV

V 0.8 .72 ≤≤− CMV

Note that the requirement on Q1/Q3 in saturation is usually more restrictive than above as Q1/Q3 do not usually reach saturation together (calculation above represents “the best case”). However, correct solution requires that we include channel-width modulation and calculate the relationship between VSD1 & VDS3 (same arguments apply to part e).

Page 31: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (31/37)

e) Find allowable range of output voltage:

1) Q3 in saturation:

V 25.05.2 )5.2(

1

13

−=+−≥≥−−=

o

OVoDS

vVvV

2) Q1/Q5 in saturation:

OVSD

OVSD

VVVV

5

1

≥≥

V 5.1 15.2

V 12

1

115

15

≤≥−=+

=≥+

o

oSDSD

OVSDSD

vvVV

VVV V 1.5 .02 1 ≤≤− ov

Page 32: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (32/37)

Exercise 10: Consider the circuit below with µnCox = 400 µA/V2, µpCox = 100 µA/V2, and Vtn = − Vpn = 0.4 V. All transistors operate at VOV = 0.2 V and ID1 = ID2 = ID3 = ID4 = ID6 = Iref = 0.2 mA a) Design the circuit (i.e., find (W/L) of all transistors) b) Find the input common-mode range c) Find the differential gain (λ = 0.2 V-1)

Page 33: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (33/37)

1) Q1 & Q2: NMOS Differential amplifier with single-ended output (1st stage)

3) Q5: Current- mirror bias for differential

2) Q6: PMOS CS amplifier (2nd stage)

5) Q3/Q4: asymmetric active load for differential amplifier

6) Q7: current-source/active load for Q6 CS amplifier

4) Qref: The reference leg of current mirror for the circuit

Page 34: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (34/37)

25)/( )/(5.0102.0 23 =⇒==× −refOVrefoxnref LWVLWCI µ

mA 4.0215 =+= DDD III

a) Find (W/L of all transistors).

Step 1: Compute all currents.

NMOS: Qref, Q1, Q2, and Q7 (all have same ID and VOV)

mA 2.067 == DD II

Step 2: Compute (W/L)s (Vov= 0.2 V)

50)/(2)/( mA 4.02 55 ==⇒== refrefD LWLWII

25)/()/()/()/( mA 2.0 721721 ====⇒==== refrefDDD LWLWLWLWIIII

mA 2.064321 ====== refDDDDD IIIIII

100)/( )/(5.0102.0 32

333 =⇒==× − LWVLWCI OVoxpµ

PMOS: Q3, Q4, and Q6 (all have same ID and VOV)

100)/()/()/( mA 2.0 643643 ===⇒=== LWLWLWIII DDD

Page 35: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (35/37)

b) Find input common mode range:

1) Q5 in saturation:

V 6.0 111 −=⇒+==− CMStnOVGSSCM VVVVVVV

V 8.0 12.0 )1( 5155 −=−≥=⇒≥−−= DSOVDDS VVVVV

2) Q1/Q3 and Q2/Q4 in saturation (because the circuit is NOT symmetric, we need to consider both cases and choose the most restrictive one).

1333

333

V 0.4 1

V 0.6 ||

DDDSD

tpOVSGSD

VVVVVVVV

==⇒−=

=+==

V 0.2V 2.0 4.0

1

1111

≤=≥−=−=

S

OVSSDDS

VVVVVV

V 2.0 8.0 1 ≤≤− SV

Similar to problem 9, we look at VS1 limits:

2A) Q1/Q3

2666

66

V 0.4 1

V 0.6 ||

DGGSG

tpOVSG

VVVVVVV

==⇒−=

=+=

V 0.2V 2.0 4.0

1

1222

≤=≥−=−=

S

OVSSDDS

VVVVVV

2B) Q2/Q4

V 6.0 2.0 ≤≤− CMV⇒≤−≤−⇒ V 0.2 0.6 .80 CMV

Page 36: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (36/37)

c) Find the differential gain (λ = 0.2 V-1):

A/V 1022.0

102.022 33

1

112

−−

×=××

===OV

Dmm V

Igg

k 25102.02.0

113

112 =

××=== −

Doo I

rrλ

A/V 1022.0

102.022 33

6

66

−−

×=××

==OV

Dm V

Ig

k 25102.02.0

113

66 =

××== −

Do I

k 25102.02.0

113

334 =

××=== −

Doo I

rrλ

k 5.12104.02.0

113

55 =

××== −

Do I

mA 2.064321 ===== DDDDD IIIII

mA 4.0215 =+= DDD III

mA 2.076 == DD II

k 25102.02.0

113

77 =

××== −

Do I

Page 37: Exercises for Differential Amplifiersaries.ucsd.edu/NAJMABADI/CLASS/ECE102/12-F/NOTES/ECE102... · 2012-12-21 · Exercise 3: The differential amplifier below should achieve a differential

F. Najmabadi, ECE102, Fall 2012 (37/37)

Q6: PMOS CS amplifier (2nd stage)

∞=

=×−=

−=

i2

3

766

R

255k)2||k25(102

)||(

x

o

oomx

o

vv

rrgvv

Q1 & Q2: NMOS Differential amplifier with single-ended output (1st stage)

255k)2||k25(102

)||||(

3

311

=×−=

−=

d

x

Loomd

x

vv

Rrrgvv

6252525 =×=×==d

x

x

o

d

od v

vvv

vvA

vx