Esercizi Amplificatori CMOS - Parte I

Esercizi Svolti e Commentati:

Microelectronics (B. Razavi, Wiley 2014)

Capitolo 7 Amplificatori CMOS

Parte I

CMOS AMPLIFIERS - CAP 7

1)

VDD = 1,8VRQ = 25KRD = ?lambda = 0w/l = 3:1M1 saturation

VTH = 0,4unCox = 200 uA/V²

VQ = VDD - IQRQ

VD = VDD - IDRD

IQRD < VTH

RD < VTH/ID = 680 ohm

VGS >= VDS - VTH

VD >= VDD - VTH

ID = 1/2 unCox w/l (VGS - VTH)² = 0,588 uA

IQ ~ 0A

2.

VQ = VTHEV

RIN = R1 || R2 = R1R2/(R1+R2) >= 20K

VTHEV = R2 VDD/(R1+R2)

ROQ = R1 || R2

ID = 1 uAw/l = 20/0,18RIN > 20K

VGS = VTH + sqrt(2ID/(unCoxw/l)) = 0,7 V = R2/(R1+R2) VDD

R1/R2 = 11/7

R1 - 20K R1/R2 = 20K > 20

R1 >= 20K . 11/7 + 20K = 51,4K

R2 >= 32,7K

3.

VDD = 1,8VRQ = 10KRD = 1KRS = 10 ohm

VDS > VOV

VD = VDD - IDRD

VOV = VGS - VTH

=> VD - VS >= VG - VS - VTH

VD >= VQ

VDD - IDRD >= VDD - VTH

IQRD <= VTH => ID <= VTH/RD = 0,4 uA

VS = IDRS = 0,04 V

VGS = 1,76V

gm = 2ID/(VGS - VTH) = 0,588 uS = (1700 ohm)^-1

4.

VDD = 1.8V

VDS = 200mV

um Cox = 200 uA/V²

VTH = 0.4V

RD = 500ohm

RS = 100ohm

A. (W/L)_min in SAT.

B. RIN > 30kO

VD > VG - VTH

VG = R2/(R1 + R2) VDD

ID = VDS/RS = 2mA => VOV ≤ 0.6

ID = 1/2 um Cox (W/L) VOV² => W/L = 2 ID / (VOV² um Cox)

=> VOV = 0.6 => (W/L)_HMW = 56

=> VGS = VG - VS = R2/(R1 + R2) VDD - 0.2 = 1.2 => R2/R1 = 2

RIN = R1 || R2 > 30k

R1R2 = 30kR1 - 30kR2 > 0

R2 ≥ 30k + 60k = 90k.

R1 > 45K

5.

VDD = 1.8V

RD = 500ohm

RS = 200ohm

W/L = 20/0.18

IR2 = ID/10

R1 || R2 - ID = 0.5mA

VGS = IR2 R2

VS = ID RS = 0.12V => VGS = VTH + sqrt(2IDS/(um Cox W/L))

=> VDS = 0.61 => VG = 0.712V

R2 = VGS/IR2 - 10 = 14.21k

VG = VDD - IR1 R1 => R1 = 21.76k

6.

V_DD = 1,8 V u_nC_ox = 200 uA/V² V_TH = 0,4 V I_D = 1 mA g_m = (100uS)^-1

g_m = 2I_D/(V_GS - V_TH) => V_GS = V_TH + 2I_D/g_m = 0,6 V = V_D

V_D = V_DD - I_DR_D     V_DS = V_GS = V_D => R_D = (V_DD - V_GS)/I_D = 1,2k

7.

V_DD = 1,8 V R_D = 2k u_nC_ox = 200 uA/V² V_TH = 0,4 V I_D = 0,5 mA k = 50/0,18

I_R = I_D/10

V_GS = V_TH + sqrt(2I_D/(u_nC_oxw/l)) = 0,534 V     R₂ = V_G/I₂ = 10,69k

V_D = V_DD - I_DR_D = 0,8 V => I₂(R₁ + R₂) = V_D => R₁ = V_S/I₂     R₂ = 5,32k

8.

V_DD = 1,8 V R_G = 20k R_D = 2k V_GS = V_DS + 100 uV V_GS = V_DS + 50 uV

u/l ? R_P ? -> PARASITIC RESISTOR.

without R_P:

V_G = V_DD = V_GS

=> V_GS - V_DS = 100 uV => I_DR_D = 100 mV

V_D = V_DD - I_DR_D = V_DS     => I_D = 50 uA     => w/l = 2I_D/(u_nC_oxV_ov²) = 0,29.

with R_P:

V_GS - V_DS = V_{R_P} = 50 uV => V_GS = V_DD - R_G * 50uV/R_P

V_DS = V_DD - (I_D - 50uV/R_P)R_D = V_GS - 50uV

=> (R_P)^-1 = 1,408 (V_DD - V_GS)/R_G => R_P = 38,3k.

9.

VDD = 1,8V

R1 = 10k

R2 = 20k

RS = 200k

RD = 1k

RP -> oo

VGS = VDS

ID = 1mA

RP < oo

VGS = VDS + Vth

WP/L?

Vt0 = VGS - Vth = 0,8V

W/T R0 :

Vg = R2/(R1+R2) VDD => (W/L) = 2ID/(Cox un) Vov²

Vs = ID RS

= 15,62

WITH Rp :

VGS = R2/(R1+R2) VDD

Vs = IO RS = (ID + IRP) RS

VD = VDD - IX RD = VDD - (ID + IRP)RD => IX = (VDD - VD)/RD = 1,4mA

=> Vs = 0,28V VGS = 0,92V VDS = 0,12V

=> VDS = RP IP = RP(IX - ID)

ID ? IF ID = 1mA => RP = 3000.

10.

V_B - I

M1 M2

L1 = L2 = 0,25 um

lambda = 0,1 V^-1

V_DS1 = V_DS2 = V_B = 0,8 V

W1, W2 : I_x = 2 I_y = 1 uA

Rout ?

I_x = 1/2 mu Cox (W/L) (V_GS - V_TH)² (1 + lambda V_DS) => W₁ = 2L I_D / [mu Cox Vov² (1 + lambda V_DS)] = 44,5 um

I_x = 2I_y => W_y = W_x / 2 = 7,25 um.

r_o = 1 / (lambda I_D)

r_o1 = 10k

r_o2 = 20k

11.

I_x = I_y = 0,6 uA

V_o1 = 1 V

V_o2 = 1 V

ass V_DS = V_B

L1 = L2 = 0,25 um

W1, W2!

r0 ?

W_x = 2L I_x / [mu Cox Vov² (1 + lambda V_DS)] = 2,76 um

W_y = 3,78 um

r0 = 1 / (lambda I_D) = 16,66k.

12.

V_DD = 1,8 V

(W/L)₁ = 10/0,18

(W/L)₂ = 30/0,18

lambda = 0,1 V^-1

V_DS = V_X = 0,9 V

mu n Cox = 200 uA/V²

mu p Cox = 100 uA/V²

V_TN = 0,4 V

V_TP = -0,4 V

I₁ = I₂ => mu n Cox (W/L)₁ (V_GS - V_TN) (1 + lambda V_DS) = (W/L)₂ mu p Cox (V_GS + |V_TP|) (1 + lambda V_DS)

V_GS,1 = V_B

V_DS,1 = V_X

V_GS,2 = V_B - V_DD

V_DS,2 = V_X - V_DD

2 (W₁/W₂) (V_B - V_TN) (1 + lambda V_X) = (V_DD - V_B - |V_TP|) (1 + lambda V_DD - lambda V_X)

0,72 V_B - 0,72 V_TN = 1,526 - 1,09 V_B => V_B = 1,0 V

13.

u = 8/a2

Vx = 1,2V ?

lambda = 0,1V^-1

VB1 = 0,25V

upCox = 100 uA/V^2

Zs = 1/gm = [sqrt(2 upCox (w/l) ID (1 + lambda VDS))]^-1 = [8,96 . 10^-3 ID]^-1

ID = 1/2 upCox (w/l) (|Vgs| - |Vthp|)^2 (1 + lambda |VDS|)

|Vgs| = |VB - Vx| => ID = 2,07 uA => Zs = 2,35 kohm.

14.

u = 10/a25

d = 0,1V^-1

VB1 = 0,2V

Vx = -1,2V

Zs = 1/gm || ro, ro >> 1/gm.

Zs ~= 1/gm

|VGS| - |Vthp| = |VB - Vx| - |Vthp| = 0,6V

ID = 0,806 uA => gm = 2ID/VOV = 0,00475 = (223 ohm)

Zs = 223 ohm

15.

(w/l)1 = 4/0,15

(w/l)2 = 10/0,2

lambda1 = 0,1V^-1

lambda2 = 0,2V^-1

Vx = 0,8V

VB?

VDD = 1,8V

ID1 = ID2 =>

ID1 = upCox (w/l)1 (VGS - Vth)^2 (1 + lambda1 VDS1)

ID2 = upCox (w/l)2 (|VGS| - |Vthp|)^2 (1 + lambda2 |VDS2|)

2 (VB - VTH)^2 (1 + lambda1 Vx) = (VDD - VB - |Vthp|)^2 (1 + lambda2 (VDD - Vx))

VB - VTH ~= VDD - VB - |Vthp|

VB ~= (VDD - |Vthp| + VTH) / 2 ~= 0,9V

16.

VDD = 1,8V

W/L = 30/0,18

l = 0,9 RD = ?

unCox = 200 uA/V²

VTH = 0,4V

A) ID = 0,5uA VQ ?

M1 saturation

B) AV = ?

VDS > VGS - VTH => VGS < VDS + VTH = VDD - ID RD + VTH = 1,2V

VGS = VTH + square root of 2ID/unCoxW/L = 0,573 ok ?

AV = - RD gm

gm = 2ID/VOV = 0,00575 = (173,3)^-1

=> |AV| = 11,4

17.

AV = 5

W/L < 20/0,18

PTOT < 1 uW

VDD = 1,8

RD = ?

AV = gm RD = 5

PTOT = VDD ID < 1 uW

=> ID < 0,55 uA

=> gm,max = 0,000495 = (202,2)^-1

=> RD = 1,02 K, MINIMUM VALUE

18.

VDD = 1,8V

AV = 10

ID = 0,5 uA

lambda1 = 0,1 V^-1

lambda2 = 0,15 V^-1

A) (W/L)1

B) (W/L)2 = 20/0,18 => VB ?

ro1 = 20K ro2 = 13,33k => ro1||ro2 = 8,08k

AV = gm2 (ro2||ro1)

=> gm2 = AV/ro2||ro1 = 0,00138 = (808,2)^-1

gm = square root of 2 unCox (W/L) ID => (W/L)1 = gm²/2ID/unCox = 26

ID1 = 1/2 unCox (W/L)(VB - VTH)² => VB = VTH + square root of 2ID/unCox/(W/L) = 0,6V