Esercizi Amplificatori CMOS - Parte III

Esercizi Svolti e Commentati:

Microelectronics (B. Razavi, Wiley 2014)

Capitolo 7 Amplificatori CMOS

Parte III

37. λ=0

(A) V_IN/V_x = R_IN/(R_S+R_IN) = 1/(1+g_m1R_S)

V_out/V_x = g_m1 · 1/g_m2 = g_m1/g_m2

=> A_v = 1/(1+g_m1R_S) · g_m1/g_m2

(B) A_v = g_m1/g_m2

(C) V_IN/V_x = R₁||(1/g_m1)/(R₁||(1/g_m1)+R_S) = 1/(1+(1+g_m1R₁)R_S)

V_out/V_x = g_m1/g_m2 => A_v = g_m1/g_m2 · 1/(1+(1+g_m1R₁)R_S)

(D) A_v = g_m1R_eq

R_eq = R_D + 1/g_m2

(E) A_v = g_m1R_eq

R_eq = R_D || 1/g_m2

38.-39. OUTPUT SWING.

V_out > V_G - |V_tp|

-g_m2r_o2 > V_G - |V_tp| => V_G < |V_tp| - g_m2r_o2

V_out > V_GG⁺ - V_thn => V_G < (V_GG⁺ - V_thn)/(g_m1r_o1)

40.

I_x = g_mV_x + V_x/r_o + V_x/R_S => I_x = g_mV_x + V_x/r_o + 1/R_S V_x

I_x/V_x = g_m + 1/r_o + 1/R_S => (1/g_m || r_o) || R_S

R_out = r_og_m/(1 + g_mr_o)

41.

R_out = (1 + g_m2r_o2)R_E + r_o

R_E = 1/g_m2 || 1/g_m2

42.

V_DD = 2V

W/L = 300

R_E = 50 ohm

P = 20 uW

R_V = ?

I_D = P/V_DD = 0,01A

A_V = R_Eg_m/(1 + g_mR_E)

A_V = 0,65

g_m = sqrt(2u_nC_oxW/L I_D) = 0,0385 = (26 ohm)^-1

43-44.

V_DD = 1,8V

I_D = 2mA

A_V = 0,85

lambda = 0

R₁ = 5K

R₂ = 220K

V_GB = R₂/(R₁ + R₂) V_DD = 1,76V

V_GS = 1,76V - I_DR_S

A_V = g_mR_S/(1 + g_mR_S) = 0,85

2I_DR_S/(V_OV(V_OV + 2I_DR_S)) = 0,85

0,3I_DR_S = 0,85V_OV = 1,496V - 0,85I_DR_S

=> R_S = 1,496V/(1,15 I_D) = 680,43 ohm

V_G ↑ V_OV ↑ R_S

45.

Av = RD / (1/gm1 + 1/gm2 || Rs)

CB

46.

Rout < 6 Ohm

P = 3 mW

lambda = 0

VDD = 1,8

1/gm = 60 => gm = 0,0165 = (60n)^-1

I = P/100 = 1,67 uA

gm² / (2IDumutex) = W/L = 383

47.

Av = 0,85

P = 2 mW

(W/L)? C source.

VDD = 1,8V

RL = 50

Av = gmRL / (1 + gmRL) = 0,85

gm = 0,85 / (0,15RL) = 0,113 ~ (8,8u)^-1

ID = 1,11 uA => W/L = 8u gm² / (2IDumutex) = 28459

48.

A) Av = gmuRL / (1/gmuRL)

RL = (RS + ro) || ro1

B) Av = gmuRL / (1 + gmuRL)

RL = ((1 + gmu2(ro))RS + Rs) || ro

C) Av = gmuRi / (1 + gmuRL)

RL = (1/gmu2 || ro2) || ro1

D) Req?

I_R1 = I_x - g_muV_gs - V_x/r_o

V_gs = R₂/(R₁+R₂) V_x

(V_x - R₂ V_x/(R₁+R₂)) 1/R₂ = I_x - g_muR₂V_x/(R₁+R₂) - V_x/r_o

R₁/R₂ V_x/(R₁+R₂) + g_mu R₂V_x/(R₁+R₂) + V_x/r_o = I_x

V_x/(R₁+R₂) (R₁/R₂ + g_muR₂) + V_x/r_o = I_x

1/(R₁+R₂) (R₁/R₂ + g_muR₂) + 1/r_o = Y_x

R_x = r_o2(R₁+R₂)/(R₂+R₁+r_o2+g_mu2r_o2R₂)

R_eq = (1 + g_mu2r_o2)R_S + R_S

A_v = g_mu1R_S/(1 + g_mu1R_S)

V_DD = 1.8V

R_C = 20K

I_D = 5uA

λ₁ = λ₂ = 0.0001V^-1

(W/L)₁ = 300/2

A_V = g_m1R_L/(1 + g_m1R_L)

R_L = r_o2 = 200KΩ

g_m = (2u_nC_ox W/L I_D)^1/2 = 0,0475 = (83Ω)^-1

=> A_v = 0,99

50.

Av = 5

Rout = 1 kOhm

RG = 10 kOhm

C1, C2 very large

lambda = 0

VDS < Vov true

VDS - Vov = 100 mV, hence not much less.

Rout = RD = 1k

VDS - RD ID - VGS + Vth = 100 mV => ID = 0,4 - 0,1 / RD = 0,3 mA.

Av = gm RD => gm = 0,00555 s = (200 ohm)^-1 w/L = 208.

51.

VDD = 1,8 V

Av = 4

Vo = 2 mV

VRS = 200 mV

Vov < 300 mV

PR1,R2 <= 5% PTOT

ID = P/VDD = 1,16 mA.

RS = VRS/ID = 169 ohm.

Av = RD / (1/gu + RS) = gm RD / (1 + gm RS) = 4

gm = 2ID/Vov = 0,0218 S = (47 ohm)^-1 Vgs = 0,5 V => Va = 0,7 V

RD = 4 (1 + gm RS) / gm = 940 kOhm.

VDS > Va or Vov = 0,1 < 300 mV

I1 = Ptot/VDD = 0,05% PTOT / VDD = 56 uA

R1 + R2 = VDD/I1 = 32,143 k

Va = R2/(R1 + R2) VDD => R2 = Va/VDD (R1 + R2) = 12,5 kOhm.

R1 = 20 kOhm

(w/L) = gu² / (2 ID mu_n Cox) = 1060.

AV = 6

CS short circuit at f = 1.5G

P = 2uW

RIN = 20K

VDD = VOV + 200mV

C1, C2 negligible at 1MHz

RIN = R1 = 20K

ID = P/VDD = 1,11mA

VD = 1,6V

VDD - IDRD = VG - VTH + 0,2B? RD = (VTH - 0,2)/ID = 180 ohm

AV = -gmu RD -> gm = AV/RD = 0.033S = (30 ohm)-1 not valid

AV = gmuRD/(1+gmuRS) -> RD - 6RS > 0 => RS < RD/6 = 30ohm

=> RS = 20 ohm

VS = 0,022V

gm = AV/(180 - 6:20) = 0.1

=> w/L = gm^2/(2IDmunCox) = 363.

C1

RIN = 1/sc1 + R1 = R1 => |SC1|-1 << R1

=> C1 >> 1/(2piR1) = 8 pf.

C1 = 160pF.

C2

(Zc)-1 >> 1/gmu||Rs

SCs + 1/gmu + 1/Rs

Cs >> 1/(2pi f) (1/gmu + 1/Rs)

Cs = 31uF.

53.

VDD = 1.8

Av = 20

p = 2mW

l = 0.1V^-1

Vout < 1.5V

M2 saturated

M1

ID = P/VDD = 2mW/1.8 = 1.11uA

ro1 = ro2 = 1/(lID) = 9K

Av = gm1 (ro2 || ro1) = gm1 (ro2/2) => gm1 = 40/ro = (0.0044 S) = (225)

VDS2 = Vb - |Vthp| < VDD - Vout M2 sat.

VIN(=VGS1) - Vth < VDS1(=Vout) => VDS1 < 1.5 => VGS1 = 1.2V

= VGS1 < 1.6V => VGS1 = 1.2V

gm1 = 2ID/Vov => (W/L)1 = gmw^2/(2uCoxID) = 43

Vb > 1.1V => Vo = 1.2V VGS2 = 0.6V VDS2 = 0.3V

(W/L)2 = 2ID/(Vov^2(1 + lVDS2)uPcox) = 135

54.

CB very large

lm = 0.5p = 0.1V^-1

VDD = 1.8V

A) Av = ?

B) Av = 15, p = 3mW

Rca = 10(ro1 || ro2)

Vout,ac = VDD/2

Av = -(Rca || ro2) gm2

ID = P/VDD = 1.7mA

ro1 = 5.9K

ro2 = ro1/2 = 2.95K

Rca = 20K

VSG2 = VDD - Vov = 0.9

VSD2 = 0.9

gm2 = 2ID/Vov = (0.0068 S) = (147)

(W/L)2 = gmw^2/(2umCoxID) = 136

gm1 = Av/1.8K = 0.0083 S = (120)

(W/L)1 = gmw^2/(2umCoxID) = 101

VDS1 = 0.9V