Esercizi stadi Cascode e specchi di corrente - Parte III

Esercizi Svolti e Commentati:

Microelectronics (B. Razavi, Wiley 2014)

Capitolo 9 Cascode Stages e

Current Mirrors

Parte III

25.

A_V = 200 I_D = 1 uA u_nth = 100 uA/V² lambda = 0.1 V^-1 ( W / L )₁ = ( W / L )₂

A_V = g_m1 r_o1 g_m2 r_o2 = g_m1² r_o²

r_o = 1 / lambda I_D = 10 k

g_m1 = sqrt(A_V) / r_o = 0,0014 S

g_m1 = sqrt(2 u_nth I_D (W / L)) => (W / L)_1,2 = g_m1² / (2 I_D u_nth) = 9,8

27.

A_V = - g_m1 [ g_m2 r_o1 r_o2 || g_m3 r_o3 r_o4 ]

g_mN = sqrt(2 I_D u_nth K_N (W / L))

r_o = 1 / lambda I_D

A_V = 500 d_n = 20,1 V^-1 = 20 / 0,18 u_nth = 400 uA/V² u_pth = 50 uA/V²

g_m2 r_o1 r_o2 = g_m2 r_o² = sqrt(2 K_N / x_N I_D³)

g_m3 r_o3 r_o4 = g_m3 r_o² = sqrt(2 K_P / x_P I_D³)

K_N = k'n_N W / L = 0,011 K_P = k'p_P W / L = 0,005

sqrt(2 I_D K_N) 1 / sqrt(lambda_P² I_D² / 2 K_P + 1 / sqrt(lambda_N² I_D² / 2 K_N)

I_D² (lambda_P² / 2 K_P + lambda_N² / 2 K_N) = 2 K_N / A_V² = 8,8*10^-8.

I_D = 1,26 uA.

28.

g_out / V_IN = - ( g_m0 + 1 / (g_o1g_o2) ) = G_u.

R_out = Ro||Rp

R_0p = (1 + g_m3 r_o3) r_o4 + r_o3 = g_m3 r_o3 r_o4 R_0N = r_o2

29.

A

Gm ≈ -gm₁

ROUT = Ron || Rp

Ron = ro1 [ (1 + gm2 r02 || Rf) + ro2 || Rp ]

≈ gm2 r02 || Rp) ro1

Rp ≈ gm3 ro1 ro3

B

Gm ≈ -gm1

Rp ≈ gm3 ro3 ro4

Rn ≈ gm2 ro2 (ro1 || Rf)

C

Gm ≈ -gm5

Rp ≈ gm3 ro3 ro4

Rn = gm2 ro2 (ro1 || ro5)

D

Gm ≈ -gm5

Ron ≈ gm2 ro2 ro1

Rop ≈ gm3 ro3 (ro4 || ro3)

30.

R2/(R1+R2) Vcc = Vt ln(I1/Is) => sensitivity dI1/dVcc => I1 = Is exp((Vcc/Vt) R2/(R1+R2))

I1 = (1/2) mu_n Cox (W/L) (R2/(R1+R2) Vcc - Vth)^2

g_m = dI/dVGE.

dVBE/dVCC : dI/dVBE = dI/dV2 = g_m dVBE/dV2.

g_m = 2I/VBE? approx Ic, => Ic linear Vcc.

31.

I1 = (1/2) mu_n Cox (W/L) (VGS - Vth)^2

dI1/dVth = - mu_n Cox (W/L) (R2/(R1+R2) VDD - Vth)(low power issue)

g_m = mu_n Cox (W/L) (VGS - Vth)

dI/dVth = dVGS/dVth * dI/dVGS = g_m dVGS/dVth.

32.

Vref / R1 = (0 - V1)/(Vgmu |Mt1|)

V1 approx Vref

V1 + Ve2 > 0 Ix > 0 => Ve2 > -V1.

33.

Vcc = 10V

VA = 50V beta = 5.

Iref = Ic1 + I1 = Ic1 + Ib2 + Ib1 = Ic1 + Ic2/beta + Ic2/beta

Iref = Ic2 (1 + 2/beta)

Vref = Vcc - Vc = Vcc - (Vce1 + Ve1) = Vcc - (Vce1 + Vce2) = Vce2 - Vce1

Vbe1 = Vce1 = Vbe2 = Vce2

Ic = Is exp(Vbe/Vt) (1 + Vce/VA)

34.

V_BE,REF + I_REF - I₁

I_C,REF = I_REF - I_x = I_REF - I_B1 - I_B2 =

= I_REF - I₁/B - I_C,REF/B

→ I_C,REF = ((B I_REF - I₁)/(B + 1)) ≈ I_REF - I₁/B

V_BE,REF + I_C,REF R_P = V_BE,2      I₁ = 2 I_REF

V_T ln(I_REF/I_S,REF) + I_C,REF R_P = V_T ln(I_C2/I_S2)

V_T ln(I_REF/I_S) - V_T ln(2 I_REF/I_S) + I_C,REF R_P = 0

V_T ln(I_REF/(2 R_eq)) = - I_C,REF R_P ⇒ R_P ≈ V_T ln(2)/I_REF

35. Rout?

R_out ≈ r_o1

R_IN,REF ?

I₁ = g_mv₁ + v₁/r_o

36. I₁ = 0.9 I_C,REF

V_BE,REF = I₁/B R_P + V_BE,1

I_C,REF = B/0.9 R_P V_T ln(10/9)

I_REF = I_C,REF + I_C(hfe)/B + I₁/B

→ R_P = ((B + 1) V_T ln(10/9)) / (0.9 (I_REF - I₁/B))

37.

(A)

I_S = A_ED_mm_i² / (ω_BN_B)

I_C α A_EI_S α (ω_B)^-1

Icopy α I_S α A_E

=> Icopy = 5I_REF.

I_REF = I_C,REF + Icopy/β + I_C,REF/β => Icopy = I_REF(5β/(β+6))

ERROR   Icopy/I_REF = 5β/(β+6).

(B)

Icopy = I_C,REF/5

=> Icopy = I_REF(β/(5β+6))

(C)

Icopy = 3/2 I_C,REF

I₂ = 5/2 I_C,REF = 5/2 · 2/3 Icopy

I_REF = I_C,REF + I_C,REF/β + Icopy/β + I₂/β

= 2/3 Icopy + 2/3 · 1/β Icopy + 1/β Icopy + 2/3 Icopy 2/5

= 2/3 Icopy(1 + 1/β + 3/2 1/β + 2/5) = 2/3 Icopy(5/2β + 7/5)

Icopy = 3/2 I_REF / (5/2β + 7/5)

38.

Icopy1 = I_REF/2.

Icopy1,2 = 6/2 I_C,REF = 6 Icopy1

I_REF = I_C,REF + I_C,REF/β + Icopy1/β + Icopy1,2/β => Icopy1 = I_REF(βu)/((β+1)u + 7βu)