Esercizi Svolti Feedback - Parte I

Esercizi Svolti e Commentati:

Microelectronics (B. Razavi, Wiley 2014)

Capitolo 12 Feedback

Parte I

1.

E = X - KY

Y = A₂E - W = XA₂ - A₁(X - YK)

Y/X = (A₂ - A₁)/(1 - A₁K)

2. E/X = ?

E = X - KY = X - K(XA₂ - EA₁)

= X - KXA₂ + KEA₁

= (1 - KA₂)/(1 - KA₁)

A)

Y = EA₁ = A₁(X - YA₂K) => Y/X = A₁/(1 + KA₂A₁)

E = X - KA₂Y = X - KA₂EA₁

=> E/X = X/(1 + KA₂A₁)

B)

y = E - W = E - A₁E = E(1 - A₁)

= (X - KY)(1 - A₁)

y/X = (1 - A₁)/(1 + (1 - A₁)K)

E = X - KY = X - K(E - EA₁)

= X - EK(1 - A₁)

E/X = 1/(1 + K(1 - A₁))

D)

y = E - W = E(1 - A₁)

= (X - P)(1 - A₁) = [X - [Y(K - 1)](1 - A₁)

y/X = (1 - A₁)/(1 + (K - 1)(1 - A₁))

E = X - Y(K - 1) = X - (EA₁ + E)(K - 1)

= X - E(1 - A₁)(K - 1)

E/X = 1/(1 - (1 - A₁)(1 - K))

3. Loop-gain - ideal components (Zin = infty, lambda = 0)

A

Vk = -k A1 Vtes

Vn = y R2 / (R1 + R2)

Y = -KAVTES

KA1 = - Vn / Vtes = R2 / (R1 + R2)

B

B = -gm Vtes R0 A2

Y = -gm Vtes R0 A2

Vk = R2 / (R1 + R2) gm(1 R0) A1 Vtes

KA1 = - Vk / Vtes = gm(1 R0 R2 / (R1 + R2)) A1

C

Y = -gm R0 A1

KA1 = gm R0 A1

D

Y / B = -gm R2 / (1 + gmR2)

=> KA1 = gm R2 / (1 + gmR2) A1

4.

X = Vo sin(wt)

Vo = 2 uV    R1/R2 = 7

A1 = 500

Y = (X - Xf)A1

=> Xf = Y R2/(R1+R2)

Y = X A1 - A1 R2/(R1+R2) Y

Y0 = V0 A1 - A1 1/(R1/R2 + 1) Y0 = 0.15 V = 150 uV

Xf0 = 1.97 uV.

X ~ Xf (good feedback).

5.

(1 - 10%) A0 = |A(jw-ib)|

w0 -> -3 dB

A(s) = A0/(1 + s/w0)

0.9 A0 = A0/sqrt(1 + (w-wo/w0)²)

w-ib = 0.46 w0

Closed-loop gain

A0/(1 + eL) 0.9 = |V/X (w-ib)|

=> wcl-ib = 0.48 (1 + L).

Bandwidth is boosted by (1+KA) in closed loop gain.

6.

K'n -> + 10%

lambda -> + 20%

e.e.q. G MIN -> < +5%.

Vout = -gm1 ro E = -gm1 ro (Vin - k Vout)

Vout/Vin = -gm1 ro/(1 - k gm1 ro) = - (1/(lambda ID)) (2 ID/Vov) 1/(1 - k (2/(lambda Vov)))

d(Vo)/Vin/dlambda = -d(lambda)/(1 + Ao k)

2.9%/(1 + Aok) < 5% -> A0k > 3.

7.

X = 0

A₀ = -g_m1r_o

V_CK = -g_m2r_o2

E = V_IN + g_m2r_o2

A₀(s) = -g_m1r_o||(sC_L)^-1 = -g_m1r_o/(1 + r_oC_Ls)

V_out = -g_m1r_o/(1 + r_oC_Ls) (V_in + g_m2r_o2)

V_out/V_in(s) = A₀/(1 + kA₀) / (1 + s/((1 + kA₀)w₀))

K A₀ = g_m1r_o/(1 + r_oC_L) · g_m2r_o2/k

DC: CLC = A₀/(1 + kA₀) = A₀/(1 + kA₀) = -g_m1r_o/(1 + g_m1r_oA

|V_out/V_in(w₀)| = A₀/(1 + kA₀)² / √(1 + 1/(1 + kA₀)²)

8.

A₀ = -g_m1(R_D || n₀)

R_D << Z(c1,c2)

K A₀ = -V_FBs/V_EBs = +g_m1(r_o||R_D) · Z_c2/(Z_c1 + Z_c2)

Z_c2/(Z_c1 + Z_c2) = C₁/(C₁ + C₂)

A = A₀/(1 + K A₀)

9.

Gx1 = A₁/(1 + kA₁) ≈ 1/k (1 - 1/(kA₁))

Gx2 = A₂/(1 + kA₂) ≈ 1/k (1 - 1/(kA₂))

kA₁ >> 1, kA₂ >> 1.

(Gx1 - Gx2)/Gx1 < 0,05 => K > 11/2100.

A₁ = 138,16

A₂ = 131,25

10.

y = d1 x - d3 x³.

A) dy/dx = d1 - 3 d3 x²

dy/dx |_x=0 = d1     dy/dx |_x=...x ~ d1

B) E.I. = d1 / (1 + K d1)

11. ANALYSE MODEL

12.

13.-14. SENSE & RETURN MECHANISM. (FEEDBACK)

A. SENSE Vout (M2) RETURN GATE (M1) VOLTAGE VOLTAGE

B. SENSE CURRENT (M2) RETURN VOLTAGE (M1)

C. SENSE Vout (M2) RETURN CURRENT (M2).

Sense voltage in gate M3.

Return voltage to gate M2.

Sense current Iout.

Return voltage to gate N1.

Sense voltage from R1/R2.

Return voltage.

15.

Iout/Vin = RL/(1/gm1 + Rs)

=> Vout/RL = Iout => Iout/Vin = 1/(1/gm1 + Rs)

Rs = 0 => G = gm1.

16.

A

neg.

B

neg.

C

II approach

Vin up Vout up neg.

Vin down

17.

A)

NEG.

1) V_in ↑ => V_out ↓ => V_gs1 ↓ => V_m1 ↓

2) BREAKING the loop.

FEEDBACK AMPLIFIER M1 CS

FEEDBACK M2 SF.

B)

POS.

V_in ↑ V_out ↓ V_ds1 ↑

FEEDBACK

FORWARD

C)

POS. V_in ↑ V_out ↓ V_gs1 ↑ => V_out ↓ NEG.

FEEDBACK FORWARD

D)

NEG.

V_in ↑ V_out ↑ V_y ↓ => V_s1 ↓

19.

C.L.G? I/O IMP?

A. d=0 B. d?0

C.L.G = A₀ / (1 + K A₀)

A₀ = g_m r_o (ro||ro)

K = 1    SENSE & RETURN VOLTAGE

R_OUT = R_OUT,0 / (1 + K A₀)

R_IN = R_IN (1 + K A₀) = ∞

R_OUT,0 = r_op || r_on

20.

A₀ = g_m1r_o1(ro||ro) · g_m5r_o5

K = 1

C.L.G = A₀ / (1 + K A₀)

R_IN = ∞

R_OUT = R_OUT,OPEN / (1 + K A₀) = r_o5 / (1 + A₀)

21.

A = g_m5r_o5

R_OUT = (1/g_m5 || r_o5)

23.

Ao = + g_m(r_o1||r_o3) g_m5r_o5

A_CL = Ao / (1 + Ao)

R_out,CL = Ro / (1 + Ao)

R_out,0 = r_os

24.

positive feeback..

I_IN up V_X down V_GS,2 up V_OUT down I_F up V_GS,2 up

negative feedback

25.

A) I_IN up V_G,2 down V_GS,2 up V_OUT up I_F down V_GS,2 up

positive feedback.

B) R_F large

V_F = V_X - g_m1R_D / (1 + g_m2(R_F + 1/g_m1))

-V_F / V_X = - g_m1R_D / (1 + g_m2(R_F + 1/g_m1))

positive feedback.

26.

R_o = I_out = I_IN - I_F

I_IN up I up V_G,2 up V_GS,1 up V_OUT up I_F down I₁ up V_GS,2 up

positive feedback

V_X / I_IN = R_D

V_OUT / V_X = R_Sg_m1 / (1 + g_m2R_S)

Ao = -R_D * (R_Sg_m1) / (1 + g_m2R_S)

K = -1/R_F

R_out,0 = R_S||(1/g_m2||r_o2)

R_out,CL = R_out,0 / (1 + KAo)

27.

(A) Rx = (1+gm2ro2)ro1 + ro2

(B) Rx = (1+gm2ro2)(ro1||ro2)

28.

Ro = Ro

RIN = 1/gm1

K = + gm2 C1/(C1+C2)

RCOUT,CL = Ro/(1+kRo)

RIN,CL = 1/gmi/(1+kRo)

29.

RM very small

VA = oo

A) Gm = Iout/VIN = gm1A1

B) K = VF/Iout = RM

Gc.L = Gm/(1+Kgm)

30

A) Gm = -gm1A1

B) K = RM

Gc.L = Gm/(1+KGm)

POSITIVE FEEDBACK