Esercizi Amplificatori Bipolari - Parte III

Esercizi Svolti e Commentati:

Microelectronics (B. Razavi, Wiley 2014)

Capitolo 5 Bipolar Amplifiers

Parte III

38

Av=? Z=10? VA=oo

RIN = rpi + (beta+1) RE

ROUT = RQ || ROOUT

Rup = (routQ || 1/gmu2) + R1

ROOUT = oo

Av = |Vout/Vin| = - Rup / (1/gmu1 + RE)

RIN = rpi + (beta+1) (1/gmu2 || rout2)

ROUT = RC

Av = - RC / (1/gmu1 + (1/gmu2 || rout2))

RIN = rpi1 + (1/gmu2 || rout2)(beta+1)

ROUT = RC

Av = - RC / (1/gmu1 + (1/gmu2 || rout2))

RIN = RB + rpi1 + (1/gmu2 || rout2)(1+beta)

ROUT = RC

Av = - RC / (1/gmu1 + (1/gmu2 || rout2)) . (rpi1 + (1/gmu2 || rout2)) / (RB + rpi1 + (1/gmu2 || rout2))

Vx = RIN / (RB + RIN)

Av = - RC / (1/gmu1 + RE + RB/(beta+1))

e

Vin R_B

Vcc

R_C

Vout

V_B

Rin = R_B + r_u1 + (p+1)(1/g_u2 || r_u2)

Rout = R_C

Av ≃ -R_C / (1/g_u1 + 1/g_u2 || r_u2 + R_S/(beta+1))

N.B. A_v,Beq = -g_u1R_C / (1 + (1/r_u1 + g_u1)R_E)

g_u1 >> 1/r_u1

= -R_C / (1/g_u1 + R_E)

39. V_A = oo

A

R_IN = r_u1 + R_E(beta+1)

R_out = R_C + (1/g_u2 || r_u2)

Av ≃ -R_out / (1/g_u1 + R_E)

B

R_IN = r_u1 + R_E(beta+1)

R_out = (1/g_u2 || r_u2)

Av ≃ -R_out / (1/g_u1 + R_E)

(C)

RIN = re1 + (1/gm3) || rpi3

ROUT = RC + (1/gm2) || rpi2

Av = - ROUT / (1/gm1 + (1/gm3) || rpi3)

(D)

RIN = rpi1 + (beta + 1)RE

ROUT = RC || rpi2

Av = - RC || rpi2 / (1/gm1 + RE)

ROUT = ro + (1 + gm1ro)RE ≈ (1 + gmRE)ro

ROUT,A = ro2 + (1 + gm2ro2)RE

RE = 1/gm2

ROUT,B = ro2 + (1 + gm2ro2)ro2

ROUT,A < ROUT,B -> CASCODE TOPOLOGY

41.

(A)

ROUT = ro1 + (1 + gm1ro1)(1/gm2 || ro2)

(B)

ROUT = ro1 + (1 + gm1ro1)RE

RE = (1/gm + RB/(beta + 1)) || ro2 ?

Rout = ro1 + (1 + gu1ra)(R1 || rpi2)

42.

(A)

Vcc = 2.5V    DC-ANALYSIS    beta = 100

C1 = OPEN    IS = 8.10^-16A    Va = oo

Ib = (Vcc - (VBE + VE))/RB = (Vcc - VBE - IERE)/RB = (Vcc - VBE)/((beta+1)RE + RB)

VBE = 0.750V   => Ib = 15.86 uA   => Ie = 1.586 uA

VBE = 0.736V   => Ib = 16.02 uA   => Ie = 1.602 uA

VBE = 0.736V   => OK

gm = Ic/Vt = (16.23,2)^-1

rpi = beta/gu = 1.63kOhm

from small-sig. C1 short.

|Av| = vout/vin = vx/vin = 8.6

|vout/vx| = Rcgu/(1 + guRE)

vx/vin = 1

(RB || Rin)

(B)

DC-ANALYSIS. C1, C2 OPEN.

Ib = (Vcc - VBE - IERE)/RB

=> Ib = (Vcc - VBE)/((beta+1)RE + RB) = 252K

VBE = 0.750 => Ia = 6.94 uA => Ic = 0.69 uA => VBE = 0.715 => Ib = 7.08 uA

=> Ic = 0.708 uA    VBE = 0.717 V

gm = (36.72)^-1 = 0.027 S

rpi = 370kOhm

RE = 0   C2 BIased

|Av| = Rcgu/(1 + guRE) . Req/(Req + Rig)

Req = RB || rpi.

c

VIN 1k C1 34k VCC1k1kRC = 10k500 2kC2

DC ANALYSIS

VCCReqVrefVref = 1,1VReq = 6,16kQ

RC = 10kVoutRE = 2,900

IB = (Vref - VBE) / ((beta+1)RE + Req) = 23,9uA

VBE = 0,7 V => IB = 6,9 uA => IC = 0,69 uAVBE = 0,715V => IB = 6,89 uA => IC = 0,689 uAVBE = 0,676 => IB = 7,00 uA => IC = 0,700 uAVBE = 0,745 => IB = 6,89 uA => IC = 0,689 uA

VBE = 0,687 V IC = 0,69 uAIB = 1,16 uA

gm = 0,00631 S = (158,5Q)^-1rpi = 15,87k

AC ANALYSIS

VIN Rsig X RthevSRthevVCC600

|Vx| / |VIN| = 5,63k / (5,63k + 1k) = 0,85

|Vout| / |Vx| = RCeq / (1 + gmRE) = 15,8

|Vx| / |VIN| = Rthev || Rin / (Rthev || Rin + Rsig)

Rin = rpi + (beta+1)RE = 65,87k

=> |Av| = 13,43.

43.

IB = 20uAbeta = 100VA = 00

IC = beta IB = 2uA

gm = IC / VT = 0,0769(32V)^-1VT = 1,307K

Av = gm RC = 38,25Rin = 1/gm = 13QRout = RC = 500Q

44. VA = oo

IC = 2 mA

beta_npn = beta_pnp = 100.

Rin = (1/gm1 || rpi1)

Rout = RC + (1/gm2 || rpi2)

Av = gm1 [RC + (1/gm2 || rpi2)]

B

Rin = (1/gm1 || rpi1)

Rout = oo.

Av = 90

C

Rin = RE || rpi1

Rout = RC + (1/gm2 || rpi2)

|Av| = gm1 (RC + 1/gm2 || rpi2)

45. A

IC = 2 mA beta = 100 VA = oo.

B

Rin = 1/gm1 || rpi1 + R2/(beta + 1)

46.

Re = 75

Rc = 1k

Rin = RE + (1/gm || rct)

47.

R1 = 10k

R2 = 2k

RC = 1k

RE = 330

VCC = 2.5V

beta = 100

Is = 8 10^-16A

VA = 00

a) Bias

b) Voltage gain

DE analysis -> CB open.

Vth = R2/(R1+R2) Vcc = 1.1V

Req = 6.67k

Req Ib = Vth - VBE - IERE

IC = (Vth - VBE)/(Req/beta + RE) = 400

VBE = 0.7V => IC = 1uA

VBE = 0.720V => IC = 0.94uA

VBE = 0.722V => IC = 0.945uA

VBE = 0.7220V

VCE = 0.722V

VCE = 0.722V

VCE = 0.7220V

VCE > VB => Q1 forward.

gm = 0.0365 = (27.5mV)^-1

rpi = 2.77k

Bias point:

VBE = 0.722V

IC = 0.945mA

IB = 9.45uA

VC = VCC - IC RC = 1.555V

VB = VBE + IERE = 1.032V

AC analysis CB short.

Av = gm RC = 36

48.

VA = oo

CB large => AC short circuit.

R2 shorted

RIN = 1/gm || rpi1

ROUT = R1

AV = gm R1

49.

VA = oo. CB large => AC gnd. R2 shorted

RIN = 1/gm1 || rpi2

ROUT = R2 || (1/gm2 || rpi2)

AV = gm1 (R2 || (1/gm2 || rpi2))

50.

VCC = 2.0V

RC = 1K

RB = 10K

RE = 500

VA < oo

beta = 100

gm = (26u)^-1

Vout/Vin = (Vout/Vx)(Vx/Vin)

|Vout/Vx| = ?

Vout = - (I1 + gm vpi)RC

I1 = (Vout - Ve)/ro

Vout = - ((Vout - Ve)/ro + gm vpi)RC

Ve = IE RE = Vx - vpi

vpi = Vx - Ve = IE rpi = (Ve/RE - Vout/RC)rpi

=> Ve = (Vx - Vout vpi)RE/(rpi + RE)

51.

V_A = oo

I_S = 2I_S2

|A_v1| = |V_OUT1 / V_IN| = R_C / (1 / g_m1 + R_E)

|A_v2| = |V_OUT2 / V_IN| = R_C / (1 / g_m + R_E)

g_m1 = I_C1 / V_T

g_m2 = I_C2 / V_T => g_m1 = g_m2 / 2 => |A_v1| = |A_v2| / 2.

g_mv_TC = -V_out / R_C => v_TC = -V_out / (g_mR_C)

I_E = R_E + V_IN = -v_TC

(g_mv_TC + v_TC / r_TC)R_E + V_IN = v_TC

=> - (g_m + 1 / r_TC)R_E (V_OUT / g_mR_C) + V_IN = -V_OUT / (g_mR_C)

[(g_m + 1 / r_TC)R_E + 1] V_OUT / (g_mR_C) + V_IN = 0

V_OUT / V_IN = - g_mR_C / ((g_m + 1 / r_TC)R_E + 1) ~ R_C / (1 / g_m + R_E).

52

Vout = -(I_c + g_mv_pi)R_C    I_c = (V_out - V_e) / r_o

V_e = - g_m v_pi / beta (r_pi + R_B)

V_out = p(v_i)

v_i = A V_out

V_in = V_E - I_ER_E

Vout / Vin = [g_m(r_pi + R_B)R_C / beta r_o + g_mR_C] / [g_m(1 + R_C/r_o)(r_pi + R_B)(1 + R_E/r_o) + (1 + 1/beta)g_m(1 + R_C/r_o)R_E - R_E/r_o[g_m(r_pi + R_B)R_C / beta r_o + g_mR_C]

r_o -> ∞

A_v = g_mR_C / g_m(r_pi + R_B) + (beta + 1)/beta g_mR_E = R_C / (r_pi + R_B + R_E)

= R_C / (R_E + R_B/(beta + 1) + 1/g_m)

52.

Vcc

R1 = 10k

R2 = 10k

R_E = 600

I_C: A_v = 0,9

V_cc

V_A =∞

R_B = R1 || R2

R_B has no effect on the voltage since X so long as v_in remains ideal

R_B != input source impedance

Vout / Vin = Vout / Vx · Vx / Vin

V_i / V_in = [r_pi + (beta + 1)R_E] / [R₁ || R₂ + r_pi + (beta + 1)R_E]

-Vout / Vx = RE

Av = R_E / (1/g_m + R_E) = RE / [1 + REu]

1/g_m = RE / |A_v| - RE |A| = RE(1 - Av² / Av)

g_m = A_v / R_E · 1 / (1 - A_v²) = (0,00795) = (126 Ω)^-1 = I_C / V_T

⇒ I_C = g_mV_T = 0,2054 µA.