(13)Compensation (cont.)p

Chapter 10

Stability and Frequency Compensation

Compensation

Basic Stabilityy

Y

X (s) = H (s) 1+ βH (s) X 1+ βH (s)

May oscillate at ω if |βH (jω) |=1 and

∠βH ( jω) = −180 (Barkhausen criteria)

Stable and Unstable Systemsy

Stability and Complex Polesy p

Basic One Pole Bode Plot

Multipole Systemsp y

Loot Locus for a 2-Pole Systemy

Phase Marging

Phase Margin (cont.)g ( )

-180 Phase Margin, θ_m = 45°

Phase Margin (cont.)g ( )

Small Signal Analysis Limitationsg y

Small signal analysis yields 65 ° of θ_m, but large

i l t i t i diff t L i l

signal transient response is different. Large signal simulation includes effects not seen in small signal

analysis analysis.

Frequency Compensationq y p

Compensation is the manipulation of gain and/or pole positions to improve phase margin.

Compensation (cont.)p ( )

Single Ended Single Stage Ampg g g p

D i t P l t O t t Dominant Pole at Output

Pre- and Post-Compensationp

Compensation (cont.)p ( )

Compensation Examplep p

Given T₀ = 5000V /V, fp1 = 2MHz, fp2 = 25MHz, fp3 = 50MHz Desire φφ_mm = 70°, find f, f f_0db0db and new ff_p1p1∴

70° =180°− tan⁻¹( f_0db

f_p1)− tan⁻¹(f_0db

f_p2) −tan⁻¹( f_0db

f_{p 2})

p p p

70° = 90 −tan⁻¹( f_0db

f_{p 2}) − tan⁻¹( f_0db

f_p2), assume 90° from f_p1 f_0db ≈ 6MHz

f_0db = T₀ f_p1 → f_p1 = f_0db /T₀ =1.2KHz f_{0db 0} f_p1 f_p1 f_{0db 0}

Fully Differential Telescopic Op- A

Amp

Cascode Current Source

I d

Impedance

Z || 1 (1+ g_{m 5}r_o5) r_o7 1+ sr_o7C_N

1 sC_L Z_out ||

sC_L = ^{o7 N L}

(1+ g_{m 5}r_o5) r_o7

1+ sr_o7C_N + 1 sC_L

o7 N L

= (1+ g_{m 5}r_o5)r_o7

1+ s (1+ g[[^{( g}_{m 5}m 5 o5r_o5)r) _o7o7C_LL + r_o7o7C_NN ]]

Compensation of Two-Stage Op Ampsp g p p

Compensation (cont.)p ( )

Miller Effect C = C_E + (1+ A ₂)C_C Miller Effect C_eq C_E + (1+ A_{v 2})C_C

f_pE = 1

2πR [C + (1+ A )C ] 2πR_out[C_E + (1+ A_{v 2})C_C]

Pro: Need a smaller capacitor

Con: Exact effect of bridging capacitor involve a zero If open-loop gain is high involve a zero. If open-loop gain is high,

zero may prevent -20db/dec curve from going all the way to 0db

Compensation (cont.)p ( )

Recall from Chap 6:

f 1

Recall, from Chap. 6:

(assume C_C includes C_GD9) f_p,in =

2π(R_S[C_E + (1+ g_mR_L )C_C ]^{+ R}L (C_C + C_L))

f _p,in ≈ 1

2πR_S[C_E + (1+ g_mR_L )C_C]

Compensation (cont.)p ( )

f R_S(1+ g_{m 9}R_L )C_C + R_SC_E + R_L(C_C + C_L)

f_p,out = ^{S m 9 L C S E L C L}

2πR_S R_L(C_EC_C + C_EC_L + C_CC_L)

f _p,out ≈ R_Sg_{m 9}R_LC_C + R_LC_C

2πR_SR_L(C_EC_C + C_CC_L ) = g_{m 9}

2π(C_E + C_L ) 2πR_SR_L(C_EC_C + C_CC_L ) 2π(C_E + C_L )

Compensation (cont.)p ( )

Recall, transfer function includes (1− s/ω_z) numerator term and

f (RHP) g_m9 f_Z(RHP) = g^m9

2πC_C

Phase and Magnitude of RHP Z

Zero

RHP Zero Removal

f_Z = g_{m 9}

2 C (1/ R ) f_Z

2πC_C(1/g_{m 9} − R_Z )

RHP Zero Removal (cont.)( )

f_Z = 1 f_Z

2πC_C(1/ g_{m 9} − R_Z )

Could set R_z = 1/g_m9, or cancel other non - dominant pole

1

C (1/g − R ) = −g^m9 C + C C_C(1/g_{m 9} R_Z ) C_L + C_E R_Z = C^L + C_E + C_C

g C ≈ C^L + C_C g C g_{m 9}C_C g_{m 9}C_C

Miller Compensation (cont.)p ( )

Temp. and Process Tracking

Miller Compensation (cont.)p ( )

g_{m 9} ∝ I_{D 9} ∝ I_D11 ∝1/ R_S

Load Capacitance Effectsp

Slewing in Two-Stage Op Ampsg g p p

Basic Two-Stage Op Ampg p p

Slewing in Two-Stage Op Ampg g p p

Other Compensation Techniques p q

Other Compensation Techniques ( t )

(cont.)

Other Compensation Techniques ( t )

(cont.)

Slewing with Common-Gate

C ti

Compensation

Other Compensation Techniques ( t )

(cont.)