EL 6033
類比濾波器 (一)
Analog Filter (I)
Lecture2: Frequency Compensation and
Multistage Amplifiers II
Instructor：Po-Yu Kuo
教師：郭柏佑
Outline


Miller Compensation in Two-Stage Amplifiers
Design of a Two-Stage Amplifier
2
Simplification for Two-Stage Amplifier

Fig. 2 is the signal representation of Fig. 1

gm1 (fig. 2) = gm1,2 (fig. 1)

gmL (fig. 2) = gmL (fig. 1)

r1, C1 (fig. 2) = equivalent output resistance (ro2//ro4),
capacitance(Cdtot,M4+Cdtot,M2+Cgtot,ML) at V1 (fig. 1)

rL, CL (fig. 2) = output resistance (roL//rob3), capacitance (CL) at Vo(fig. 1)
3
Hybrid-π Model of Two-Stage Amplifier




Fig. 2 is the Hybrid-π Model of two-stage amplifier
Hybrid-π Model is used to derive small-signal transfer function(Vo/Vin) of the
two-stage amplifier
When convert to Hybrid-π Model, the circuit is linear with approximation
To understand frequency compensation, small signal model must be
obtained
4
Question#1

What is the Hybrid-π Model and dc gain of this circuit?
5
Answer



Dc gain of the first gain stage is +ve
Dc gain of the second gain stage is –ve
Overall dc gain=-gm1gmLr1rL (-ve gain!)
6
Why We Need Frequency Compensation?




Frequency compensation relates certain circuit specifications with design
parameters

Circuit specifications: unity-gain bandwidth (BW), phase margin (PM)
and CL

Design parameters: gm1, gmL, Cm
gm √I, and area of Cm dominates the chip area of amplifier
Frequency compensation can optimize BW and PM by using minimum
current consumption (gm) and smallest chip area (Cm) for a particular CL
DC gain specification decides the values of r1 and rL
7
Concept of Bode Plots (1)
Transfer Functions:
A(s)=s
A(s)=1/s
A(s)=1/(1+s/p)
(LHP pole @ p)
8
Concept of Bode Plots (2)
A(s)=1+s/z
(LHP zero @ z)
Both magnitude and phase
Increase!!
A(s)=1-s/z
(RHP zero @ z)
Magnitude increases but phase
decrease!!
9
Miller Compensation in Two-Stage Amplifier (1)

C 
g m1 g mL r1rL 1  s m 
g mL 
Vo


Vin

1  sCm g mL r1rL 1  s CL
g mL






Numerator:

DC gain

Zero: (1-as) → RHP zero, (1+as) → LHP zero

1 RHP zero exits → phase margin degradation
Denominator:

Poles: (1+as+bs2+…), all coefficient terms (a, b, …) should be positive
(LHP poles); otherwise amplifier is unstable

2 LHP poles exist
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Miller Compensation in Two-Stage Amplifier (2)

C 
g m1 g mL r1rL 1  s m 
g mL 
Vo


Vin

1  sCm g mL r1rL 1  s CL
g mL









DC gain = gm1gmLr1rL= A1x AL
RHP zero (zRHP): gmL/Cm
p-3dB = 1/CmgmLr1rL
p2 = gmL/CL
UGF = DC gain x p-3dB = gm1/Cm
11
Miller Compensation in Two-Stage Amplifier (3)

Stability (phase margin of the amplifier):
 UGF 
 UGF 
1  UGF 
  tan1 



PM  180  tan 1 

tan



 p2 
 z RHP 
 p 3dB 
 UGF 
 UGF 
  tan1 

 90  tan1 
p
z
 2 
 RHP 
g
g
C 
 90  tan1  m1  L   tan1  m1
 g mL C m 
 g mL




PM>45∘to preserve stability

PM>60∘to preserve stability and achieve better settling time

The presence of RHP zero degrades stability
What is the relationship of gm1, gmL and Cm in order to achieve stability?


12
Dimension Condition of Cm

If RHP zero neglected

Case 1: PM=60 ∘
g
C 
tan1  m1  L   30
 g mL C m 
1.73g m1C L
Cm 
g mL
g
UGF  0.58 mL
 CL

& PM=90∘
g m1C L
g mL
UGF 

Recall: Single-Stage Amplifier, UGF=gmL/CL
Case 2: PM=45 ∘
Cm 




g mL
CL
BW of amplifier trades with the stability (PM)
In most textbook: Cm=2gm1CL/gmL & UGF=0.5gmL/CL, then PM=63.4∘
13
Question

As mentioned previously, UGF=gm1/Cm, do you think is it the best way
to increase UGF of the amplifier by decreasing Cm? From equation,
decreasing Cm does not increase the power consumption and
decreases the chip area. Then you should ask yourself “does it have
any free lunch in the world”?
14
Question




UGF increases due to the increase in
p-3dB.
However, p2 does not change, p2 is
smaller than UGF and PM is much
smaller than 45 ∘
Stability problem arises!
No Free Lunch!!!
15
Solution(1)

What is the frequency domain behavior if we increase gm1 only based
on UGF=gm1/Cm?

Again, UGF increases but the amplifier suffersfrom the stability
problem!!
16
Correct approach to increase BW

How to enhance UGF without hurting stability (PM)?

Step1:
gmL ↑
Both p2 and zRHP move to higher freq.
PM ↑ with same BW

Step2: :
Cm ↓ according to Cm=2gm1CL/gmL
BW ↑
Rule of Thumb:
Larger current should be allocated to the output stage for
UGF enhancement!!
gmL >> gm1!!
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Effect of RHP Zero

By taking RHP zero into consideration and assume Cm=2gm1CL/gmL;
then
 BW 
 BW
  tan 1 
PM  90  tan 1 
 p2 
 z RHP
g 
 63.4  tan 1  m1 
 g mL 







If gmL=gm1, then PM=18.4 ∘(instability)
If gmL=10gm1, then PM=57.7 ∘(stability degradation)
gmL >> gm1 to preserve stability due to RHP zero!
larger gmL implies larger power consumption.
Miller compensation is not suitable for low-power design due to the
presence of RHP zero!
RHP zero removal techniques Low-Power design!!
18
Miller Compensation with Null Resistor
Add Extra resistor

 1


g m1 g mL r1rL 1-sCm 
 Rm 
Vo

 g mL


Vin


1  sCm g mL r1rL 1  s C L 
g mL 





No change in pole locations!
Rm is used to improve PM as
zRHP is removed by Rm = 1/gmL
PM = 63.4∘
low-power design condition
19
Dimension Condition of Rm


LHP zero is generated if Rm > 1/gmL
If the LHP zero is used to cancel p2, then Rm is set as
 C
Rm  1  L
 Cm


 1

g

1  2 mL

g

g m1
 mL 
 1

g
 mL
Both zRHP and p2 are cancelled
Rm cannot be too large since very large Rm causes open circuit and no
pole-splitting effect due to Miller compensation (Rm < r1/10)
Rule of Thumb:
1/gmL ≤ Rm < r1/10
20
Miller Compensation Implementation of Two-Stage Amplifier
Vgs,ML


If Rm is implemented by transistor(s), then the transistor(s)should be
placed between the drain of M4 and Cm to ensure the transistor(s)
always in the triode region!
Vgs,ML should be equal to Vgs,M3 and Vgs,M4 for minimizing the
systematic offset voltage.
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Outline


Miller Compensation in Two-Stage Amplifiers
Design of a Two-Stage Amplifier
22
Design Example (1)



If the specification is given as

CL=10pF

UGF > 3MHz

PM > 60 ∘

DC Gain > 80dB

SR > 2.5V/μs

Power Consumption < 160W

Supply Voltage = 2V
Designer’s job is

choose Rm, Cm, (W/L)i, Li, I to meet specifications!!
What are the relationship between designer’s job and the
specifications?
23
Design Example (2)

Recall: for PM ≈ 63.4 ∘, then UGF=gmL/2CL & Cm=2gm1CL/gmL

gmL is fixed (415 μA/V) & Rm=2.4 kΩ

Assume gm1=150 μA/V, Cm is fixed at 7.2 pF

Theoretical UGF=3.3 MHz (gmL/(2π)2CL)

Further assume r1=1.3 MΩ and ro=200 kΩ

Theoretical dc gain=84 dB

Use Hybrid-π model to verify the bandwidth, dc gain and phase
margin performances by using Hspice or Cadence
24
Design Example (3)

After choose the value of each circuit parameter such as

gm1, gmL, CL, Cm … etc.
Verify the performance of Hybrid-π model

Simulate the Hybrid-π model in Hspice or Spectre

25
Design Example (4)

AC Simulation Results of Hybrid-π model in Spectre
26
Design Example (5)





SR is the change rate of output voltage when time change →
Ideally, SR is infinity
SR=min(IMb3/CL, IMb2/Cm) ≈ IMb2/Cm (in most cases)
Systematic Offset Requirement (Vgs,M4=Vgs,ML (W/L)ML/(W/L)M4=2IML/IMb2)
and total power consumption (Itot ≈IMb2+IML) → fix(W/L)ML and IML (need
iterations)

Make sure (W/L)ML and IML meet ro and dc gain requirements
Iterations of above steps are necessary until all specificationsare met.
27
Design Flow(1)

Step1: Make sure all transistors work in correct region
Simulate the common mode result

Then check if the all transistor work in sat. region

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Design Flow(2)

Transistor status
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Design Flow(3)

Step2: Start AC simulations
30
Design Flow(4)

Step3: Start transient response analysis
31
Performance Summary
32
Tips of Simulations



DC Analysis: make sure all transistors operating in the
saturation region, and check the lowest supply voltage to
achieve the required input common-mode range.
AC Analysis and Pole-Zero Analysis: check dc gain,
BW, stability (phase margin, pole and zero locations) and
power consumption
Transient Analysis: check step response of the
amplifier (slew rate and settling time). It should be noted
that the input step amplitude should be within the input
common-mode range of the amplifier.
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