inf3410 — fall 2016 - uio.no · for integrated circuits which normally have a pure low-pass...
TRANSCRIPT
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INF3410 — Fall 2016
Excerpt of Sedra/Smith Chapter 9: FrequencyResponse of Basic CMOS Amplifiers
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Content
High Frequency Small Signal Model of MOSFETs (book9.2)
High Frequency Response of CS and CE Amplifiers (book9.3)
Toolset for Frequency Analysis and Complete CSAnalysis (9.4)
CG and Cascode HF response (Book 9.5)
Source follower HF response (Book 9.6)
Differential Amp HF Analysis (book 9.7)Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 2
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Content
High Frequency Small Signal Model of MOSFETs (book9.2)
High Frequency Response of CS and CE Amplifiers (book9.3)
Toolset for Frequency Analysis and Complete CSAnalysis (9.4)
CG and Cascode HF response (Book 9.5)
Source follower HF response (Book 9.6)
Differential Amp HF Analysis (book 9.7)Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 3
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MOSFET IC transfer function
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 4
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BW and GB(P)
GB=BW∗ AM
GB: gain bandwidth product, BW: bandwidth, AM:mid-band gain. There is usually a trade-off between BWand AM. If this trade off is inversely proportional, the GBis constant, e.g. in opamp feedback configurations.
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 5
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BW and GB(P) for CMOS integrated Circuits
For integrated circuits which normally have a purelow-pass characteristics (i.e. no explicit AC-coupling atthe input) you can substitute AM with ADC, i.e. the gainat DC. And:
BW = fH = f−3dB
Where fH is the high frequency cutoff and is thefrequency at whic point AM i reduced by -3dB, i.e. thesignal power is reduced by 1
2 , as 10 log1012 = 3.0
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 6
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MOSFET ’Parasitic’ Capacitances Illustration
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 7
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MOSFET ’Parasitic’ Capacitances Equations
Cgs = CoxW(2
3L+ Lov) (9.22)
Cgd = CoxWLov (9.23)
Csb/db =Csb0/db0
r
1+VSB/DBV0
(9.24/9.25)Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 8
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High Frequency Small Signal Model (1/2)
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 9
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High Frequency Small Signal Model (2/2)
With only the two most relevant parasitic capacitors.
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 10
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Unity Gain Frequency fT
Short circuit current gain. A measure for the best casetransistor speed.
Neglecting the current through Cgd:
io
ii=
gm
s(Cgs +Cgd)(9.28)
fT =gm
2π(Cgs +Cgd)(9.29)Excerpt of Sedra/Smith Chapter 9: Frequency Response
of Basic CMOS Amplifiers 11
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Trade-off fT vs Ao (i.e. GB)
fT =gm
2π(Cgs +Cgd)(9.29)
≈3μnVov
4πL2
A0 = gmro (7.40)
≈2
λVov
=2L
λL︸︷︷︸
const
VovExcerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 12
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Summary CMOS HF Small Signal Model
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 13
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Content
High Frequency Small Signal Model of MOSFETs (book9.2)
High Frequency Response of CS and CE Amplifiers (book9.3)
Interrupt: Transfer Function and Bode Plot
Toolset for Frequency Analysis and Complete CSAnalysis (9.4)
CG and Cascode HF response (Book 9.5)
Source follower HF response (Book 9.6)
Differential Amp HF Analysis (book 9.7)
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 14
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CS Amplifier HF small signal model
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 15
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Using the Miller Effect
Note the simplyfying assumption that vo = gmrovgs, i.e.neglecting feed forward contributions of igd which willstill be very small around fH and makes the following aquite exact approximation of the dominant pole’sfrequency fP ≈ fH
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 16
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CS transfer function dominant Rsig(1/4)
vgs(s(Cgs +Ceq) +1
R′sig) = vsig
1
R′sigExcerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 17
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CS transfer function dominant Rsig(2/4)
vgs
vsig=
1R′sig
s(Cgs +Ceq) +1R′sig
vgs
vsig=
1
1+ sR′sig(Cgs +Ceq)Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 18
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CS transfer function dominant Rsig(3/4)
vo
vsig=
AM
1+ sω0
(9.51)
ωH =1
R′sig(Cgs +Ceq)(9.53)
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 19
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CS transfer function dominant Rsig(4/4)
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 20
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Interrupt: Transfer Function and Bode Plot
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 21
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Transfer Function
The transfer function H(s) of a linear filter isÉ the Laplace transform of its impulse reponse h(t).É the Laplace transform of the differential equation
describing the I/O realtionship that is then solvedfor Vout(s)
Vin(s)
É (this lecture!!!) the circuit diagram solved quitenormally for Vout(s)
Vin(s)by putting in impedances Z(s) for
all linear elements according to some simple rules.
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 22
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Impedances of Linear Circuit Elements
resistor: R
capacitor: 1sC
inductor: sLIdeal sources (e.g. the id = gmvgs sources in small signalmodels of FETs) are left as they are.
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Transfer Function in Root Form
Transfer functions H(s) for linear electronic circuits canbe written as dividing two polynomials of s.
H(s) =a0 + a1s+ ...+ amsm
1+ b1s+ ...+ bnsn
H(s) is often written as products of first order terms inboth nominator and denominator in the following rootform, which is conveniently showing some properties ofthe Bode-plots. More of that later.
H(s) = a0
(1+ sz1)(1+ s
z2)...(1+ s
zm)
(1+ sω1
)(1+ sω2
)...(1+ sωn
)Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 24
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Bode Plots
Plots of magnitude (e.g. |H(s)|) in dB and phase e.g.∠H(s) or ϕ) vs. log(ω).
In general for transfer functions withonly real poles and no zeros (purelow-pass): a) ω→ 0+ |H(s)| is constantat the low frequency gain and∠H(s) = 0o b) for each pole as ωincreases the slope of |H(s)| incresesby − 20dB
decade c) each pole contributes-90o to the phase, but in a smoothtransistion so that at a frequencyexactly at the pole it is exactly -45o
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 25
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General rules of thumb to use real zeros andpoles for Bode plots (1/3)
a) find a frequency ωmid with equalnumber k of zeros and poles wherez1, ..., zk, ω1, ..., ωk < ωmid ⇒
|H(s)| ≈ K|z(k+1)...|zm|
|ω(k+1)|...|ωn|
∠H(s) ≈ 0o
and the gradient of both |H(s)| and∠H(s) is zero
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 26
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General rules of thumb to use real zeros andpoles for Bode plots (2/3)
b) moving from ωmid in the magnitudeplot at each |ωi| add -20dB/decade tothe magnitude gradient and for each|zi| add +20dB/decadec) moving from ωmid in the phase plotto higher frequencies at each ωi add-90o to the phase in a smoothtransition (respectively −45o right atthe poles) and vice versa towardslower frequencies.
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 27
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General rules of thumb to use real zeros andpoles for Bode plots (3/3)
d) For the zeros towards higherfrequencies if the nominater is of theform (1+ s
zi) add +90o and if its of the
form (1− szi) (refered to as right half
plain zero as the solution for s of0 = (1− s
zi) is positive) add -90o to the
phase in a smooth transition (i.e.respectively ±45o right at the zeros)and vice versa towards lowerfrequencies.
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 28
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CS Frequency Response, Dominant CL (1/2)
vo(s(Cgd +CL) +GL) + gmvgs = vgssCgd
vo
vgs=
sCgd − gms(Cgd +CL) +GL
= −gmRL1− sCgdgm
s(Cgd +CL)RL + 1(9.65)
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 29
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CS Frequency Response, Dominant CL (2/2)
ωz =gm
Cgd(9.66)
ωp =1
(Cgd +CL)RL(9.67)
ωz
ωp= gmRL
�
1+CL
Cgd
�
(9.68)
ωt =gm
CL +Cgd(9.69)
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 30
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Content
High Frequency Small Signal Model of MOSFETs (book9.2)
High Frequency Response of CS and CE Amplifiers (book9.3)
Toolset for Frequency Analysis and Complete CSAnalysis (9.4)
CG and Cascode HF response (Book 9.5)
Source follower HF response (Book 9.6)
Differential Amp HF Analysis (book 9.7)Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 31
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Notation in this Book
A(s) = AMFHs
FH(s) =(1+ s
ωz1)...(1+ s
ωzn)
(1+ sωp1
)...(1+ sωpm
)
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 32
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Dominant Pole Approximation
If ωp1 < 4ωp2 and ωp1 < 4ωz1 then
A(S) ≈1
1+ sωp1
ωH ≈ ωp1
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 33
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An Approximation Without a Dominant Pole
2nd order example
|FH(ωH)|2 =1
2=
(1+ω2H
ω2z1)(1+
ω2H
ω2z2)
(1+ω2H
ω2p1)(1+
ω2H
ω2p2)
=
1+ω2H
�
1ω2z1+ 1
ω2z2
�
+ω4H
�
1ω2z1ω
2z2
�
1+ω2H
�
1ω2p1
+ 1ω2p2
�
+ω4H
�
1ω2p1ω
2p2
�
⇒ ωH ≈1
s
1ω2p1
+ 1ω2p2− 2
ω2z1− 2
ω2z2
(9.76)
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 34
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An Approximation Without a Dominant Pole
general:
ωH ≈1
s
1ω2p1
+ 1ω2p2...+ 1
ω2pm− 2
ω2z1− 2
ω2z2... − 2
ω2zn
(9.77)
If ωp1 is much smaller than all other pole- andzero-frequencies this reduces to the dominant poleapproximation.
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 35
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Open-Circuit Time Constants Method
ωH ≈1
∑
iCiRi
Where Ci are all capacitors in the circuit and Ri is theresistance seen by Ci when the input signal source iszeroed and all other capacitors are open circuited.
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 36
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Open-Circuit Time Constants MethodExample CS Amp
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The Difficult One is Rgd
ix = −vgs
Rsig
=vgs + vx
RL+ vgsgm
=vx
RL− ixRsig
�
1
RL+ gm
�
Rgd =vx
ix=�
RL +Rsig (1+ gmRL)�
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Open Circuit Time Constant
τH = RsigCgs +RLCL +�
RL +Rsig (1+ gmRL)�
Cgd
= Rsig�
Cgs + (1+ gmRL)Cgd
�
+RL�
Cgd +CL
�
(9.88)
Previously:
ωH =1
R′sig(Cgs + (1+ gmRL)Cgd)(9.53)
ωH =1
(Cgd +CL)RL(9.67)
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Comparing Approximations
If you combine the prviously transfer functions for vgsvsig
derived from (9.46) and vovgs
from (9.65) as A(s) =vgsvsig
vovgs
you get both of these previous ωH as poles and cancompute the combined ωH according to (9.77):
τH =1
ωH≈r
[R′sig(Cgs +Ceq)]2 + [(Cgd +CL)RL]2 (9.77)
So the geometric mean rather than the sum ...
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 40
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Content
High Frequency Small Signal Model of MOSFETs (book9.2)
High Frequency Response of CS and CE Amplifiers (book9.3)
Toolset for Frequency Analysis and Complete CSAnalysis (9.4)
CG and Cascode HF response (Book 9.5)
Source follower HF response (Book 9.6)
Differential Amp HF Analysis (book 9.7)Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 41
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CG Amplifier HF Response
NOTE: no Miller effect!Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 42
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CG Amplifier HF Response T-model
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 43
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CG Amplifier HF Response without ro
τp1 = Cgs
�
Rsig||1
gm
�
τp2 = (Cgd +CL)RL
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 44
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CG Amplifier open circuit time-constant withro for Cgs
τgs = Cgs
�
Rsig||ro +RL
gmro
�
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 45
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CG Amplifier open circuit time-constant withro for Cgd +CL
τgd = (Cgd +CL)�
Rsig||(ro +Rsig + gmroRsig)�
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 46
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CG Amplifier HF Response Conclusion
No Miller effect that would cause low impedance at highfrequencies, but due to low input resistance theimpedance is already low at DC ⇒ low AM for Rsig > 0
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 47
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Cascode Amplifier HF Response
τgs1 = Cgs1Rsigτgd1 = Cgd1
�
(1+ gm1Rd1)Rsig +Rd1�
where Rd1 = ro1||ro2 +RL
gm2ro2
τgs2 = (Cgs2 +Cdb1)Rd1
τgd2 = (CL +Cgd2)(RL||(ro2 + ro1 + gm2ro2ro1))
τh ≈ τgs1 + τgd1 + τgs2 + τgd2
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 48
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Cascode Amplifier HF ResponseRearranging τh grouping by the threenodes’ resistors:
τh ≈ Rsig�
Cgs1 +Cgd1(1+ gm1Rd1)�
+Rd1(Cgd1 +Cgs2 +Cdb1)
+(RL||Ro)(CL +Cgd2)
Thus, if Rsig > 0 and terms with Rsig aredominant one can either get largerbandwidth at the same DC gain than a CSamplifier when RL ≈ ro or get more DCgain at the same bandwith than a CSamplifier when RL ≈ gmr2
oor increase both
bandwith and DC gain to less than theirmaximum by tuning RL somewhereinbetween.
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 49
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Cascode Amplifier HF ResponseRearranging τh grouping by the threenodes’ resistors:
τh ≈ Rsig�
Cgs1 +Cgd1(1+ gm1Rd1)�
+Rd1(Cgd1 +Cgs2 +Cdb1)
+(RL||Ro)(CL +Cgd2)
With Rsig ≈ 0 one can trade higher BW forreduced ADC or higher ADC for reduced BWcompared to a CS amp, keeping the unitygain frequency (i.e. the GB) constant.
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 50
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Cascode vs CS
CS:
ADC = −gm(ro||RL)τH = Rsig
�
Cgs + (1+ gm(ro||RL))Cgd
�
+ (ro||RL)�
Cgd +CL
�
Cascode:
ADC = (−gm1(RO||RL)where RO = gm2ro2ro1
τH = Rsig�
Cgs1 +Cgd1(1+ gm1Rd1)�
+Rd1(Cgd1 +Cgs2 +Cdb1)
+(RL||Ro)(CL +Cgd2)
where Rd1 = ro1||ro2 +RL
gm2ro2
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 51
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Content
High Frequency Small Signal Model of MOSFETs (book9.2)
High Frequency Response of CS and CE Amplifiers (book9.3)
Toolset for Frequency Analysis and Complete CSAnalysis (9.4)
CG and Cascode HF response (Book 9.5)
Source follower HF response (Book 9.6)
Differential Amp HF Analysis (book 9.7)Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 52
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Source Follower HF Response
A(s) = AM1+
�
sωz
�
1+ b1s+ b2s2= AM
1+�
sωz
�
1+ 1Q
sω0
+ s2
ω20
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 53
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Source Follower Frequency ResponsePossibilities
ωp1,p2 =
− 1Qω0±È
1ω2
0Q2 − 4 1
ω20
2Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 54
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Intuition for Resonance/Instability
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 55
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Dependence on Q-factor (1/2)
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2Re(A)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Im(A)
One Real Pole ωp1
=1/1000
1/A(j ω)A(j* ω)|j*[10 0 ,10 5 ]
A(j* ω)|j* ω=j*10 0
A(j* ω)|j* ω=j*10 3
j*ω=j*10 5
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2Re(A)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Im(A)
Two Identical Real Poles: ω0
=1/1000 and Q=0.5
1/A(j ω)A(j* ω)|j*[10 0 ,10 5 ]
A(j* ω)|j* ω=j*10 0
A(j* ω)|j* ω=j*10 3
j*ω=j*10 5
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 56
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Dependence on Q-factor (2/2)
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2Re(A)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Im(A)
Two Complex Poles: ω0
=1/1000 and Q=0.707
1/A(j ω)A(j* ω)|j*[10 0 ,10 5 ]
A(j* ω)|j* ω=j*10 0
A(j* ω)|j* ω=j*10 3
j*ω=j*10 5
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2Re(A)
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Im(A)
Two Complex Poles: ω0
=1/1000 and Q=1.0
1/A(j ω)A(j* ω)|j*[10 0 ,10 5 ]
A(j* ω)|j* ω=j*10 0
A(j* ω)|j* ω=j*10 3
j*ω=j*10 5
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 57
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Content
High Frequency Small Signal Model of MOSFETs (book9.2)
High Frequency Response of CS and CE Amplifiers (book9.3)
Toolset for Frequency Analysis and Complete CSAnalysis (9.4)
CG and Cascode HF response (Book 9.5)
Source follower HF response (Book 9.6)
Differential Amp HF Analysis (book 9.7)Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 58
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HF Analysis of Current-Mirror-Loaded CMOSAmp (1/2)
Neglecting ro in currentmirror:
GM = gm1+ s Cm
2gm3
1+ s Cmgm3
ωp2 =gm3
Cm
ωz =2gm3
Cm
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 59
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HF Analysis of Current-Mirror-Loaded CMOSAmp (2/2)
vo = vidGMZo
vo
vid= gm3Ro
1+ s Cm2gm3
1+ s Cmgm3
!
1
Ro +1
sCLRo
!
ωp1 =1
CLRo
And ωp1 is usually clearlydominant.
Excerpt of Sedra/Smith Chapter 9: Frequency Responseof Basic CMOS Amplifiers 60