Subthreshold Operation and gm/Id designSubthreshold Operation and gm/Id design Author Michael H....
Transcript of Subthreshold Operation and gm/Id designSubthreshold Operation and gm/Id design Author Michael H....
M.H. Perrott
Analysis and Design of Analog Integrated CircuitsLecture 16
Subthreshold Operation and gm/Id Design
Michael H. PerrottApril 1, 2012
Copyright © 2012 by Michael H. PerrottAll rights reserved.
M.H. Perrott
A Closer Look at Transconductance
Assuming device is in strong inversion and in saturation:
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Id
Vgs
Id_op
ΔV
VTH Vgs_op
Vds > ΔV
M1
Id
Vgs
NMOS
g
s
d
gm = ΔVgs
ΔId
Vgs_op
ID =μnCox
2
W
L(Vgs − VTH)2(1 + λVds)
⇒ gm =δIdδVgs
≈ μnCoxW
L(Vgs−VTH) ≈
s2μnCox
W
LId
⇒ gm ≈Id
q2μnCoxW/L√
Id≈ 2Id
(Vgs − VTH)
M.H. Perrott
Unity Gain Frequency for Current Gain, ft
Under fairly general conditions, we calculate:
ft is a key parameter for characterizing the achievable gain·bandwidth product with circuits that use the device 3
|Id/Iin|
f0dB
M1
Id
Iin
NMOS
g
s
d
ft
Vds
M.H. Perrott
Current Density as a Key Parameter
Current density is defined as the ratio Id/W:- We’ll assume that current density is altered by keeping
Id fixed such that only W varies Maintains constant power ro (i.e., 1/gds = 1/(Id)) will remain somewhat constant
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Id
W
M1
Id
Id
WL
W
M.H. Perrott
Investigating Impact of Current Density
For simplicity, let us assume that the CMOS device follows the square law relationship
- This will lead to the formulations:
- These formulations are only accurate over a narrow region of strong inversion (with the device in saturation)
- However, the general trends observed from the above expressions as a function of current density will provide useful insight
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ID ≈μnCox
2
W
L(Vgs − VTH)2
gm ≈2Id
Vgs − VTHVgs − VTH ≈
s2L
μnCox
µIdW
¶
M.H. Perrott
10−7
10−6
10−5
10−4
10−3
0
0.5
1
1.5x 10
−6 Gate Overdrive versus Current Density
Vgs−
Vth
(Vol
ts)
10−7
10−6
10−5
10−4
10−3
0
2
4
6
8x 10
−3 Transconductance versus Current Density
gm (1
/Ohm
s)
10−7
10−6
10−5
10−4
10−3
0
1
2
3
4x 10
11 ft versus Current Density
ft (H
z)
Current Density Id/W (Amps/micron)
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Investigate the Impact of Increasing Current Density
gm decreases
ft increases
Gate overdrive increases
gm ≈2Id
Vgs − VTH
ft =1
2π
gm
Cgs∝√W
W
Vgs − VTH ≈s
2L
μnCox
µIdW
¶
W decreased with fixed Id
M.H. PerrottHigher gm (more gain)
10−7
10−6
10−5
10−4
10−3
0
0.5
1
1.5x 10
−6 Gate Overdrive versus Current Density
Vgs−
Vth
(Vol
ts)
10−7
10−6
10−5
10−4
10−3
0
2
4
6
8x 10
−3 Transconductance versus Current Density
gm (1
/Ohm
s)
10−7
10−6
10−5
10−4
10−3
0
1
2
3
4x 10
11 ft versus Current Density
ft (H
z)
Current Density Id/W (Amps/micron)
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Transconductance Efficiency Versus ft
gm decreases
ft increases
Gate overdrive increases
gm ≈2Id
Vgs − VTH
ft =1
2π
gm
Cgs∝√W
W
Vgs − VTH ≈s
2L
μnCox
µIdW
¶
Higher ft (faster speed)
M.H. PerrottWeak
10−7
10−6
10−5
10−4
10−3
0
0.5
1
1.5x 10
−6 Gate Overdrive versus Current Density
Vgs−
Vth
(Vol
ts)
10−7
10−6
10−5
10−4
10−3
0
2
4
6
8x 10
−3 Transconductance versus Current Density
gm (1
/Ohm
s)
10−7
10−6
10−5
10−4
10−3
0
1
2
3
4x 10
11 ft versus Current Density
ft (H
z)
Current Density Id/W (Amps/micron)
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Transistor “Inversion” Operating Regions
gm decreases
ft increases
Gate overdrive increases
gm ≈2Id
Vgs − VTH
ft =1
2π
gm
Cgs∝√W
W
Vgs − VTH ≈s
2L
μnCox
µIdW
¶
Moderate Strong
M.H. Perrott
Key Insights Related to Current Density
Current density sets the device operating mode- Weak inversion (subthreshold): highest gm efficiency
Achieves highest gm for a given amount of current, Id- Strong inversion: highest ft
Achieves highest speed for a given amount of current, Id- Moderate inversion: compromise between the two Often the best choice for circuits that do not demand the
highest speed but cannot afford the low speed of weak inversion (subthreshold operation)
Key issue: validity of square law current assumption
- The above is only accurate over a narrow range of strong inversion (i.e., the previous plots are inaccurate) General observations above are still true, though
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ID =μnCox
2
W
L(Vgs − VTH)2(1 + λVds)
M.H. Perrott
A Proper Model for Subthreshold Operation
Drain current:
- Where:
- Note: channel length modulation, i.e., , is ignored here10
ID = ID0W
LeVgs/(nVt)
³1− e−Vds/Vt
´
n =Cox + Cdepl
Cox≈ 1.5
S D
GVGS<VTH
VDS
ID
Vt =kT
q≈ 26mV at T = 300K
ID0 = μnCox(n− 1)V 2t e−VTH/(nVt)
M.H. Perrott
Saturation Region for Subthreshold Operation
Saturation occurs at roughly Vds > 100 mV
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⇒ ID = ID0W
LeVgs/(nVt)
³1− e−Vds/Vt
´≈ ID0
W
LeVgs/(nVt)
0 0.1 0.2 0.3 0.4 0.50
20
40
60
80
100
120
140Drain current Versus Vds and Vgs
I d (nA
)
Vds (Volts)
Vgs = 0.4V
Vgs = 0.42V
Vgs = 0.44V
M.H. Perrott
Transconductance in Subthreshold Region
Assuming device is in subthreshold and in saturation:
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Id
Vgs
Id_op
Vds > 100mV
M1
Id
Vgs
NMOS
g
s
d
gm = ΔVgs
ΔId
Vgs_op
Vgs_op
⇒ gm =δIdδVgs
≈ ID0W
LeVgs/(nVt)
1
nVt=
IdnVt
Recall for strong inversion : gm ≈2Id
(Vgs − VTH)
ID ≈ ID0W
LeVgs/(nVt)
gm purely afunction of Id!
M.H. Perrott
Comparison of Strong and Weak Inversion for gm
Assumption: Id is constant with only W varying Strong inversion formulation predicts ever increasing
gm with reduced overdrive voltage
- Reduced current density leads to reduced overdrive voltage and therefore higher gm
Weak inversion formulation predicts that gm will hit a maximum value as current density is reduced
- Note that the area of the device no longer influences gmwhen operating in weak inversion (i.e., subthreshold)
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gm =≈ IdnVt
gm ≈2Id
(Vgs − VTH)
M.H. Perrott
vgs gmvgsCgs rogmbvs
dg
s
dg
s
Hybrid- Model in Subthreshold Region (In Saturation)
Looks the same in form as for strong inversion, but different expressions for the various parameters
- We can use the very same Thevenin modeling approach as in strong inversion We just need to calculate gm and gmb differently
gm ≈µ1
n
¶IdVt
gmb ≈µn− 1n
¶IdVt
ro ≈1
λId
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M.H. Perrott
vgs gmvgsCgs ind2rogmbvs
dg
s
dg
s
Noise for Subthreshold Operation (In Saturation)
i2nd = 4kTγgdso∆f +Kf
f
g2mWLC2ox
∆f
Recall transistor drain noise in strong inversion:
In weak inversion (i.e., subthreshold):Thermal noise 1/f noise
i2nd = 2kTngm∆f +Kf
f
g2mWLC2ox
∆f
Thermal noise 1/f noise15
M.H. Perrott
Strong Inversion Versus Weak Inversion
Strong inversion (Vgs > VTH)- Poor gm efficiency (i.e., gm/Id is low) but fast speed- Need Vds > (Vgs – VTH) = V to be in saturation- Key device parameters are calculated as:
Weak inversion (Vgs < VTH)- Good gm efficiency (i.e., gm/Id is high) but slow speed- Need Vds > 100mV to be in saturation- Key device parameters are calculated as:
Moderate inversion: compromise between the two
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gm ≈2Id
(Vgs − VTH)gmb ≈
γgm
2q2|ΦF |+ VSB
ro ≈1
λId
gm ≈µ1
n
¶IdVt
gmb ≈µn− 1n
¶IdVt
ro ≈1
λId
Thevenin Modeling Techniques Can Be Applied to All Cases
M.H. Perrott
gm/Id Design
gm/Id design is completely SPICE based- Hand calculations of gm, ro, etc. are not performed
Various transistor parameters are plotted in terms of gm/Id- Low gm/Id corresponds to strong inversion- High gm/Id corresponds to weak inversion
Once a given value of gm/Id is chosen, it constrains the relationship between W, L, ft, etc. such that the sizing of devices becomes a straightforward exercise
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M.H. Perrott
Useful References Related to gm/Id Design
Prof. Bernhard Boser’s Lecture:- B. E. Boser, "Analog Circuit Design with Submicron
Transistors," IEEE SSCS Meeting, Santa Clara Valley, May 19, 2005, http://www.ewh.ieee.org/r6/scv/ssc/May1905.htm
Prof. Boris Murmann’s Course Notes: - https://ccnet.stanford.edu/cgi-
bin/course.cgi?cc=ee214&action=handout_view&V_section=general See Slides 45 to 67 in particular
Prof. Reid Harrison’s paper on a low noise instrument amplifier:- http://www.ece.utah.edu/~harrison/JSSC_Jun_03.pdf
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M.H. Perrott 19
Summary
CMOS devices in saturation can be utilized in weak, moderate, or strong inversion- Each region of operation involves different expressions
for drain current as a function of Vgs and Vds- It is best to use SPICE to calculate parameters such as gm, gmb, ro due to the complexity of the device model in encompassing these three operating regions gm/Id methodology is one such approach
- Weak inversion offers large gm/Id but slow speed, and strong inversion offers fast speed but lower gm/Id- Moderate inversion offers the best compromise between achieving reasonable gm/Id and reasonable speed
Thevenin modeling approach is valid for all operating regions once gm, gmb, and ro are known