Subthreshold Operation and gm/Id designSubthreshold Operation and gm/Id design Author Michael H....

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M.H. Perrott Analysis and Design of Analog Integrated Circuits Lecture 16 Subthreshold Operation and g m /I d Design Michael H. Perrott April 1, 2012 Copyright © 2012 by Michael H. Perrott All rights reserved.

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:

2

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

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

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

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