Noise Tutorial:
Low-frequency CMOS
Analog Design
May 16, 2002
Rafael J. Betancourt-Zamora
http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
Outline
•
Goals
•
Noise Overview
•
Resistor Noise Sources
•
MOSFET Noise Sources
•
Noise of Common Amplifier Topologies
•
Examples
•
Summary
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
Goals
•
Grasp the importance of noise performance in low-frequency
instrumentation design
•
Understand the types of noise present in integrated resistors
and MOS devices.
•
Understand the noise trade-offs of different circuit topologies
•
Perform simple noise analysis
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
Outline
•
Goals
•
Noise Overview
•
Resistor Noise Sources
•
MOSFET Noise Sources
•
Noise of Common Amplifier Topologies
•
Examples
•
Summary
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
What is noise?
• Unwanted disturbance that interferes with a desired signal
• External: power supply & substrate coupling, crosstalk, EMI, etc.
• Internal: random fluctuations that result from the physics of the devices or materials
• Smallest detectable signal, signal-to-noise ratio (SNR), and dynamic range are
determined by noise
2
P signal
Vrms, signal
SNR = -------------------- = ------------------------2
P noise
Vrms, noise
We will look at internal noise sources and how
they affect key performance metrics.
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
What does it look like?
WHITE NOISE (GAUSSIAN DISTRIBUTION)
V (t )
FUZZ ≈ 6σ
P (x )
t
MEAN
m = 0
STANDARD DEVIATION
σ = V rms
2
V rms ⁄ Hz
VARIANCE
σ
2
2
= V rms
(Normalized Noise Power)
Noise Power Spectral Density
f
• Random process with zero mean
• Instantateous amplitude is unpredictable
• Average noise power can be measured (variance)
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
Types of Noise
THERMAL
• thermal excitation of charge carriers
2
V rms ⁄ Hz
2
V rms
---------- =
∆f
4kTR , V2/Hz
f
SHOT
• fluctuations in dc current flow through junctions
2
A rms ⁄ Hz
2
I rms
----------- =
∆f
2qI DC , A2/Hz
f
FLICKER
2
V rms ⁄ Hz
• traps in semiconductors
1⁄f
f
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
2
V rms
------------- =
∆f
2
K f ⋅ V DC 2
, V /Hz
-------------------f
T = Temperature, K
k = 1.38 x 10-23 J/K
q = 1.6 x 10-19 C
Kf = Flicker coefficient
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Noise Tutorial: Low-frequency CMOS Analog Design
Outline
•
Goals
•
Noise Overview
•
Resistor Noise Sources
•
MOSFET Noise Sources
•
Noise of Common Amplifier Topologies
•
Examples
•
Summary
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
Resistor Thermal Noise
2
*
Vn
NOISE DENSITY
NOISE POWER
∆fn
2
Vn
In
R
2
Vn
------ =
∆f
2
R
2
, V /Hz
In
------- =
∆f
= 4kTR∆fn , V2
In =
4kTR
2
2
= NOISE BANDWIDTH
2
4kT
---------------- , A /Hz
R
4kT
2
---------------- ∆fn, A
R
1kΩ @ 25 ºC
• Integrate noise PSD over frequency to arrive at total noise power.
4.05
nV ⁄ H z
• For white noise, multiply PSD by noise bandwidth.
4.05
pA ⁄ H z
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
Noise Bandwidth
A v (f )
2
No. Poles
1
∆f n =
π
--- ⋅ ∆f 3dB
2
2
∆f n =
1.22 ⋅ ∆f 3dB
3
∆f n =
1.16 ⋅ ∆f 3dB
2nd-order BPF
∆f n =
π
--- ⋅ ∆f 3dB
2
2
A vo
f3dB
f
∆fn
2
Avo
∞
⋅ ∆f n =
∫
NOISE BW
2
A v(f) df
0
• Noise bandwidth is defined for a brickwall transfer function
• Noise bandwidth is not the same as 3dB bandwidth
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
kT/C Limit
2
R
Vo
2
Vn
2
4kTR∆fn , V2
2
1 
4kTR  ----------
Vo =
*
∆fn =
∆f 3dB =
C
Vo =
π
-- ⋅ ∆f 3dB
2
1
--------------2 πRC
}
∆f n =
1
----------4R C
4RC
2
Vo =
kT
------ , V2
C
• Total noise power is independent of R
• Capacitor is noiseless, but accumulates noise from resistor
• This fundamental limit is important for sampled systems
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
Combining Noise Sources
SERIES
*
PARALLEL
2
I n 1 R2
R1
2
V n1
2
R1
*
2
I n2
2
Vn 1 + Vn 2
2
R1+R2
R1||R2
2
In1 + In 2
2
*
V n2
R2
2
Vo =
4kT ( R1 + R2 )∆fn
I 2o
=
4kT ( ---1----
+ ---1---- )∆fn
R1 R2
• For uncorrelated noise sources, just add the noise powers.
• For correlated sources use:
2
Vo
=
2
V n1
c ≤1
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
+
2
V n2
+ 2cV n1 V n2
CORRELATION COEFFICIENT
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Noise Tutorial: Low-frequency CMOS Analog Design
Resistor Flicker (1/f) Noise
2
2
K f ⋅ VD C 2
Vnf
------- = --------------------- , V /Hz
∆f
f
1⁄f
2
V n ⁄ Hz
THERMAL
f
fL
fH
fc
fH K ⋅ V2
f H
DC
2
f

--------------------- df = K f ⋅ V DC ln  -----  , V2
fL
f
fL
2
Vnf =
∫
2
Vn f =
2.3 K f ⋅ V 2DC
FOR fH/fL = 10
• Constant noise power per decade
• fc defines 1/f corner frequency where
flicker noise is equal to thermal noise.
TOTAL RESISTOR NOISE
2
Vo =
 f H
4kTR∆fn + K f ⋅ V 2DC ln  -----
 fL 
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
, V2
ASSUMES
∆fn » f c
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Noise Tutorial: Low-frequency CMOS Analog Design
Noise Index
2
Vn f
------------2
V DC
2
=
DEC
K f ⋅ V D C ln ( 10 )
----------------------------------------------------2
VD C
2.3Kf ⋅ 10
=
2.3 K f
–6
, µV/V/decade
NI
=
2
V nf
NI ⋅ V DC 1 2
= ----------------------------⋅ -- , V /Hz
– 12 f
2.3 × 10
2
2
2
fc
2
NI ⋅ VDC
, Hz
= -------------------------------------------------– 12
2.3 × 10
⋅ 4kTR
• Normalize 1/f noise per Volt, per decade of frequency
• Can express 1/f noise PSD and fc in terms of NI
20log(NI)
poly -15dB
ndiff -10dB
-5dB
pdiff
[Motchenbacher’93]
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
Outline
•
Goals
•
Noise Overview
•
Resistor Noise Sources
•
MOSFET Noise Sources
•
Noise of Common Amplifier Topologies
•
Examples
•
Summary
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
Noise Sources in a MOSFET
DRAIN THERMAL NOISE
2
In d
------- =
∆f
γ=
4kTγg m , A2/Hz
{
2
Vn g
2
I nd
* *
2/3, long channel
2-3, short channel
• due to limited channel conductance
GATE THERMAL NOISE
2
V ng
-------- = 4kTRG ⁄ 3 , V2/Hz
∆f
• can be neglected with good layout
FLICKER NOISE
2
Kf
Vnf
2
------- = ------------------- , V /Hz
∆f
WLC o xf
2
Vn f
FOR NMOS
– 24 2
K f = 1.2 × 10
V ⋅F
[Razavi’01]
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
Rds = 1/gm
Long Channel
Saturation
SPICE
AF
2
K
⋅
I
In f
f DC , A2/Hz
------ = ----------------------2
∆f
L EFF C o x f
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Noise Tutorial: Low-frequency CMOS Analog Design
Input-referred Noise
2
Vn f
2
2
I nd
*
2
Vn i
Vn i
*
*
2
I ni
• Define equivalent input voltage and
INPUT-REFERRED DRAIN NOISE
current noise sources.
• MOSFET input current noise source
can be neglected at low frequencies.
2
In d
Vn d
-------- = ------∆f
∆f
2
2
⁄ gm
=
4kTγ
--------------------gm
, V 2/Hz
TOTAL INPUT-REFERRED NOISE
2
Vni
------- =
∆f
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
Kf
4kTγ
2
--------------------- + ------------------- , V /Hz
g m WLCox f
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Noise Tutorial: Low-frequency CMOS Analog Design
1/f Corner Frequency
2
V n ⁄ Hz
1⁄f
THERMAL
f
fc
4kTγ
--------------------gm
=
Kf
---------------------WLCox fc
Kf ⋅ g m
f c = --------------------------------------4kTγWLCox
• Defines where flicker noise is equal to thermal noise.
• Need to increase transistor area to reduce flicker noise.
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
Outline
•
Goals
•
Noise Overview
•
Resistor Noise Sources
•
MOSFET Noise Sources
•
Noise of Common Amplifier Topologies
•
Examples
•
Summary
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
Current Source
VDD
2
I n1
------- =
∆f
I1
2
VBIAS
Vo
M1
2
I n1
2
Vo
-------∆f
2
=
In 1 2
------- ⋅ ro , V2/Hz
∆f
=
4kTγg mr o , V2/Hz
2
Vo
-------∆f
4kTγg m , A2/Hz
2
• Neglected flicker noise of M1
• Need to minimize output current noise
• Minimize gm for low noise
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
Common Source
VDD
2
I n1
------- =
∆f
2
I nL
RL
2
Vo
2
Vn f
Vi
M1
2
InL
--------- =
∆f
4kTγg m , A2/Hz
2
4kT
---------------- , A /Hz
RL
2
Kf
Vnf
------- = ------------------- , V2/Hz
∆f
WLC o xf
2
I n1
*
2
Vn i
------- =
∆f
Kf
4kTγ
4kT
--------------------- + ------------------- + ---------------------g m WLC o xf g 2 R
, V2/Hz
m L
• Neglected output conductance of M1
• Maximize gm , R L, and transistor area for low noise
• For a differential pair, double the noise
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
Source Follower
2
Vn 1
Vi
VDD
2
V n1
-------- =
∆f
M1
*
2
Vo
VBIAS
M2
Kf
4kTγ
2
--------------------- + ------------------- , V /Hz
g m1 WLCox f
2
I n2
------- =
∆f
4kTγg m , A2/Hz
2
I n2
2
I n2  1
2
2  2
------- ⋅  --------------- || r o1 || ro2
2
2

Vni
V n1 ∆f  g m1
------- = -------- + --------------------------------------------------------------------------- , V2/Hz
∆f
2
∆f
Av
WHERE
Av ≈ 1
• Neglected flicker noise of M2
• VERY NOISY, AVOID
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
Common Gate
VDD
2
InL
--------- =
∆f
2
I nL
RL
2
Vo
VBIAS
M1
2
Vn f
/Hz
2
Kf
Vnf
------- = ------------------- , V2/Hz
∆f
WLC o xf
2
I n1
------- =
∆f
4kTγg m
, A2/Hz
2
I n1
*
2
V ni
------- =
∆f
Kf
4kTγ
4kT
2
--------------------- + ---------------------- + ------------------- , V /Hz
g m g 2 R WLCox f
m L
2
2
g
Kf
Ini
2
4kT
m
------- = ---------------- + ------------------- , A /Hz
∆f
RL WLCo x f
Vi
Rs = 1/gm
(neglecting
body effect)
4kT
---------------- , A
RL
2
• Neglected output conductance of M1
• Define equivalent input voltage and current noise sources
• Input noise current cannot be ignored
• Transfers load current noise to the input
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
Cascode
VDD
2
I nL
RL
2
Vo
VBIAS
M2
2
Kf
4kTγ
2
--------------------- + ------------------- , V /Hz
g m1 WLCox f
2
2
4kT
---------------- , A /Hz
RL
V n1
-------- =
∆f
2
InL
--------- =
∆f
I n2
2
V ni
------- =
∆f
2
Vn 1
Vi
Kf
4kTγ
4kT 2
--------------------- + ------------------- + -------------------------- , V /Hz
g m1 WLCox f g 2 R
m1 L
M1
*
• Neglected output conductance of M1
• M2’s noise contribution is negligible at low frequencies
• Capacitance at M1’s drain increase noise at high frequencies
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
Outline
•
Goals
•
Noise Overview
•
Resistor Noise Sources
•
MOSFET Noise Sources
•
Noise of Common Amplifier Topologies
•
Examples
•
Summary
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
Variable Resistor
Vi
kRT
2
In1
Vo
(1-k)RT
2
In 2
Vo
kRT
(1-k)RT
2
In
WORST CASE
k = 0.5
Req = RT/4
2
2
4kT
---------------- , A /Hz
Req
2
16kT
2
--------------------- , A /Hz
RT
In
------=
∆f
In
------- =
∆f
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
Req = k(1-k)RT
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Noise Tutorial: Low-frequency CMOS Analog Design
Inverting Operational Amplifier Filter
C
2
Vn 1
R1
R2
1. SHORT Vna
2
------- =
∆f
Avi = – R2
------Avni = 1 + R2
------R1
R1
1
1
∆f n = π
-- ⋅ ----------------= ------------2 2πR2 C
4R2 C
Vo
*
3. APPLY SUPERPOSITION
2
Vn i
V n2
2
------- = V n 1 + -------2
∆f
Avi
2. SHORT Vi
 A2 
2  vni
V na  -----------
 2 
 Avi 
where
+
2
V na
2
Vn i
• Low frequency operation
*
-
2
• Ignore 1/f noise
2
Vn 2
*
Vi
• Negligible input noise current
2
Vni
2
Vn 2
2
2
------- = V n 1 + -------+ V na  1 + R1
------- 

R2
2
∆f
A vi
FOR LARGE GAIN, R2 >> R1
2
= V na  1 + R1
------- 

2
R2
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
2
2
Vni
Vn 2
2
2
------- = V n 1 + -------- + V na , V2/Hz
2
∆f
A vi
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Noise Tutorial: Low-frequency CMOS Analog Design
Outline
•
Goals
•
Noise Overview
•
Resistor Noise Sources
•
MOSFET Noise Sources
•
Noise of Common Amplifier Topologies
•
Examples
•
Summary
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
What We Learned
NOISE IN GENERAL
• Thermal noise and flicker noise are the major sources of headaches
for low-noise, low frequency instrumentation design in CMOS.
• Noise bandwidth is not the same as 3dB bandwidth.
• Capacitors are noiseless, but they can accumulate noise.
• Resistors can have flicker noise, but only when there is current flow.
NOISE IN MOSFETS
• Need to increase transistor area to reduce flicker noise.
• Current source: Minimize gm for low noise current
• Common source: Maximize g m , RL, and transistor area for low noise
• Source follower: VERY NOISY, AVOID
• Common gate: Transfers load current noise to the input. Input noise
current cannot be ignored.
• Cascode: Cascode transistor noise contribution is negligible at low
frequencies.
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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Noise Tutorial: Low-frequency CMOS Analog Design
References
1. P.H. Garret, “Analog I/O Design: Acquisition, Conversion,
Recovery,” Reston Publishing, Reston, VA, pp. 264, 1981.
2. D.A. Johns, K. Martin “Analog Integrated Circuit Design,”
John Wiley & Sons, pp. 706, 1997.
3. C.D. Motchenbacher, J.A. Connelly, “Low Noise Electronic
System Design,” John Wiley & Sons, pp. 422, 1993.
4. B. Razavi, “Design of Analog CMOS Integrated Circuits,”
McGraw-Hill, pp. 684, 2001.
© 2002, Rafael J. Betancourt-Zamora • http://www.betasoft.org/
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