Brownian Entanglement: Entanglement in classical brownian motion
Lecture 7: Noise · 5 , Spring 2004 9 ENE 5400 Thermomechanical (Brownian) Noise For mechanical...
Transcript of Lecture 7: Noise · 5 , Spring 2004 9 ENE 5400 Thermomechanical (Brownian) Noise For mechanical...
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ENE 5400 , Spring 2004 1
Lecture 7: Noise
Basics of noise analysis Thermomechanical noise
Air damping Electrical noise
Interference noise» Power supply noise (60-Hz hum)» Electromagnetic interference
Electronics noise» Thermal noise» Shot noise» Flicker (1/f) noise
Calculation of total circuit noise
Reference: D.A. Johns and K. Martin, Analog Integrated Circuit Design, Chap. 4,John Wiley & Sons, Inc.
ENE 5400 , Spring 2004 2
Why Do We Care?
Noise affects the minimum detectable signal of a sensor Noise reduction:
The frequency-domain perspective: filtering» How much do you see within the measuring bandwidth
The time-domain perspective: probability and averaging Bandwidth Clarification
-3 dB frequency Resolution bandwidth
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ENE 5400 , Spring 2004 3
Time-Domain Analysis
White-noise signal appears randomly in the time domain with an average value of zero
Often use root-mean-square voltage (current), also thenormalized noise power with respect to a 1-ΩΩΩΩ resistor, as defined by:
Signal-to-noise ratio (SNR) = 10⋅⋅⋅⋅ Log (signal power/noise power) Noise summation
∫∫ ++=+=
+=T
nnrmsnrmsn
T
nnrmsn
nnn
dtVVT
VVdttVtVT
V
tVtVtV
0 212
)(22
)(10
221
2)(
21
2)]()([
1
)()()(
∫
∫
=
=
T
nrmsn
T
nrmsn
dttIT
I
dttVT
V
0
2/12)(
0
2/12)(
])(1
[
])(1
[
0, for uncorrelated signals
ENE 5400 , Spring 2004 4
Frequency-Domain Analysis
A random signal/noise has its power spread out over the frequency spectrum
Noise spectral density is the average normalized noise power over a 1-Hz bandwidth (unit = volts-squared/Hz or amps-square/Hz)
It is common to use root spectral density (e.g., V/√√√√Hz)
Vn2(f)
f
Spectral density
1
10
100µV2/Hz
f
Vn(f)
Root spectral density
1
3.16
10µV/√Hz
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ENE 5400 , Spring 2004 5
Filtered Noise
Root-mean-squared voltage can also be obtained in the frequency domain using
Total mean-squared value from a filter output is:
You should avoid designing circuits with a larger bandwidth than your signal actually requires
∫∞
=0
22)( )( dffVV nrmsn
dffVfjAV nrmsno )()2( 2
0
22)( ∫
∞= π
A(s)Vn
2(f) Vno2(f) = |A(j2ππππf)|2Vn
2(f)
Input noise output noise
filter
ENE 5400 , Spring 2004 6
Example: Filtered Noise
Vn(f)
20nV/√Hz
101 100 103 104 f (Hz)101 100 103 104
|A(j2πf)|, dB
0
A(s) = 1/(1 + s/2πfo)
f (Hz)
fo = 103
=2)rms(noV
4
ENE 5400 , Spring 2004 7
Noise Bandwidth
Noise bandwidth fx = ππππfo/2
fo
|A(j2πf)|
0
A(s) = 1/(1 + s/2πfo)
f (Hz)fo
|Abrick(j2πf)|
0
f (Hz)
fx
=
ENE 5400 , Spring 2004 8
Minimum Detectable Signal
Usually expressed in VOLTS Min. detectable signal = root spectral density ×××× / sensitivityBW
g V/√√√√Hz √√√√Hz V/gFor example:
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ENE 5400 , Spring 2004 9
Thermomechanical (Brownian) Noise
For mechanical sensors, thermomechanical noise due to air damping is the ultimate performance limit
A dimensionless parameter, namely, the quality factor, is commonly used to describe the damping and also the Brownian noise
ENE 5400 , Spring 2004 10
First: Understand 2nd-order Lumped-Parameter System
Transfer function x(s)/F(s) = 1/(ms2 + bs + k) = 1/[m(s2 + 2ξωξωξωξωns + ωωωωn2)]
Damping ratio ξξξξ = b / (2mωωωωn)» Underdamped (ξξξξ < 1, overshoot), critically damped (ξξξξ = 1),
and overdamped (ξξξξ > 1) Natural frequency ωωωωn = (k/m)1/2
Damped natural frequency ωωωωd = (1 - ξξξξ2)1/2ωωωωn
k
M
b
Fx ms2 + bs + k
1Force F(s) Displacement x(s)
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ENE 5400 , Spring 2004 11
Cont’d: Transient-Response Characteristics
Reduce rise time tr⇒ have to increase ωωωωn
Reduce maximum overshoot Mp⇒ have to increase ξξξξ Settling time ts ∝∝∝∝ 1/(ξξξξ ωωωωn)
Mp
tr t
Dis
pla
cem
ent
z(t)
tr
ENE 5400 , Spring 2004 12
Second: Definition of Quality Factor (Q) The most fundamental definition of Q is that it is proportional to
the ratio of energy stored to the energy lost, per unit time: Q ≡≡≡≡ (energy stored) / (average energy dissipated per radian);
dimensionless Q is related to the damping ratio: Q = 1 / (2ξξξξ) Q can be measured using the half-power points on frequency
spectrum: Q = ωωωωn / (ωωωωh2 – ωωωωh1) Note that the amplitude is amplified by Q times at resonance
A
ampl
itude
Q⋅A
ωωh1 ωh2ω
QA/√2
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ENE 5400 , Spring 2004 13
Derivation: Q = 1/2ξξξξ
Take the 2nd-order mechanical system for example:
The rate of change of energy with time: (the minus sign indicateenergy dissipation)
Assume a simple harmonic motion of x(t) = X⋅⋅⋅⋅sin(ωωωωdt), the dissipated energy per cycle is:
)(tfkxxbxm ====++++++++ &&&
=×= velocityforcedt
dW
2
2/2
0
1 ξξξξωωωωωωωω
ωωωωππππωωωωππππ
−−−−====
========∆∆∆∆ ∫ ====
nd
dtXbdtxFW
d&
ENE 5400 , Spring 2004 14
Cont’d
The stored total energy of the system:
The quality factor :
22
1
22
2
1
2
1
2
1d
for
max mXmvkXW ω===
<<ξ44 344 21
=Q
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ENE 5400 , Spring 2004 15
Verify Q and Half-Power Bandwidth
The amplitude at resonant is amplified by Q times of the amplitude (Xo) at d.c.
Am
plit
ud
e (X
/Xo) Q
Q/√√√√2
∆ω∆ω∆ω∆ω
ωωωωωωωωn
12 hh
nQωωωωωωωω
ωωωω−−−−
≈≈≈≈
ωωωωh2ωωωωh1
ENE 5400 , Spring 2004 16
Derivations
Standard 2nd-order system:
At ωωωω = ωωωωn, the resonant amplitude is Q; at ωωωωh = r⋅⋅⋅⋅ωωωωn, the amplitude is Q/√√√√2. Therefore:
Also, we already know that:
++++
−−−−
====++++++++
====
nn
nn
n
jss
sG
ωωωωωωωωξξξξ
ωωωωωωωωωωωωξωξωξωξω
ωωωω
21
12
)(
2
222
2
(((( )))) (((( ))))222 21
12
)(rr
QsG
hjsξξξξ
ωωωω++++−−−−
============
ξξξξ21====Q
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ENE 5400 , Spring 2004 17
Cont’d
Therefore we get:
Q
rr
r
r
rr
hh
nnhh
nhh
nnnhh
========−−−−
====−−−−∴∴∴∴
====++++====−−−−====−−−−
++++≈≈≈≈++++++++−−−−====
−−−−≈≈≈≈++++−−−−−−−−====
====−−−−++++−−−−−−−−
ξξξξωωωωωωωωωωωωξωξωξωξωωωωωωωωω
ωωωωωωωωωωωωξωξωξωξωωωωωωωωωωωωωωωωω
ξξξξξξξξξξξξξξξξ
ξξξξξξξξξξξξξξξξ
ξξξξξξξξ
21
,2
2
4
211221
211221
0)81()42(
1212
12
2221
222
21
22
2222
2221
2224
Q
ENE 5400 , Spring 2004 18
Example: A Parallel RLC Tank Circuit
The impedance is zero at DC and at infinitely high frequency:
At the resonant frequency ωωωωn= 1/(LC)0.5, the reactive terms are cancelled; Z = R
Peak energy stored in the network at resonance is:
=Z
IinR C L
+
_
Vout
=totE
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ENE 5400 , Spring 2004 19
Cont’d
The average power dissipated in the resistor at resonance:
The quality factor Q is:
How does the bandwidth (here it means ∆ω∆ω∆ω∆ω at the half-power bandwidth) relate to Q; Let ωωωω = ωωωωn + ∆ω∆ω∆ω∆ω, the -3 dB point, then
2/2
11
]2[1
12
3 RCj
RLC
Lj
R
Z
nn
dB ====∆∆∆∆++++
≈≈≈≈∆∆∆∆++++∆∆∆∆++++
====−−−−
ωωωωωωωωωωωωωωωωωωωω
RIP pkave2
21====
RCQCL
RPE
Q
n
ave
totn
ωωωω
ωωωω
====
========/
Large R is good in parallel network
ENE 5400 , Spring 2004 20
Cont’d
Therefore we get ∆ω∆ω∆ω∆ω= 1 / (2RC) The bandwidth ωωωωh2 – ωωωωh1 = 2⋅⋅⋅⋅∆ω∆ω∆ω∆ω = 1 / RC We can verify that:
QC/L
R
RC/
LC/
hh
n ===ω−ω
ω1
1
12
11
ENE 5400 , Spring 2004 21
Air-Damping Analysis
ENE 5400 , Spring 2004 22
Couette Damping
Motion-induced shear stress ττττ sets up a velocity gradient in the fluid between two plates Viscosity is the proportionality constant that relates ττττ = η⋅η⋅η⋅η⋅∂∂∂∂v/∂∂∂∂x;
(ηηηηair = 77 µµµµN⋅⋅⋅⋅s/m2 @ 25 °°°°C) Incompressible laminar flow; density change is negligible Linear velocity profile is valid for small gaps
dt
dx
x
B
B
xxB
Vg
AF
g
V
y
v
x
η=
η=∂∂
η=τ
g
Movable plate
Linear velocity profile
x
yVx
Shear stress acting on the plate:
Total force acting on the plate:
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ENE 5400 , Spring 2004 23
Squeeze-Film Damping
Damping caused by squeezing air in and out of plates The distribution of gas pressure between plates is governed by the
compressible gas-film Reynolds equation; assuming a small plate motion, the linearized Reynolds equation is:
V
FBzg
gK
K.
)g
z(
t)
P
P(
t)
P
P(
gP
n
.n
eff
aaeff
a
λ=
+η=η
∂∂=∆
∂∂−∆∇
η
1591
22
63891
12
Knudsen number:
Effective gas viscosity:
Mean free path of air (70 nm @1atm and 25 °C)
Atmospheric pressurePressure change
(η = 22.6 µN⋅s/m2 @1atm and 25 °C)
ENE 5400 , Spring 2004 24
Cont’d
The equation can be solved by the eigenfunction method, assuming that ∆∆∆∆p(x,t) = ∆∆∆∆p(x)e-ααααt in an one-dimensional problem, followed by separation of variables
Squeeze-film damping exhibits damper (low-frequency) and spring-like (high-frequency) behaviors
The solution can be simulated by an equivalent circuit model Lmn and Rmn depend on the plate size (w and l), gap, and effective
viscosity (m,n are odd numbers)
effmn
amn
lwzg
wnlmmnR
lwPzg
mnL
ηηηηππππ
ππππ
3
3622222
42
)(768)(
)()(
64)(
)(
−−−−++++====
−−−−====L11
R11
L13
R13
L31
R31
Lmn
Rmn
+
_
velocity
force
( ) ( )∫ ∆=1
0dxt,xPwlPtFforcetotal a
Ref: J. J. Blech, “On isothermal squeeze films,”J. Lubrication Tech., vol. 105, pp. 615-620, 1983.
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ENE 5400 , Spring 2004 25
Derivation of Brownian Noise
ENE 5400 , Spring 2004 26
Brownian Motion
The zigzag motion performed by suspended particles in a liquid or a gas; named after the English botanist Robert Brown Fluctuations from thermodynamic equilibrium
Derived by Einstein and Smoluchowski (using different methods from kinetic theory) in 1906 Experimentally verified by Perrin, Svedberg, Seddig, and
others A simpler derivation can follow Langevin’s method
Zigzag particle motion
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ENE 5400 , Spring 2004 27
Thermomechanical Noise
A good reference can be found in T.B. Gabrielson, “Mechanical-thermal noise in micromachined acoustic and vibration sensors,”IEEE Trans. on Electron Devices, ED-40(5), p. 903-909, 1993
Brownian noise force:
=∆ 2nF
kb: Boltzmann’s constant (1.38 ×××× 10-23 J/K)T : temperature in Kelvinb : damping coefficient (unit = N/(m/s))∆∆∆∆f: measuring bandwidth (unit = Hz)
ENE 5400 , Spring 2004 28
( )1totalFrbrm +−= &&&
22
2
22
2
1
2
1mvFr
dt
drb
dt
rdm total +⋅=+
Multiply by r:
A particle of mass M with arbitrary acceleration:
Equation (2) becomes:
Here is the Derivation
Other influences combinedFriction force
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ENE 5400 , Spring 2004 29
Tkmvmvmv
tconsBoltzmannKJk
Tkmvmv
byx
b
byx
====++++====
××××====
========
−−−−
222
23
22
21
21
21
)tan(/103803.121
21
21
Using the Equipartition theorem:
Therefore:
Cont’d
ENE 5400 , Spring 2004 30
By integration from t = 0 to t = ∆∆∆∆t to obtain the traveled distance:
Cont’d
fb
Tk
r
fb
Tk
tb
Tkrrr
b
b
bo
∆=∆∴
∆=∆=−=∆
=
4
44
0
222
To get the Brownian force, we first get the noise velocity:
fTbkvbFthen
t
bTk
tr
v
bn
b
∆∆∆∆====∆∆∆∆⋅⋅⋅⋅====∆∆∆∆∆∆∆∆
====∆∆∆∆∆∆∆∆====∆∆∆∆
4
/42
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ENE 5400 , Spring 2004 31
Noise Models for Circuit Elements
Thermal noise is due to thermal excitation of charge carriers in a conductor; occurs in all resistors above absolute zero temperature Was first observed by J.B. Johnson (Johnson noise) and
analyzed by H. Nyquist in 1928 a white noise
Shot noise occurs in pn junctions because the dc bias current is not continuous and smooth due to pulsed current caused by individual flow of carriers Was first studied by W. Schottky in 1918 Depend on bias current; also a white noise
Flicker noise (1/f noise) arises due to release of trapped charges in semiconductors A significant noise source in MOS transistors
ENE 5400 , Spring 2004 32
Equivalent Resistor Model with Added Noise
Spectral density functions:
=
=
)f(I
)f(V
R
R
2
2
* VR2(f)
R (noiseless)
R (noiseless) IR2(f)=R (noisy) =
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ENE 5400 , Spring 2004 33
Forward-Biased Diodes
Spectral density functions:
Small-signal resistance of the diode rd does not contribute any thermal noise:
=
=
)f(V
)f(I
d
d
2
2
d
bd
qTkVsD qI
TkreII bD ====⇒==== )//(
* Vd2(f)
rd (noiseless)
rd (noiseless) Id2(f)=diode (noisy) =
ENE 5400 , Spring 2004 34
MOS Transistors in the Active Region
Spectral density function is the sum of the flicker (1/f) noise and thermal noise 1/f noise voltage source and thermal noise voltage source
The equivalent voltage source of 1/f and thermal noise sources
=
=
)f(V
)f(V
t
f
2
2
( ) ( )fVfV)f(V tfi222 +=
Vi2(f)*=
noisy
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ENE 5400 , Spring 2004 35
Operational Amplifiers (Op-amps)
Noise is modeled using three uncorrelated input-referred noise sources: Vn
2(f), In+2(f), and In-
2(f) Op-amps with MOSFET input transistors can ignore current
noises
*Vn
2(f)In+
2(f)
In-2(f)
+
_ Noiseless op-amp
ENE 5400 , Spring 2004 36
Capacitors and Inductors
Capacitors and inductors do not generate any noises; however they accumulate noise generated by other noise sources The rms noise voltage Vno across a capacitor is equal to
(kbT/C)1/2 , regardless of the resistance seen across it (why?)
VR2(f)=4kbTR *
C
Vno
R (noiseless)
( ) =2rmsnoV
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ENE 5400 , Spring 2004 37
Cont’d
The rms noise current Ino across inductor is equal to (kbT/L)1/2 , regardless of the resistance seen across it
Ino
LR (noiseless)IR
2
ENE 5400 , Spring 2004 38
Sampled Signal Noise
The sample-and-hold circuit for A/D or D/A conversion, when turned off, the noise as well as the desired signal is held on the capacitor C
Over-sampling of the signal helps to reduce the noise level Signal values add linearly, whereas their noise values add as the
root of the sum of squares (i.e., neq = (n12 + n2
2 + n32+ …)0.5 )
C
Vin Vout
φclk
Ron
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ENE 5400 , Spring 2004 39
Op-amp Example
2
2221
21 21
)]()()([)(ff
fnnfnno CfRj
RfIfIfIfV
π+++= −
+
_Vi
R1
Rf
Cf
Vo
R2
R2
Low-pass filter
R1
+
_ Rf
Cf
Vno(f)
Inf
**In+
In-
VnVn2
In1
Equivalent noise model
2
1222
22
222 21
/1)]()()([)(
ff
fnnnno RfCj
RRfVfVRfIfV
π++++= +
)()()( 22
21
2 fVfVfV nonono +=
ENE 5400 , Spring 2004 40
Example: CMOS Differential Input Pair
Goal: to obtain an equivalent input-referred voltage noise by summing up all the contributed noises at the output, then dividing by the gain of the circuit
)1
()()3
16()
3
16()(
3
2
1
3
1
2
mm
mb
m
bneq gg
gTk
g
TkfV +=
*
* *
* *
Vn1 Vn2
Vn3 Vn4
Vn5
Q1 Q2
Q3 Q4
Q5
Vin+ Vin-
VDD
VSS
Vno
omn
no
n
no
omn
no
n
no
RgV
V
V
V
RgV
V
V
V
34
4,
3
3,
12
2,
1
1,
==
==
Noise output due to various sources:
21
ENE 5400 , Spring 2004 41
3
5
5
5,
2 m
m
n
no
g
g
V
V=
Due to symmetry, the drain of Q2 will track that of Q1:
)()(2)()(2)( 23
23
21
21
2 fVRgfVRgfV nomnomno +=
(can be neglected)
Output noise value:
Cont’d
Output value can be related back to an equivalent input noise value by dividing it by the gain, gm1Ro
213
23
21
2 )/)((2)(2)( mmnnneq ggfVfVfV +=
ENE 5400 , Spring 2004 42
Diioximi
mibni
IL
WCg
gTkfV
)(2
)1
)(3
2(4)(2
µ=
=
Thermal noise (1/f noise neglected) of each transistor can besubstituted:
Can increase gm1 (more current) to minimize thermal noise contribution.Remember the price you pay for noise reduction is POWER!!
Cont’d