1

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

3

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:

5

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)

6

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

7

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

9

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

10

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:

12

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.

13

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

16

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) =

17

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

18

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

19

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

20

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