noise analysis and modeling

8/2/2019 Noise Analysis and Modeling

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Prof. Tai-Haur Kuo, EE, NCKU, Tainan City, Taiwan , Analog IC Design, 20114-1

Noise Analysis and Modeling Circuit noise

1. interference noise

2. inherent noise

Interference noise

result from interaction between circuit and outside world or

between different parts of circuit itself.

examples :

(i) power supply noise on ground wires.

(ii) electromagnetic interference between wires.

can be reduced by careful circuit wiring or layout

Inherent noise

refers to random noise signals that can be reduced butnever eliminated since this noise is due to fundamental

properties of circuits.


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Noise Analysis and Modeling (Cont.)

examples :

thermal noise and flicker noise

only moderately affected by circuit wiring or layout, suchas using multiple contact to change resistance value of atransistor. However, inherent noise can be significantlyreduced through proper circuit design, such as changing

the circuit structure or increasing bias current.

Only inherent noise will be discussed in the following


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Time-Domain Analysis

Assumption : All noise signals have a mean value of zero.This assumption is valid in most physical systems.

Root mean square(rms) voltage value is defined as

rms current is defined as

Typically, a longer T gives a more accurate rmsmeasurement

Normalized noise power, pdiss

Vn(t) is applied to a 1 resistor

or Pdiss =

2/1T

0

2

nn(rms) dt](t)VT

1

[V

2

n(rms)

2n(rms)

diss V1

VP

2

n(rms)

2

n(rms) I

1I

2/1T

0

2nn(rms) dt](t)I

T

1[I


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Time-Domain Analysis (Cont.)

Signal-to noise ratio(SNR), dBSNR=10

Example : normalized signal power =

normalized noise power =

dBm

dB units relate the relative ratio of two power levels

For dBm units, all power levels are referenced by1mW

Examples : 1mW = 0dBm and 1W = -30dBm

It is common to reference the voltage level to either a50 or 75 resistorexample:

]V

Vlog[]

V

Vlog[SNR

)rms(n

)rms(x

)rms(n

)rms(x2010

2

2

dBm

mW

V rmsn

1

50log10

2)(

dBmmW

I rmsn

1

50log10

)(2

or 75

or

when Vx(rms)=Vn(rms), SNR=0dB

]log[powernoise

powersignal

2)rms(xV

2

)rms(n

V


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Noise Summation

Noise sources Vn1(t),Vn2(t),Vn3(t),.

Total noise Vn0(t)= Vn1(t)+Vn2(t)+Vn3(t)+

Example : summation of 2 noise sources

- Voltage noises - Current noises)t(ino

)t(in1 )t(in2

)t(V 1n

)t(V 2n

)t(Vno

)rms(n)rms(n)rms(n)rms(n

T T

nn)rms(n)rms(nnn)rms(no

VCVVV

dt)t(V)t(VT

VVdt)t(V)t(VT

V

2122

21

0 021

22

21

2

212

2

21

)t(V)t(V)t(Vnnno 21


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Noise Summation (Cont.)

Correlation coefficient

where -1C1

C = 1; the two noise signals are fully correlatedC = 0 ; the two noise signals are fully uncorrelated

Typically, different inherent noise sources are uncorrelated

For two uncorrelated noise signals

V2no(rms)=V2n1(rms)+ V

2n2(rms)

For two fully correlated noise signals

V2no(rms)=[Vn1(rms) Vns(rms)]2

To reduce overall noise, concentrate on the reduction oflarge noise signals.

)rms(n)rms(n

Tnn

VV

dt)t(V)t(VTC

21

0 211


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Frequency-Domain Analysis

Units of Hz(rather than radians/sec) are commonly used Noise spectral density

periodic signals(e.g. sinusoid) have power at distinct

frequency random signals have their power spread out over the

frequency spectrum

Example

Time-domain signal

0 2.0 4.0 6.0 8.0

3.0

2.01.00

1.02.0

)(sTime

)(fVn


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Frequency-Domain Analysis (Cont.)

Spectral density

Root spectral density

Vertical axis is a measureof the normalized noise

power over 1 Hz

bandwidth at eachfrequency point

1.0 100.1 100 000,1

0.1

16.3

10

6.31

Hz

V

)(fVn

1.0 100.1 100 000,1

0.1

10

100

000,1

Hz

)V( 2

Hz

)(2 fVn

Hz


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Frequency-Domain Analysis (Cont.)

Resolution Bandwidth(RBW) V2/Hz use 1Hz bandwidth => normalized Mean-squared value of a random signal at a precise

frequency is zero.

Random-noise power must be measured over a specificbandwidth.

Example1: Normalized power between 99.5Hz and100.5Hz is 10(v)2---shown in previous page

Example2: Mean-squared value of noise power at100Hz is 1(v)2 when 0.1Hz is used

=> Mean-squared value measured at 100Hzis directly proportional to the bandwidth of the

bandpass filter used for measurement Total mean-squared power

0

22 df)f(VV n)rms(n

0

22 df)f(II n)rms(n


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Noise Types in CMOS transistor

Two major sources Thermal, or white noise

flat spectrum density

Vn(f)=Vnw is a constant Flicker, or 1/f noise

Spectrum density is inversely proportional to frequencyVn2(f)= /f where Kv is a constant.

The intersection of flicker and white noise curves iscalled 1/f noise corner

Spectral density

2vK

26262n )101(

f

)102.3()f(V

2.3V)f(V nwn HzV

1.0 0.1 10 100

0.1

2.3

10

)(fVn

Hz

Hz

V

1/f noisedomnates

White noisedominates

Hz

V

1,000


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Noise in MOSFET

Thermal noise ( white noise caused by random thermalmotion of electron) Noise power

Real Resistor Rmean square VnT:

f: bandwidth in which the noise is measured, in Hz4kT, at room temperature, is equal to 1.661020VC

MOSFET

If the device is in saturation, then

frequencyfkTR4V2

nT

m

gR2

31

f cant be infinite, could be assumed up to

several hundred MHz for MOSFET. Since,

for very high frequency (1012Hz), other

physical phenomena enter which cause

to decrease with increasing frequency.

fVmm

nT

gkT

g

i

nT 3822 )(

~VnT

inT2nV


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Noise in MOSFET (Cont.)

Flicker Noise (1/f)In an MOS transistor, extra electron energystates exist at boundary between the Si and SiO2. These can

trap and release electrons from the channel, and henceintroduce noise. Since the process is relatively slow, most of

the noise energy will be at low frequency.

Noise power

f

f

f

WLC

kV

ox

nf

2

Thermal noise + Flicker noise

where G is the noise conductance

The mean squares of the noise currents are added, since thedifferent noise mechanism are statistically independent.

infgWLfC

K

g

KTiii m

oxm

nfnTn

222

3

8


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Filtered Noise

Noise amplification and filtering

spectral density

: input noise spectral density

: output noise spectral density

root spectral density

)()()( fVfjAfV nino222

2)(fVni)(fVno

)(fVni2

)()()( fVfjAfV nino222 2

)()()( fVfjAfV nino 2

)s(A


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Filtered Noise (Cont.)

Total output mean-squared value

Summation of multiple filtered uncorrelated noise sources

Example : 3 sources

)f(V 1n

)f(V 2n

)f(V 3n

21

321

222 )](|)(|[)(,,

fVfjAfVi niino

)s(A1

)s(A2

)s(A3

Uncorrelated

noise sources

0

222 2 df)f(V)fj(AV ni)rms(no

n

i

niino )f(V)j(A)f(V1

222 2


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Noise Bandwidth

The noise bandwidth of a given filter is equal to the frequencyspan of a brick wall filter that has the same output noise rmsvalue that the given filter has when white noise is applied toboth filters. (Peak gains are the same for the given and brick-

wall filters.) Example : a 1st-order lowpass response with a 3 dB

bandwidth of fo(Such a response would occur

from a RC filter with )

input signal Vni(f)=Vnw (White noise)

1. For the response A(s)= ,

2. For brick-wall filter with fx bandwidth,

3. From 1 and 2, noise bandwidth fx=

RC2

1fo

of2

s1

1

2fV

df

)f

f(1

VV o

2nw

0 2

o

2nw2

)rms(no

2

fo

xnw

f

nw)rms(no fVdfVVx 2

0

22


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Noise Bandwidth (Cont.)

)f(Vni

KR 1

F159.0C RC2

1f0

)f(Vno

|)f2j(A| brick

100

fo10

f oof

ox f2

f

0

20

)(dB

|)f2j(A|

100

fo10

ofof of10

0

20

)(dB

Hz

nv

0.1 10 100 310 410

20

-20dB/decade

f

)(fVn

0f2s1

1)s(A


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Noise Bandwidth (Cont.)

For a RC filter

KT/C noise

SC sampling circuit

circuit noise model

Vi=0 is assumed

RCfo 2 1 RCfx 4 1and

iV

oV

clock

iV oV

noiseless resistor

eqR

C

KT

CR

KTRV

eq

eq

)rms(o 4

42

C

eq2R KTR4V


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Approximate Noise Calculation

Piecewise integration of noise

simplify integration formulas

integrate noise power

in different frequency

regions and then

add together

example)f(Vni )f(Vno

)f(Vno

)Hz

nV(

1N 2N 3N 4N

)Hz(frequency

curvef

1

)f(Vni

)Hz

nV(202

200

)Hz(frequency

200

20

|)f2j(A| )(dB

765432 101010101010101

765432 101010101010101

765432 1010101010101012

02200

)s(A


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Approximate Noise Calculation (Cont.)

The noise power in the N4 region is quite close to the totalnoise power. Thus, in practice, there is little need to find the

noise contributions in N1~N3 regions. Such an observationleads us to the 1/f noise tangent principle.

)(....

)()()()()(

nv1088510331106310841

dffVdffVdffVdffVV

21

9855

21

100

1

10

100

10

10 10

2no

2no

2no

2normso

3 4

3 4


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Approximate Noise Calculation (Cont.)

1/f noise tangent principle

To determine the frequency region or regions that

contribute to dominant noise, lower a 1/f noise line until

it touches the spectral density curve --- The total noisecan be approximated by the noise in the vicinity of the1/f line.

The reason this simple rule works is that a curveproportional to 1/x results in equal power over eachdecade of frequency. Therefore, by lowering this

constant power/frequency curve, the largest power

contribution will touch it first.


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Noise Models for Circuit Elements

Three main noise mechanisms1. Thermal noise

white noise

2. Shot noise occurs in pn junctions

white noise

3. Flicker noise 1/f noise

Resistor noise

thermal noise is the major noise source spectral density VR

2(f) or IR2(f)


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Noise Models for Circuit Elements (Cont.)

1. Series voltage noise sourceVR2(f)=4KTR

where K is Boltzmamns constant (1.38x10-23JK-1)

T is temperature in Kelvin's

R is the resistance value

2. Parallel current noise source

Diode noise

short noise

Vd2(f)=2KTrd

where

Capacitors and inductors do not generate noise

R

KT

R

(f)V(f) R

R

42

22 I

DDT

SDST

SDT

DDDd

qKTVV)ln(VVrIIII

I

I

I

II

1

D

d

dd

qr

(f)V(f) II 2

2

22


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Noise Models for Circuit Element (Cont.)

OPAMPs

Bipolar OPAMP CMOS OPAMP

(All noise sources are uncorrelated)

)(fIn2

)(fVn2

)(fIn2

NoiselessNoiseless

)(fVn2


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Element Noise Models

Resistor

R

(Noiseless)

Diode

Forward(Biased)

R

KTfR

42 )(I

Dd

q

KTr

I

D2d KTr2)f(V

Ddq(f) II 22 D

dqI

KTr

(Noiseless)

R (Noiseless)

KTR)f(VR 42

R

(Noiseless)


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)

)f(f

Kq((f)

)g

KT(r(f)V

CBBi

mbi

22

2

2

2

14

IIII

(Activeregion)

MOSFET

(Activeregion)

)f(V 2g

)f(I2d

)f(V 2i

Noiseless

foxWLC

K

mg

1)

3

2(KT4)f(

2iV

simplified model for low andmoderate frequencies

BJT

)f(I2i

Noiseless)f(V2

i

Opamp )(Noiseless)(),(),( fIfIfV nnn

Value depends on opamp

typically, all uncorrelated

md

ox

g

gkTfI

fWLC

KfV

)3

2(4)(

)(

2

2

)(fIn2

)(fVn2

)f(In2


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OPAMP example a lowpass filter equivalent noise model

assuming all noise sources are uncorrelated

using superposition1. due to , , and

2. due to , , and

3.

Noise Analysis Examples

)f(V2 1no )f(n1I )f(nfI )f(nI

)f(V2 2no )f(nI )f(V 2n )f(Vn

)f(V)f(V)f(V

2

2no

2

1no

2

no

fC

2R

1R

1nI nI

nI

fR

nfI

nV2nV

)f(Vno

1RfR

fC

2R

iV oV


27/32


Noise Analysis Examples (Cont.)

Vno12 (f)

Vno2

2(f)

ff

ftotal

f

f

totaloCSR

R

SCR

V

11

1II

2

122

2

2

2

22

221

1)]()()([)(ff

f

nnnnoCfRj

RR

fVfVRffV

I

22

0

21

22)rms(no)rms(nono)rms(no VVdf)f(VV

1R fR

fC

totalVoV

oV

totalI

fC

fR

ff

f

totaloCSR

RR

VV1

1 1

2

f2

n

2

nf

2

n1

2

no1j21

R(f)I(f)I(f)I(f)V

ffCfR


28/32



CMOS examples a differential input stage

assuming

Q1 and Q2 are identicalQ3 and Q4 are . identical

3

5

5

343

121

23

2

1

m

m

n

no

omn

no

n

no

omn

no

n

no

gg|

VV|)(

Rg|V

V||

V

V|)(

Rg|

V

V||

V

V|)(

(The drain of Q2 will

track that of Q1)

inV

inV

biasV

42 dsdso r//rR

)output(

552

1nm Vg 552

1nm Vg

55 nmVg

Q5

Q2Q1

Q3 Q4

2nV1nV

4nV3nV

5nV DDV

oV

SSV

5nV


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Since (3) is relative small compared to the others, it can beignored.

Equivalent input noise

For the white noise portion, i.e. thermal noise

)f(n

V)oRmg()f(nV)oRmg()f(noV

23

232

21

212

2

2)

1mg

3mg

)(f(2

3nV2)f(21nV22)oR1mg(

)f(2noV

)f(2neqV

,mi

g

1

3

2KT4f

2thermalniVassuming ))(()()(

)s(A )s(A

)(

)(

fA

fVno equivalent

Input noise

iV iVoV

)f(Vno

)()()()()(

3m

g

12

1m

g3m

gKT

3

16

1m

g

1KT

3

16f2

thermalneqV

oV


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gm1 should be made as large as possible to minimize

thermal noise contribution.

For flicker (1/f) noise portion

assuming

Diiimi )L

w(Coxg I2

]LW

LK)(

LW

K[

fC)f(V

fCLW

K)f(V

p

n

ox

ker)flic(neq

oxii

iker)flic(ni

231

13

11

12

2

2

. (4.101)

p

n

nnker)flic(neq)

LW(

)L

W(

)f(V)f(V)f(V1

32

3

2

1

2

22


31/32



Recall that the first term in (4.101) is due to the p-channelinput transistors, Q1 and Q2, and the second term is due tothe n-channel loads, Q3 and Q4 . We note some points for 1/f

noise here:1. For L1=L3 , the noise of the n-channel loads dominate

since and typically n-channel transistors have

larger 1/f noise than p-channel transistors (i.e. , K3 > K1).2. Taking L3 longer greatly helps due to the inverse squared

relationship in the second term of (4.101). This limits the

signal swings somewhat, but it may be a reasonably

trade-off where low noise is important.

3. The input noise is independent of W3, and therefore we

can make it large to maximize signal swing at the output.

pn


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4. Taking W1 wider also helps to minimize 1/f noise.(Recall that it helps white noise, as well.)5. Taking L1 longer increases the noise because the

second term in (4.101) is dominant. Specifically, this

decreases the input-referred noise of the p-channeldrive transistors , which are not the dominate noisesources, but it also increases the input-referred noise ofthe n-channel load transistors, which are the dominant

noise sources !Total rms input noise, Vneq(rms.)

2 , integrated from f1 to f2.

)ff)](gm

()gm

gm(kT)

gm(kT[ 12

3

2

1

3

1

1

3

161

3

16

)]Lw

L)((a

Lw

a[

p

nn

p

2

31

1

112

1

2;f

f

c

kawhere

ox

ii ln

dffVfVVf

fflicneqthermalneqrmsneq

2

1

2 )()( ker)()()(

noise analysis and modeling

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