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Reference: [1]

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5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

5.2.Noise types

In order to reduce errors, the measurement object and the measurement system should be matched not only in terms of output and input impedances, but also in terms of noise.

The purpose of noise matching is to let the measurement system add as little noise as possible to the measurand.

We will treat the subject of noise matching in Section 5.4. Before that, we have to describe the most fundamental types of noise and its characteristics (Sections 5.2 and 5.3).

Reference: [1]

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

5.2.1. Thermal noise

Thermal noise is observed in any system having thermal losses and is caused by thermal agitation of charge carriers.

Thermal noise is also called Johnson-Nyquist noise. (Johnson, Nyquist: 1928, Schottky: 1918).

An example of thermal noise can be thermal noise in resistors.

Reference: [1]

vn(t)

6s

2s

Vn rms

f(vn)

2R(t)

en2

White (uncorrelated) noise

t

f

0

0

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

Example: Resistor thermal noise

vn(t)

T 0

R

V

t

Normal distribution

according to the

central limit

theorem

C

enC

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

A. Noise description based on the principles of thermodynamics and statistical mechanics (Nyquist, 1828)

To calculate the thermal noise power density, en2( f ), of a resistor, which is in thermal equilibrium with its surrounding, we temporarily connect a capacitor to the resistor.

Real resistor

R

Ideal, noiseless resistor

en

Noise source

From the point of view of thermodynamics, the resistor and the capacitor interchange energy:

mv2

2

mi vi 2

2

mv2

2

kT

2

In thermal equilibrium:

=

=3

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

Illustration: The law of equipartition of energy

Each particle has three degrees of freedom

mivi 2

2

CV2

2

CV2

2

kT

2

In thermal equilibrium:

=

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

Illustration: Resistor thermal noise pumps energy into the capacitor

Each particle has three degrees of freedom

mivi 2

2

enR

CV2

2

kT

2

In thermal equilibrium:

=

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

Since the obtained dynamic first-order circuit has a single degree of freedom, its average energy is kT/2.

This energy will be stored in the capacitor:

enC ( f )

enR ( f )

H( f ) =

Real resistor

R

C

Ideal, noiseless resistor

enC

Noise source

According to the Wiener–Khinchin theorem (1934), Einstein (1914),

CsnC2

kT

kT

CvnC(t)2

= = snC2 = .

2

2

C

2

enR 2(f)H(j2pf)2ej2pf tdf

snC2 = RnC (0) =

0

kT

enR2(f)

1

= .

= enR2(f)

df =

C

4RC

1+ (2pfRC)2

0

Power spectral density of resistor noise:

enR2(f)= 4kTR [V2/Hz].

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

enR P(f)2

enR(f)2

1

R = 50 W,

C= 0.04 fF

0.8

0.6

0.4

0.2

f

1 GHz

10 GHz

100 GHz

1 THz

10 THz

100 THz

SHF

EHF

IR

R

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

B. Noise description based on Planck’s law for blackbody radiation (Nyquist, 1828)

hf

ehf /kT-1

enR P2(f)= 4R [V2/Hz].

A comparison between the two Nyquist equations:

Zero-point energyf(t)

eqn (f)2

enR (f)2

8

6

Quantum noise

4

2

0

f

SHF

EHF

IR

R

1 GHz

10 GHz

100 GHz

1 THz

10 THz

100 THz

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

C. Noise description based on quantum mechanics (Callen and Welton, 1951)

The Nyquist equation was extended to a general class of dissipative systems other than merely electrical systems.

hf

ehf /kT-1

hf

2

eqn2(f)= 4R + [V2/Hz].

Ratio, dB

120

100

80

60

40

20

f, Hz

0

102

104

106

108

1010

1012

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

The ratio of the temperature dependent and temperature independent parts of the Callen-Welton equation shows that at 0 K there still exists some noise compared to the Nyquist noise level at Tstrd= 290 K (standard temperature: kTstrd = 4.0010-21)

2

ehf /kT-1

10Log dB.

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

D. Equivalent noise bandwidth, B

An equivalent noise bandwidth,B,is defined as the bandwidth of an equivalent-gain ideal rectangular filter that would pass as much noise power as the filter in question.

By this definition, the Bof an ideal filter is its actual bandwidth.

For practical filters, Bis greater than their 3-dB bandwidth. For example, an RC filter has B= 0.5pfc, which is about 50% greater than its 3-dB bandwidth.

As the filter becomes more selective (sharper cutoff characteristic), its equivalent noise bandwidth, B, approaches the 3-dB bandwidth.

Reference: [4]

=en in2H(f)2 df

0

1

1 + (f / fc )2

= en in2

df

= en in20.5p fc

Vn o rms2= en in2B

0

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

Example: Equivalent noise bandwidth of an RC filter

R

en o(f)

Vn o rms2=en o2(f)df

0

C

en in

1

2pRC

fc = = D f3dB

en o2

en o2

B=0.5pfc = 1.57fc

en in2

en in2

Equal areas

Equal areas

0.5

fc

B

1

0.5

0.1

fc

f /fc

0.01

0.1

1

10

100

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

Example: Equivalent noise bandwidth of an RC filter

R

en o

1

C

en in

f /fc

1

2pRC

fc = = D f3dB

1

0

2

4

6

8

10

Butterworth filters:

1

1+ ( f /fc )2n

H( f )2 =

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

Example: Equivalent noise bandwidth of higher-order filters

First-order RC low-pass filterB= 1.57fc.

Two first-order independent stages B= 1.22fc.

- second order B= 1.11fc.
- third order B= 1.05fc.
- fourth order B= 1.025fc.

Noise voltage:

At room temperature:

Vn rms = 4kTRDfn[V].

en = 0.13R[nV/Hz].

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

Amplitude spectral density of noise:

en = 4kTR[V/Hz].

Examples:

Vn rms = 4kT1MW1MHz = 128 mV

Vn rms = 4kT1kW1Hz = 4 nV

Vn rms = 4kT50W1Hz = 0.9 nV

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

1) First-order filtering of the Gaussian white noise.

E. Normalization of the noise pdf by dynamic networks

Input noise pdf

Input and output noise spectra

Output noise pdf

Input and output noise vs. time

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

1) First-order filtering of the Gaussian white noise.

Input noise pdf

Input noise autocorrelation

Output noise pdf

Output noise autocorrelation

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

2) First-order filtering of the uniform white noise.

Input noise pdf

Input and output noise spectra

Output noise pdf

Input and output noise vs. time

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

2) First-order filtering of the uniform white noise.

Input noise pdf

Input noise autocorrelation

Output noise pdf

Output noise autocorrelation

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

F. Noise temperature, Tn

Different units can be chosen to describe the spectral density of noise: mean square voltage (for the equivalent Thévenin noise source), mean square current (for the equivalent Norton noise source), and available power.

en2 = 4kTR[V2/Hz],

in2 = 4kT/R[I2/Hz],

en2

na = kT[W/Hz].

4R

It is a common practice to characterize other, nonthermal sources of noise, having available power that is unrelated to a physical temperature, in terms of an equivalent noise temperature Tn:

na2( f )

Tn ( f ) .

k

Then, given a source's noise temperature Tn,

na2( f ) kTn ( f ) .

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

Any thermal noise source has available power spectral density na( f ) kT ,where T is defined as the noise temperature, T = Tn.

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

Example: Noise temperatures of nonthermal noise sources

Cosmic noise: Tn= 1 … 10 000 K.

Environmental noise: Tn(1 MHz) = 3108 K.

T

Vn2(f)= 320p2(l/l)kT = 4 kT RS

l << l

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

G. Thermal noise in capacitors and inductors

Ideal capacitors and inductors do not dissipate power and then do not generate thermal noise.

For example, the following circuit can only be in thermal equilibrium if enC = 0.

R

C

enR

enC

Reference: [2], pp. 230-231

f > 0

f > 0

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

R

C

enR

enC

In thermal equilibrium, the average power that the resistor delivers to the capacitor, PRC, must equal the average power that the capacitor delivers to the resistor, PCR. Otherwise, the temperature of one component increases and the temperature of the other component decreases.

PRC is zero, since the capacitor cannot dissipate power. Hence, PCR should also be zero: PCR= [enC(f)HCR(f)]2/R = 0, where HCR(f)= R/(1/j2pf+R). Since HCR(f) 0, enC (f)= 0.

Reference: [2], p. 230

=enR2H(f)2 df

0

VnC rms2

R

C

VnC

enR

1

2pRC

= 4kTR 0.5p

kT

C

VnC rms2=

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

H. Noise power at a capacitor

Ideal capacitors and inductors do not generate any thermal noise. However, they do accumulate noise generated by other sources.

For example, the noise power at a capacitor that is connected to an arbitrary resistor value equals kT/C:

= 4kTRB

Reference: [5], p. 202

R

C

VnC

enR

kT

C

VnC rms2=

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

The rms voltage across the capacitor does not depend on the value of the resistor because small resistances have less noise spectral density but result in a wide bandwidth, compared to large resistances, which have reduced bandwidth but larger noise spectral density.

To lower the rms noise level across a capacitors, either capacitor value should be increased or temperature should be decreased.

Reference: [5], p. 203

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise

Home exercise:

Some feedback circuits can make the noise across a capacitor smaller than kT/C, but this also lowers signal levels.

Compare for example the noise value Vn erms in the following circuit against kT/C. How do you account for the difference? (The operational amplifier is assumed ideal and noiseless.)

1nF

1k

Vn erms

Vn o rms

1k

1pF

vs

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

5.2.2. Shot noise

R

Shot noise (Schottky, 1918) results from the fact that the current is not a continuous flow but the sum of discrete pulses, each corresponding to the transfer of an electron through the conductor. Its spectral density is proportional to the average current and is characterized by a white noise spectrum up to a certain frequency, which is related to the time taken for an electron to travel through the conductor.

In contrast to thermal noise, shot noise cannot be reduced by lowering the temperature.

I

ii

www.discountcutlery.net

Reference: Physics World, August 1996, page 22

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

Illustration: Shot noise in a conductor

R

i

I

t

Reference: [1]

I

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

Illustration: Shot noise in a conductor

R

i

I

t

Reference: [1]

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

A. Statistical description of shot noise

We start from defining n as the average number of electrons passing a cross-section of a conductor during one second, hence, the average electron current I = qn.

We assume then that the probability of passing through the cross-section two or more electrons simultaneously is negligibly small. This allows us to define the probability that an electron passes the cross-section in the time interval dt = (t, t+dt) as P1(dt) = ndt.

Next, we derive the probability that no electrons pass the cross-section in the time interval (0, t+dt):

P0(t+dt) = P0(t) P0(dt) = P0(t) (1-ndt).

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

This yields

with the obvious initiate state P0(0) = 1.

dP0

dt

= -nP0

The probability that exactlyone electrons pass the cross-section in the time interval (0, t+dt)

P1(t+dt) = P1(t) P0(dt) + P0(t) P1(dt)

= P1(t) (1-ndt) + P0(t) ndt .

This yields

with the obvious initiate state P1(0) = 0.

dP1

dt

= -nP1 +nP0

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

In the same way, one can obtain the probability of passing the cross section N electrons, exactly:

dPN

dt

= -nPN+nPN -1

.

PN(0) = 0

By substitution, one can verify that

(nt)N

N!

PN (t) = e- n t,

which corresponds to the Poisson probability distribution.

(1t)N

N!

PN (t) = e-1 t

N=10

N=20

N=30

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

Illustration: Poisson probability distribution

0.12

0.1

0.08

0.06

0.04

0.02

t

0

10

20

30

40

50

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

The average number of electrons passing the cross-section during a time interval t can be found as follows

(nt)N

N!

(nt)N-1

(N -1)!

S

S

Nt= e- nt = nt e- nt = nt ,

n=1

n=0

and the squared average number can be found as follows:

(nt)N

N!

(nt)N

N!

S

S

Nt2= Nt2 e- nt= [N (N -1) + N ] e- nt

n=0

n=0

(nt)N-2

(N -2)!

S

= nt +(nt)2 e- nt = nt +(nt)2.

n=2

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

The variance of the number of electrons passing the cross-section during a time interval t can be found as follows

sN2= Nt2 - ( Nt )2 = nt.

We now can find the average current of the electrons, I, and its variance, irms2:

I = it= (q/t) Nt= qn,

irms2=(q/t)2 sN2= (q/t)I .

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

B. Spectral density of shot noise

Assuming t=1/2 fs,we finally obtain the Schottky equation for shot noise rms current

In2=2 qI fs .

Hence, the spectral density of the shot noise

in(f)= 2 qI .

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise

C. Shot noise in resistors and semiconductor devices

In devices such as tunnel junctions the electrons are transmitted randomly and independently of each other. Thus the transfer of electrons can be described by Poisson statistics. For these devices the shot noise has its maximum value at 2qI.

Shot noise is absent in a macroscopic, metallic resistor because the ubiquitous inelastic electron-phonon scattering smoothes out current fluctuations that result from the discreteness of the electrons, leaving only thermal noise.

Shot noise may exist in mesoscopic (nm) resistors, although at lower levels than in a tunnel junction. For these devices the length of the conductor is short enough for the electron to become correlated, a result of the Pauli exclusion principle. This means that the electrons are no longer transmitted randomly, but according to sub-Poissonian statistics.

Reference: Physics World, August 1996, page 22

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise

5.2.3. 1/f noise

Thermal noise and shot noise are irreducible (ever present) forms of noise. They define the minimum noise level or the ‘noise floor’. Many devises generate additional or excessnoise.

The most general type of excess noise is 1/for flicker noise. This noise has approximately 1/f spectrum (equal power per decade of frequency) and is sometimes also called pink noise.

1/f noise is usually related to the fluctuations of the devise properties caused, for example, by electric current in resistors and semiconductor devises. Curiously enough, 1/f noise is present in nature in unexpected places, e.g., the speed of ocean currents, the flow of traffic on an expressway, the loudness of a piece of classical music versus time, and the flow of sand in an hourglass.

No unifying principle has been found for all the 1/f noise sources.

Reference: [3]

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise

In electrical and electronic devices, flicker noise occurs only when electric current is flowing.

In semiconductors, flicker noise usually arises due to traps, where the carriers that would normally constitute dc current flow are held for some time and then released. Although both bipolar and MOSFET transistors have flicker noise, it is a significant noise source in MOS transistors, whereas it can often be ignored in bipolar transistors.

References: [4] and [5]

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise

An important parameter of 1/f noise is its corner frequency, fc, where the power spectral density equals the white noise level. A typical value of fc is 100 Hz to 1 kHz.

in 2(f), dB

Pink noise

White noise

f,decades

fc

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise

Flicker noise is directly proportional to the density of dc (or average) current flowing through the device:

a AJ2

f

in2(f)= ,

where a is a constant that depends on the type of material, and A is the cross sectional area of the devise.

This means that it is worthwhile to increase the cross section of a devise in order to decrease its 1/f noise level.

References: [4] and [5]

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise

For example, the spectral power density of 1/f noise in resistors is in inverse proportion to their power dissipating rating. This is so, because the resistor current density decreases with square root of its power dissipating rating:

1 W, 1 W

1 A

a AJ2

f

in 1W(f)2=

in 1W2(f), dB

White noise

f,decades

fc

a AJ2

9f

in 9W2(f)=

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise

For example, the spectral power density of 1/f noise in resistors is in inverse proportion to their power dissipating rating. This is so, because the resistor current density decreases with square root of its power dissipating rating:

1 W, 1 W

1 A

a AJ2

f

in 1W2(f)=

1/3 A

in 1W2(f), dB

1/3 A

1 A

1/3 A

1 W, 9 W

White noise

f,decades

fc

a AJ2

9f

in 9W2(f)=

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise

For example, the spectral power density of 1/f noise in resistors is in inverse proportion to their power dissipating rating. This is so, because the resistor current density decreases with square root of its power dissipating rating:

1 W, 1 W

1 A

a AJ2

f

in 1W2(f)=

1/3 A

in 1W2(f), dB

1/3 A

1 A

1/3 A

1 W, 9 W

White noise

f,decades

fc

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise

Example: Simulation of 1/f noise

Input Gaussian white noise

Input noise PSD

Output 1/f noise

Output noise PSD

5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise

Example: Simulation of 1/f noise

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