1 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

1

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.

5.2. Noise types

5. SOURCES OF ERRORS. 5.2. Noise types

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 in Sections 5.2 and 5.3 the

most fundamental types of noise and its characteristics.

Influence

Measurement System

Measurement Object

Mat

chin

g

+ xx

2MEASUREMENT THEORY FUNDAMENTALS. Contents

5. Sources of errors

5.1. Impedance matching

5.4.1. Anenergetic matching

5.4.2. Energic matching

5.4.3. Non-reflective matching

5.4.4. To match or not to match?

5.2. Noise types

5.2.1. Thermal noise

5.2.2. Shot noise

5.2.3. 1/f noise

5.3. Noise characteristics

5.3.1. Signal-to-noise ratio, SNR

5.3.2. Noise factor, F, and noise figure, NF

5.3.3. Calculating SNR and input noise voltage from NF

5.3.4. VnIn noise model

5.4. Noise matching5.4.1. Optimum source resistance

5.4.2. Methods for the increasing of SNR

5.4.3. SNR of cascaded noisy amplifiers

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.

4

vn(t)

tf (vn)

vn(t)


R V

6

Vn

Example: Resistor thermal noise

T 0

2R()

0

White (uncorrelated) noise

en2

f0

Normal distribution according to thecentral limit theorem

5

C enC


To calculate the thermal noise power density, enR2( f ), of a

resistor, which is in thermal equilibrium with its surrounding, we

temporarily connect a capacitor to the resistor.

R

Ideal, noiseless resistor

Noise source

Real resistor

A. Noise description based on the principles of

thermodynamics and statistical mechanics (Nyquist, 1828)

From the point of view of thermodynamics, the resistor and the

capacitor interchange energy:

enR


m v

2

2

Each particle has three degrees of freedom

mivi 2

2

mi vi 2

2=

m v 2

2= 3

k T2

In thermal equilibrium:

x

z

Illustration: The law of the equipartition of energy

y


Illustration: Resistor thermal noise pumps energy into the capacitor

Each particle (mechanical equivalents of electrons in the resistor) has three degrees of freedom

CV

2

2

mivi 2

2

C VC 2

2=

k T2


The particle (a mechanical equivalent of the capacitor) has a single degree of freedom

x

y

z


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:

R

Ideal, noiseless resistor

Noise source

Real resistor

C enC

H( f )= enC ( f )

enR ( f )

C VC 2

2=

k T2


enR

9

kTC


= =nC 2 =

C nC 2

2

kT

2

kT

C

C vnC (t) 2

2

0

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

(1914),

enR 2( f ) H(j2 f)2

e j 2 f d f

nC 2 RnC () =

1 d f0

enR2( f )

1) +2 f RC(2

enR2( f )

4 RC

enR2( f ) = 4 k T R [V2/Hz].

Power spectral density of resistor noise:

C VC 2

2=

10

SHF EHF IR R

10 GHz 100 GHz 1 THz 10 THz 100 THz 1 GHz

1

0.2

0.4

0.6

0.8

enR P( f )2

enR( f )2

f


enR P 2( f ) = 4 R [V2/Hz] .

B. Noise description based on Planck’s law for blackbody

radiation (Nyquist, 1828)

h f

eh f /k T 1

A comparison between the two Nyquist equations:

R = 50 ,C f 0.04 = F

= R 50 ,C = 0.04 f F


The Nyquist equation was extended to a general class of

dissipative systems other than merely electrical systems:

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

h f

eh f / k T 1

Zero-point energy f(T)

h f

2

C. Noise description based on quantum mechanics

(Callen and Welton, 1951)

eqn ( f )2

enR ( f )2

SHF EHF IR R

10 GHz 100 GHz 1 THz 10 THz 100 THz 1 GHz f0

2

4

6

8

Quantum noise

12

The ratio of the temperature dependent and temperature

independent parts of the Callen-Welton equation shows that at 0

K and f 0 there still exists some noise compared to the Nyquist

noise level at Tstd = 290 K (standard temperature: k Tstd =

4.001021)

10 Log [dB]2

eh f / k Tstd 1

f, Hz

Remnant noise at 0 K, dB

103 106 109

-20

-40

-60

-80

-100

-120

0


100

13

An equivalent noise bandwidth, B , is defined as the bandwidth

of an equivalent-gain ideal rectangular filter that would pass as

much power of white noise as the filter in question:

D. Equivalent noise bandwidth, B


B d

f.

0

IA( f )I2

IAmax I2

IA( f )I2

IAmax I2

f

1

0.5

B

ff

B

linear scale

Equal areas Equal areas

Lowpass Bandpass

0

14

R

C

en o( f )

fc = = f3dB 1

2 RC


=en in2 0.5 fc

Vn o 2 = en o

2( f ) d f 0

= en in2H( f )2 d f

0

Example: Equivalent noise bandwidth of an RC filter

=en in2

1

1) + f / fc (2d f

0

Vn o 2 = en in

2 B

enR

15

fc

2 4 6 8 10

1

Equal areas

1

0.5

f /fc

B = 0.5 fc 1.57 fc R

C

en o

0


en o2

en in2

0.01 0.1 1 10 100

0.1

1

f /fc

fc

B

0.5 Equal areas

en o2

en in2

fc = = f3dB 1

2 RC

Example: Equivalent noise bandwidth of an RC filter

en in

16

Two first-order independent stages B = 1.22 fc.

Butterworth filters:

H( f )2= 1

1 ) +f / fc (2n

Example: Equivalent noise bandwidth of higher-order filters

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


second order B = 1.11 fc.

third order B = 1.05 fc.

fourth order B = 1.025 fc.

17

Amplitude spectral density of noise, rms/Hz0.5:

en = 4 k T R [V/Hz].

Noise voltage, rms:

Vn = 4 k T R [V].


en = 0.13R [nV/Hz].

At room temperature:

18

Vn = 4 k T 1k 1Hz 4 nV

Vn = 4 k T 50 1Hz 0.9 nV

Vn = 4 k T 1M 1MHz 128 V

Examples:


19

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

Input noise pdf Input and output noise spectra

Output noise pdf Input and output noise vs. time


E. Normalization of the noise pdf by dynamic networks

20

Input noise pdf Input noise autocorrelation

Output noise pdf Output noise autocorrelation


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

21

Input noise pdf Input and output noise spectra

Output noise pdf Input and output noise vs. time


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

22

Input noise pdf Input noise autocorrelation

Output noise pdf Output noise autocorrelation


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

23

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.

F. Noise temperature, Tn


R

en( f )

R in( f )

R na( f )

en2 = 4 k T R [V2/Hz],

in2 = 4 k T/ R [I2/Hz],

na k T [W/Hz] . en

2

4 R en( f )

24

Any thermal noise source has available power spectral density

na( f ) k T , where T is defined as the noise temperature, T Tn.

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:

Tn ( f ) .

na ( f )

k

Then, given a source's noise temperature Tn,

na ( f ) kTn ( f ) .


na( f )

en( f )

Nonthermal sources of noise

25

l

Example: Noise temperatures of nonthermal noise sources

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

Antenna noise temperature:

T

vn2( f ) = 320 2(l/)2

k T = 4 k Ta R

l


R= 80 2(l/)2 is the radiation resistance.

Reference: S. I. Baskakov.

26

Example: Antenna noise temperature, Ta (sky contribution only)


101 103 105 107100 102 104 106 108 109

Frequency, MHz

A

nten

na

noi

se t

empe

ratu

re, T a

(K)

100

101

102

103

104

300

Quantum noise limitTQ h f / k

O2

H2O

S. Okwit, “An historical view of the evolution of low-noise concepts and techniques,” IEEE Trans. MT&T, vol. 32, pp. 1068-1082, 1984.

Galactic noise limitTG 100 2.4

27

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.

G. Thermal noise in capacitors and inductors

R C

Reference: [2], pp. 230-231


enR enC

28

Reference: [2], p. 230

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 where

HCR( f ) R /(1/j2f+R). Since HCR( f ) , enC ( f ) .

R C

f f


enR enC

29

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:


R

C VnC


H. Noise power at a capacitor

VnC 2

= enR2H( f )2 d f

0

4 k T RB

4 k T R 0.5 1

2 RC

VnC 2

k T

C

enR

30

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.



VnC 2

kT

C

R

C VnC

enR

32

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.

Reference: Physics World, August 1996, page 22

5.2.2. Shot noise

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

D

I

ii

www.discountcutlery.net

33

D

Reference: [1]

I

t

i


Illustration: Shot noise in a diode

34

D

Reference: [1]

I

t

i

Illustration: Shot noise in a diode


I

35

We start from defining n as the average number of electrons

passing the pn junction of a diode during one second, hence,

the average electron current I = q n.

We assume then that the probability of passing through the

junction two or more electrons simultaneously is negligibly

small. This allows us to define the probability that an electron

passes the junction in the time interval dt = (t, t + d t) as P1(d t)

= n d t.

Next, we derive the probability that no electrons pass the

junction in the time interval (0, t + d t):

P0(t + d t ) = P0(t) P0(d t) = P0(t) [1 P1(d t)] = P0(t) P0(t) n d t.

A. Statistical description of shot noise


36

This yields

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

The previous equation, P0(t + d t ) = P0(t) P0(t) n d t, yields

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


= n P0

d P0

d t

The probability that exactly one electron passes the junction in

the time interval (0, t + d t)

P1(t + d t ) = P1(t) P0(d t) + P0(t) P1(d t)

= P1(t) (1 n d t) + P0(t) n d t .

= n P1 + n P0

d P1

d t

37

In the same way, one can obtain the probability of passing the

junction electrons, exactly:


= n PN + n PN 1

d PN

d t

PN (0) = 0

.

which corresponds to the Poisson probability distribution.

PN (t) = e n t ,)n t( N

N !

By substitution, one can verify that

38

N = 10


Illustration: Poisson probability distribution

10 20 30 40 50

0.02

0.04

0.06

0.08

0.1

0.12

PN (t) = e 1 t)1 t( N

N !

t

N = 20 N = 30

0

39

The average number of electrons passing the junction during a

time interval (0,) can be found as follows


N = N e n = n e n = n,

)n( N

N !

and the squared average number can be found as follows:

N = 0

)n( N 1

)N 1( !N = 1

N 2

= N 2 e n = [N (N 1) + N ] e n

)n( N

N !N = 0

N = 0

)n( N

N !

=nne n = nn. n = 2

)n ( N 2

)N 2( !

=e n

40

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

variance, irms2:


I = i = = q n,

in rms2 .

The variance of the number of electrons passing the junction

during a time interval can be found as follows

N2 = N

2 N)2 = n.

q N

q N

2 q

2 n

2

q I

41

Hence, the spectral density of the shot noise


in rms2 = 2 q B.

in( f ) = 2 q .

B. Spectral density of shot noise

Assuming = 1/( 2 B), we finally obtain the Schottky equation

for shot noise rms current


C. Shot noise in resistors and semiconductor devices

In devices such as diode 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 in2( f ) = 2 q I.

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 does exist in mesoscopic (nm) resistors, although at

lower levels than in a diode 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

43

The most general type of excess noise is 1/f or flicker noise.

This noise has approximately 1/f power 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 device

properties caused, for example, by electric current in resistors

and semiconductor devices.

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.

Reference: [3]

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 devices generate additional or excess noise.

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

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

44

References: [4] and [5]

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 bipolar, JFET, and MOSFET transistors have flicker

noise, it is a significant noise source in MOS transistors,

whereas it can often be ignored in bipolar transistors (and some

modern JFETs).


45

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 ff is 100 Hz to 1 kHz.


f, decades

in ( f ), dB

ff

White noise

Pink noise

10 dB/decade

46

References: [4] and [5]

Flicker noise is directly proportional to the dc (or average)

current flowing through the device:


in2( f )

where Kf is a constant that depends on the type of material.

Kf I

f


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:

f, decadesff

White noise

in 1W2( f ), dB

10 dB/decade

Pink noise

1 1 W1 Ain 1W

2( f ) Kf I

f






1 1 W1 Ain 1W

2( f ) Kf I

f

1 9 W

1/3 A

1/3 A

1/3 A

in 9W2( f )

Kf (I / 3)

3 f

f, decades

in 1W2( f ), dB

ff

White noise

Pink noise


f, decadesff





White noise

in 1W2( f ), dB

Pink noise

in 9W2( f )

Kf I

9 f

1 1 W1 A

1 9 W

1/3 A

1/3 A

1/3 A

in 1W2( f )

Kf I

f


Example: A simulation of 1/f noise

Input Gaussian white noise Input noise PSD

Output 1/f noise Output noise PSD


Example: Simulation of 1/f noise

Filter

0 +1 ij

1kR

10kfc C

1/(2*pi*x) RC

j(2 pi i ) j(2 pi i )RC

1u

1

0

Real

0

100000

2 1

20

52Next lecture

Next lecture:

Documents

1 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