FOUNDATION 1: CIMI REFERENCE MODEL. CIMI Reference Model - Core.
Reference: [1]
description
Transcript of Reference: [1]
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.
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]
5.2. Noise types
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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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.
3
vn(t)
tf(vn)
vn(t)
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
R V
6
Vn rms
Example: Resistor thermal noise
Normal distribution according to thecentral limit theorem
T 0
2R()
0
White (uncorrelated) noise
en2
f0
4
C enC
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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.
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:
en
55. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
Illustration: The law of equipartition of energy
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:
65. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
C V 2
2=
k T2
In thermal equilibrium:
Illustration: Resistor thermal noise pumps energy into the capacitor
Each particle has three degrees of freedom
CV
2
2
mivi 2
2
75. 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:
R
Ideal, noiseless resistor
Noise source
Real resistor
C enC
H( f )= enC ( f )
enR ( f )
enR
C V 2
2=
k T2
In thermal equilibrium:
8
kTC
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
= =nC 2 =
C nC 2
2
kT
2
kT
C
C vnC (t) 2
2
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 f
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:
9
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
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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
105. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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
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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: k Tstrd = 4.001021)
10 Log dB. 2
eh f /k T 1
f, Hz
Ratio, dB
102 104 106 108 1010 1012
20
40
60
80
100
120
0
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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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 B of an ideal filter is its actual bandwidth.
For practical filters, B is greater than their 3-dB bandwidth. For
example, an RC filter has B = 0.5 fc, 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.
D. Equivalent noise bandwidth, B
Reference: [4]
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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R
C
en o( f )
fc = = f3dB 1
2 RC
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
en in
=en in2 0.5 fc
Vn o rms2 = 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 rms2 = en in
2 B
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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
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
en in
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
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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.
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
second order B = 1.11 fc.
third order B = 1.05 fc.
fourth order B = 1.025 fc.
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Amplitude spectral density of noise:
en = 4 k T R [V/Hz].
Noise voltage:
Vn rms = 4 k T R fn [V].
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
en = 0.13R [nV/Hz].
At room temperature:
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Vn rms = 4 k T 1k 1Hz 4 nV
Vn rms = 4 k T 50 1Hz 0.9 nV
Vn rms = 4 k T 1M 1MHz 128 V
Examples:
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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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
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
E. Normalization of the noise pdf by dynamic networks
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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
1) First-order filtering of the Gaussian white noise.
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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.
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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.
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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 = 4 k T R [V2/Hz],
in2 = 4 k T/ R [I2/Hz],
na = k T [W/Hz] . en
2
4 R
F. Noise temperature, Tn
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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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 ) .
na2( 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
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Example: Noise temperatures of nonthermal noise sources
Cosmic noise: Tn= 1 … 10 000 K.
Environmental noise: Tn(1 MHz) = 3108 K.
T
Vn2( f ) = 320 2(l/) k T = 4 k T R
l
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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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
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
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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
enR enC
f f
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
27
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:
Reference: [5], p. 202
R
C VnC
enR
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
H. Noise power at a capacitor
VnC rms 2
= enR2H( f )2 d f
0
4 k T RB
4 k T R 0.5 1
2 RC
VnC rms 2
kT
C
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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
VnC rms 2
kT
C
R
C VnC
enR
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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 rms in the following
circuit against kT/C. How do you account for the difference?
(The operational amplifier is assumed ideal and noiseless.)
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
Home exercise:
1nF
1k
1k 1pF
Vn rmsVn o rms
vs
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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
R
I
ii
www.discountcutlery.net
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Reference: [1]
R
I
t
i
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
Illustration: Shot noise in a conductor
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Reference: [1]
I
t
iR
Illustration: Shot noise in a conductor
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
I
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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 = q n.
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 + d t) as
P1(d t) = n d t.
Next, we derive the probability that no electrons pass the cross-
section in the time interval (0, t + d t):
P0(t + d t ) = P0(t) P0(d t) = P0(t) (1 n d t).
A. Statistical description of shot noise
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
34
This yields
with the obvious initiate state P1(0) = 0.
This yields
with the obvious initiate state P0(0) = 1.
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
= n P0
d P0
d t
The probability that exactly one electrons pass the cross-
section 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
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In the same way, one can obtain the probability of passing the
cross section electrons, exactly:
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
= 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
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N = 10
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
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
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The average number of electrons passing the cross-section
during a time interval can be found as follows
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
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( !
38
We now can find the average current of the electrons, I, and its
variance, irms2:
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
I = i = (q /N= q n,
irms2 = (q /N
2= (q /.
The variance of the number of electrons passing the cross-
section during a time interval can be found as follows
N2 = N
2 N)2 = n.
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Hence, the spectral density of the shot noise
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
In2 = 2 q fs.
in( f ) = 2 q .
B. Spectral density of shot noise
Assuming = 1/ 2 fs, we finally obtain the Schottky equation for
shot noise rms current
405. 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 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 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
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The most general type of excess noise is 1/f or 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.
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 devises 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.
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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 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.
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise
43
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.
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise
f, decades
in 2( f ), dB
fc
White noise
Pink noise
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References: [4] and [5]
Flicker noise is directly proportional to the density of dc (or
average) current flowing through the device:
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise
in2( f ) , a A J
2
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.
455. 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:
f, decadesfc
White noise
1 Ain 1W ( f )2 a A J
2
f
1 1 W
in 1W2( f ), dB
465. 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 1 W1 Ain 1W
2( f ) a A J 2
f
1 9 W
1/3 A
1/3 A
1/3 A
1 A in 9W2( f ) a A J
2
9 f
f, decades
in 1W2( f ), dB
fc
White noise
475. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise
f, decadesfc
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:
White noise
1 A
1 9 W
1/3 A
1/3 A
1/3 A
1 A in 9W2( f ) a A J
2
9 f
in 1W2( f ) a A J
2
f
1 1 W
in 1W2( f ), dB
485. 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
495. SOURCES OF ERRORS. 5.2. Noise types. 5.2.3. 1/f noise
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
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