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Transcript
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
1
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
2
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]
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
3
Example: Resistor thermal noise
vn(t)
T0
vn(t)
6s
Vn rms
R
V
t
f(vn)
2s
Normal distribution
according to the
central limit
theorem
2R(t)
en2
White (uncorrelated) noise
0
f
0
t
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
4
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
enC
C
en
Noise source
From the point of view of thermodynamics, the resistor and the
capacitor interchange energy:
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
5
Illustration: The law of equipartition of energy
Each particle has
three degrees
of freedom
mivi 2
2
m v2
2
In thermal equilibrium:
mi vi 2 m v 2
kT
=
=
3
2
2
2
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
6
Illustration: Resistor thermal noise pumps energy into the capacitor
Each
particle
has three
degrees of
freedom
mivi 2
2
CV 2
2
In thermal equilibrium:
CV2 kT
=
2
2
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
7
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:
H( f ) =
Real resistor
R
Ideal, noiseless resistor
enC
C
enR
Noise source
In thermal equilibrium:
CV2 kT
=
2
2
enC ( f )
enR ( f )
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
C snC 2
kT
C vnC (t) 2
=
=
2
2
2
8
 snC
2
kT
=
.
C
According to the Wiener–Khinchin theorem (1934), Einstein
(1914),

snC 2 = RnC (0) =  enR 2( f ) H(j2p f)2 e j 2p f t d f
0
= enR

2( f )
kT
enR2( f )
1
d
f
=
=
.
 1+ (2p f RC)2
C
4 RC
0
Power spectral density of resistor noise:
enR2( f ) = 4 k T R [V2/Hz].
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
9
B. Noise description based on Planck’s law for blackbody
radiation (Nyquist, 1828)
enR P
2( f ) =
4R
hf
eh f /k T - 1
[V2/Hz].
A comparison between the two Nyquist equations:
enR P( f )2
enR( f )2
1
R = 50 W,
C = 0.04 f F
0.8
0.6
0.4
0.2
1 GHz
10 GHz
SHF
100 GHz
EHF
1 THz
10 THz
IR
100 THz
R
f
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
10
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.
eqn
eqn ( f )
enR ( f )2
2( f ) =
4R
hf
eh f /k T - 1
+
hf
[V2/Hz].
2
Zero-point energy
2
8
6
4
Quantum noise
2
0
1 GHz
10 GHz
SHF
100 GHz
EHF
1 THz
10 THz
IR
100 THz
R
f
 f(t)
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
11
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 = 4.0010-21)
10 Log
2
eh f /k T
-1
dB.
Ratio, dB
120
100
80
60
40
20
0
102
104
106
108
1010
1012
f, Hz
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
12
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 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 p 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.
Reference: [4]
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
13
Example: Equivalent noise bandwidth of an RC filter
R
en in
en o( f )
C

Vn o rms2 =  en o2( f ) d f
0

=  en in2H( f )2 d f
0
1
fc =
= D f3dB
2p RC

= en in
2
1
 1 + ( f / f )2 d f
c
0
= en in2 0.5 p fc
Vn o rms2 = en in2 B
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
14
Example: Equivalent noise bandwidth of an RC filter
R
en o
en o2
en in2
B = 0.5 p fc = 1.57 fc
1
C
en in
Equal areas
0.5
fc
fc =
1
= D f3dB
2p RC
0
en o2
en in12
1
2
4
6
8
10
f /fc
B
Equal areas
0.5
0.1
fc
0.01
0.1
1
10
100
f /fc
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
15
Example: Equivalent noise bandwidth of higher-order filters
First-order RC low-pass filter
B = 1.57 fc.
Two first-order independent stages
B = 1.22 fc.
Butterworth filters:
H( f
)2
=
1
second order
B = 1.11 fc.
third order
B = 1.05 fc.
fourth order
B = 1.025 fc.
1+ ( f / fc )2n
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
Amplitude spectral density of noise:
en =  4 k T R [V/Hz].
Noise voltage:
Vn rms =  4 k T R D fn [V].
Examples:
Vn rms =  4 k T 1MW 1MHz = 128 mV
Vn rms =  4 k T 1kW 1Hz = 4 nV
Vn rms =  4 k T 50W 1Hz = 0.9 nV
16
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
17
E. Normalization of the noise pdf by dynamic networks
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
18
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
19
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
20
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 = 4 k T R [V2/Hz],
in2 = 4 k T/ R [I2/Hz],
en2
na 
= k T [W/Hz].
4R
21
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
22
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:
na2( f )
Tn ( f ) 
.
k
Then, given a source's noise temperature Tn,
n a 2 ( f )  k Tn ( f ) .
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
23
Example: Noise temperatures of nonthermal noise sources
Cosmic noise:
Tn= 1 … 10 000 K.
Environmental noise:
Tn(1 MHz) = 3108 K.
T
Vn ( f )2
= 320 p2(l/l) k T = 4 k T RS
l << l
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
24
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
enR
C
enC
Reference: [2], pp. 230-231
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
25
R
enR
C
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.
f>0
f>0
Reference: [2], p. 230
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
26
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:

VnC rms =  enR2H( f )2 d f
2
R
0
VnC
enR
C
= 4 kTR B
= 4 k T R 0.5 p
VnC rms 2 =
1
2p RC
kT
C
Reference: [5], p. 202
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
27
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.
R
VnC
enR
C
VnC rms 2 =
kT
C
Reference: [5], p. 203
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.1. Thermal noise
28
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 e rms 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 e rms
1k
vs
1pF
Vn o rms
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
29
5.2.2. Shot noise
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.
R
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
30
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
31
Illustration: Shot noise in a conductor
R
i
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 = 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 crosssection in the time interval (0, t + d t):
P0(t + d t ) = P0(t) P0(d t) = P0(t) (1- n d t).
32
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
33
This yields
d P0
dt
= - n P0
with the obvious initiate state P0(0) = 1.
The probability that exactly one electrons pass the crosssection 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 .
This yields
d P1
dt
= - n P1 + n P0
with the obvious initiate state P1(0) = 0.
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
34
In the same way, one can obtain the probability of passing the
cross section N electrons, exactly:
d PN
dt
= - n PN + n PN -1
.
PN (0) = 0
By substitution, one can verify that
PN (t) =
(n t) N
N!
e- n t ,
which corresponds to the Poisson probability distribution.
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
35
Illustration: Poisson probability distribution
(1 t) N
PN (t) =
N!
e- 1 t
0.12
0.1
0.08
0.06
N = 10 N = 20
0.04
N = 30
0.02
0
10
20
30
40
50
t
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
36
The average number of electrons passing the cross-section
during a time interval t can be found as follows

(nt ) N
n =0
N!
S
Nt =
e- nt = nt e- nt

(nt ) N-1
S (N -1) ! = nt ,
n =1
and the squared average number can be found as follows:

Nt 2 =
S
n =0
Nt 2
(nt ) N
N!
= nt + (nt )2


S
e- nt =
n =0
(n t ) N-2
S (N -2) !
n =2
[N (N -1) + N ]
(nt ) N
e- nt = nt + (nt )2.
N!
e- nt
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
The variance of the number of electrons passing the crosssection 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 = q n,
irms2 = (q /t )2 sN2 = (q /t ) I .
37
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
38
B. Spectral density of shot noise
Assuming t = 1/ 2 fs, we finally obtain the Schottky equation for
shot noise rms current
In2 = 2 q I fs .
Hence, the spectral density of the shot noise
in( f ) =  2 q I .
5. SOURCES OF ERRORS. 5.2. Noise types. 5.2.2. Shot noise
39
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
40
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 excess noise.
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.
No unifying principle has been found for all the 1/f noise sources.
Reference: [3]
41
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]
42
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
43
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:
in
2( f )
2
a
A
J
=
,
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]
44
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:
1A
1 W, 1 W
in 1W ( f )2 =
a AJ2
f
in 1W2( f ), dB
White noise
f, decades
fc
45
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:
1A
1 W, 1 W
in 1W2( f ) =
a AJ2
f
1/3 A
in 1W2( f ), dB
1A
in 9W
1/3 A
2( f )
a AJ2
=
9f
1/3 A
1 W, 9 W
White noise
f, decades
fc
46
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:
1A
1 W, 1 W
in 1W2( f ) =
a AJ2
f
1/3 A
in 1W2( f ), dB
1A
in 9W
1/3 A
2( f )
a AJ2
=
9f
1/3 A
1 W, 9 W
White noise
f, decades
fc
47
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
48
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49
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