Transcript Measurement Theory Principles

5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.1. Signal-to-noise ratio, SNR
1
5.3. Noise characteristics
5.3.1. Signal-to-noise ratio, SNR
The signal-to-noise ratio is the measure for the extent to which
a signal can be distinguished from the background noise:
SNR 
S
N
where Sin is the signal power, and Nin is the noise power.
Reference: [4]
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.1. Signal-to-noise ratio, SNR
2
A. Signal-to-noise ratio at the input of the system, SNRin
Measurement object
ZS=RS + jXS
Measurement system
SNRin
vS
vin
Noiseless
Zin=Rin + jXin
RL
Sin
SNRin 
Nin
It is usually assumed that the signal power, Sin, and the noise
power, Nin, are dissipated in the noiseless input impedance of
the measurement system.
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.1. Signal-to-noise ratio, SNR
3
Example: Calculation of SNRinsrc
Measurement object
Measurement system
ZS=RS + jXS
Noiseless
SNRin
vS
Zin=Rin + jXin
vin
1) Sin =
Vs 2 Zin
Zs + Zin
3) SNRin =
Vs 2
Vn
2
, 2) Nin =
2
=
Vs 2
4 k T Rs B
Note that SNRin is not a function of Zin.
Vn 2 Zin
Zs + Zin
.
2
RL
,
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.1. Signal-to-noise ratio, SNR
4
B. Signal-to-noise ratio at the output of the system, SNRo
1) The system is noisy.
Measurement object
Measurement system
ZS=RS + jXS
vS
Noisy
SNRin
Gp
Power gain,
SNRo
Ap
Sin Ap
Sin
So
=

SNRo 
msr
(N
+N
)
A
Nin
No
in
in
p
SNRo  SNRin
RL
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.1. Signal-to-noise ratio, SNR
5
2) The system is noiseless.
Measurement object
ZS=RS + jXS
vS
SNRo
Measurement system
Gp
src
Noiseless
SNRin
Power gain,
Ap
So
Sin Ap
Sin
 src =
=
No
Nin Ap
Nin
SNRosrc = SNRin
SNRosrc
RL
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF
6
5.3.2. Noise factor, F, and noise figure, NF
Noise factor is used to evaluate the signal-to-noise degradation
caused by the measurement system (H. Friis, 1940s).
Measurement object
Measurement system
ZS=RS + jXS
vS
SNRin
Gp
F
SNRin
SNRo
Noisy
Power gain,
SNRo
Ap
RL
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF
7
The signal-to-noise degradation is due to the additional noise,
Nomsr , which the measurement system contributes to the load.
Measurement object
ZS=RS + jXS
vS
F
Measurement system
SNRin
Gp
SNRin
SNRo
Noisy
Power gain,
SNRo
Ap
So /Nosrc _____
No
Sin /Nin _______
=
=
=
So /No
Nosrc
So /No
msr
src + N msr
N
N
o
o
o
= 1 + _____
= __________
1
src
Nosrc
No
RL
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF
8
Noise factor is also used to evaluate the noise contribution of
the measurement.
Measurement object
ZS=RS + jXS
vS
Measurement system
SNRin
Gp
F
Noisy
Power gain,
No
Nosrc
SNRo
Ap
RL
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF
9
Example: Calculation of noise factor
Measurement object
Measurement system
RS
vin
Gv
enS
F 
No
No
src
=
=
vo
Voltage gain,
Av
Vno2/RL
4 kTRsBn (GV AV)2 /RL
Vno2
4 kTRs Bn (GV AV)2
RL
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF
10
Conclusions:
F =
Vno2
4 kTRS Bn (G AV)2
The following three characteristics of noise factor can be seen
by examining the obtained equation:
1. It is independent of load resistance RL,
2. It does depend on source resistance Rs,
3. If the measurement system were completely noiseless,
the noise factor would equal one.
References: [2]
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF
11
C. Noise figure
Noise factor expressed in decibels is called noise figure (NF):
NF  10 log F
Due to the bandwidth term in the denominator
F =
Vno2
4 kTRS Bn (G AV)2
there are two ways to specify the noise factor: (1) a spot noise,
measured at specified frequency over a 1-Hz bandwidth,or (2)
an integrated, or average noise measured over a specified
bandwidth.
References: [2]
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF
12
E. Measurement of noise factor
We will consider the following methods for the measurement of
noise factor: (1) the single-frequency method, and (2) the white
noise method.
1) Single-frequency method. According to this method, a
sinusoidal test signal vs is increased until the output power
doubles. Under this condition the following equation is satisfied:
Measurement object
Measurement system
RS
vS
vin
Gv
vo
Voltage gain,
Av
RL
References: [2]
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF
Measurement object
Measurement system
RS
vS
vin
Gv
vo
Voltage gain,
1) (Vs GV AV)2 + Vno2 = 2 Vno2
VS = 0
2) Vno2
3) F =
13
Av
RL
VS = 0
= (VS GV AV)2
Vs = 0
Vno2
No
*
VS = 0
=
(VS GV AV)2
4 kTRS Bn (GV AV
)2
=
VS2
4 kTRS Bn
References: [2]
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF
F=
14
Vs2
4 kTRs Bn
The disadvantage of the single-frequency meted is that the
noise bandwidth of the measurement system must be known.
A better method of measuring noise factor is to use a white
noise source.
2) White noise method. This method is similar to the previous
one. The only difference is that the sinusoidal signal generator
is now replaced with a white noise current source:
References: [2]
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF
Measurement object
Measurement system
vin
RS
iS ( f
)
Voltage gain,
it = 0
No
= 2 Vno2
it = 0
Av
RL
it = 0
= (is Rs G AV)2 Bn
Vno2
3) F =
vo
Gv
1) (is Rs G AV)2 Bn + Vno2
2) Vno2
15
*
it = 0
=
(is Rs G AV)2 Bn
4 kTRs Bn (G AV
)2
=
is2 Rs
4 kT
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF
F =
iin2 Rs
4 kT
The noise factor is now a function of only the test noise signal,
the value of the source resistance, and temperature. All of
these quantities are easily measured.
Neither the gain nor the noise bandwidth of the measurement
system need be known.
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5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. Vn- In noise model
17
5.3.3. Vn- In noise model
The actual network can be modeled as a noise-free network
with two noise generators, en and in, connected to its input
(Rothe and Dahlke, 1956):
Measurement object
RS
Measurement system
en
vS
in
Noiseless
Rin
Av
vo
RL
In a general case, the en and in noise generators are correlated.
Reference: [2]
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. Vn- In noise model
Measurement object
RS
18
Measurement system
en
vS
in
Noiseless
Rin
vo
Av
RL
The en source represents the network noise that exists when Rs
equals zero, and the in source represents the additional noise
that occurs when Rs does not equal zero,
The use of these two noise generators plus a complex
correlation coefficient completely characterizes the noise
performance of the network.
At relatively low frequencies, the correlation between the en and
in noise sources can be neglected.
Reference: [2]
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. Vn- In noise model
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Example: Input voltage and current noise spectra
(ultralow noise, high speed, BiFET op-amp AD745)
en
in
Reference: www.analog.com
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. Vn- In noise model
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A. Total input noise as a function of the source impedance
Measurement object
RS
Measurement system
en
vS
in
Noiseless
Rin
vo
Av
Assuming no correlation between the noise sources, the total
equivalent noise voltage reflected to the source location can
easily be found:
RL
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. Vn- In noise model
Measurement object
RS
21
Measurement system
en
Noiseless
vs
Rin
in
Measurement object
vo
Av
RL
Measurement system
RS
en
vS
in Rs
Noiseless
Av
enS =  4 kT Rs + en 2 + 2r Vn In + (in Rs)2
vo
RL
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. Vn- In noise model
22
We now can connect an equivalent noise generator in series
with the input signal source to model the total input voltage of
the whole system.
We assume that the correlation coefficient in the previous
equation r = 0. (For the case r  0, it is often simpler to
analyze the original circuit with its internal noise sources.)
Measurement object
enS
vS
Measurement system
RS
vo
Voltage gain,
Av
RL
vnS =  4 kT Rs + en 2 + (in Rs)2
Reference: [7]
5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. Vn- In noise model
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B. Measurement of en and in
Measurement system
en
en = (Vn o / B) / AV
in
Noiseless
vn o
Av
RL
Measurement system
en o >> (4 kT Rt + en2)0.5
in RS = en o / AV
in = en o / (AV RS )
en
Rt
in
Noiseless
Av
vn o
RL
5. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR
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5.4. Noise matching: maximizing SNR
Measurement errors can occur due to the undesirable
interaction between the measurement system and:
the object under test.
Environment
Measurement
Object
Influence
Measurement
System
Matching
x+D x
Matching
Disturbance
y +Dy1
Observer
Influence
The purpose of noise matching is to let the measurement
system add as little noise as possible to the measurand.
5. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR
Let us first find the noise factor F and the signal-to-noise ratio
SNRo of the measurement system as a function of the source
resistance:
2 + (i R )2
N
N
4kTR
+
e
o
nS
S
n
n S
____ = __________________
= f ( Rs )
F = ____
=
Nosrc
N src
4kTRS
2
2
V
V
s
s
= f ( Rs )
SNRo = SNRnS  ____2 = _________________
4kTRS+ en2 + (in RS)2
Nn s
We then will try and maximize the SNRo at the output of the
measurement system by matching the source resistance.
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5. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR. 5.4.1. Optimum source resistance
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5.4.1. Optimum source resistance
B = 1 Hz, en = 2 nV/Hz0.5, in = 20 pA /Hz0.5
100
2
VS
SNR = _________________
4kTRS+ en2 + (in RS)2
RS for maximum SNR
vn in , nV/Hz0.5
4kTRS + en2 + (in RS)2
__________________
F=
4kTRS
en
vS = en·1 Hz0.5
Source
noise
1
 4kTRS
RS for minimum F
en = in Rn 
en
RS opt =
in
in RS
10
0.1
100
F 0.5, dB
101
102
103
104
101
102
103
104
20
10
SNR 0.5, dB
0
-10
RS opt is called
the optimum source resistance
(also noise resistance).
-20
-30
100
RS , W
5. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR. 5.4.1. Optimum source resistance
F 0.5, dB
4kTRS + en2 + (in RS)2
__________________
F=
4kTRS
Vs2
_________________
SNR =
4kTRS+ en2 + (in RS)2
27
20
10
SNR 0.5, dB
0
-10
-20
-30
100
101
102
103
RS , W
It is important to note that the source resistance that maximizes
SNR is RS =  0, whereas the source resistance that minimizes
F is RS =  Rn.
We can conclude therefore, that for a given RS, SNR cannot be
increased by connecting a resistor to RS.
104
5. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR. 5.4.1. Optimum source resistance
28
Adding a series resistor, R, increases the total source
resistance, RsS = Rs + R, and decreases the SNR.
Measurement object
RS
vS
F 0.5, dB
4kTRS + en2 + (in RS)2
__________________
F=
4kTRs
VS2
_________________
SNR =
4kTRS+ en2 + (in RS)2
20
10
SNR 0.5,
dB
0
-10
-20
-30
100
101
102
RS , W
103
104
5. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR. 5.4.1. Optimum source resistance
29
Adding a series resistor, R, increases the total source
resistance, RsS = Rs + R, and decreases the SNR.
Measurement object
RS
+
R
vS
F 0.5, dB
4kTRS + en2 + (in RS)2
__________________
F=
4kTRS
2 2
VsV
S
_________________
____________________
==
SNR 
2 (i (i
4kTR
en2e+
)2S)2
4kTRsS++
n RnsR
n +
20
10
SNR 0.5,
dB
0
-10
-20
-30
100
101
102
RS , W
103
104
5. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR. 5.4.1. Optimum source resistance
30
Adding a parallel resistor, R, decreases by the same factor both
the input signal and the source resistance seen by the
measurement network, and therefore decreases the SNR.
Measurement object
RS
vs
F 0.5, dB
4kTRS + en2 + (in RS)2
__________________
F=
4kTRS
VSs2
_________________
SNR =
4kTRS+ en2 + (in RS)2
20
10
SNR 0.5,
dB
0
-10
-20
-30
100
101
102
RS , W
103
104
5. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR. 5.4.1. Optimum source resistance
31
Adding a parallel resistor, R, decreases by the same factor both
the input signal and the source resistance seen by the
measurement network, and therefore decreases the SNR.
Measurement object
RS
vS
RS  = RS / [(RSIIR)R ]
R
vS= vS / [(RSIIR)R ]
F 0.5, dB
4kTRS + en2 + (in RS)2
__________________
F=
4kTRS
Vs2VS  2
_________________
____________________
SNR 
==
2
2
4kTR
en2++en(i2 n+R(i
4kTR
s+S
s)n RS )
20
10
SNR 0.5,
dB
0
-10
-20
-30
100
101
102
RS , W
103
104
5. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR. 5.4.1. Optimum source resistance
Conclusions.
The noise factor can be very misleading: the minimization of F
does not necessarily leads to the maximization of the SNR.
This is referred to as the noise factor fallacy.
32
5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Methods for the increasing of SNR
5.4.2. Methods for the increasing of SNR
Methods for the increasing of SNR are base on the following
relationship:
1
SNRo =
SNRin
F
The strategy is simple: to increase SNRo, keep SNRin constant
while decreasing the noise figure.
 SNRo =
1
SNRin
F
The SNR at the output will increase because the relative noise
power contributed by the system to the load will decrease.
33
5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Methods for the increasing of SNR
34
A. Noise reduction with parallel input devices
This method is commonly used in low-noise OpAmps to
increase SNR is to connect several active devices in-parallel:
Measurement system
vin
en
in
io sc
gm vin
ro
Rin
k
en
in
gm vin
Rin
ro
Reference: [7]
5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Methods for the increasing of SNR
35
Home exercise: Prove that the following network is equivalent to the
previous one.
Equivalent measurement system
vin
en/k 0.5
k 0.5 in
io sc
gm vin
Rin
k
ro
Reference: [7]
5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Methods for the increasing of SNR
36
Thanks to parallel connection of input devices, it is possible to
decrease the ratio, (en / in )p = (en / in )single / k , with no change in
RS and vS, and hence in the SNRin.
Measurement object
RS
Equivalent measurement system
vin
en/k 0.5
vS
k 0.5 in
io sc
k gm vin
Rin
k
en
SNRo = SNRo max and F = Fmin at RS =
k in
ro

en / in
k=
RS
Note that SNRo cannot be improved if the RS is too large.
Reference: [7]
5.4.1. Methods
Optimum for
source
resistanceof SNR
5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2.
the increasing
37
Home exercise: Prove that
SNRo p = k SNRo single at F min
Reference: [7]
5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Methods for the increasing of SNR
38
B. Noise reduction with an input transformer
Measurement object
RS
Measurement system
1: n
vin
en
vo
vS
in
AV
n2 RS
n vS
SNRin
(n VS )2

F

= const,
SNRin =
2
4 kT n Rin
SNRo
1
 SNRo =
SNRin .
F
RL
5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Methods for the increasing of SNR
39
Example: Noise reduction with an ideal input transformer
RS
B = 1 Hz, en = 2 nV/Hz0.5, in = 20 pA /Hz0.5
100
1: n
RS n2
vS n
1
 SNRo =
SNRin
F
F
SNR1: n = SNR
Fmin
vn in , nV/Hz0.5
vS
in Rs
10
en
vS = en·1 Hz0.5
Source
noise
1
 4kTRsB
RS for minimum F
0.1
100
F 0.5, dB
101
102
103
104
20
10
SNR1: n = n2 SNRF min
RS opt
2
n = R
S
SNR 0.5, dB
SNR1: n0.5
0
-10
-20
SNR F min 0.5
-30
100
101
102
RS, W
103
104
5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Methods for the increasing of SNR
Example: Noise reduction with a non-ideal input transformer
RS
R1
1: n
R2
vS
(Rs + R1 ) n2 + R2
vS n
40
5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.3. SNR of cascaded noisy amplifiers
41
5.4.3. SNR of cascaded noisy amplifiers
Our aim in this Section is to maximize the SNR of a three-stage
amplifier.
RS
enS1
enS2
AV 1
enS3
AV 2
AV 3
vO
vS
Reference: [4]
5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.3. SNR of cascaded noisy amplifiers
RS
enS1
enS2
AV 1
42
enS3
AV 2
AV 3
vO
vS
VS2
1) SNRin 
Vno2 / AV12 AV22 AV32
2) Vno 2 = [enS12 AV12 AV22 AV32 + enS22 AV22 AV32 + enS32 AV32 ] B
VS2 / B
3) SNRin 
enS12 + enS22 /AV12 + enS32 /AV12 AV22
Conclusion: keep AV1 >> 1 to neglect the noise contribution of
the second and third stages.
Reference: [4]
Next lecture
43
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