1 LECTURE 7. Contents 5.Sources of errors 5.1.Impedance matching 5.4.1.Non-energetic matching 5.4.2.Energetic matching 5.4.3.Non-reflective matching 5.4.4.To

1LECTURE 7. Contents

5. Sources of errors

5.1. Impedance matching

5.4.1. Non-energetic matching

5.4.2. Energetic matching

5.4.3. Non-reflective matching

5.4.4. To match or not to match?

5.2. Basic 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

2

5.3. Noise characteristics

Reference: [4]

The signal-to-noise ratio is the measure for the extent to which

a signal can be distinguished from the background noise:

SNR SN

where S is the signal power, and N is the noise power.

5.3.1. Signal-to-noise ratio, SNR

5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.1. Signal-to-noise ratio, SNR

3

SNRin Sin

Nin


A. Signal-to-noise ratio at the input of the system, SNRin

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.

Measurement object Measurement system

vS

ZS=RS + jXS

Zin=Rin + jXin

Noiseless

RL

SNRin


Example: Calculation of SNRin

1) Sin ,VS

2 Zin

(ZS + Zin)2 2) Nin

,

Vn 2 Zin

(ZS + Zin)2

3) SNRin VS

2

Vn 2

VS 2

4 k T RS N


vS

ZS=RS + jXS

Zin=Rin + jXin

Noiseless

RL

SNRin

Note that SNRin is not a function of Zin.

5

SNRo srcSNRin

SNRo src SNRin



vS Power gain, Ap

Noiseless

SNRo src So

No src

ZS=RS + jXS

1) The measurement system is noiseless.

Sin Ap

Nin Ap

Sin

Nin

RL

B. Signal-to-noise ratio at the output of the system, SNRo

6

SNRo

SNRo SNRin



vS Power gain, Ap

Noisy

SNRo So

No

ZS=RS + jXS

2) The measurement system is noisy.

SNRin

Sin Ap

(Nin+Nin msr) Ap

Sin

Nin

RL

7

SNRo

Noise factor is used to evaluate the signal-to-noise degradation

caused by the measurement system (H. T. Friis, 1944).

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

5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.2. Noise factor, F, and noise figure, NF

F SNRin

SNRo


vS Power gain, Ap

NoisyZS=RS + jXS SNRin

RL

8

SNRoSNRin

The signal-to-noise degradation is due to the additional noise,

No msr , which the measurement system contributes to the load.



vS Power gain, Ap

NoisyZS=RS + jXS

F SNRin

SNRo

SNRo src

SNRo

So /No src

So /No

No

No src

No src + No msr

No src

RL

No msr

No src


RS


F No

No src

Vno2/RL

4 kTRS NB )GV AV(2 /RL

Vno2

4 kTRS NB )GV AV(2

vovin

Example: Calculation of noise factor

Voltage gain, AVRL

enS GV

Here and below, we assume that the reactance in the source

output impedance is compensated by the properly chosen input

impedance of the measurement system (noise tuning).


F Vno

2

4 kTRS B )G AV(2

The following three characteristics of noise factor can be seen

by examining the obtained equation:

1. It is independent of the load resistance RL,

2. It does depend on the source resistance RS,

3. If the measurement system was completely noiseless,

the noise factor would equal one.

Reference: [2]

Conclusions:


Noise factor expressed in decibels is called noise figure )NF(:

NF 10 log F

Due to the bandwidth term in the denominator

there are two ways to specify the noise factor: (1) a spot noise,

measured at specified frequency over a 1Hz bandwidth, or (2)

an integrated, or average noise measured over a specified

bandwidth.

C. Noise figure

F Vno

2

4 kTRS NB )G AV(2

Reference: [2]


Reference: [2]

We will consider the following methods for the measurement of

noise factor: (1) the single-frequency method, and (2) the white

noise method.

E. Measurement of noise factor

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:

RS


vS

vovin

Voltage gain, AvGv

RL


Reference: [2]

RS


vS

vovin

1) (VS GV AV)2 + Vno

2 2 Vno

2

VS 0 VS 0

2) Vno2

)VS GV AV(2

VS 0

3) F No src

Vno2

VS 0 (VS GV AV)2

4 kTRS NB )GV AV(2

VS

2

4 k T RS NB

Voltage gain, AVGV

RL


Reference: [2]

F

The disadvantage of the single-frequency method 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 source:

VS2

4 k T RS NB



in ) f (

vovin

1) (in RS GR AV)2 NB + Vno2

2 Vno2

in 0 in 0

2) Vno2

)in RS GR AV(2 B in 0

3) F No src

Vno2

in 0 (in RS GR AV)2 NB

4 kTRS NB )GR AV(2

in

2 RS

4 k T

RS

Voltage gain, AvGR

RL

16


F

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.

The noise bandwidth of the measurement system should not be

known.

in2

RS

4 k T

The standard reference temperature is T0 = 290 K for that

k T0= 4.001021. (H. T. Friis: NF, Pa, and T0.)

17

Reference: [2]

5.3.3. VnIn 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):

RS


vS

vo

Rin

Noiseless

AV RL

In a general case, the en and in noise generators are correlated.

en

in

5. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. VnIn noise model

18

Reference: [2]

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 a linear network.

RS


vS

vo

Rin

Noiseless

AV RL

en

in


19

Reference: www.analog.com

Example: Input voltage and current noise spectra (ultralow-noise, high-speed, BiFET op-amp AD745)

en

in


20

The total equivalent noise voltage reflected to the source

location can easily be found if we apply the following

modifications to the input circuit:

A. Total input noise as a function of the source impedance

RS


vS

vo

Rin

Noiseless

AV RL

en


in

21

en at S = 4 kT RS + en 2 + 2 en in + )in RS(2

RS


vS

vo

en

in Rin

Noiseless

AV

RS


vS

vo

in Rs

Noiseless

AV

RL

RL

en


Rin

22

RS


vS

vo

Voltage gain, AV

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 0. (For the case 0, it is often simpler to

analyze the original circuit with its internal noise sources.)

en at S

en at S = 4 kT RS + en 2 + )in RS(2

RL

Reference: [7]


Noiseless


B. Measurement of en and in

Measurement system

Noiseless

AV

1) Rt = 0 4 kT Rt + )in Rt(2 0

2) en = vn o) f ( / AV

1) (in Rt)2 >> 4 kT Rt + en

2

2) in Rt vn o) f ( / AV

3) in [vn o) f ( / AV ] / Rt

Measurement system

Noiseless

AVRt

vn o

RL

vn o

RL

en

in

en

in

Rin

Rin

245. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR

5.4. Noise matching: maximizing SNR

The purpose of noise matching is to let the measurement

system add as little noise as possible to the measurand.

Influence

Measurement System

Measurement Object

Mat

chin

g

+ xx

255. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR

where, Nat S and SNRat S are the noise power and the signal-to-

noise ratio at the source location.

We then will try and maximize the SNRo at the output of the

measurement system by matching the source resistance.

VS2

Nat SSNRo = SNRat S

VS2

4kTRS+ en2 + )in RS(2 f ) RS

(

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: F = f ) RS ( and SNRo = f ) RS (.

4kTRS + en2 + )in RS(2

4kTRS

No

No srcF

Nat S

NR f ) RS

(S

26

F 0.5, dB

SNR 0.5, dB

5.4.1. Optimum source resistance

1

10

100

0.1101 102 103 104100

e n at

S ,

nV/H

z0.5

en = 2 nV/Hz0.5, in = 20 pA /Hz0.5

RS min F

RS max SNR

5. SOURCES OF ERRORS. 5.4. Noise matching: maximization of SNR. 5.4.1. Optimum source resistance

RS , 101 102 103 104100

-30

-20

-10

0

10

20

Measurement

system noise

en = in Rn

RS opt =en

in

RS opt is called

the optimum source resistance

)also noise resistance.)

VS2

4kTRS+ en2 + )in RS(2SNR


4kTRSF

vS = en1 Hz0.5

in RS

en

4kTRS

Source noise

27

SNR 0.5, dB

F 0.5, dB

RS , 101 102 103 104100

-30

-20

-10

0

10

20

It is important to note that the source resistance that maximizes

SNR is RS max SNR 0, whereas the source resistance that

minimizes F is RS min F RS opt .

For a given RS, SNR cannot be increased by connecting a

resistor to RS.

Vs2



4kTRSF


28

RS

Measurement object

vS

SNR 0.5, dB

F 0.5, dB

RS , 101 102 103 104100

-30

-20

-10

0

10

20

VS2



4kTRSF


Adding a series resistor, R, increases the total source

resistance up to RS opt = RS + R and (!) decreases SNR.

RS RS opt

29

VS2



4kTRSF

SNR 0.5, dB

RS

Measurement object

vS

+R


F 0.5, dB

RS , 101 102 103 104100

-30

-20

-10

0

10

20

4kTRS opt + en2 + )in RS opt (2

4kTRS opt F

VS2

4kTRS+ en2 + )in RS(2SNR VS

2


RS R RS opt

Adding a series resistor, R, increases the total source

resistance up to RS opt = RS + R and (!) decreases SNR.

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 SNR.

RS

vS


VS2



4kTRSF

Measurement object

SNR 0.5, dB

F 0.5, dB

SNR 0.5, dB

RS , 101 102 103 104100

-30

-20

-10

0

10

20

RS RS opt

31

SNR 0.5, dB

F 0.5, dB

SNR 0.5, dB

RS , 101 102 103 104100

-30

-20

-10

0

10

20

VS2



4kTRSF

RS

Measurement object

vS

R


VS2


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 SNR.

RS / [)RS R(/R] RS opt

VS / [)RS R(/R] VS

k )RS R(/R > 1

F 0.5, dB

SNR 0.5, dB

RS , 101 102 103 104100

-30

-20

-10

0

10

20

k 2

k 2

4kTRS opt + en2 + )in RS opt (2

4kTRS opt F

VS2

4kTRS+ en2 + )in RS(2SNR VS

2

4kTRS k + en2

k 2 + )in RS(2SNR

32

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 (erroneous belief).


335. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Methods for the increasing of SNR

Methods for the increasing of SNR are based on the following

relationship:

SNRo = SNRin

1

F

The strategy is simple: to increase SNRo, keep SNRin constant

while decreasing the noise figure:

SNRo = SNRin

1

F

The SNR at the output will increase because the relative noise

power contributed by the measurement system will decrease.

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 the SNR, several active devices are connected

in-parallel:


Reference: [7]

io scvin

k

Measurement system

Rin ro

Rin ro

gm vin

gm vin

en

in

en

in

35

Reference: [7]

Home exercise: Prove that the following network is equivalent to the

previous one.

io sc

Equivalent measurement system

vin

Rin

k

k gm vin


en/k 0.5

in k 0.5

ro

k

36

io sc

Reference: [7]

Measurement object

vS

k= en / in

RS

Equivalent measurement system

vin

Rin

k

RS

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

vS and RS, and hence in the SNRin.

Note that SNRo cannot be improved if the RS is too large.

SNRo = SNRo max and F = Fmin at RS =en

k in

k gm vin


en/k 0.5

in k 0.5

ro

k

37

Reference: [7]Reference: [7]

Home exercise: Prove that

SNRo p = k SNRo single at F min


38

SNRin (n VS )2

4 kT n2 RS

const,

SNRo = SNRin .1

F

RS

Measurement object

vin1: n

n vS

n2 RS

vo

Measurement system

AV

vS

F SNRin

SNRo

RL

B. Noise reduction with an input transformer


en

in Rin

39

SNRo )1: n( = SNRo

FFmin

Example: Noise reduction with an ideal input transformer

1

10

100

0.1101 102 103 104100

e n at

S ,

nV/H

z0.5

B = 1 Hz, en = 2 nV/Hz0.5, in = 20 pA /Hz0.5

en

in Rs

RS n2

vS n

RS

vS

1: n

F 0.5, dB

RS, 101 102 103 104100

-30

-20

-10

0

10

20

SNR 0.5, dB SNRo )1: n(

0.5

1

FSNRo = SNRin

Measurement

system noise

Source noise

RS for minimum F

SNRo )1: n( = n2 SNRo F min

n2= RS opt

RS

4kTRsB


vS = en1 Hz0.5

F 0.5Fmin 0.5

SNRo 0.5 SNRo )1: n(

0.5

SNRo F min0.5

40

To prove that

let us consider the following. At noise matching, the amplifier

sees Rs=Rn, thus the total noise at the amplifier output remains

the same: No )1:n( = No F min . The transformer, however, increases

the amplifier input voltage: vin )1:n( = n2 vin F min. Therefore:


SNRo )1: n( = n2 SNRo F min

SNRo )1: n( n2 SNRo F min

vin )1:n(2

No )1:n(

n2 vin F min

2

No F min

41

1

10

100

0.1101 102 103 104100

e n at

S ,

nV/H

z0.5

B = 1 Hz, en = 2 nV/Hz0.5, in = 20 pA /Hz0.5

en

in Rs

RS n2

vS n

RS

vS

1: n

F 0.5, dB

RS, 101 102 103 104100

-30

-20

-10

0

10

20

SNR 0.5, dB

Measurement

system noise

Source noise

RS for minimum F

4kTRsB


vS = en1 Hz0.5

F 0.5Fmin 0.5

SNRo 0.5 SNRo )1: n(

0.5

SNRo F min0.5

Note that:

n2 F

Fmin

SNRo )1: n( = SNRo

FFmin

F/Fmin n SNRo )1: n(0.5

42

vS n

(RS + R1 )n2 + R2

RS

vS

1: n R1 R2

Example: Noise reduction with a non-ideal input transformer


43

Reference: [4]

Our aim in this section is to maximize the SNR of a three-stage

amplifier.

RS

vS

AV 1 AV 2 AV 3

vO

enS1 enS2 enS3

5.4.3. SNR of cascaded noisy amplifiers

5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.3. SNR of cascaded noisy amplifiers

44

Reference: [4]

2) Vno 2 = [enS1

2 AV1

2 AV22 AV3

2 + enS22 AV2

2 AV32 + enS3

2 AV32 ] NB

1) SNRo SNRat S

VS 2

Vno2

/) AV12 AV2

2 AV32(

3) SNRo VS

2 / NB

enS12

+ enS22

/AV12 + enS3

2 /AV1

2 AV2

2

Conclusion: keep AV1 >> 1 to neglect the noise contribution of

the second and third stages.

5. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.3. SNR of cascaded noisy amplifiers

RS

vS

AV 1 AV 2 AV 3

vO

enS1 enS2 enS3

45Next lecture

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1 LECTURE 7. Contents 5.Sources of errors 5.1.Impedance matching 5.4.1.Non-energetic matching 5.4.2.Energetic matching 5.4.3.Non-reflective matching 5.4.4.To