1 5.3. noise characteristics reference: [4] the signal-to-noise ratio is the measure for the extent...

1

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 Smsr is the signal power, and Nmsr 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

2

Reference: [4]

SNRmsr .Smsr

Nmsr

It is usually assumed that the signal power, Smsr, and the noise

power, Nmsr, are dissipated in the noiseless input impedance of

the measurement system.


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

Measurement object Measurement system

Noiseless

Smsr

Rs

Sin Rmsr


Example: Calculation of SNR at the input of a measurement system

1) Smsr ,Vin rms

2 Rin

Rs + Rmsr 2

2) Nmsr ,Vn rms

2 Rin

Rs + Rmsr2

3) SNRmsr Vin rms

2

Vn rms2

Vin rms2

4 k T R fn


Noiseless

Smsr

Rs

Sin Rmsr

4

Reference: [4]


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

Rs


RL

Sin Power gain, AP

Noisy

GP

SNRo

SNRo .So

No

5

Reference: [4]

SNRo*

.


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

Rs


RL

Sin Power gain, AP

Noiseless

GP

SNRo*

So

No*

6

Noise factor is used to compare at the output the noise

contribution of a system (amplifier) against the noise power

delivered by the source (H. Friis, 1940s):

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

Rs


RL

Sin Power gain, AP

Noisy

GP

SNRo

7






Rs


RL

Sin

SNRo*

Power gain, AP

Noiseless

GP

F SNRo

*

SNRo

SNRmsr*

SNRmsr

*

SNRo

note that SNRo*

= SNRmsr* since the measurement system is

noiseless.

8






F .

SNRmsr

SNRo

Rs


RL

Sin

SNRo

Power gain, AP

Noisy

GP

SNRmsr

9

F ,No

No*


where No is the total noise power at the output, and No* is the

noise power at the output of the same but noiseless system (the

output noise comes only from the source).

A. Another definition for noise factor

F SNRmsr

SNRo

Smsr /Nmsr

*

So /No

(So /AP (/)No

*/AP )

So /No


Rs


RL

No

Power gain, AP

Noisy

GP

Illustration:

ens


Rs


RL

No*

Power gain, AP

Noiseless

GP

Illustration:

F ,No

No*

ens


Rs


RLens

F No

No*

Vno2/RL

4 kTRsBn )GV AV(2 /RL

Vno2

4 kTRs Bn )GV AV(2

VoVmsr

Example: Calculation of noise factor

Voltage gain, AVGV


F Vno

2

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]

Conclusions:


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

References: [2]

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 Bn )G AV(2


References: [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 Vin (rms) is increased until the output

power doubles. Under this condition the following equation is

satisfied:

Rs


RL

Vin

VoVmsr

Voltage gain, AVGV


References: [2]

Rs


RL

Vin

VoVmsr

1) (Vin GV AV)2 + Vno

2 2 Vno

2

Vin 0 Vin 0

2) Vno2

)Vin GV AV(2

Vin 0

3) F No

*

Vno2

Vin 0 (Vin GV AV)2

4 kTRs Bn )GV AV(2

Vin

2

4 kTRs Bn

Voltage gain, AVGV


References: [2]

FVin

2

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:



RL

iin) f (

VoVin

1) (iin Rs G AV)2 Bn + Vno2

2 Vno2

it 0 it 0

2) Vno2

)iin Rs G AV(2 Bn it 0

3) F No

*

Vno2

it 0 (iin Rs G AV)2 Bn

4 kTRs Bn )G AV(2

iin

2 Rs

4 kT

Rs

Voltage gain, AVGV

19

iin2

Rs

4 kT


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.

Neither the gain nor the noise bandwidth of the measurement

system need be known.


1. Increasing the source resistance may decrease the noise

factor, while increasing the total noise in the circuit.

2. If a purely reactive source is used, noise factor is

meaningless, since the source noise is zero, making the

noise factor infinite.

3. When the measurement system noise is only a small part of

of the source thermal noise (as with some low-noise FETs),

the noise factor requires taking the ratio of two almost equal

numbers. this can produce inaccurate results.

References: [2]

The concept of noise factor has three major limitations:

D. Limitations of the noise factor concept

F Vno

2

4 kTRs Bn )G AV(2

21

Noise factors varies with the bias conditions, frequency, and

temperature as well as source resistance, and all of these

should be defined when specifying and comparing noise

factors.

Knowing the noise factor for one value of source does not allow

the calculation of the noise factor at other values of resistance.

This is because both the source noise and measurement

system noise vary as the source resistance is changed.


References: [2]

Noise factor is usually specified for matched devices and is a

popular figure of merit in RF applications.

225. SOURCES OF ERRORS. 5.3. Noise characteristics. 5.3.3. Two source noise model

Reference: [2]

5.3.3. Two source noise model

A more recent (1956) approach and one that overcomes the

limitations of noise factor, is to model the noise in terms of an

equivalent noise voltage and current.

The actual network can be modeled as a noise-free network

with two noise generators, en and in, connected to its input:

Rs


RL

Vin

VoVmsr en

in Rmsr

Noiseless

AV


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 the network.

At relatively low frequencies, the correlation between the en and

in noise sources can be neglected.

Rs


RL

Vin

VoVmsr en

in Rmsr

Noiseless

AV


Reference: www.analog.com

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

en in


Assuming no correlation between the noise sources, the total

equivalent input noise voltage of the whole system can be

found by superposition:

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

Rs


RL

Vin

VoVmsr en

in Rmsr

Noiseless

AV


Vn in rms = 4kTRsB + Vn rms2 + )In rms

Rs(2 .

Rs


RL

Vin

VoVmsr en

in Rmsr

Noiseless

AV

Rs


RL

Vin

VoVmsr

in Rs

Noiseless

AV

en



Rs(2 .

Rs


RL

Vin

Vo

Voltage gain, AV

Vmsr

We now can connect an equivalent noise generator in series

with input signal voltage source to model the total input voltage

of the whole system.

Vn in

28

Vn in

Measurement

system noise



Rs(2 .

Example: Total equivalent input noise voltage as a function of Rs

1

10

100

0.1101 102 103 104100

Vn

in r

ms,

nV/H

z0.5

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

Rs,

in Rs

en

4kTRsB

Rs

Source noise


B. Measurement of en and in

Measurement system

RL

Vn oen

in

Noiseless

AV

en = )Vn o / B( / AV

Vn o rms >> )4 kT Rt B + Vn2(0.5

in Rs = )Vn o / B( / AV

in = )Vn o / B( / AV Rs

Measurement system

RL

Vn oen

in

Noiseless

AVRt

305. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.1. Optimum source resistance

5.4. Low-noise design: noise matching

Let us first find the signal-to-noise ratio SNR and the noise

factor F of the measurement system as a function of the source

resistance.

We next will try and maximize the SNR at the output of the

measurement system by matching the source resistance.

5.4.1. Maximization of SNR

Rs


RL

Vin

VoVmsr en

in

Noiseless

AV

31

SNR 0.5, dB

1

10

100

0.1101 102 103 104100

v n in

rm

s, nV

/Hz0.

5


in Rs

Vin = en1 Hz0.5

Rn for minimum F

Rs for maximum SNRvn in rms = [4kTRs

+ en rms2

+ (in rms Rs)2]0.5


en = in RnF 0.5, dB

Rs, 101 102 103 104100

-30

-20

-10

0

10

20

Measurement

system noise

Rn =en

in

A. Noise resistance Rn

en

Source noise 4kTRsB

Rn is called noise resistance


SNRmsr =(n Vin rms)2

4 kT n2 Rmsr

=const,

SNRo = SNRmsr .1

F

Rs

Measurement object

Vmsr1: n

n2 Rs

n Vin

RL

Vo

in

Measurement system

en

AV

Vin

B. Transformer coupling

F SNRmsr

SNRo

33

Example: Transformer coupling


1

10

100

0.1101 102 103 104100

v n in

rm

s, nV

/Hz0.

5


en

in Rs

Vin = en1 Hz0.5Rs n2

Vin n

Rs

Vin

1: n

SNR1: n = SNRF

Fmin

SNR1: n = n2 SNRmin F

F 0.5, dB

Rs, 101 102 103 104100

-30

-20

-10

0

10

20

SNR 0.5, dB SNR1: n

0.5

SNRmin F 0.5

SNRo = SNRmsr

1

F

Measurement

system noise

Source noise

Rn for minimum F

n2= Rn

Rs

4kTRsB

34

C. Parallel connection of input stages


Rs


RL

Vin

VoVmsr en

in

Noiseless

AV

en k / in k = en k 0.5/ in k 0.5

Rs =en / in

k

en

in

Noiseless

AV

k

k= en / in

Rs


Example:

36

D. SNR of cascaded noisy amplifiers


Reference: [4]

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

amplifier.

For the sake of simplicity, let us assume that all the stages are

identical in terms of noise, and their en >> in Rs.

Rs

Vin

AV 1 AV 2 AV 3

Voenen en


Reference: [4]

2) Vno rms2 = [)4kT Rs+en

2( AV12 AV2

2 AV32 + en

2 AV22 AV3

2 + en2 AV3

2 ] B

1) SNRin Vin

2

Vno2

/ AV12 AV2

2 AV32

3) SNRin Vin

2 / B

(4kT Rs+en2 + )en

2/AV12 + en

2/AV12

AV22

Conclusion: keep AV1 > 5 to neglect the noise contribution of the

second and third stages.

Rs

Vin

AV 1 AV 2 AV 3

Voenen en

385. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.2. Noise in diodes

5.4.2. Noise in diodes

ID

indrd

ID

end

rd

ID

2) in d 2 = 2 q ID = 2 k T / rd

1) rd k T

q ID

3) en d 2 = )2 k T / rd ( rd

2 = 2 k T rd

, ID k T

q rd

395. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.3. Noise in bipolar transistors

5.4.3. Noise in bipolar transistors

inb

enb

inc

C

r

v

rb

C

gmv

ro

B C

IB IC

A. Small-signal equivalent circuit

enb2 = 4 k T rb

inb2 = 2 q IB

inc2 = 2 q IC


inb

enb

incr

v

rbB C

in o

B. Mid-frequency noise model

gmv

Rs

Rs


3) en = in o /Gv Ag , Gv = r /)rb+ r(, Ag = gm

Rs=

4) in2 = in o /Gi Ag

2 , Gi = r

Rs=

1) in o2 = {[enb r /) rb+ r(] gm}2 +[ inb )rbIIr( gm]2 + inc

2

Rs=

2) in o2 = ) inb r gm(2 + inc

2

Rs

inb

enb

incr

v

rbB Cgmv

Rs in o


5) en2 = enb

2+ )inb rb(

2 + [inc) rb+ r(/ gm r]2

6) in2 = inb

2+ [inc

/) gm+ r(]2

inb

enb

incr

v

rbB Cgmv

Rs in o


5) en2 = enb

2+ )inb rb(

2 + [inc) rb+ r(/ gm r]2

6) in2 = inb

2+ [inc

/) gm+ r(]2

in

IC

IB en

in

en

r

v

rbB Cgmv

Rs in o

445. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.4. Noise in FETs

5.4.4. Noise in FETs

due to the thermal noise of the base resistance,

the shot noise in both the collector and base currents,

and the flicker noise of the base current:

The noise in bipolar transistors are:

BJT en2 = 4kT rB + 2qIC re

2 in2 = 2qIB + a A IB / f

in

ID

IG en

455. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.5. Noise in differential and feedback amplifiers

.

5.4.5. Noise in differential and feedback amplifiers

465. SOURCES OF ERRORS. 5.4. Low-noise design: noise matching. 5.4.6. Noise measurements

.

5.4.6. Noise measurements

47Next lecture

Next lecture:D (diode)

FID flicker noise

RS thermal noise associated with RS

SID shot noise

TOT total noise

J (JFET)

FID flicker noise

RD thermal noise associated with RD

RG thermal noise associated with RG


SID shot noise

TOT total noise

M (MOSFET)

FID flicker noise

RB thermal noise associated with RB




SID shot noise

TOT total noise

48Next lecture

Next lecture:

J (JFET)

FID flicker noise




SID shot noise

TOT total noise

49Next lecture

Next lecture:

M (MOSFET)

FID flicker noise





SID shot noise

TOT total noise

50Next lecture

Next lecture:

Q (BJT)

FIB flicker noise


RC thermal noise associated with RC

RE thermal noise associated with RE

SIB shot noise associated with base current

SIC shot noise associated with collector current

TOT total noise

1 5.3. noise characteristics reference: [4] the signal-to-noise ratio is the measure for the extent...

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