reader et8017 electronic instrumentation, chapter5

113

Chapter 5

• How should noise signals be specified?• How are noise-equivalent input signals calculated?• How is the noise-related detection limit calculated?• How much noise is generated by passive and active components?• What is noise matching and how can this concept be used in design?

5.1 IntroductionThe random error in a signal is generally referred to as noise. Unlike the system-atic errors discussed in Chapters 2-4, random errors are not reproducible. Thisimplies that the amplitude at a particular moment in time cannot be predicted.For this reason, unlike deterministic errors, random errors cannot be corrected orcompensated for.

A practical system is typically prone to two sources of noise:• Internally generated noise and • Unwanted external signals coupled into the system (also referred to as Elec-

tro-Magnetic Interference (EMI)).

Internal noise in a system is generated by dissipative components. This could beresistors in an electronic circuit or a damper in a mechanical system. Noise isspecified in terms of statistic parameters. A practical noise spectrum is usuallycomposed of a frequency-dependent part (the 1/f noise) and a frequency-inde-pendent part (the white noise), as shown in Fig. 5.1.

DETECTION LIMIT DUE TO NOISE AND ELECTRO-MAGNETIC INTERFERENCE

Electronic Instrumentation R.F. WolffenbuttelChapter 5: DETECTION LIMIT DUE TO NOISE AND ELECTRO-MAGNETIC INTERFER-ENCE

114

Noise generation is not restricted to the electrical domain. The noise spectrum inFig. 5.1. is actually the noise in a Micro Electro-Mechanical System (MEMS)caused by squeeze-film mechanical damping. Sources of internal noise are dis-tributed throughout the system and the task is to calculate the equivalent input-referred noise sources. In this chapter internal noise is considered frequency-independent (white), unless otherwise explicitly mentioned.

External noise is coupled into a system at the node that is the most susceptible tothis particular source of noise (usually the input). Obviously, this susceptibilityneeds to be minimised. A notorious source of EMI is the mains voltage, but alsothe internal clock of the digital part of a system is a well-known source of EMI inthe analog front-end of that system.

When calculating the effect of noise on the detection limit, any source of noiseshould be referred to the input.

5.2 Equivalent input noise sourcesThe noise voltage is the square root of the noise power and is defined as:

This is basically the rms-value of the signal. A noise voltage is usually indicatedas un, and the component referred to is included in the subscript (e.g. noise in R1:un,R1). Noise is usually specified in terms of the squared noise voltage and isreferred to as the noise power, Pn [V2] or [A2]. A specification in terms of ‘noisepower’ is fundamentally incorrect, since neither [A2] nor [V2] has the unit ofpower [W]. Nevertheless, this approach for specifying noise is universally used.

Figure 5.1, Typical noise spectrum of a critically damped mechanical structure.

(5-1)

10-5

10-4

10-3

10-2

10-1

100

101

-500

-450

-400

-350

-300

-250

-200

Mag

nitu

de [d

B]

Frequency [Hz]

( )2 21limT

n n n nTT

u u t dt u Pt→∞

−

= = =∫

Section 5.2Equivalent input noise sources

115

Figure 5.1 indicates the frequency-dependent of noise. Noise is, for this reason,often specified in terms of the noise spectral power, sn, which is the noisepower per unit of bandwidth. The noise spectral power in a noise voltage is sn,u[V2/Hz]. The variable sn indicates a specification in terms of noise spectralpower, while the u in the subscript indicates that it is a voltage noise source thatcould also include an identification of the component (e.g. noise spectral powerin resistor R1 is represented by a series voltage source (see next section), whichis specified as: sn,uR1). This convention, with the obvious modifications, is alsoused for a noise current, sn,i [A2/Hz].

The noise spectral power sn is the noise power per unit bandwidth,which is [V2/Hz] in the case of a noise voltage and [A2/Hz] in the caseof a noise current.

Similar to the description of offset in a system using input-referred sources ofoffset, the noise of a system is specified in terms of input-referred equivalentsources of noise that represent the effect of all the distributed internal sources ofnoise in the system, as shown in Fig. 5.2.

Similar to the case of offset, the non-ideal (noisy) components and the sources ofnoise are indicated by a hatched patch or a gray-scale inner area.

Calculating the rms-amplitude of the equivalent input noise sources in a systemas a function of the distributed noise sources, is based on the propagation of sto-chastic errors. The noise power of the equivalent noise sources should be equalto the total noise power of the independent noise sources that it represents.

Unlike the case of offset, where the amplitudes of the distributed sources are lin-early added to obtain the equivalent input offset, the calculation of the equivalentstochastic error is based on the linear addition of the noise powers of the distrib-uted sources of error (= the linear addition of the variances of the stochastic sig-nals). Thus:

Figure 5.2, System with (a) distributed noise sources and (b) equivalent inputnoise sources.

Rg Un,eq

RiUg=0 In,eq

+

_

(b)

Rg

RiUg=0+

_

(a)


116

The calculation of the noise powers of the input-referred sources ofnoise of a system is based on the linear addition of the noise powers(i.e. the squared amplitudes) of all the distributed noise sources, eachreferred to the input.

The equivalent noise voltage of the statistically-independent noise voltagesources un1 and un2 in series is expressed as:

Similarly, the equivalent noise current of two statistically-independent noise cur-rent sources in parallel is calculated using:

The noise power of two voltage sources in series is equivalent to thenoise power of one voltage source, with the total noise power beingequal to the sum of the noise powers of the two sources. Similarly, twonoise current sources in parallel are represented by one currentsource with a noise power equal to the sum of the noise powers ofthese two sources.

Example 5.1An AC Volt meter with an rms reading is specified for a total equivalentinput noise voltage unm= 1 mV. This instrument is used for noise measure-ment and gives a unt= 4 mV reading. What is the noise voltage of thesource, uns, assuming matched noise bandwidths?

Solutionunt

2 = unm2+ uns

2.Hence: uns

2 = unt2 - unm

2= (4×10-3)2- (1×10-3)2= 15×10-6 V2→uns= √15 mV.

Expressions 5.2 and 5.3 are applied when calculating of the equivalent noisesources at the input terminals as a function of statistically-independently distrib-uted noise sources in a measurement system.The statistic independence of thesources of noise is a fundamental requirement, as is demonstrated by the noiseparadox shown in Fig. 5.3. Two resistors are connected in series with a noisecurrent, in. Two approaches are available to calculate the equivalent noise volt-

(5-2)

(5-3)

2 2, 1 2n eq n nu u u= +

2 2, 1 2n eq n ni i i= +

Section 5.2Equivalent input noise sources

117

age that represents the thermal noise of the resistors and the noise voltage gener-ated across these resistors due to the noise current.

The first is based on adding the noise power generated at resistor R1 caused byin2 to the noise power representing the thermal noise in the resistor itself, un,R

2.Applying the same approach to R2 and combining the results yields un1,eq

2.

The second approach combines R1 and R2 for the calculation of the total noisevoltage due to in and results in un2,eq:

These solutions are not identical. The first approach assumes that the noise volt-ages across R1 and R2 due to in are statistically independent, but these are actu-ally dependent since they originate from the same source. Therefore, un1,eq(option (a)) is incorrect.

Two equivalent input noise sources are required for the full specification of thenoise in a system: an equivalent input noise voltage plus an equivalent inputnoise current. Similar to the case of offset, these two equivalent input sourcescan be calculated using two independent expressions. These are derived at twoextreme conditions:

1. Short-circuited input and2. Open-circuited input.

This procedure is used to calculate the equivalent input noise sources as a func-tion of distributed sources for an amplifier composed of two cascaded sections,each with specified equivalent input noise sources, as shown in Fig. 5.4.

Figure 5.3, Noise paradox.

(5-4)

R2

R1in

unR1

unR2

R2

R1

un,eq

( )

( )( ) ( )

2 2 2 2 2 2 21, , 1 , 2 1 2

2 2 2 2 2, 1 , 2 1 2

2 2 2 22, , 1 , 2 1 2

n eq n R n R n n

n R n R n

2n eq n R n R n

a u = u +u +i R +i R =

u +u +i R + R

b u = u +u +i R + R


118

An open input results in the equivalent input noise current:

A short-circuited input yields the equivalent input noise voltage:

This result confirms the benefits of having a high-gain first stage.

5.3 Specifying noise in components

5.3.1 Noise in passive componentsA resistor, R, exhibits thermal noise.

The thermal noise in a resistor, R, is described by a noise voltagesource unR with noise spectral power sn,uR= 4kBTR [V2/Hz] in serieswith a noise-free, resistor, R. Alternatively, this noise is described by anoise current source inR with sn,iR= 4kBT/R [A2/Hz] in parallel to R.

Figure 5.4, Model used to calculate the equivalent input noise sources for twogain sections in cascade (a) with distributed sources of noise and (b) withequivalent input sources of noise.

(5-5)

(5-6)

un1

U1 Ri

Ro

G1U1in1

+

+

un2

in2

++

_

u2a

+

_

un,eq

U1 Ri

Ro

G1U1in,eq

+

+

+

_

u2b

+

_

(a)

(b)

2 2 2 22 1 1 2 2 2 2 22 2 2 2 2 22 , 1 , 1 2 2

12 22 2

( ) ( )

( )a n i n o n

n n ob n eq i n eq n

i

a b

u i R G i R uu i Ru i R G i i

G Ru u

= + + +

= → = +

=

2 2 2 22 1 1 2 2 2 2 22 2 2 2 2 22 , 1 , 1 2

12 22 2

( ) ( )

( )a n n o n

n n ob n eq n eq n

a b

u u G i R uu i Ru u G u u

Gu u

= + + +

= → = +

=

Section 5.3Specifying noise in components

119

The noise spectral power is the noise power per unit bandwidth, where kBdenotes the Boltzmann constant and T is the absolute temperature. The two rep-resentations are shown in Fig. 5.5.

It should be noted that any dissipating component is a thermal noise generator(irrespective of signal domain). The resistive part of an impedance is the noise-generating part. Therefore, the general formulation of the noise spectral power inan impedance is:

The noise spectral power is independent of the frequency range from the (1/f)-regime up to very high frequencies and is often referred to as white noise. Thenoise behaviour of electronic components is often specified in terms of the noisespectral density, which is the square root of the noise spectral power:

The noise spectral density in a resistor, R, is: √sn,uR [V/√Hz], withnoise rms-amplitude unR= (sn,uR×B)1/2 [V]. Similarly: √sn,iR [A/√Hz],with inR= (sn,iR×B)1/2 [A].

The total noise power is equal to the noise spectral power integrated over thenoise bandwidth, which is defined as an abrupt cut-off. This is a theoretical fil-ter with a constant transfer function in the passband, a perfect suppression of thesignal in the stopband, and an infinitely narrow transition interval in between.This filter response can be approximated in a high-order filter, but not in thefirst- or second-order filters that are commonly used in analog circuits. There-fore, a conversion factor is required, which can be calculated using Fig. 5.6.

The basic idea is that the total noise power at the output of the two filters must beequivalent. Therefore, it is sufficient to calculate the total output noise for bothfilters and to equate the results:

Figure 5.5, Thermal noise in resistor.

(5-7)

Un

InR = RR OR

( )

( )

2,

2,

2n uZ n B

2Bn iZ n

s u f = 4k T Z [ /Hz]V4k Ts i f = [ /Hz]AZ

=

=


120

In a first-order low-pass filter with a -3 dB cut-off frequency at ωc=1/τ and anon-attenuated transfer at DC (|Ho|2=1), the noise bandwidth results in:

which implies that a conversion factor equal to π/2 is required. Therefore, if thenoise power of a resistor R passes through a noise-free first-order low-pass filter(LPF) with a cut-off frequency at fc [Hz], the noise bandwidth B= πfc/2 [Hz] andthe output total noise power is equal to 2πkBTRfc.

A capacitor is not a dissipating element. Nevertheless, a capacitor, C, does affectthe noise power. The dielectric loss needs to be taken into consideration toexplain this effect. Dielectric loss is represented by a parallel resistor, Rp, with aresistance that increases with the quality factor of the capacitance. The noisespectral power is expressed as: sn,iR= 4kBTRp, which is represented by a noisecurrent source in parallel to the terminals, as shown in Fig. 5.7. The resultingnoise voltage in series with the capacitor follows as:

Figure 5.6, Calculation of noise bandwidth (solid curve) from a practical filterfrequency response (dashed line).

(5-8)

(5-9)

Figure 5.7, Equivalent total noise source in a capacitor.

ωc

H0

0lin ωB

H(ω)

( ) ( )212 22

n o0 0o

P = H B = H d B H dH

ω ω ω ω∞ ∞

→ =∫ ∫

( )( ) [ ]

0

/c20

1dB = = arctg = = rad s ,2 21+π πω ωτ ω

τ τωτ

∞∞

∫

inRp = RpC C= C

unC

RpC=


121

Note that the factor (2π)-1 is introduced to allow the noise spectral power to beexpressed in [W/(rad/s)] rather than [W/Hz].

The capacitor results in a noise voltage unC, with a total noise power of kBT/C[V2], irrespective of the actual value of the dielectric loss resistor Rp. Therefore,the total noise over an unlimited bandwidth does not depend on the quality of thecapacitor. This surprising result is due to the fact that, although the impedance isproportional to Rp, in2 is inversely proportional to Rp and the bandwidth isinversely proportional to RpC (see lower line in Eqn. 5-10). However, the operat-ing bandwidth is limited in a realistic circuit and a high value for Rp does matterin a low-noise circuit design.

A similar discussion applies to a practical inductor with the losses represented bya series resistor Rs, as shown in Fig. 5.8. The noise spectral power is representedby a noise voltage source in series with the inductor and results in a noise currentexpressed by:

The non-ideal inductance results in a noise current source inL, with total noisepower of kBT/L [A2] that is independent of the series resistance Rs. Similar to the

(5-10)

Figure 5.8, Equivalent total noise source in an inductor.

(5-11)

inL

Rs Rs=

un

=L L

=

Rs

LL


122

capacitor, the quality of the inductance is important for low-power circuit designin cases of a limited frequency band of operation.

The next section shows that noise in active components is represented by severalvoltage noise sources and/or current noise sources. An electronic circuit is com-posed of several active and passive components. The overall noise performanceof the circuit is specified in terms of an equivalent input voltage noise source incombination with an equivalent input current noise source. These are calculatedas a function of the distributed sources of noise that are due to the componentsused.

5.3.2 Modeling noise in active componentsThe noise characteristics of active devices are described by distributed noisesources within the device, which are closely related to the lumped componentsthat describe the operating mechanism of the active component. Resistors in theequivalent circuit diagram usually represent connection wires resistances or non-active (bulk) parts of the device are sources of thermal noise.

The operation of active devices involves controlled electric charge transportthrough a channel, as reflected in the voltage-to-current characteristic. An elec-trical current is defined as the amount of charge passing a point in that channelper unit of time. Statistical fluctuations in this charge transport result in a noisecurrent that is proportional to the current (= average amount of charge trans-ported per unit time) and is referred to as shot noise, with a noise spectral powerexpressed as:

Charge transport through a semiconductor junction is modelled as asource of shot noise with a noise spectral power sn,i= 2qId.

The equivalent circuit diagram of a forward-biased diode is shown in Fig. 5.9and includes a series bulk resistance, which gives rise to thermal noise as repre-sented by the noise voltage un, with a noise spectral power of sn,us= 4kBTRs, anda parallel impedance, Zp, which is composed of the small-signal dynamic resis-tance, rd, in parallel to a very large leakage resistor, Rp> 10 MΩ, and the diffu-sion capacitance Cd. Thus: Zp= rd/(1+jωrdCd), with rd expressed as:

(5-12)

(5-13)

2, 2 [ /Hz]An is = qI

( ) ( )( )ln ln ln,

B d Bd s

sd Bd

d d d d

k T I k T I Iq IU k Tqr =I I I qI

∂ ∂ − ∂ = = =

∂ ∂ ∂


123

where Is is the Id-independent saturation current. The thermal noise in the bulkresistance, sn,us, dominates the noise performance in a forward-biased diode(typical values for (sn,us)1/2 are in the range 10-100 nV/√Hz).

It should be noted that a dynamic resistor in a small-signal equivalentcircuit diagram is not a dissipating component, but rather the lin-earised voltage-to-current characteristic at a certain operating pointthat is set by biasing. Consequently, it is not a source of thermal noise.

The equivalent circuit diagram of the reverse-biased diode is shown in Fig. 5.10.Although the same components are included in this circuit as are in the forward-biased diode, their values are significantly different. The differential resistor rd isin the reverse-biased diode set by the negative value of the voltage applied.

Figure 5.9, Sources of noise in a forward-biased diode.

Figure 5.10, Sources of noise in a reverse-biased diode.

Id

U+

Ud

(b)

uns

ind

Rs

Id

U+

Rs

Ud

Zp Zp

Id

Ud=U_

(b)

uns

ind

Rs Rs

Id

Ud=U_

ZpZp


124

In a reverse-biased diode -Ud (kBT/q). Therefore:

Hence, rd→ ∞. The charge storage in a reverse-biased junction is determined bythe space charge layer boundaries of the depleted region and is specified by Cband Zp= Rp/(1+jωRpCb). Since Rp Rs, uns is insignificant and the noise is set bythe diode shot-noise, ind. Due to the very low leakage currents in state-of-the-artdiodes at moderate reverse voltage values, (sn,id)1/2= (2qId)1/2= 1-100 fA/√Hz,typically. It should be noted that Zp is capacitive over most of the practical fre-quency range. The noise performance of a (photo)diode is discussed in moredetail in Chapter 7.

Also the noise sources in a bipolar transistor result from the operating mecha-nism. The small-signal hybrid-π equivalent circuit diagram of the bipolar transis-tor, including noise sources, is shown in Fig. 5.11. The differential resistance rb'c→ ∞ when disregarding the Early effect.

Minority charge carriers are injected into the base region from the emitter. Somerecombine in the base layer, while others diffuse towards the base-collectedspace charge region to be collected in the collector. Those charge carriers thatactually reach the depleted area around the reverse-biased base-collector junc-tion are accelerated giving rise to a collector current. Those charge carriers thatrecombine in the non-depleted part of the base region give rise to a base current.

The injection and recombination of charge carriers are subject to statistical fluc-tuations. Therefore the noise performance is represented by shot noise sourcesconnected to the base and collector, with the noise spectral power equal to: sn,ib=2qIb1 and sn,ic= 2qIc, respectively.

The operation of the practical bipolar transistor is affected by resistors in serieswith the base, collector and emitter and are due to bulk resistance of the doped

(5-14)

Figure 5.11, Sources of noise in a bipolar transistor.

exp1 0 0,

sBd s

d d d d

qUIk TI I=

r U U U

− ∂ ∂ ∂ × = = →

∂ ∂ ∂

gmub'e

C

E

B

unb1 Rbb'

B

rb'e

ub'

ue

inc

rb'c

Cb'c

Cb'e

C

E

B'

inb1


125

layers and contact resistance. Especially the base series resistance, Rbb’, isimportant for the noise performance of the transistor. The noise spectral power ofthis resistance is described by: sn,ub1= 4kBTRbb’.

The source of shot noise due to the collector leakage current, Ico, is in parallel toCb'c and is represented by: sn,ico= 2qIs, but is generally disregarded. At low fre-quencies (ω< rb’eCb’e), the effect of the collector current shot noise at the inputresults from the basic device operation in: unb2= inc/gm. The equivalent inputnoise voltage is obtained by adding the noise powers of un1 and un2

The operation of the (junction) field effect transistor (JFET) is based on the gate-source voltage-controlled conductance of a channel between drain and source.The equivalent circuit of the JFET, including sources of noise, is shown in Fig.5.12. The resistive channel gives rise to a thermal noise source in parallel to thedrain and source contact with a noise spectral power that is proportional to thechannel conductance: ind= 4kBT/Rch. It can be shown that the channel conduc-tance is proportional to the trans-conductance of the transistor, Rch= αch/gm, withαch as a constant (αch= 1 for biasing below pinch-off and αch= 3/2 for biasingbeyond pinch-off). Usually an amplifier based on the JFET is biased abovepinch-off and the noise spectral power is equal to: sn,id= 4kBT×[2/(3gm)].

In addition the gate leakage current has to be taken into consideration, which is asource of shot noise ing, with a noise spectral power of sn,ig= 2qIg. Due to thesimilar operating mechanism, this modelling also applies to the metal-oxidesemiconductor field effect transistor (MOSFET).

The actual values of the equivalent input noise sources usually result in lessfavourable numbers for bipolar transistors. However, it should be mentioned thatthis modelling applies to the white noise only. The frequency at which the 1/fnoise spectral power is equal to the white noise spectral power is typically muchlower for a bipolar transistor (100 Hz range for a bipolar transistor versus about100 kHz for a MOSFET), which favours a bipolar transistor-based amplifier in

Figure 5.12, Sources of noise in a JFET.

D

S

G

Cgd

gmugs

G

rgs

ug

us

rgd

Cgs

D

S

Ind

rdsIng


126

low-frequency applications. These are highly specialised cases, therefore, thediscussion in the next sections remains restricted to white noise.

5.4 Calculating the equivalent input noise sources

5.4.1 Noise in OPAMP-based amplifiersThe equivalent input noise sources in an OPAMP are specified as un when inseries with any of the inputs and in when in parallel to the inputs, as shown inFig. 5.13a. Note that the polarity of the noise source is not relevant, therefore, itis not important whether the equivalent input noise voltage is included in the dia-gram in series with the inverting or the non-inverting input.

The specification of the equivalent input noise current in parallel to the inputnodes results in a complication. In the case of a circuit with both a resistor inseries with the inverting input and a resistor in series with the non-invertinginput, as shown in Fig. 5.13b, the noise current results in noise voltages in serieswith each of these resistors. It is important to note that these are correlated (seenoise paradox) and that these noise voltages should be linearly subtracted. Thenoise voltage that would be due to the lowest resistor is often disregarded toavoid this problem.

The equivalent input noise sources of an OPAMP-based circuit need to be calcu-lated as a function of those of the OPAMP and the other components in the cir-cuit. Calculation of the input current noise source in a circuit for voltage read-outis only relevant when the impedance of the signal voltage source cannot be disre-garded. A source resistance Rg gives, in addition to un,eq

2 (and the thermal noiseof the signal source itself), an additional noise power in,eq

2×Rg2. Similarly, for

read-out of an ideal current source (Rg→ ∞) the equivalent input noise voltageneeds not be considered. Otherwise the noise power, un,eq

2/Rg2, should be added

to in,eq2.

As a first example the equivalent input noise current is calculated for the trans-impedance circuit shown in Fig. 5.14. The signal source is assumed to be ideal.The equivalent input noise sources of the opamp are specified as: un1 and in1.

Figure 5.13, Equivalent input noise sources in an OPAMP (a) generalrepresentation and (b) effect of in in the case of input series resistances.

R2un

in +

--

(a)un

in +

--R1

(b)

Section 5.4Calculating the equivalent input noise sources

127

Consequently, the distributed (uo1) and equivalent (uo2) systems are comparedwith open inputs only. The transfer function of uo/un1 is identical to that of thenon-inverting amplifier with gain Gv= 1 due to the ideal current source. Thenoise power at the output is expressed as:

The equivalent input current source follows as:

Assume the following noise specifications for the operational amplifier: theequivalent input noise voltage spectral power (sn,u1)1/2= 4 nV/√Hz and theequivalent noise current spectral power (sn,i1)1/2= 1 pA/√Hz. For a trans-imped-ance Rm= Uo/Ii= 104 V/A (Rf= 10 kΩ), the results is an equivalent input spectralnoise power (4kBT= 1.65×10-20 J) equal to:

The second example uses the same technique to calculate of the equivalent inputnoise sources for the non-inverting voltage amplifier shown in Fig. 5.15 as afunction of OPAMP specifications and the noise of the resistors in the circuit.

Figure 5.14, Trans-impedance amplifier with distributed noise sources.

(5-15)

(5-16)

(5-17)

Figure 5.15, Non-inverting voltage amplifier with distributed noise sources.

Rf

un1in1

Ii+

--

Uo1

unR

2 2 2 2 2 2 2 21 1 1 2 ,o n n f nR o n eq fu = u + i R + u = u = i R

2 22 21, 12 2

n nRn eq n

f f

u ui = + i +R R

9 20 4-26 2

, 8

,

4 10 1.65 10 10 281×10 [A /Hz]10

1.68 pA Hz

2

-24ni eq 4

ni eq

s = + + =1010

s = /

− − × × ×→

ui

Rs

Rf

+

--

uo1

un1

un,Rs

in1

un,Rf


128

Short-circuiting of the input results in an output noise voltage, uo1, for the repre-sentation with distributed noise sources in the circuit, while the equivalent inputnoise sources result in uo2:

Equating uo1 and uo2 yields:

Dimensioning the voltage amplifier for a gain Gv= 100 using Rs= 1 kΩ and Rf=99 kΩ and again applying an OPAMP with the following noise specifications:(sn,u1)1/2= 4 nV/√Hz and (sn,i1)1/2= 1 pA/√Hz, yields:

5.4.2 Noise in transistor circuitsObviously, the generic approach for calculating of equivalent input noise sourcesas a function of distributed noise sources is also applicable at the componentlevel. The equivalent input noise sources of the common-source (CS) gain stageshown in Fig. 5.16 can be found using the FET noise sources presented in Fig.5.12 in Section 5.3.

Short-circuiting the input yields for R«rds:

Hence for uo1= uo2 results:

(5-18)

(5-19)

(5-20)

(5-21)

(5-22)

2 2 2 2 21 1 1 , ,

22 22 ,

22 2s f s f s f f

o n n n Rs n Rfs s f s s

s fo n eq

s

R + R R .R R + R Ru = u +i +u +u

R R + R R R

R + Ru = u

R

×

2 2 2 2 2, 1 1 , ,

2 2 2

s f f sn eq n n n Rs n Rf

s f s f s f

R R R Ru = u + i + u + uR + R R + R R + R

,

,

.

. 5.77 nV Hz

2 218 24 20

nu eq

220 20

nu eq

1k 99k 99s =16 10 +10 +1 65 10 1k +1k +99k 100

11 65 10 99k = 3331 10 s = /100

− − −

− −

Ω× Ω × × × Ω× Ω Ω

× × Ω× × →

( )2 2 21 2 2 2

2 2 2 2 22 2

2 ,2 2 2

2

o nd nRgd

m gdo n eq

gd

Ru = i + i1 + R C

g R + R Cu = u

1 + R C

ω

ωω

,

2 2 2 22 nd nR nd nRn eq 2 2 2 2

m gd m

i + i i + iu =g + C gω


129

Open input leads to:

Hence, it follows that:

This is basically the circuit used as the building block for the input stage ofOPAMPs which thus determines the OPAMP noise specifications in terms ofequivalent OPAMP input noise sources.

Figure 5.16, Calculation of equivalent input noise sources in a CS gain stage.

(5-23)

(5-24)

D

S

G

R

VDDCgd

gmugs

G

rgs

ug

us

rgd

Cgs

D

S

ind inR

R

rdsing

=Cgd

gmugs

G

rgs

ug

us

rgd

Cgs

D

S R

rds

in,eq

un,equo2

uo1

Rg

VSS

Ui

( )

,

22 2gd2m

2 2 2 2 2gs gd gs2 2 2 2

o1 ng nd nR2 2 2 2gd gs gd gs2 2 2 2

2 2 2 2gd gs gd gs

22 2gd2m

2 2 2 2gs gd gs2 2

o2 n eq2 2gd gs2 2

2 2gd gs

Cg R + RC C C Ru = i i +i

C C C C1+ R 1+ R

C C C C

Cg R + RC C C

u iC C

1+ RC C

ω

ω ω

ω

ω

++

+ +

+=

+

( ) ( )( ) ( ),

22 2 22 2 2gs gdgs nd nR gs2 2 2 2 2n eq ng ng nd nR2 22 2 2 2

mgd gsm gs gd

C + i +iC C Ci = i + i + i +i

gg + + C CC C

ω ω

ω


130

5.4.3 Noise in the instrumentation amplifierThe Wheatstone bridge has evolved over the previous chapters from a DC-driven circuit into an AC-driven differential circuit. One conclusion drawn inchapter 3 is that the Wheatstone bridge should be AC-driven, to avoid anunfavourable detection limit due to the equivalent input offset sources of theread-out circuits. Moreover, in Chapter 4 the benefits of a differential excitationare outlined to avoid excessive CMRR requirements of the read-out. As a conse-quence, the detectivity in the differentially-driven, AC-operated Wheatstonebridge is noise limited. The noise performance of the instrumentation amplifiercould be analysed using Fig. 5.17.

The OPAMP equivalent input noise sources are indicated as unx and inx where xrefers to the number in OAx. The aim is to find an expression for the equivalentinput noise sources of the entire instrumentation amplifier: un,eq and in,eq. TheWheatstone bridge is the signal source of the instrumentation amplifier. The sig-nals and components should be expressed in terms of the Thevenin equivalent:ug and Rg, as shown in Fig. 5.18a. Since the instrumentation amplifier is usuallydesigned to provide a high CMRR, it is reasonable to disregard the effect of OA3and only consider the distributed noise sources in the differential-to-differentialinput stage (OA1-OA2).

The instrumentation amplifier is assumed to be a voltage amplifier only. Conse-quently the trans-impedance, Zm= uo1/iid→ ∞. The output noise voltage at open-input can, therefore, not be used to calculate in,eq. From Figs. 5.17 and 5.18b theequivalent input noise current results in:

Figure 5.17, Noise in an instrumentation amplifier for read-out of an AC-operated Wheatstone bridge with differential excitation.

+

--

uo1

R2

+

--

R7R6

+

--

R5

R4

R1

R3+

--uexc

2

OA1

OA2

uexc2

+

--

Ro-∆R

Ro+∆R

Ro+∆R

Ro-∆R

uid

un1

un2

un3

in1

in2

in3

OA3


131

Short-circuiting of the differential input of the instrumentation amplifier circuitwith distributed noise sources results in:

Figure 5.18, (a) Wheatstone bridge to Thevenin equivalent and (b) overallequivalent input noise sources of the instrumentation amplifier.

(5-25)

+

--uexc

2

uexc2

+

--

Ro-∆R

Ro+∆R

Ro+∆R

Ro-∆R

uid

+

--

uo2

R2

+

--

R7R6

+

--

R5

R4

R1OA1

OA2

un,eq

(a)

in,eq OA3

+

--uexc

2

uexc2

+

--

Ro-∆R

Ro+∆R

Ro+∆R

Ro-∆R

uidinR

inR

inR

inR

∆RRo

Ro/2

uid

(2inR2)1/2

uexc (2inR2)1/2

Ro/2

∆RRo

Ro/2

uiduexc

Ro/2(2unR

2)1/21

(2unR

2)1/21

∆RRo

Rg=Ro uiduexc

unR

=

=

=

=

uid

(b)

R3

ug=

,2 2 2n eq n 1 n 2i = i + i


132

The equivalent input noise voltage is:

Equation 5-27 assumes that the noise in R5 and R7 can be disregarded in the caseof a sufficiently high differential voltage gain, Gdd= (R5+R6+R7)/R6, in the dif-ferential-to-differential pre-amplifier. This is also a requirement for a highCMRR.

Equations 5-25 and 5-27 indicate that the noise analysis in an instrumentationamplifier is rather straightforward, since only the components in the differentialinput circuit need to be considered.

5.4.4 Noise in the charge amplifierThe charge amplifier has already been introduced in Chapter 2 as a very suitablecircuit for the read-out of capacitive sensors. The charge amplifier is not used tomeasure the amplitude of an unknown small signal, but rather to measure thevalue of a (sensor) capacitance, Cs, using a well-defined excitation signal, ui.Conventionally the charge amplifier uses the ratio Cs/Cf 1, with Cf as the feed-back capacitance, as shown in Fig. 5.19.

A large resistor Rf is connected in parallel to Cf to limit the output DC level thatwould otherwise result from offset (see also section 3.3.3). The charge amplifieris based on an OPAMP with an open-loop gain of A(ω)= Ao/(1+jωτv). For opera-

(5-26)

(5-27)

Figure 5.19, Charge amplifier with Rf and A(ω).

( ) ( )

,

2 22 2 2 2 2 22 5 6 7 2o1 n1 n2 nR6 nR5 nR7

1 6 1

22 22 5 6 7o2 n eq

1 6

R R R R Ru = u +u u u uR R R

R R R Ru = uR R

+ +× + + +

+ +×

( ),

22 2 2 2 2 2 2 2 26n eq n1 n2 nR6 nR5 nR7 n1 n2 nR6

5 6 7

Ru = u +u u u u u +u uR R R

+ + + + + +

Is+

--

uo

Cs

Rf

Cf

ui

+

--

If

A(ω)


133

tion at high frequencies (ω> τv), the open-loop transfer function simplifies toA(ω)≈ Ao/(jωτv). The feedback transfer function can be derived using Fig. 5.19:

The frequency range for proper operation of the charge amplifier is 1/(RfCs)< ω<ωT, with ωT= Ao/τv, as shown in Fig. 5.20. Since Cs/Cf 1, the charge amplifieroperates as an attenuator with the output signal uo= -Cs×ui/Cf as a measure of asensor capacitance, Cs. The intended feedback function, -Cs/Cf, is within theOPAMP open-loop modulus plot up to the unity-gain frequency, ωT (i.e. the loopgain remains much smaller than 1 up to the unity-gain frequency). Consequentlythe OPAMP is capable to provide this function almost up to ωT.

The essential advantages of operating the charge amplifier at a frequency closeto the unity-gain frequency are:

(5-28)

Figure 5.20, Modulus plot of the charge amplifier.

( )( ) ( )

( )

( )( )

( )

s

f f fs f

f f

f s

2 f s ff f

f s f s

2 f ff f f f

11

1

1 (1 ) 1

s i

i so

oo

v

o

vi v

o o

v v v

o o o

i u u j C

u j R C u u j R Cu u i Z

j R CAu A u u u

ju j R C

R C Cuj R C

A Aj R C j R C

R Cj R C j R C jA A A

ω

ω ωω

ωωτ

ωττω ω

ω ωτ τ τω ω ω ω

−

− −−

+ − −

= −

+ − −− = +

= − −

−=

+ + + −

− −

≈ =

+ + − + +

1/τvA0

log ω

ωT = Ao/τv

uoui

[dB]

CsCf

1RfCf

1RfCs

0


134

1. Low transducer impedance and 2. Maximum spectral distance to the main source of interference: capacitive

coupling of the mains voltage.

As is indicated in section 2.6.2, a user is primarily interested in the change incapacitance, ∆C, and not in its nominal value, Cso. A differential capacitive sen-sor can be read-out using a charge amplifier with a second sinewave of inversepolarity (180o out-of-phase), as shown in Fig. 2.20. The transfer function uo/ui=2(∆C/Cf).

The detection limit of the AC-operated charge amplifier is determined by noise.Figure 5.21 shows the charge amplifier with the distributed noise sourcesincluded. The calculation of the equivalent input noise sources is complicated bythe capacitive impedances. The charge amplifier is basically operated at the fre-quency of the excitation voltage. However, this signal is usually modulated bythe dynamics of the non-electrical signal as measured by the sensor. Therefore,the charge amplifier is assumed to operate in a specified frequency rangebetween fmin and fmax.

The TOTAL equivalent noise voltage AT THE OUTPUT is derived using Fig.5.21 and results in:

Note that the total capacitance between excitation sources and inverting input isequal to 2Cso. This equation contains a frequency-independent part and a fre-quency-dependent part. The noise-equivalent change in the capacitance, ∆Cdet, iscalculated by referring the output noise back to the input:

Figure 5.21, Noise in the differential charge amplifier.

(5-29)

+

--

Uo1Cso-∆C

Rf

Cf

Ui

+

--A(ω)

un1

in1

unR

Ui

+

--

Cso+∆C

( ) ( )( ) ( ) ( )( )

( ) ( )

22s f

n,uo n,ui n,ii n,iR fs

2 2f s f f

n,ui n,ii n,iRf f f f

/ 2/ 2

1 21 1

Z Zs s s s Z

Z

j R C C Rs s sj R C j R C

ω ωω

ω

ωω ω

+= + + =

+ + + + + +


135

Example 5.2The charge amplifier shown in Fig. 5.21 is used to read-out a differentialcapacitive sensor. For this amplifier, which includes distributed noisesources, the offset is disregarded. With respect to the other specifications,the OPAMP is assumed to be ideal (no equivalent input offset or biassources, an infinite CMRR and an infinite open-loop gain). Determine thedetection limit when considering the following data:•Cs= 10 pF•Cf= 100 pF•Rf= 500 kΩ,•fmin= 99 kHz and•fmax= 101 kHz.Equation 5-29 applies, although a simplification is possible:

The spectral noise components are specified as:•4kBT= 1.65×10-20 J•ini= 0.1 pA/√Hz•uni= 2 nV/√Hz.

Equation 5-31 indicates that the noise spectrum is composed of a fre-quency-independent part (the first term) plus a frequency-dependent part(the second term). The total noise power due to the frequency-independent(white noise) part amounts to:

(5-30)

(5-31)

(5-32)

( )

o, min

deto,min det

/

22

no

noi f

f i

u uC uu u C CC u

ε

ε

=

∆= − → ∆ =

( )2 2

, , , ,

10 1

2 12

f f

s fn uo n ui n ii n iR

f f

R C

C Cs s s s

C f C

ω π

π

= →

+= + +

( )2

218 3 14 2,

24 10 1.2 2 10 1.15 10 Vs f

n uif

C Cs f

C− − +

× ∆ = × × × × = ×


136

The frequency-dependent part results in:

The frequency-dependent noise is dominant. The total noise amounts to:uno= (3.33×10-14)1/2= 182 nV. The detection limit for the measurement of∆Cs due to noise can be calculated in the case of excitation with ui= 5 Vrmsand an inaccuracy specification ε= 5%, as:

The detection limit reduces with (noise) bandwidth.

5.5 Noise matchingSo far the emphasis has been on the noise added to the input signal by the read-out circuits. The assumption has been that the signal source is free of noise.Moreover, the equivalent input noise sources have been derived without consid-ering the relation between the calculated equivalent input voltage source, un,eq,and the equivalent input current source, in,eq. A closer inspection reveals thatthere is a ratio between un,eq

2 and in,eq2 that would give an optimum noise per-

formance at a certain value of the source resistance, Rg.

The starting point for any design of read-out circuits should be to add as little aspossible to the measurement uncertainty. Hence, the added noise power shouldbe as small as possible. The equivalent requirement is that the Signal-to-NoiseRatio (SNR) should be reduced as little as possible. Hence, any approach tonoise reduction should be without input signal attenuation. The extent to which aread-out circuits satisfies this requirement is specified in terms of the Noise Fac-tor, NF.

The noise factor, NF, is defined as the ratio between the SNR at thesource without the circuit connected, and the SNR at the output withsource connected.

(5-33)

(5-34)

max

min

, ,22 2 2 2

min max

26 20 5 3 314 2

2 20

4 / 4 / 1 1/4 4

10 1.65 10 / 5 10 10 10 2.18 10 V4 10 99 101

fn ii B f n ii B f

f ff

s k T R s k T Rdf f

C C f fπ π

π

− − − −−

−

+ + = − =

+ × ×− = × ×

∫

( )

9o,min no

f ,min 10 8so i s,min

f i

/ 182.5 10 /0.05 3.65 V

2 10 365 10 /10 36.5aF2

o

u uC uCu u C

C u

ε µ−

− −

= = × =

∆= − → ∆ = = × × =

Section 5.5Noise matching

137

In equation:

The noise factor NF= 1 when no noise is added. Note that H(ω) is the transferfunction of the read-out; H(ω)2= Ps(output)/Ps(source). Since Pn(source) and theequivalent input noise of the read-out circuit, Pn(read-out), are assumed to benon-correlated, Pn(source+read-out)= Pn(source)+ Pn(read-out). The noise fac-tor is generally specified in [dB]. Noise matching is achieved at a minimumnoise factor.

For calculation of the minimum noise factor, consider the read-out of a signalsource with a noisy source resistance Rg, as shown in Fig. 5.22. The noise perfor-mance of the read-out is specified using un,eq and in,eq, whereas the spectralnoise distribution of the source is defined by the resistance and is equal to: sn,Rg=4kBTRg. The additional noise spectral power due to read-out is described by:sn(read-out)= sn,ueq+ sn,ieqRg

2. Hence:

The challenge is to optimise un,eq and in,eq for optimum noise performance, sincethe value of Rg can in principle not be modified by the designer. The conditionfor minimum noise factor follows from:

(5-35)

Figure 5.22, System with (a) distributed noise sources and (b) equivalent inputnoise sources.

(5-36)

( )( )( )

( ) ( )

( ) ( )( )

( )( )

) (

s

nsource

soutput2

n n

n n n

n n

P sourceP sourceSNRNF = = = P outputSNR

H P source P readout

P source + P readout P readout

= 1+P source P source

ω +

(b)

Rg

RiUg

(a)

Rgun,eq

Riin,eq

un,Rg

Ug

2 2 2, ,n eq n eq g

g

u +i RNF = 1+

4kTR


138

Noise matching is achieved when this condition has been met. The associatedvalue of the noise factor is:

Note that the noise performance remains primarily determined by the noisepower, as defined by equivalent input sources. Only after the total noise powerhas been reduced to a minimum through design should noise matching be pur-sued.

Noise matching involves a redistribution of noise over un,eq and in,eqthrough design. This step should be preceded by a noise analysis aim-ing for minimum noise generated within, or injected into the read-outcircuit.

An incorrect approach for noise performance optimisation would be the insertionof a series resistance Rs, as shown in Fig. 5.23.

The result is an additional noise voltage, un,Rs, with noise spectral power sn,Rs=4kBTRs in series with un,eq and, hence, noise matching at a higher noise level:

(5-37)

(5-38)

Figure 5.23, Counter-productive attempt at noise matching using a seriesresistor.

(5-39)

( )( )

2 2 2 2 2

2

×∂→

∂B g n g B n n g 2 n

g,opt2g nB g

4k TR 2i R - 4k T u +i RNF u= = 0 =RR i4k TR

2

minn

B g

2uNF =1+4k TR

Rgun,eq

Riug

in,eq

un,Rg un,RsRs

( ) ( )( ) ( ) ( )

2 2

2 2 2 2

→

++

2n B g B s n n g s

2 2

B g s n n n ng s g ss

B g g B g

source+readout = 4k TR +4k TR +u +i R + RP

4k T R + R +u +i u iR + R R + RRNF = = 1+4k TR R 4k TR

Section 5.6External sources of error

139

Adding Rs results in a higher noise level. Moreover, the attenuation of the sourcesignal, ug, to the input, ui, is increased, which gives rise to a further reduction ofthe SNR.

Noise matching is possible without increasing the noise level by using an idealtransformer, as shown in Fig. 5.24. Basically the transformer is used for imped-ance conversion. Using a winding ratio 1:m causes the equivalent input noisevoltage to be transformed into the input divided by a factor m: u’n

2= un,eq2/m2.

Similarly, the equivalent input noise current is transformed into the input by mul-tiplication by a factor m: i’n,eq

2= m2×in,eq2. Using Eqn 5-37 yields noise match-

ing at:

Unfortunately, transformers are large, bulky and perform poorly in terms oflosses due to core and resistive part of the coils. Although transformers based onstripline inductors are used in RF analog circuit design, these are, as a generalrule, poorly compatible with CMOS circuit fabrication.

5.6 External sources of errorThis section describes measurement errors that are due to undesirable interactionbetween the system and its operating environment, which is referred to as Elec-tro-Magnetic Interference (EMI). These sources of error can be deterministic(e.g. air temperature) and/or random (e.g. capacitive coupling of EM fields). Theadvantage of random errors is the relatively simple identification of signal fluc-tuations in the system. Usually, it more difficult to recognise a deterministic errorfrom the system output signal, such as thermal drift (although unexpected sensi-tivity of the system to temperature could be detected from the effect of daily tem-perature variations).

There are a multitude of causes for an external source of error. Some of the mostfrequently experienced sources of external error are listed here:• The change in air temperature affects the measurement system via the tem-

perature coefficient of the various components and sub-systems. Moreover,

Figure 5.24, Noise matching using a transformer.

(5-40)

Rgun,eq

RiUgin,eq

+

_1:m

Rgu'n,eq

Rii'n,eq

+

_1:m

Ug

2 2 2n n42

g2 2 22gnn

/u m u= =mRi i Rm

→


140

the Seebeck effect causes an additional error in the case of temperature gradi-ents with electrical interconnections composed of different metals and solderbonds at different temperatures.

• As mentioned already in Section 4.2.3, capacitive coupling between themeasurement system and other electrical systems could give rise to an injec-tion of noise. Power supply cables, igniters in combustion engines and thyris-tor control systems for electrical engines are notorious sources of EMI.

• Magnetic induction can also be a source of external error, as shown in Fig.5.25. A magnetic field is generated by an electric conductor that supplies acurrent. An error voltage is induced in when the magnetic field passes a wind-ing formed by another conductor. The induced voltage increases with theenclosed area, S, of this winding and with the rate of change in magneticinductance: un= S(∂B/∂t).

Figure 5.25, Magnetic injection of noise.

Figure 5.26, (a) Additive error due to ground current and (b) star connection.

Zg

B ZiUiUg

+

_

+_

un

RiUg

+

_

RcIgnd

RiUg

+

_

Rc2Rc1

RcRc

Ignd

(a)

(b)

Rg

Rg


141

• Safety regulations usually require each instrument in a measurement and con-trol setup to be connected to ground level. The problem arises when the zeroinput terminal (also referred to as the ‘low terminal’, ‘signal ground’ or ‘sig-nal return’) of each signal source or instrument in the system is also locallyconnected to ground level. Each connection to ground has a finite cableresistance and many units are locally connected. In a worst-case scenario, themost remote unit supplies a large ground current, resulting in a ground loopvoltage, as shown in Fig. 5.26a. Basically, an offset error equal to Ignd×Rc isgenerated, where Ignd denotes the ground current and Rc is the part of thecable resistance between two units. The most effective technique to avoidsuch an error is to use only one common node as the safety ground and toconnect each instrument to this node with a separate cable. This star connec-tion, shown in Fig. 5.26b, is without additive error, but does introduce a scaleerror equal to: 2Rc/(Ri+2Rc), which can be disregarded in the case of a propervoltage read-out: Ri» Rc. Another approach to avoid errors due to ground cur-rent is to use a read-out with a differential input. Any ground current resultsin a common-mode voltage that is subsequently suppressed using a highCMRR.

Instrumentation techniques are available to minimise the effect of EMI. It shouldbe emphasised that there are fundamentally two issues, which combined deter-mine the noise coupled into the system:• The first is the amount of external noise present that should be minimised.• The second is the susceptibility of the system to a particular external source

of noise that should be minimised.

Consequently, a well-designed electronic system should emit as little noise aspossible over the entire spectral range and should simultaneously be robustenough to withstand exposure to a certain level of external noise power. Thesetwo aspects are included in the EMI specifications, which are subject to nationaland international regulations. As complex electronic systems operate in environ-ments with many sub-systems, complying with EMI requirements by meetingthe low emissions and low-susceptibility design criteria is paramount for ensur-ing overall system operation and thus safety.

Since the EMI environment of such a complex system largely depends on thenoise generated within the system itself, the emissions and susceptibility of eachsub-system is reasonably predictable and can be optimised through design.Therefore, EMI issues should be included at an early stage in the design.


142

The weakest link in a measurement setup is the electrical interconnectionbetween the signal source and instrument, as these are outside the respective cas-ings. The problems associated with capacitive coupling have already been dis-cussed in Example 4.2 (Section 4.2.3). Shielding is used to isolate the cablesfrom this coupling. Applying a conductive shield, as shown in Fig. 5.27, isolatesthe measurement system from the capacitive coupling of external noise andinterference. To effectively drain the injected charge, the shield should be suffi-ciently electrically conductive and should be connected either to ground poten-tial or to a fixed-value voltage source with a very low source impedance.

A similar isolation technique can be applied to reduce the magnetic couplingwhich uses a ferro-magnetic shield. Magnetic shields are usually more difficultto use, since a relatively thick layer of a material of high magnetic permeabilityis required for effective shielding, which complicates handling.

A major problem in conductive shielding is the large capacitance between theinner signal cable and the grounded outer shield. This cable capacitance affectsthe signal transfer ui,g/ug:

A conductive shield can be used to reduce the capacitive coupling to less than5% (Cc’= 0.05×Cc). The noise level at ui due to un (denoted as ui,n) is reduced bya factor 20. However, the available –3dB bandwidth of ui for the read-out of ug(denoted as ui,g) is also reduced. Assuming a resistive input impedance of theread-out, Zi= Ri, and source, Zg= Rg yields:

Figure 5.27, Shielding as a technique for reducing capacitive coupling.

(5-41)

Zg

Ziuiug

+

_ unIn

Cc

+

_

+

_

Za

, ,

// //// ,// // //

g i ai ai g g i n n

i a g g i a c

Z Z ZZ Zu u u uZ Z Z Z Z Z Z

= =+ +


143

Basically for Ri Rg, the pole in ui,g/ug shifts from ω= (RgCi)-1 without shield-ing, to ω= [Rg(Ci+Ca)]-1 with shielding. EMI indeed reduces proportionally toCc‘ for ω< (RgCa)-1.

Example 5.3A voltage-to-current converter with trans-conductance Gm= 1 A/V is usedfor the read-out of a sensor signal ug with source resistance Zg= Rg= 2 kΩ.This voltage-to-current converter is composed of a trans-conductance cir-cuit, with gm= 10 mA/V, and is followed by a current amplifier, with Gi=100, as shown in Fig. 5.28. The noise performance of both the trans-con-ductance circuit and the current amplifier is specified in terms of equivalentinput noise sources. Moreover, both are prone to capacitive coupling fromthe 230 Vrms/50 Hz power supply voltage.

1. Calculate the equivalent spectral noise sources at the input of the overallvoltage amplifier. The following specifications are given:•un1= 1 nV/√Hz, •un2= 100 nV/√Hz, •in1= 0.5 pA/√Hz, •in2= 10 pA/√Hz, •Ri1= Ro1= Ro2= 100 kΩ and •Ri2= 10 Ω (4kBT= 1.65×10-20 J).

(5-42)

Figure 5.28, Noise in a two-stage trans-impedance amplifier.

( )

'

, , '

'

1 ,1 1

i gc

i g i gi nig c

i g i gg i g na c a

i g i g

R Rj C

u R RuR j R CR R R Ru R R uj C j C CR R R R

ωω

ω ω

+= × =

+ + + ++ +

un1

Ri1

gmui1

ui1 ii2

ug in1

io

Ro1

un2

Ri2

Giii2

in2

230 V / 50 Hz

Ro2

Cc1 Cc2Rg

gm Gi


144

At short-circuited input:

At open input:

Hence:

2. Calculate the detection limit due to noise of the read-out, if the noisebandwidth is between 1 Hz and 1 kHz and the inaccuracy is: ε= 1%.

The source noise is composed of 1/f noise plus white noise. The latter is dueto only the source resistance. The 1/f noise spectral power reduces by 10dB/decade at an increasing frequency and is equal to that of the white noisecomponent at f= 100 Hz, as is shown in Fig. 5.29.

(5-43)

(5-44)

(5-45)

(5-46)

2 2 2 21 11 1 2 2

1 2 1 2 1 2

2 2 12 ,

1 2

2 21 2

12 2 2

o oo n m i n i n i

o i o i o i

2

oo n eq m i

o i

o o

R Ri = u g G +u G +i GR R R R R R

Ri = u g GR R

i = i

× × × × + + +

× × +

2 2 2 21 11 1 1 2 2

1 2 1 2 1 2

2 2 12 , 1

1 2

2 21 2

12 2 2

o oo n i m i n i n i

o i o i o i

2

oo n eq i m i

o i

o o

R Ri = i R g G +u G +i GR R R R R R

Ri =i R g GR R

i =i

× × × × × + + +

× × × +

14 222 2 2 2 18 18, 1 2 2 6 4

1

,

14 222 2 2 2 26 26, 1 2 2 16 6

1 1 1

,

1 1 10 1010 2 1010 10

1.4nV/ Hz

1 1 10 1025 10 25 1010 10

0.5pA/ Hz

2 2

n eq n n nm o m

n eq

2 2

n eq n n nm o i m i

n eq

u =u +u +i +g R g

u

i =i +u +i +g R R g R

i

− −− −

−

− −− −

= + = × →

=

= × + × →

=

( ) ( )2 2 2 18 26 6, , ,

15 2

2 15 4 2,min ,min

2 10 25 10 4 10 999

3 10 V/ 3 10 /10 5.5µV

n readout n eq n eq g

g n g

P u i R B +

P P V uε

− −

−

− −

= + = × × × × =

× →

= = × → =


145

3. Calculate the total noise spectral power of the source in the frequencyband in between 1 Hz and 1 kHz.

4. Calculate the noise factor of the read-out:NF=1+Pn,read-out/Pn,source= 1010log(1+ 30/482)= 1010log(1.06)= 0.26 dB.

The 230 Vrms/50 Hz power supply voltage is capacitively coupled to theinput of both the trans-conductance and the current amplifier via Cc1 andCc2, respectively.

5. Calculate the equivalent input noise voltage, un,eq, in series with ug dueto capacitive coupling for Cc1= Cc2= 1 pF and the resulting detection limit,Ui,det, for an inaccuracy specification ε= 1%.

Figure 5.29, Noise spectrum of the source impedance Zg.

(5-47)

(5-48)

100

4kBTRg=33x10-18 V2/Hz

0

log f [Hz]

Pn(f)[dB] 1/f noise

20 dB

1 10 1k 10k

white noise

( ) ( ) ( )( )( )( )

1kHz 100Hz

,1Hz 1Hz

18 16 2

4k 100 /

4k 1kHz 1Hz 100 ln 100Hz ln 1Hz

33 10 999 100ln 100 482 10 V

n source B g

B g

P TR df df f

TR

+− −

= + =

− + − =

× = ×

∫ ∫

( )

( )( )

1 1 711 1

1 1 1

1 1 2 22

1 2 2 1 2 2 2

8 7 8n,eq

i,det n,eq

: 2 10 11

1 1 1:

10 2 10 10 230 152µV

/ 15.2mV

g i g cic g c

n g i c g c

i o i cc

n o i c o i m m i c m

R R j R CuC = R Cu R R Z j R C

u R R CCu R R Z R R g g R Z g

u

u u

ωπ ω

ω

ω

π π π

ε

−

− − −

= ×+ +

= × × =+ +

× → = × + × × =

= =


146

Note that the injected EMI voltages are correlated and, consequently,should be linearly added. Detectivity is limited by EMI at the input. There-fore, shielding techniques should be applied to ensure that the detectionlimit is determined by internal noise.

5.7 Exercises5.1 A signal source generates, apart from the signal of interest, white noise witha spectral density of (sn,ug)1/2= 300 nV/√Hz. A Volt meter is used for read-outwith a specified equivalent input noise voltage unui with (sn,ui)1/2= 100 nV/√Hz.Calculate the total noise un,tot in the case of a 10 MHz noise bandwidth.

Solution:Addition of noise spectral power densities: (3×10-7)2 + (10-7)2 = 10-13 V2/Hz. Inthe case of a 10 MHz bandwidth: un,tot= √(10-13×107) = 1 mV.

5.2 An AC Volt meter with rms reading is specified for a total equivalent inputnoise voltage unm= 1 mV. This instrument is used for noise measurement andgives a unt= 4 mV reading. What is the noise voltage of the source, uns, assumingmatched noise bandwidths?

Solution:unt

2 = unm2+ uns

2, Hence, uns2 = unt

2 - unm2= (4×10-3)2- (1×10-3)2=

15×10-6 V2→ unt= √15 mV.

A voltage amplifier is constructed using a voltage-to-current converter (trans-conductance), gm1, and is put in cascade with a current-to-voltage converter(trans-impedance amplifier), Ri2, as shown in Fig. 5.30, to give an overall trans-conductance equal to: Gv= gm1×Ri2.

Figure 5.30, Voltage amplifier using a trans-conductance circuit in series with atrans-impedance circuit, both with equivalent input noise sources included.

un1

Ri1

gmui1

ui1 ii2

uiin1

uo1

Ro1

un2

Ri2

Rmii2

in2

Ro2

Section 5.7Exercises

147

The specifications of the equivalent input noise sources of the sub-systems inFig. 5.30 are: • (sn,u1)1/2= 0.5 nV/√Hz, (sn,u2)1/2= 10 nV/√Hz, • (sn,i1)1/2= 2 pA/√Hz, (sn,i2)1/2= 10 pA/√Hz. • Ri1= Ro1= 1 ΜΩ, Ri2= Ro2= 100 Ω, gm1= 100 mA/V and Zm= Rm= 10 Ω.5.3 Calculate the equivalent input noise sources, un,eq and in,eq for a noise band-width of B= 10 kHz.

Solution:A short-circuited input results in:

Similarly for the open input:

Hence:

The equivalent input noise voltage of the second stage, un2, is not significant inthis setup. A noise bandwidth of B= 10 kHz yields: un,eq= 51 nV and in,eq= 224pA.

(5-49)

(5-50)

(5-51)

2 2 2 21 11 1 2 2

1 2 1 2 1 2

2 2 12 ,

1 2

2 21 2

12 2 2

o oo n m m n m n m

o i o i o i

2

oo n eq m m

o i

o o

R Ru = u g R +u R +i RR R R R R R

Ru = u g RR R

u = u

× × × × + + +

× × +

2 2 2 21 11 1 1 2 2

1 2 1 2 1 2

2 2 12 , 1

1 2

2 21 2

12 2 2

o oo n i m m n m n m

o i o i o i

2

oo n eq i m m

o i

o o

R Ru =i R g R +u R +i RR R R R R R

Ru =i R g RR R

u =u

× × × × × + + +

× × × +

2 22 2 2 2 20 16 22, 1 2 2 5

1

20 2

2 22 2 2 2 24 17 22, 1 2 2 11

1 1 1

2

1 1 1 125 10 10 1010 0.1

26 10 /

1 1 1 14 10 10 1010 10

5 10

2 2

n eq n n nm o m

2 2

n eq n n nm i o m i

u = u +u +ig R g

V Hz

i = i +u +ig R R g R

− − −

−

− − −

−

= × + + =

×

= × + + =

× 4 2 /A Hz


148

reader et8017 electronic instrumentation, chapter5

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