113Chapter 5DETECTION LIMITDUE TO NOISE ANDELECTRO-MAGNETICINTERFERENCE• 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.
114 Electronic Instrumentation R.F. Wolffenbuttel Chapter 5: DETECTION LIMIT DUE TO NOISE AND ELECTRO-MAGNETIC INTERFER- ENCE -200 -250 Magnitude [dB] -300 -350 -400 -450 -500 -5 -4 -3 -2 -1 0 1 10 10 10 10 10 10 10 Frequency [Hz] Figure 5.1, Typical noise spectrum of a critically damped mechanical structure. Noise generation is not restricted to the electrical domain. The noise spectrum in Fig. 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 to this particular source of noise (usually the input). Obviously, this susceptibility needs to be minimised. A notorious source of EMI is the mains voltage, but also the internal clock of the digital part of a system is a well-known source of EMI in the analog front-end of that system. When calculating the effect of noise on the detection limit, any source of noise should be referred to the input. 5.2 Equivalent input noise sources The noise voltage is the square root of the noise power and is defined as: T 1 u n ( t ) dt = T →∞ t ∫ un = lim 2 un = 2 Pn (5-1) −T This is basically the rms-value of the signal. A noise voltage is usually indicated as 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 is referred to as the noise power, Pn [V2] or [A2]. A specification in terms of ‘noise power’ is fundamentally incorrect, since neither [A2] nor [V2] has the unit of power [W]. Nevertheless, this approach for specifying noise is universally used.
Section 5.2 115 Equivalent input noise sourcesFigure 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. Rg Rg Un,eq + + Ug =0 Ri Ug =0 In,eq Ri _ _ (a) (b)Figure 5.2, System with (a) distributed noise sources and (b) equivalent inputnoise sources.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:
116 Electronic Instrumentation R.F. Wolffenbuttel Chapter 5: DETECTION LIMIT DUE TO NOISE AND ELECTRO-MAGNETIC INTERFER- ENCE The calculation of the noise powers of the input-referred sources of noise of a system is based on the linear addition of the noise powers (i.e. the squared amplitudes) of all the distributed noise sources, each referred to the input. The equivalent noise voltage of the statistically-independent noise voltage sources un1 and un2 in series is expressed as: un ,eq = un1 + un 2 2 2 (5-2) Similarly, the equivalent noise current of two statistically-independent noise cur- rent sources in parallel is calculated using: in,eq = in1 + in2 2 2 (5-3) The noise power of two voltage sources in series is equivalent to the noise power of one voltage source, with the total noise power being equal to the sum of the noise powers of the two sources. Similarly, two noise current sources in parallel are represented by one current source with a noise power equal to the sum of the noise powers of these two sources. Example 5.1 An AC Volt meter with an rms reading is specified for a total equivalent input 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 the source, uns, assuming matched noise bandwidths? Solution unt2 = unm2+ uns2. Hence: uns2 = unt2 - unm2= (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 noise sources at the input terminals as a function of statistically-independently distrib- uted noise sources in a measurement system.The statistic independence of the sources of noise is a fundamental requirement, as is demonstrated by the noise paradox shown in Fig. 5.3. Two resistors are connected in series with a noise current, in. Two approaches are available to calculate the equivalent noise volt-
Section 5.2 117 Equivalent input noise sources unR2 unR1 R1 R1 in un,eq R2 R2Figure 5.3, Noise paradox.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,R2.Applying the same approach to R2 and combining the results yields un1,eq2.The second approach combines R1 and R2 for the calculation of the total noisevoltage due to in and results in un2,eq: ( a ) un21,eq = un2,R1 +un2,R 2 +in2 R12 +in2 R22 = un ,R1 + un ,R 2 +in ( R12 + R2 ) 2 2 2 2 (5-4) ( b ) un22,eq = un2,R1 +un2,R 2 +in2 ( R1 + R2 ) 2These 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.
118 Electronic Instrumentation R.F. Wolffenbuttel Chapter 5: DETECTION LIMIT DUE TO NOISE AND ELECTRO-MAGNETIC INTERFER- ENCE un1 un2 + + + + Ro + in1 U1 Ri G1U1 in2 u2a (a) _ _ un,eq + + + Ro + in,eq U1 Ri G1U1 u2b (b) _ _ Figure 5.4, Model used to calculate the equivalent input noise sources for two gain sections in cascade (a) with distributed sources of noise and (b) with equivalent input sources of noise. An open input results in the equivalent input noise current: u2 a = (in1Ri G1 ) 2 + (in 2 Ro )2 + un 2 2 2 un 2 + in 2 Ro2 2 2 u2 b = (in ,eq Ri G1 ) 2 2 → in ,eq = in1 + 2 2 (5-5) G12 Ri2 u2 a = u2 b 2 2 A short-circuited input yields the equivalent input noise voltage: u2 a = (un1G1 )2 + (in 2 Ro ) 2 + un 2 2 2 un 2 + in 2 Ro2 2 2 u2 b = (un ,eqG1 )2 2 → un ,eq = un1 + 2 2 (5-6) G12 u2 a = u2 b 2 2 This result confirms the benefits of having a high-gain first stage. 5.3 Specifying noise in components 5.3.1 Noise in passive components A resistor, R, exhibits thermal noise. The thermal noise in a resistor, R, is described by a noise voltage source unR with noise spectral power sn,uR= 4kBTR [V2/Hz] in series with a noise-free, resistor, R. Alternatively, this noise is described by a noise current source inR with sn,iR= 4kBT/R [A2/Hz] in parallel to R.
Section 5.3 119 Specifying noise in componentsThe 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. Un R = OR R In RFigure 5.5, Thermal noise in resistor.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: s n ,uZ = u n ( f ) = 4k B T Z [ V 2 /Hz] 2 4k B T (5-7) s n ,iZ = in ( f ) = 2 [ A 2 /Hz] ZThe 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:
120 Electronic Instrumentation R.F. Wolffenbuttel Chapter 5: DETECTION LIMIT DUE TO NOISE AND ELECTRO-MAGNETIC INTERFER- ENCE H0 H(ω) 0 ωc B lin ω Figure 5.6, Calculation of noise bandwidth (solid curve) from a practical filter frequency response (dashed line). ∞ ∞ 1 Pn = H o B = ∫ H (ω ) dω → B = ∫ H (ω ) d ω 2 2 2 2 (5-8) 0 Ho 0 In a first-order low-pass filter with a -3 dB cut-off frequency at ωc=1/τ and a non-attenuated transfer at DC (|Ho|2=1), the noise bandwidth results in: ∞ ∞ dω 1 π π B= ∫ 2 = arctg (ωτ ) = = ω c [ rad / s ] , (5-9) 0 1+ (ωτ ) τ 0 2τ 2 which implies that a conversion factor equal to π/2 is required. Therefore, if the noise 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] and the output total noise power is equal to 2πkBTRfc. unC C = C Rp = C Rp in = C Rp Figure 5.7, Equivalent total noise source in a capacitor. A capacitor is not a dissipating element. Nevertheless, a capacitor, C, does affect the noise power. The dielectric loss needs to be taken into consideration to explain this effect. Dielectric loss is represented by a parallel resistor, Rp, with a resistance that increases with the quality factor of the capacitance. The noise spectral power is expressed as: sn,iR= 4kBTRp, which is represented by a noise current source in parallel to the terminals, as shown in Fig. 5.7. The resulting noise voltage in series with the capacitor follows as:
Section 5.3 121 Specifying noise in components (5-10)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. un Rs Rs Rs inL = = = L L L LFigure 5.8, Equivalent total noise source in an inductor.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: (5-11)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
122 Electronic Instrumentation R.F. Wolffenbuttel Chapter 5: DETECTION LIMIT DUE TO NOISE AND ELECTRO-MAGNETIC INTERFER- ENCE capacitor, the quality of the inductance is important for low-power circuit design in cases of a limited frequency band of operation. The next section shows that noise in active components is represented by several voltage noise sources and/or current noise sources. An electronic circuit is com- posed of several active and passive components. The overall noise performance of the circuit is specified in terms of an equivalent input voltage noise source in combination with an equivalent input current noise source. These are calculated as a function of the distributed sources of noise that are due to the components used. 5.3.2 Modeling noise in active components The noise characteristics of active devices are described by distributed noise sources within the device, which are closely related to the lumped components that describe the operating mechanism of the active component. Resistors in the equivalent 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 transport through 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 channel per unit of time. Statistical fluctuations in this charge transport result in a noise current 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 power expressed as: sn,i = 2qI [A2/Hz] (5-12) Charge transport through a semiconductor junction is modelled as a source of shot noise with a noise spectral power sn,i= 2qId. The equivalent circuit diagram of a forward-biased diode is shown in Fig. 5.9 and 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, and a 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: k BT I d ∂ ln ∂ k BT ( ln ( I d ) − ln ( I s ) ) ∂U d q Is = q k T (5-13) rd = = = B , ∂I d ∂I d ∂I d qI d
Section 5.3 123 Specifying noise in componentswhere 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). U+ U+ Id Id Ud Ud uns Rs (b) Rs Zp ind ZpFigure 5.9, Sources of noise in a forward-biased diode.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. _ _ Ud=U Ud=U Id Id uns Rs (b) Rs ind Zp ZpFigure 5.10, Sources of noise in a reverse-biased diode.
124 Electronic Instrumentation R.F. Wolffenbuttel Chapter 5: DETECTION LIMIT DUE TO NOISE AND ELECTRO-MAGNETIC INTERFER- ENCE In a reverse-biased diode -Ud (kBT/q). Therefore: qU − ∂ I s exp 1 ∂I k BT = ∂ I s × 0 → 0, (5-14) = d = rd ∂U d ∂U d ∂U d Hence, rd→ ∞. The charge storage in a reverse-biased junction is determined by the space charge layer boundaries of the depleted region and is specified by Cb and Zp= Rp/(1+jωRpCb). Since Rp Rs, uns is insignificant and the noise is set by the diode shot-noise, ind. Due to the very low leakage currents in state-of-the-art diodes 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 more detail 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 rbc → ∞ when disregarding the Early effect. Cbc unb1 Rbb C B C rbc B ub Cbe B inc gmube inb1 rbe E ue E Figure 5.11, Sources of noise in a bipolar transistor. Minority charge carriers are injected into the base region from the emitter. Some recombine in the base layer, while others diffuse towards the base-collected space charge region to be collected in the collector. Those charge carriers that actually reach the depleted area around the reverse-biased base-collector junc- tion are accelerated giving rise to a collector current. Those charge carriers that recombine 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 sources connected 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 series with the base, collector and emitter and are due to bulk resistance of the doped
Section 5.3 125 Specifying noise in componentslayers 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 toCbc 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 Cgd D G D rgd ug rds Ing G Ind Cgs rgs gmugs S us SFigure 5.12, Sources of noise in a JFET.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
126 Electronic Instrumentation R.F. Wolffenbuttel Chapter 5: DETECTION LIMIT DUE TO NOISE AND ELECTRO-MAGNETIC INTERFER- ENCE low-frequency applications. These are highly specialised cases, therefore, the discussion in the next sections remains restricted to white noise. 5.4 Calculating the equivalent input noise sources 5.4.1 Noise in OPAMP-based amplifiers The equivalent input noise sources in an OPAMP are specified as un when in series with any of the inputs and in when in parallel to the inputs, as shown in Fig. 5.13a. Note that the polarity of the noise source is not relevant, therefore, it is not important whether the equivalent input noise voltage is included in the dia- gram in series with the inverting or the non-inverting input. -- -- R1 in + in + un R2 un (a) (b) Figure 5.13, Equivalent input noise sources in an OPAMP (a) general representation and (b) effect of in in the case of input series resistances. The specification of the equivalent input noise current in parallel to the input nodes results in a complication. In the case of a circuit with both a resistor in series with the inverting input and a resistor in series with the non-inverting input, as shown in Fig. 5.13b, the noise current results in noise voltages in series with each of these resistors. It is important to note that these are correlated (see noise paradox) and that these noise voltages should be linearly subtracted. The noise voltage that would be due to the lowest resistor is often disregarded to avoid 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-out is only relevant when the impedance of the signal voltage source cannot be disre- garded. A source resistance Rg gives, in addition to un,eq2 (and the thermal noise of the signal source itself), an additional noise power in,eq2×Rg2. Similarly, for read-out of an ideal current source (Rg→ ∞) the equivalent input noise voltage needs not be considered. Otherwise the noise power, un,eq2/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.
Section 5.4 127 Calculating the equivalent input noise sources -- Rf unR Ii + Uo1 in1 un1Figure 5.14, Trans-impedance amplifier with distributed noise sources.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: u o 1 = u n 1 + i n21 R 2 + u nR = u o 2 = i n2, eq R 2 2 2 f 2 2 f (5-15)The equivalent input current source follows as: 2 u n1 u2 in2, eq = + in21 + nR (5-16) R2 f R2fAssume 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: 2 4 ×10−9 1.65 ×10−20 ×104 sni,eq = +10-24 + = 281×10-26 [A2/Hz]→ 4 10 108 (5-17) sni,eq =1.68 pA/ HzThe 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. Rf -- un,Rf Rs un,Rs uo1 + in1 un1 uiFigure 5.15, Non-inverting voltage amplifier with distributed noise sources.
128 Electronic Instrumentation R.F. Wolffenbuttel Chapter 5: DETECTION LIMIT DUE TO NOISE AND ELECTRO-MAGNETIC INTERFER- ENCE 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 input noise sources result in uo2: 2 2 2 R + Rf Rs .R f R + Rf Rf u =u s 2 2 2 +in1 × s 2 2 +un ,Rs + un ,Rf o1 n1 R +R Rs s f Rs Rs (5-18) 2 R + Rf uo 2 = un ,eq s 2 2 Rs Equating uo1 and uo2 yields: 2 2 2 RR Rf Rs u 2 = u +i s f 2 2 2 + un , Rs 2 + un , Rf (5-19) n , eq n1 Rs + R f n1 Rs + R f Rs + R f 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: 2 2 1kΩ× 99kΩ 99 snu,eq = 16 × 10−18 +10−24 −20 +1.65 × 10 × 1kΩ× + 1kΩ +99kΩ 100 (5-20) 2 1 1.65 × 10−20 × 99kΩ× = 3331 × 10 −20 → snu,eq = 5.77 nV/ Hz 100 5.4.2 Noise in transistor circuits Obviously, the generic approach for calculating of equivalent input noise sources as a function of distributed noise sources is also applicable at the component level. The equivalent input noise sources of the common-source (CS) gain stage shown 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: ( ) 2 2 u o1 = R i n2d + i n2R 1 + ω 2 R 2C gd 2 (5-21) 2 g m R 2 + ω 2 R 2C gd 2 2 2 u = u 1 + ω 2 R 2 C gd o2 2 n ,e q Hence for uo1= uo2 results: 2 2 2 2 2 ind + inR ind + inR (5-22) u n ,eq = g + ω C gd 2 m 2 2 gm2
Section 5.4 129 Calculating the equivalent input noise sources VDD Cgd R G D uo1 D rgd G ug rds ing Rg ind inR Cgs rgs gmugs R Ui S us S VSS Cgd = un,eq G D uo2 rgd ug rds in,eq Cgs rgs gmugs R us SFigure 5.16, Calculation of equivalent input noise sources in a CS gain stage.Open input leads to: 2 gm R 2 2 2 C gd +R 2 ω 2Cgs C gd + C gs 2 ( ) 2 2 R2 2 uo1 = ing + i 2 +i 2 2 2 C gd C gs C gd C gs nd nR 2 2 1+ ω 2 R 2 2 1+ ω 2 R 2 2 C gd + C gs 2 C gd + C gs 2 (5-23) gm R 2 2 C2 + R 2 2 gd 2 ω 2C gs 2 C gd + C gs 2 uo 2 = 2 2 2 in ,eq 2 2 C gd C gs 1+ ω R 2 C gd + C gs 2Hence, it follows that: 2 2 ( ω 2Cgs ( C gs + C gd ) ind +inR 2 2 ) ω 2Cgs (i ) 2 2 2 in,eq = ing + 2 ing + 2 2 +inR (5-24) g ( C gd + C gs ) + ω C C 2 2 nd 2 m 2 2 gs 2 gd g mThis 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.
130 Electronic Instrumentation R.F. Wolffenbuttel Chapter 5: DETECTION LIMIT DUE TO NOISE AND ELECTRO-MAGNETIC INTERFER- ENCE 5.4.3 Noise in the instrumentation amplifier The Wheatstone bridge has evolved over the previous chapters from a DC- driven circuit into an AC-driven differential circuit. One conclusion drawn in chapter 3 is that the Wheatstone bridge should be AC-driven, to avoid an unfavourable detection limit due to the equivalent input offset sources of the read-out circuits. Moreover, in Chapter 4 the benefits of a differential excitation are outlined to avoid excessive CMRR requirements of the read-out. As a conse- quence, the detectivity in the differentially-driven, AC-operated Wheatstone bridge is noise limited. The noise performance of the instrumentation amplifier could be analysed using Fig. 5.17. un1 + R1 R2 OA1 Ro-∆R -- Ro+∆R -- in1 -- uexc OA3 + 2 R5 + u o1 R6 in3 R7 -- un3 uid uexc Ro-∆R Ro+∆R -- 2 + R3 un2 OA2 + R4 in2 Figure 5.17, Noise in an instrumentation amplifier for read-out of an AC- operated Wheatstone bridge with differential excitation. The OPAMP equivalent input noise sources are indicated as unx and inx where x refers to the number in OAx. The aim is to find an expression for the equivalent input noise sources of the entire instrumentation amplifier: un,eq and in,eq. The Wheatstone 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 usually designed to provide a high CMRR, it is reasonable to disregard the effect of OA3 and only consider the distributed noise sources in the differential-to-differential input 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 the equivalent input noise current results in:
132 Electronic Instrumentation R.F. Wolffenbuttel Chapter 5: DETECTION LIMIT DUE TO NOISE AND ELECTRO-MAGNETIC INTERFER- ENCE 2 2 R R + R6 + R7 2 u = 2 × 5 2 o1 R1 R6 2 2 ( R2 2 un1 +un2 + unR6 + unR5 + unR7 R1 2 ) ( ) 2 (5-26) R R + R6 + R7 2 uo2 = 2 × 5 2 un ,eq R1 R6 The equivalent input noise voltage is: 2 u2 n,eq = u +u + u 2 n1 2 n2 2 nR6 + R6 2 unR5 + unR7 R5 + R6 + R7 2 ( ) un1 +un2 + unR6 2 2 2 (5-27) Equation 5-27 assumes that the noise in R5 and R7 can be disregarded in the case of 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 high CMRR. Equations 5-25 and 5-27 indicate that the noise analysis in an instrumentation amplifier is rather straightforward, since only the components in the differential input circuit need to be considered. 5.4.4 Noise in the charge amplifier The charge amplifier has already been introduced in Chapter 2 as a very suitable circuit for the read-out of capacitive sensors. The charge amplifier is not used to measure the amplitude of an unknown small signal, but rather to measure the value 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. Rf Cs Cf If -- + Is uo A(ω) + ui -- Figure 5.19, Charge amplifier with Rf and A(ω). A large resistor Rf is connected in parallel to Cf to limit the output DC level that would otherwise result from offset (see also section 3.3.3). The charge amplifier is based on an OPAMP with an open-loop gain of A(ω)= Ao/(1+jωτv). For opera-
Section 5.4 133 Calculating the equivalent input noise sourcestion 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: is = ( ui − u− ) jωCs u− (1 + jω Rf Cf ) − ( ui − u− ) jω Rf C s uo u− − is Z f = (1 + jω Rf Cf ) Ao uo = A (ω )( u+ − u− ) − u− jωτ v uo − jω Rf Cs (5-28) = ui τ R τ ( C + Cf ) 1 + jω v + Rf Cf − ω 2 f v s Ao Ao − jω Rf Cs − jω Rf Cs ≈ = τ RCτ τ 1 + jω v + Rf Cf − ω 2 f f v (1 + jω Rf Cf ) 1 + jω v Ao Ao Ao 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. uo 1/τv A0 ui [dB] 1 RfCs ωT = Ao/τv 0 log ω Cs Cf 1 RfCfFigure 5.20, Modulus plot of the charge amplifier.The essential advantages of operating the charge amplifier at a frequency closeto the unity-gain frequency are:
134 Electronic Instrumentation R.F. Wolffenbuttel Chapter 5: DETECTION LIMIT DUE TO NOISE AND ELECTRO-MAGNETIC INTERFER- ENCE 1. Low transducer impedance and 2. Maximum spectral distance to the main source of interference: capacitive coupling of the mains voltage. Cso+∆C Rf unR -- Cf Ui -- + + Cso-∆C Uo1 A(ω) + Ui -- in1 un1 Figure 5.21, Noise in the differential charge amplifier. As is indicated in section 2.6.2, a user is primarily interested in the change in capacitance, ∆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 inverse polarity (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 sources included. The calculation of the equivalent input noise sources is complicated by the capacitive impedances. The charge amplifier is basically operated at the fre- quency of the excitation voltage. However, this signal is usually modulated by the dynamics of the non-electrical signal as measured by the sensor. Therefore, the charge amplifier is assumed to operate in a specified frequency range between fmin and fmax. The TOTAL equivalent noise voltage AT THE OUTPUT is derived using Fig. 5.21 and results in: 2 Z (ω ) / 2 + Z f (ω ) + ( sn,ii + sn,iR ) ( Z f (ω ) ) = 2 sn,uo = sn,ui s Z s (ω ) / 2 (5-29) 2 1 + jω Rf ( 2Cs + Cf ) 2 Rf sn,ui + ( sn,ii + sn,iR ) 1 + jω Rf Cf 1 + jω Rf Cf Note that the total capacitance between excitation sources and inverting input is equal to 2Cso. This equation contains a frequency-independent part and a fre- quency-dependent part. The noise-equivalent change in the capacitance, ∆Cdet, is calculated by referring the output noise back to the input:
Section 5.4 135 Calculating the equivalent input noise sourcesuo, min = uno / ε ∆Cdet u (5-30)uo,min = ( − ) 2 ui → ∆Cdet = C f no Cf 2ε uiExample 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: ωRf C f = 10π 1→ 2 2 2C + C f 1 (5-31) sn,uo = sn,ui s C + ( sn,ii + sn,iR ) 2π f C f f 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: 2 2C + C f × ∆f = 4 × 10 × (1.2 ) × 2 × 10 = 1.15 × 10 V 2 sn ,ui s −18 3 −14 2 (5-32) C f
136 Electronic Instrumentation R.F. Wolffenbuttel Chapter 5: DETECTION LIMIT DUE TO NOISE AND ELECTRO-MAGNETIC INTERFER- ENCE The frequency-dependent part results in: s n ,ii + 4 k B T / R f f max s n ,ii + 4 k B T / R f 1 1 ∫ = − = 2 df / f 4π C 2 2 f f min 4π C f 2 2 f min f max (5-33) 10 − 26 + 1.65 × 10 / 5 × 10 5 10 − 3 10 − 3 − 20 − 14 − = 2.18 × 10 V 2 4π 2 × 10 − 20 99 101 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 Vrms and an inaccuracy specification ε= 5%, as: uo,min = uno / ε = 182.5 × 10−9 / 0.05 = 3.65 µ V ∆Cs Cu (5-34) uo = ( − ) 2 ui → ∆Cs,min = f o,min = 10−10 × 365 × 10−8 /10 = 36.5aF Cf 2ui The detection limit reduces with (noise) bandwidth. 5.5 Noise matching So 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 that there is a ratio between un,eq2 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 as possible to the measurement uncertainty. Hence, the added noise power should be as small as possible. The equivalent requirement is that the Signal-to-Noise Ratio (SNR) should be reduced as little as possible. Hence, any approach to noise reduction should be without input signal attenuation. The extent to which a read-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 the source without the circuit connected, and the SNR at the output with source connected.
Section 5.5 137 Noise matchingIn equation: Ps ( source ) SNRsource Pn ( source ) NF = = = SNRoutput Ps ( output ) H (ω ) Pn ( source) + Pn ( readout ) 2 (5-35) Pn ( source ) + Pn ( readout ) P ( readout ) = 1+ n Pn ( source ) Pn ( source )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. Rg Rg un,eq un,Rg Ri in,eq Ri Ug Ug (a) (b)Figure 5.22, System with (a) distributed noise sources and (b) equivalent inputnoise sources.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,ieqRg2. Hence: 2 2 2 un ,eq +in ,eq Rg NF = 1+ (5-36) 4kTRgThe 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:
138 Electronic Instrumentation R.F. Wolffenbuttel Chapter 5: DETECTION LIMIT DUE TO NOISE AND ELECTRO-MAGNETIC INTERFER- ENCE = ( ∂NF 4k BTRg × 2in Rg - 4k BT un +in Rg 2 2 2 2 ) = 0 → R2 = 2 un2 (5-37) ( 4kBTRg ) g,opt ∂Rg 2 in Noise matching is achieved when this condition has been met. The associated value of the noise factor is: 2 2un (5-38) NFmin = 1+ 4kBTRg Note that the noise performance remains primarily determined by the noise power, as defined by equivalent input sources. Only after the total noise power has 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,eq through design. This step should be preceded by a noise analysis aim- ing for minimum noise generated within, or injected into the read-out circuit. An incorrect approach for noise performance optimisation would be the insertion of a series resistance Rs, as shown in Fig. 5.23. Rg un,Rg Rs un,Rs un,eq in,eq Ri ug Figure 5.23, Counter-productive attempt at noise matching using a series resistor. 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: P n ( source + readout ) = 4k BTRg +4k BTRs +un +in ( Rg + Rs ) → 2 2 2 (5-39) 4k BT ( Rg + Rs ) +un +in ( Rg + Rs ) Rs un + in ( Rg + Rs ) 2 2 2 2 2 2 NF = = 1+ + 4k BTRg Rg 4k BTRg
Section 5.6 139 External sources of errorAdding 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. Rg un,eq Rg un,eq + + Ug in,eq Ri in,eq Ri _ _ Ug 1:m 1:mFigure 5.24, Noise matching using a transformer.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’n2= un,eq2/m2.Similarly, the equivalent input noise current is transformed into the input by mul-tiplication by a factor m: i’n,eq2= m2×in,eq2. Using Eqn 5-37 yields noise match-ing at: 2 2 2 u n/ m u (5-40) = R 2 → m4 = 2 n 2 g 2 2 m in inR gUnfortunately, 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,
140 Electronic Instrumentation R.F. Wolffenbuttel Chapter 5: DETECTION LIMIT DUE TO NOISE AND ELECTRO-MAGNETIC INTERFER- ENCE the Seebeck effect causes an additional error in the case of temperature gradi- ents with electrical interconnections composed of different metals and solder bonds at different temperatures. • As mentioned already in Section 4.2.3, capacitive coupling between the measurement 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. Zg un + + Ug B Ui Zi _ _ Figure 5.25, Magnetic injection of noise. • 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 a current. An error voltage is induced in when the magnetic field passes a wind- ing formed by another conductor. The induced voltage increases with the enclosed area, S, of this winding and with the rate of change in magnetic inductance: un= S(∂B/∂t). Rg + Ug Ri (a) _ Ignd Rc + Rg Ug Ri (b) _ Rc Rc Ignd Rc1 Rc2 Figure 5.26, (a) Additive error due to ground current and (b) star connection.
Section 5.6 141 External sources of error• Safety regulations usually require each instrument in a measurement and con- trol setup to be connected to ground level. The problem arises when the zero input 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 locally connected to ground level. Each connection to ground has a finite cable resistance and many units are locally connected. In a worst-case scenario, the most remote unit supplies a large ground current, resulting in a ground loop voltage, as shown in Fig. 5.26a. Basically, an offset error equal to Ignd×Rc is generated, where Ignd denotes the ground current and Rc is the part of the cable resistance between two units. The most effective technique to avoid such an error is to use only one common node as the safety ground and to connect 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 scale error equal to: 2Rc/(Ri+2Rc), which can be disregarded in the case of a proper voltage 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 results in a common-mode voltage that is subsequently suppressed using a high CMRR.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 Electronic Instrumentation R.F. Wolffenbuttel Chapter 5: DETECTION LIMIT DUE TO NOISE AND ELECTRO-MAGNETIC INTERFER- ENCE Cc Zg + + Za + ug In ui Zi _ un _ _ Figure 5.27, Shielding as a technique for reducing capacitive coupling. The weakest link in a measurement setup is the electrical interconnection between 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 cables from this coupling. Applying a conductive shield, as shown in Fig. 5.27, isolates the measurement system from the capacitive coupling of external noise and interference. 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 coupling which uses a ferro-magnetic shield. Magnetic shields are usually more difficult to use, since a relatively thick layer of a material of high magnetic permeability is required for effective shielding, which complicates handling. A major problem in conductive shielding is the large capacitance between the inner signal cable and the grounded outer shield. This cable capacitance affects the signal transfer ui,g/ug: Z i // Z a Z g // Z i //Z a ui , g = u g , ui , n = un (5-41) Z i // Z a + Z g Z g // Z i //Z a + Z c A conductive shield can be used to reduce the capacitive coupling to less than 5% (Cc’= 0.05×Cc). The noise level at ui due to un (denoted as ui,n) is reduced by a 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 the read-out, Zi= Ri, and source, Zg= Rg yields:
Section 5.6 143 External sources of error Ri Rg jω Cc ui ,g Ri 1 ui ,n Ri + Rg = × , = jωRgCc (5-42) ug Ri + Rg Ri Rg un RR 1 + jω Ca 1 + jω i g ( Cc + Ca ) Ri + Rg Ri + RgBasically 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 5.3 A voltage-to-current converter with trans-conductance Gm= 1 A/V is used for 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 equivalent input noise sources. Moreover, both are prone to capacitive coupling from the 230 Vrms/50 Hz power supply voltage. 230 V / 50 Hz Cc1 Cc2 gm Gi Rg un1 un2 ui1 ii2 io gmui1 Giii2 Ri1 Ro1 Ri2 Ro2 ug in1 in2 Figure 5.28, Noise in a two-stage trans-impedance amplifier. 1. Calculate the equivalent spectral noise sources at the input of the overall voltage 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).