Single JFET Front-End Amplifier for Low Frequency Noise

Article Single JFET Front-End Ampliﬁer for Low Frequency Noise Measurements with Cross Correlation-Based Gain Calibration

Graziella Scandurra , Gino Giusi and Carmine Cioﬁ *

Department of Engineering, University of Messina, 98166 Messina, Italy; [email protected] (G.S.); [email protected] (G.G.) * Correspondence: ccioﬁ@unime.it

Received: 10 September 2019; Accepted: 18 October 2019; Published: 21 October 2019

Abstract: We propose an open loop voltage amplifier topology based on a single JFET front-end for the realization of very low noise voltage amplifiers to be used in the field of low frequency noise measurements. With respect to amplifiers based on differential input stages, a single transistor stage has, among others, the advantage of a lower background noise. Unfortunately, an open loop approach, while simplifying the realization, has the disadvantage that because of the dispersions in the characteristics of the active device, it cannot ensure that a well-defined gain be obtained by design. To address this issue, we propose to add two simple operational amplifier-based auxiliary amplifiers with known gain as part of the measurement chain and employ cross correlation for the calibration of the gain of the main amplifier. With proper data elaboration, gain calibration and actual measurements can be carried out at the same time. By using the approach we propose, we have been able to design a low noise amplifier relying on a simplified hardware and with background noise as low as 6 nV/√Hz at 200 mHz, 1.7 nV/√Hz at 1 Hz, 0.7 nV/√Hz at 10 Hz, and less than 0.6 nV/√Hz at frequencies above 100 Hz.

Keywords: low noise voltage ampliﬁer; cross correlation; noise measurements

1. Introduction Low frequency noise measurements (LFNMs) are among the most sensitive tools for the investigation of the quality and reliability of electron devices and systems [1]. While LFNMs are especially useful in the investigation of the quality and reliability of electron devices [2–6] and in the investigation of the conduction mechanisms of advanced electron devices [7–9], the application of LFNMs in the development of advanced sensing approaches has been suggested [10–12]. Performing meaningful noise measurements, however, requires that the background noise (BN) of the measurement chain be much smaller than the noise to be detected. This is especially true in the case of low frequencies (f < 1Hz) where the low frequency noise introduced by the instrumentation usually sets the lower limit of the background noise that can be obtained. It is worth noting that cross correlation approaches that allow reaching a BN level well below that of the amplifier employed in the measurement are barely effective at very low frequencies because of the very long measurement time that would be required for averaging out the uncorrelated noise sources [13–15]. It is for this reason that it is important to design noise preamplifiers with the lowest possible level of BN at very low frequencies. In the case of voltage noise measurements, the BN of the system can be reduced to the equivalent input voltage noise (EIVN) of the voltage amplifier directly connected to the device under test (DUT). A possible approach to the design of very low noise voltage amplifiers is to add a differential stage based on discrete, very low noise devices in front of a solid-state high-gain stage. In this way, the entire system can be regarded as

Electronics 2019, 8, 1197; doi:10.3390/electronics8101197 www.mdpi.com/journal/electronics Electronics 2019, 8, 1197 2 of 14 a very low noise operational amplifier (OA) whose gain can be set and stabilized by means of passive feedback [16–19]. This approach has the added advantage that because of the differential input stage, coupling to the DUT down to DC is possible, which can be relevant in the field of LFNMs, where the frequencies of interest may extend well below 1 Hz [20]. While the lowest level of EIVN are obtained by employing BJT based input stages, JFET input stages are preferred in all those cases in which noise is to be measured on nodes where a significant DC component is present. This is often the case in the field of noise measurements on electron devices, since measurements need to be performed on biased DUTs. Using BJTs, AC coupling down to the tens of mHz range would be made quite difficult because of the large bias currents and equivalent input noise current of these devices [20]. When compared to a common source (CS), open loop, single transistor stage, however, the EIVN of a low noise amplifier employing a differential JFET input stage is at least doubled because both transistors in the input stage equally contribute to the BN [18]. Moreover, obtaining stability when dealing with an operational amplifier based on a discrete component input stage is never an easy task, and frequency compensation often results in reduced bandwidth and BN increase at higher frequencies [18,19]. On the other hand, the voltage gain of an open loop CS stage directly depends on the characteristics of the active device so that known and stable gain cannot be guaranteed by design. This means that calibration measurements with known noise sources must be performed in order to obtain the actual gain of the amplifier, or gain adjustment circuitry needs to be added in order to set the gain to the desired value. In order to address these issues, we propose, in this paper, an approach that aims to take full advantage of the noise performances and simplicity of design of an open loop low noise amplifier based on a CS JFET stage while avoiding, at the same time, the need for time-consuming measurements for gain calibration and/or gain adjustment. As it will be shown in the following, by resorting to auxiliary OA based amplifiers operating in parallel with the main CS JFET low noise amplifier, the value of the gain of the main amplifier can be obtained by cross correlation while noise measurements on the device or system under investigation are in progress.

2. Proposed Approach We start by analyzing the properties of the circuit in Figure1 that can be regarded as the cascade of a JFET CS amplifier followed by an ideal transimpedance amplifier (TIA) stage with gain AR. In the actual circuit, the TIA is implemented with an OA in shunt-shunt feedback configuration [21]. The (ideal) circuit in Figure1 can be regarded as a good representation of the actual circuit as long as the input impedance of the TIA is much smaller than the impedance of the capacitor CD at all frequencies of interest. We assume a dual supply operation (that is required, in any case, for a simple implementation of the TIA). In our design we employ an IF3601 large area JFET by InterFet, capable of providing transconductance gains (gm) in the order of a few tens of mA/V with a bias drain current in the order of a few mA [18]. This device is characterized by an equivalent input noise that can be as low as 0.3 nV/√Hz at 100 Hz. As is characteristic of JFET devices, DC and AC parameters are quite spread. The datasheet for IF3601 lists a pinch off voltage in the interval from 2 up to 0.35 V, while − − only a minimum value for the saturation current at VGS = 0 is given. Assuming that the JFET is in the active region of operation and that the gate current is negligible, in the circuit configuration in Figure1, we have VGS + VSS ID = − . (1) RS

While the actual value of ID depends on the characteristic of the particular JFET used in the circuit, the fact that V is certainly smaller than 2 V (typically in the order of a few hundred mV − GS for currents in the order of a few mA) means that for VSS equal to 6 V or more, the bias current (ID) can be set with a reasonable error (typically less than 10%) by the ratio VSS/RS. In other words, with the simple bias configuration in Figure1, the e ffect of parameter dispersion is modest, as far as the bias point is concerned. Note, however, that this does not imply in any way that we can ensure, by design, a predictable and known gain for the amplifier, since even for the same bias current, ElectronicsElectronics 20192019,, 88,, x 1197 FOR PEER REVIEW 33 of of 14 14 transconductance gain can change significantly depending on the particular device being used. When employingthe transconductance the amplifier gain in canFigure change 1 as significantlypart of a noise depending measurement on the system, particular the deviceknowledge being of used. the actualWhen gain employing is, of course, the amplifier necessary. in In Figure principle,1 as part one ofcould a noise perform measurement calibration system, measurements the knowledge once each of singlethe actual amplifier gain is is, built of course, or change necessary. some circuit In principle, parameter one until could the perform desired calibrationgain is obtained measurements [22]. However, once sinceeach singlethe gain amplifier can change is built with or changethe temperature some circuit and parameter also because until of the ageing desired of the gain components, is obtained [the22]. onlyHowever, way to since ensure the accuracy gain can changein actual with measurements the temperature is to andfrequently also because repeat of the ageing calibration of the components, procedure. Beforethe only discussing way to ensure the approach accuracy we in actual propose measurements for addressing is to this frequently issue, we repeat need the to calibrationdevote our procedure.attention toBefore the frequency discussing response the approach that can we proposebe obtained for addressingfrom the amplifier this issue, inwe Figure need 1. to Indeed, devote since our attention we are interestedto the frequency in an amplifier response to that be canused be in obtained LFNM applications, from the amplifier we need in to Figure extend1. Indeed,the lower since frequency we are limitinterested well below in an amplifier1 Hz. The to small be used signal in input–output LFNM applications, transfer we function need to of extend the circuit the lower in Figure frequency 1 can belimit calculated well below starting 1 Hz. from The the small small signal signal input–output equivalent transfer circuit in function Figure of2, where the circuit all the in Figurerelevant1 can noise be sourcescalculated are startingalso included. from theAs far small as the signal transfer equivalent function circuit is concerned, in Figure we2, wherecan assume all the all relevant noise sources noise inactivesources areand also we included.obtain: As far as the transfer function is concerned, we can assume all noise sources inactive and we obtain: 1+ = = ∙ ∙ ∙ , (2) V O1 1+gmAR 1+sτA 1+sτD 1+1 + sτ S AV(s) = = , (2) V 1 + g R ·1 + sτ ·1 + sτ ·1 + sτ where I m S A D 0S where =; =; =; =′;′ = RS . (3) τA = RACA; τD = RDCD; τS = RSCS; τ0S = R0SCS; R0S =1+ . (3) (1 + gmRS) From Equation (2), it is apparent that since (’s < s), a constant frequency response is obtained From Equation (2), it is apparent that since (τ0s < τs), a constant frequency response is obtained for frequencies above the one corresponding to the pole with the largest magnitude. As far as the for frequencies above the one corresponding to the pole with the largest magnitude. As far as the pole pole corresponding to A is concerned, since the resistance RA can be in the order of a few M, its corresponding to τA is concerned, since the resistance RA can be in the order of a few MΩ, its frequency frequency can be easily set below a few tens of mHz by employing for CA capacitances in the tens of can be easily set below a few tens of mHz by employing for CA capacitances in the tens of µF range. F range. In the case of the pole frequencies corresponding to D and, especially, ’s, the situation is In the case of the pole frequencies corresponding to τD and, especially, τ0s, the situation is quite quite different. With reference to Figure 1 and Equation (1), RS must be in the order of 1 k in order different. With reference to Figure1 and Equation (1), RS must be in the order of 1 kΩ in order to to obtain bias currents in the order of a few mA. As a consequence, RD must also be in the same order obtain bias currents in the order of a few mA. As a consequence, RD must also be in the same order of magnitude (with RD conveniently lower that RS in order to ensure that the JFET operates in the of magnitude (with RD conveniently lower that RS in order to ensure that the JFET operates in the active region). When JFETs such as the one employed in our design are biased with a drain current active region). When JFETs such as the one employed in our design are biased with a drain current in in the order of a few mA, the transconductance gain gm is typically in the order of a few tens of mA/V. the order of a few mA, the transconductance gain gm is typically in the order of a few tens of mA/V. This means that the equivalent resistance RS’ (the resistance seen toward the source of the JFET) may This means that the equivalent resistance RS0 (the resistance seen toward the source of the JFET) may be as low as a few tens of ohms. In this situation, the only way to obtain a value of ’S in the order of be as low as a few tens of ohms. In this situation, the only way to obtain a value of τ0S in the order of a a few seconds or tens of seconds (to obtain a pole frequency well below 1 Hz) is to resort to a capacitor few seconds or tens of seconds (to obtain a pole frequency well below 1 Hz) is to resort to a capacitor value in the order of 1 F or more for CS (few tens of mF in the case of CD for obtaining D >> 10 s). value in the order of 1 F or more for CS (few tens of mF in the case of CD for obtaining τD >> 10 s).

VDD

RD C TIA V J D O1 1 A i C R t A i ID t V I CS

RS RA

-VSS

Figure 1. Reference schematic of a JFET CS stage-based amplifier. The second stage (transimpedance amplifier (TIA)) represents the idealized behavior of an operational amplifier-based transimpedance

ampliﬁer with gain AR.

Electronics 2019, 8, x FOR PEER REVIEW 4 of 14

Figure 1. Reference schematic of a JFET CS stage-based amplifier. The second stage (transimpedance amplifier (TIA)) represents the idealized behavior of an operational amplifier-based transimpedance amplifier with gain AR. Electronics 2019, 8, 1197 4 of 14

gmvGS GS D CD TIA VO1 enj v CA I RA RS RD it e e C e nRA nRS S nRD i nt ARit

Figure 2. Small signal equivalent circuit of the amplifier in Figure1. All relevant noise sources are Figure 2. Small signal equivalent circuit of the amplifier in Figure 1. All relevant noise sources are shown in the figure. The noise sources enRX account for the thermal noise introduced by the resistances shown in the figure. The noise sources enRX account for the thermal noise introduced by the resistances RX; the source enj is used to model the noise introduced by the JFET; the source int is used to model the RX; the source enj is used to model the noise introduced by the JFET; the source int is used to model the noise introduced by the transimpedance amplifier (TIA). noise introduced by the transimpedance amplifier (TIA). While up to a few years ago, employing capacitances in the order of 1 F would have been largely impractical,While up supercapacitors to a few years ago, in veryemploying compact capacitanc size arees nowadays in the order easily of 1 availableF would have with been capacitances largely impractical,ranging from supercapacitors a few mF to in a very few F.compact Unlike size other are high nowadays specific easily capacitance available devices with capacitances (electrolytic rangingcapacitors), from supercapacitors a few mF to a have few beenF. Unlike shown other to be high compatible specific withcapacitance instrumentation devices (electrolytic intended for capacitors),LFNM applications supercapacitors [23–25]. have Note been that becauseshown to of thebe compatible circuit configuration with instrumentation in Figure1, the intended DC voltage for LFNMdrop across applications the capacitor [23–25].C Noteis V thatthat, because in the of casethe circuit of the JFETconfiguration we employ in Figure in our design,1, the DC is typicallyvoltage S − GS dropbelow across 1 V, the and capacitor this allows CS is for −V employingGS that, in the supercapacitors case of the JFET in we the employ order of in a our few design, farads is with typically a bias belowvoltage 1 wellV, and below this theallows maximum for employing rated value. supercapacitors As far as the BNin the is concerned,order of a forfew the farads sake ofwith simplicity, a bias voltagewe will well limit below our analysis the maximum at frequencies rated value. well aboveAs far theas the lower BN frequencyis concerned, corner for ofthe the sake amplifier. of simplicity, In this wesituation, will limit we our can analysis replace all at capacitorsfrequencies in well Figure abov2 withe the short lower circuits, frequency obtaining, corner for of the the passband amplifier. input In this situation, we can replace all capacitors in Figure 2 with short circuits, obtaining, for the passband output signal gain AVPB, input output signal gain AVPB, VO1 AVPB = = gmAR. (4) VI = =. (4) Note that if the TIA is obtained employing the classical approach of an OA in shunt-shunt feedback configuration,Note that if the the gain TIAAR iscoincides obtained with employing the value the of theclassical feedback approach resistance of anR ROAbetween in shunt-shunt the output and the inverting input of the OA [21]. In the assumption of all capacitors replaced by short circuits, feedback configuration, the gain AR coincides with the value of the feedback resistance RR between thethe output noise sources and theenRA invertingand enRS input(due of to the the OA resistances [21]. InR theA and assumptionRS, respectively) of all capacitors do not contribute replaced to by the output noise. Assuming all noise sources are uncorrelated, the power spectral density (PSD) S of short circuits, the noise sources enRA and enRS (due to the resistances RA and RS, respectively) doENO not contributethe noise atto the the output output of noise. the amplifier Assuming can beall writtennoise sources as are uncorrelated, the power spectral density (PSD) SENO of the noise at the output of the amplifier can 2 be written as 2 AR 2 SENO = SENJ gmAR + SEND + SIT AR , (5) RD | | =|| + +|| , (5) where SEND is the PSD of the noise source enRD due to the thermal noise introduced by the resistance where SEND is the PSD of the noise source enRD due to the thermal noise introduced by the resistance RD and SIT is the PSD of the equivalent input current noise source int at the input of the transimpedance RD and SIT is the PSD of the equivalent input current noise source int at the input of the transimpedance amplifier. It must be noted that in most cases, SIT essentially coincides with the PSD of the current noise amplifier. It must be noted that in most cases, SIT essentially coincides with the PSD of the current source due to the thermal noise of the feedback resistance RR used in the transimpedance amplifier. noiseTherefore, source we due can to write the thermal noise of the feedback resistance RR used in the transimpedance amplifier. Therefore, we can write 4 kT 4 kTR S = 4 kTR ; S = = R , (6) END D END 2 RR R 4 4 R =4 ; = = , (6) where k is the Boltzmann constant and T the absolute temperature. whereThe k is performancesthe Boltzmann of constant a low noise and amplifier T the absolute are better temperature. appreciated in terms of the equivalent input noise, whose PSD SENI can be obtained from Equation (5) by dividing by the modulus of the passband gain squared. We obtain, using also Equation (6), Electronics 2019, 8, x FOR PEER REVIEW 5 of 14

The performances of a low noise amplifier are better appreciated in terms of the equivalent input noise, whose PSD SENI can be obtained from Equation (5) by dividing by the modulus of the passband Electronics 2019, 8, 1197 5 of 14 gain squared. We obtain, using also Equation (6),

4 4 = SENI = + 4kTRD + 4kTRR . (7) SENI = || = SENJ +|| +|| . (7) 2 2 2 gmAR gmRD gmRR The contribution of the noise due to the resistance decreases as the value of the resistance increases.The contribution As we haveof noted the noise before, due R toD is the typically resistance in decreasesthe order asof the1 k value while of the gm resistanceis in the order increases. of a few tens of mA/V. This means that the contribution of the second term in Equation (7) is equivalent As we have noted before, RD is typically in the order of 1 kΩ, while gm is in the order of a few tens of tomA the/V. thermal This means noise that of the a resistance contribution of ofa thefew second ohms termthat is in well Equation below (7) 1 is nV/ equivalent√Hz, and to therepresents, thermal especiallynoise of a at resistance low frequencies, of a few ohmsa very that small is well(even below negligible) 1 nV/ √contributionHz, and represents, with respect especially to the atnoise low introducedfrequencies, by a the very JFET. small As (even far as negligible) the contribution contribution of the withthird respectterm is toconcerned, the noise from introduced the previous by the argument,JFET. As far it asfollows the contribution that it is sufficient of the third to term design is concerned, the transimpedance from the previous amplifier argument, with a feedback it follows resistancethat it is su inffi excesscient toof design10 k to the make transimpedance its contribution amplifier negligible. with By a feedbackfollowing resistancethe design in guidelines excess of we10 khaveΩ to discussed make its contribution so far, therefore, negligible. it is possible By following to design the a design voltage guidelines amplifier wewhose have equivalent discussed input so far, voltagetherefore, noise it is reduces possible to to the design equivalent a voltage input amplifier voltage whose noise of equivalent a single low input noise voltage JFET noise device. reduces to the equivalentAs we have input noted voltage before, noise the ofmain a single problem low noisewith JFETthe design device. we propose is that the passband voltageAs gain we have is directly noted proportional before, the main to the problem transconductance with the design gain wegm of propose the JFET is that and, the therefore, passband a largevoltage spread gain isin directlythe actual proportional value of the to gain thetransconductance is to be expected depending gain gm of the on JFETthe particular and, therefore, device a used large inspread otherwise in the nominally actual value identical of the gainrealizations. is to be expectedWe believe depending that for the on theapproach particular we devicepropose used to be in feasibleotherwise in nominallyLFNM applications, identical realizations. a sufficiently We simple believe and that reliable for the approach approach for we proposethe calibration to be feasible of the amplifierin LFNM gain applications, must be adevised. sufficiently To this simple end, and we reliable propose approach the approach for the schematically calibration of illustrated the amplifier in Figuregain must 3. be devised. To this end, we propose the approach schematically illustrated in Figure3.

LN A1 A2 A3 V VO1 O2 V O3

Figure 3. Amplifier configuration for gain calibration. A1 is the low noise (LN) amplifier in Figure1; Figure 3. Amplifier configuration for gain calibration. A1 is the low noise (LN) amplifier in Figure 1; A2 and A3 are operational amplifier (OA)-based amplifiers with stable and known gain. A2 and A3 are operational amplifier (OA)-based amplifiers with stable and known gain.

With reference to Figure3, the amplifier A1 is a very low noise amplifier designed according to the With reference to Figure 3, the amplifier A1 is a very low noise amplifier designed according to guidelines discussed above. The other two amplifiers (A2 and A3) are nominally identical OA-based thevoltage guidelines amplifiers discussed with known above. gain. The Sinceother thetwo three amplifiers amplifiers (A2 and have A their3) are inputs nominally connected identical together, OA- based voltage amplifiers with known gain. Since the three amplifiers have their inputs connected it is mandatory that the current noise at the inputs of the amplifiers A2 and A3 be as low as possible together,in order notit is to mandatory degrade the that equivalent the current input noise current at the noiseinputs (EICN) of the ofamplifiers the overall A2 and system. A3 be This as low can as be possibleobtained in by order resorting not to to degrade either JFETthe equivalent or MOSFET input input current OAs thatnoise are (EICN) characterized of the overall by extremely system. This low canlevels be ofobtained EICN. Typically,by resorting low to noise either JFET JFET input or MOSFET OAs are input characterized OAs that byare acharacterized lower level of by flicker extremely noise lowwith levels respect of toEICN. low Typically, noise MOSFET low noise input JFET OAs. input However, OAs are as itcharacterized will become by clear a lower in the level following, of flicker for noisethe approach with respect we propose to low noise to be MOSFET effective, input we are OAs. interested However, in a as low it will level become of voltage clear noise in the in following, the white for the approach we propose to be effective, we are interested in a low level of voltage noise in the noise region for A2 and A3. Suppose now that we can perform cross spectra measurements among whiteany two noise outputs region in for Figure A2 and3. Let A3 us. Suppose also assume now thatthat suchwe can cross perform spectra cross estimation spectra is measurements performed in among any two outputs in Figure 3. Let us also assume that such cross spectra estimation is a frequency range in which the voltage gains AV1, AV2, and AV3 of the amplifiers are constant, and performedtheir values in can a frequency be well represented range in which by real the numbers. voltage Withoutgains AV loss1, AV of2, generality,and AV3 of wethecan amplifiers assume are the constant, and their values can be well represented by real numbers. Without loss of generality, we gains to be positive numbers. Since amplifiers A2 and A3 are identical, AV2 = AV3 = AV. With these can assume the gains to be positive numbers. Since amplifiers A2 and A3 are identical, AV2 = AV3 = AV. assumptions, we would obtain for the cross spectra S12 and S23 between outputsVO1 and VO2 and With these assumptions, we would obtain for the cross spectra S12 and S23 between outputsVO1 and outputs VO2 and VO3, respectively, VO2 and outputs VO2 and VO3, respectively, 2 S12 = AV1AV2SI = AV1AVSI; S23 = AV2AV3SI = AVSI, (8) = = ; = = , (8) where SI is the power spectral density of the voltage noise at the input of the system. Note that the contribution of the equivalent input voltage noise (EIVN) sources at the inputs of the amplifiers is

Electronics 2019, 8, 1197 6 of 14

completely rejected. Since AV is known, independently of the value of SI, Equation (8) can be used to estimate the voltage gain AV1 of the low noise ampliﬁer as follows:

S12 AV1 = AV . (9) S23

Because of Equation (8), it might be proposed that since the cross spectra approach allows for complete rejection of the contribution of the EIVN of the amplifiers, employing just the two amplifiers A2 and A3 and estimating the cross spectrum S23 is all we need to obtain the PSD of the input noise, without the trouble of resorting to the design in Figure1. The fact is, however, that the process of estimating cross spectra involves the averaging of the estimated spectra over several time records of the signal to be analyzed. It is indeed the very process of averaging that leads to the cancellation of the uncorrelated noise sources, allowing only the correlated ones to emerge as the number of averages increases [14]. As a general rule, the magnitude of the uncorrelated noise decreases with the square root of the number of averages, i.e., with the number of time records being elaborated. When using a discrete Fourier transform (DFT)-based spectrum analyzer, the duration of each time record is the inverse of the resolution bandwidth ∆f that is also the minimum frequency (other than DC) at which the DFT and, hence, spectra and cross spectra can be estimated. Actually, as has been discussed in detail in [26], in the low frequency range, where flicker noise is dominant, the resolution bandwidth needs to be much smaller than the minimum frequency of interest. Suppose now that one is interested in estimating a spectrum at 1 kHz. Employing a very low noise voltage amplifier based on IF3601 [18], 18 2 one can obtain, at that frequency, a BN less than 1 nV/√Hz (10− V /Hz). If we want to employ a cross correlation configuration using only amplifiers A2 and A3 in the figure, we can obtain an estimate of the time that is required to obtain an equivalent BN of 1 nV starting from the knowledge of their EIVN. Let us assume that A2 and A3 are based on the MOSFET input OA TLC070 (the one that is actually used in our design). The manufacturer lists, for these devices, an EIVN of 7 nV/√Hz (49 10 18 V2/Hz) × − at 1 kHz that is 49 times the EIVN of the low noise amplifier. This means that in order to reduce the equivalent BN by means of cross correlation to the same level of the low noise amplifier, we need about 2500 averages (492). Assuming ∆f = 100 Hz << 1 kHz, one time record lasts 10 ms, so that the required measurement time is about 25 s, which can be considered a quite manageable time in most applications. Things change dramatically, however, if we are interested in very low frequencies as in the case of LFNMs on electron devices. Let us assume that fmin = 1 Hz is the minimum frequency of interest (although LFNMs often extend well below 1 Hz). The EIVN of an IF3601-based low noise amplifier can be as low as 1.4 nV/√Hz or 2 10 18 V2/Hz [18]. As far as the TLC070 is concerned, although the × − manufacturer only lists the equivalent input noise down to 10 Hz, since we are well within the flicker region, we can extrapolate the noise at 1 Hz to be about 90 nV/√Hz ( 8 10 15 V2/Hz), that is, 4 103 ≈ × − × times larger than that of the low noise amplifier. As before, we assume that we can employ ∆f = fmin/10 = 100 mHz, resulting in a record duration of 10 s. With a factor of 4 103 between the noise of the × TLC070 and the one of a good low noise voltage amplifier, we would therefore require a measurement time of 160 106 s (about 5 years) to obtain, by cross correlation between the outputs of A and A , a × 2 3 BN equivalent to that of A1. This clearly demonstrates that at very low frequencies, cross correlation is unpractical if the uncorrelated noise to be reduced is large compared to the desired BN. It is therefore important to design voltage amplifiers characterized by intrinsically low equivalent input voltage noise. At the same time, as long as we restrict to frequencies above a few hundred Hz, a few tens of seconds are usually sufficient for obtaining a good estimate of the quantities in Equation (8). In light of these considerations, the approach we propose can be quite effective. As long as we can assume a flat gain for the amplifier A1 at all frequencies of interest, we can perform cross spectra measurement in a conveniently high frequency range so that a correct estimate of the cross spectra in Equation (9) can be obtained in a matter of a few tens of seconds; immediately afterwards, once the actual gain of the low noise amplifier is known, noise measurement at the output VO1 can be started down to the lowest frequency of interest. Typical noise measurements down to frequencies below 1 Hz can last several Electronics 2019, 8, 1197 7 of 14 minutes or several tens of minutes so that the initial measurement session for the calibration of the gain has a very low impact on the overall measurement time, especially because it can be performed with the actual device to be characterized already connected to the measurement system. Moreover, as we shall demonstrate in the section devoted to the experimental results, if proper advanced elaboration approaches are used for spectral estimation, the gain estimation can be obtained as part of the very same measurement session employed for the estimation of the noise produced by the DUT. A key factor for the application of the approach we propose is the determination of the measurement time required for the correct estimation of the gain according to Equation (9). As we have noted above, the smaller the SI is compared to the equivalent input noise of the amplifiers, the longer the measurement time for obtaining the correct estimation of the gain A1. When performing spectra and cross spectra measurements employing a DFT-based spectrum analyzer, we obtain their estimate at discrete frequencies fk = k∆f, with k being an integer number. Let S11M(fk) and S23M(fk) be the estimates of the cross spectra S11 and S23 in Equation (9) after averaging over M records. Since we assume AV1 to be constant, we can obtain an improved estimate of this parameter by averaging Equation (9) over several frequencies fk as follows:

kmax AV X S12M( fk) AV1,M = . (10) M S22M( fk) k=kmin

As will be shown in the next section, the quantity AV1,M can be monitored vs. M (i.e., vs. measurement time) until it stabilizes, at which point its value can be assumed to be the correct estimate of the gain of the low noise ampliﬁer.

3. System Design and Experimental Results In order to test the effectiveness of the approach we propose, we designed the system reported in Figure4 following the guidelines discussed in the previous section. The complete component list is reported in Table1. A single AC coupling filter ( CA, RA) is used for removing the DC across the DUT connected to the input. The amplifier A1 is the actual implementation of the circuit in Figure1, while A2 and A3 are identical two-stage voltage amplifiers with a passband gain of 1111 (61 dB). This is the result of the first (OA2) and second (OA3) stage, having a gain of 11 and 101, respectively. The higher frequency corner of the amplifier A2 is set by the second stage to about 100 kHz (the gain bandwidth product of OA TLC070 used for OA2 and OA3 is 10 MHz). The AC coupling network CA2–RA2, that removes the DC introduced by the offset of the first stage, introduces a low frequency corner at about 16 mHz that, in combination with the input AC filter (CA, RA), sets the lower frequency corner for the response from the input to the output VO2. Since the auxiliary amplifiers A2 and A3 are only used for gain calibration (frequency range of interest above a few hundred Hz) the time constant CA2RA2 is not critical. From the point of view of the input of the system, the connection of the inputs of the amplifiers A2 and A3 to the gate of the JFET results in a negligible contribution in terms of current noise (the equivalent input current noise for the TLC070 is 0.6 fA/√Hz) and in a contribution of a few tens of pF to the input capacitance of the system that is, however, dominated by the JFET (the typical gate to source and gate to drain capacitances for the IF3601 are 300 and 200 pF, respectively). As far as the low noise amplifier is concerned, the JFET is biased in such a way as to obtain a drain current of about 2.5 mA and a drain to source voltage of about 2 V, sufficient to ensure operation in the active region. In these bias conditions, we expect to obtain a transconductance in the range from 20 to 50 mA/V. As far as the frequency response is concerned, there are three pole frequencies to be set, one of which—the one associated with CS—depends on the value of the transconductance gm. With a target passband extending well below 1 Hz, we must ensure that all pole frequencies are well below this limit. With the values of the parameters in Table1, the pole frequency associated to CS remains below 10 mHz regardless of the value of gm in the range from 20 to 50 mA/V. The pole associated to CA is set at 2 mHz, while the pole associated to CD is set at 10 mHz with these values, regardless of Electronics 2019, 8, 1197 8 of 14

Electronics 2019, 8, x FOR PEER REVIEW 8 of 14 the actual frequency of the pole due to CS, we can expect an essentially ﬂat response above 200 mHz. responseDepending above on the 200 value mHz. of Dependinggm, with RR on= 50the k valueΩ, the of passband gm, with gainRR = of50 thek low, the noise passband ampliﬁer gain rangesof the lowfrom noise 60 up amplifier to 68 dB ranges (from 1000from up 60 toup 2500). to 68 dB (from 1000 up to 2500).

VDD R RD R CD

VO1 CA J1 OA1 VI

RS C RA S

-VSS A1

R3 R4 R2 R1 VO2

C OA A2 R 3 OA2 A2 A2 VO3 A 3

Figure 4. Detailed schematic of the low noise amplifier (A1) together with the two identical Figure 4. Detailed schematic of the low noise amplifier (A1) together with the two identical auxiliary auxiliary amplifiers (A2, A3) required for gain calibration by means of cross correlation measurements. amplifiers (A2, A3) required for gain calibration by means of cross correlation measurements. The The detailed component list is reported in Table1. detailed component list is reported in Table 1. Table 1. List for the circuit in Figure3. Table 1. List for the circuit in Figure. 3. Component Type Value Component Type Value J1 Low noise discrete JFET IF3601 1 VDD,JV SS 4 Low Ah, lead noise acid discrete batteries JFET IF3601 6 V OA1,V OA2,DD, V OA3SS MOSFET4 Ah, input, lead acid low noise batteries OA TLC070 6 V OA1, OA2,RA OA3 MOSFET1% metallic input, film low resistor noise OA 3.3 TLC070 MΩ R 2 5 kΩ, 1/8 W, 0.1% metallic film in parallel 1 2.5 kΩ SA R × 1% metallic film resistor 1 3.3 M RD 3 5 kΩ, 1/8 W, 0.1% metallic film in parallel 1.67 kΩ RS 2× × 5 k, 1/8 W, 0.1% metallic film in parallel1 2.5 k RR 0.1% metallic film resistor 50 kΩ D 1 CRA 3 × 5 k, 1/8 W,Polyester, 0.1% metallic 66 V film in parallel 221.67µF k CRSR 0.1%Supercapacitor, metallic film 5V resistor 503 F k CCDA Supercapacitor,Polyester, 1266 VV 10 22 mF F R , R 0.1% metallic film resistor 1 kΩ 1 CS3 Supercapacitor, 5V 3 F R2 0.1% metallic film resistor 10 kΩ CD Supercapacitor, 12 V 10 mF R4 0.1% metallic film resistor 100 kΩ 1 3 RCA,2 R 0.1%Polyester, metallic 66film V resistor 10 1µ kF RAR22 1%0.1% metallic metallic film film resistor resistor 1 10 M Ωk R4 1 Resistors are in0.1% parallel metallic to minimize film flicker resistor noise [17]. 100 k CA2 Polyester, 66 V 10 F The actual prototypeRA2 that was implemented1% metallic and used film for resistor testing is shown in Figure 1 M 5. The yellow line superimposed to the1 picture Resistors represents are in parallel the to wiring minimize that flicker needs noise to be [17]. added to connect the gate of the JFET J1 to the inputs of the amplifiers A2 and A3 as in Figure4. Removing the yellow connection allowedThe actual to test prototype each amplifier that was independently implemented of and the used others. for testing A block is shown diagram in ofFigure the measurement5. The yellow linesetup superimposed is also shown to inthe Figure picture6. represents The amplifier, the wiring together that with needs the to battery be added pack to forconnect its supply the gate and of the JFET J1 to the inputs of the amplifiers A2 and A3 as in Figure 4. Removing the yellow connection allowed to test each amplifier independently of the others. A block diagram of the measurement setup is also shown in Figure 6. The amplifier, together with the battery pack for its supply and the

Electronics 2019, 8, 1197 9 of 14 ElectronicsElectronics 2019 2019, 8,, x8 ,FOR x FOR PEER PEER REVIEW REVIEW 9 of9 of14 14

DUT,DUT,the as DUT,as is istypically typically as is typically the the case case thein in LFNMs, case LFNMs, in LFNMs,is isenclosed enclosed is enclosedin in aluminum aluminum in aluminum box box that that acts box acts as that as a shielda actsshield aswith with a shield respect respect with toto interferencesrespect interferences to interferences coming coming from from coming the the environment. fromenvironment. the environment. Th The DUTse DUTs used The used DUTsfor for the usedthe test test forexperiments experiments the test experiments are are simple simple are resistors,resistors,simple but resistors,but in in the the case but case inof theofnoise noise case measurement ofmeasurement noise measurement on on electron electron on devices, electrondevices, the devices,the bias bias system system the bias for for systemthe the device device for the wouldwoulddevice also also would typically typically also typicallybe be contained contained be contained in in the the shielded inshielded the shielded box. box. Three box.Three ThreeBNC BNC connectors BNC connectors connectors are are used areused used to to allow toallow allow connection from the outputs of the amplifiers (VO1, VO2, and VO3) to the input of the acquisition and connectionconnection from from the the outputs outputs of of the the amplifiers ampliﬁers ( (VVOO1,1 ,VVO2O, 2and, and VOV3O) 3to) tothe the input input of ofthe the acquisition acquisition and and elaborationelaborationelaboration system. system. system.

GND GNDGND GNDGND GND V V VO1 OV3 OV2 VO1 O3 O2 R R4 DR R R4 D CD R CD R

OA3 J1 OA1 OA3 J1 OA1 R3R 3 R R SRS ARA R AR2A2 C CA2 AC CA2 A

CSC VDDV S DD R GND 2R OA2 GND 2 OA2 -VSS -VSS R VIV 1R1 I AA A A GNDGND 3 3 A2 2 A1 1

Figure 5. Top view of the implementation of the circuit in Figure 4. All components of amplifiers A1 FigureFigure 5. 5.TopTop view view of of the the implementation implementation of of the the circuit circuit in in Figure Figure 4.4 .All All components components of of amplifiers ampliﬁers AA1 1 and A2 are labeled with the same names used in Figure 4. The yellow path shows the connection that andand AA2 are2 are labeled labeled with with the the same same names names used used in in Figure Figure 4.4 .The The yellow yellow path path shows shows the the connection connection that that needs to be made to connect the gate of the JFET to the inputs of A2 and A3. needsneeds to to be be made made to to connect connect the the gate gate of of the the JFET JFET to to the the inputs inputs of of AA2 and2 and A3A. 3.

AluminumAluminum Box Box

VO3 VO3 V V DDV O2V 6 V DD O2 6 V V O1V GNDGND O1 GND GND DUT 6 V6 V DUT -VSS amplifier enclosure -VSS amplifier enclosure PCI-4462PCI-4462

Figure 6. Block diagram of the measurement setup. Two lead acid 6 V batteries are used to supply the FigureFigure 6. 6.Block Block diagram diagram of ofthe the measurement measurement setup. setup. Two Two le adlead acid acid 6 V 6 batteriesV batteries are are used used to tosupply supply the the amplifier. The batteries, the amplifier and the DUT are all located inside an aluminum box that acts as amplifier.amplifier. The The batteries, batteries, the the amplifier amplifier and and the the DUT DUT ar ear alle all located located inside inside an an aluminum aluminum box box that that acts acts a shield against external interferences. as asa shielda shield against against ex externalternal interferences. interferences. As shown in Figure6, for the acquisition and elaboration of the output signals, we resorted to a As shown in Figure 6, for the acquisition and elaboration of the output signals, we resorted to a NationalAs shown Instruments in Figure 4-channel 6, for the DSA acquisition board (PCI and 4462)elaboration and dedicated of the output software signals, developed we resorted around to thea National Instruments 4-channel DSA board (PCI 4462) and dedicated software developed around the Nationalpublic domain Instruments library 4-channel QLSA [27 DS]. WhileA board the (PCI development 4462) and of dedicated dedicated software software developed is not mandatory, around as the we public domain library QLSA [27]. While the development of dedicated software is not mandatory, as publichave domain discussed library in the QLSA previous [27]. paragraph,While the development resorting to theof dedicated QLSA library software in this is specificnot mandatory, application as we have discussed in the previous paragraph, resorting to the QLSA library in this specific wecan have be extremely discussed convenient. in the previous As discussed paragraph, in [27 ],resorting QLSA operates to the veryQLSA much library like ain large this number specific of application can be extremely convenient. As discussed in [27], QLSA operates very much like a large applicationconventional can DFT be extremely spectrum conven analyzersient. all As receiving discussed the in same [27], input QLSA signal, operates but withvery eachmuch one like operating a large number of conventional DFT spectrum analyzers all receiving the same input signal, but with each numberwith the of same conventional record length DFT butspectrum in a di ffanalyzerserent frequency all receiving range the and, same hence, input with signal, a diff erentbut with resolution each one operating with the same record length but in a different frequency range and, hence, with a onebandwidth. operating In with particular, the same at higherrecord frequencies,length but in the a resolutiondifferent frequency bandwidth ra isnge larger, and, resulting hence, with in time a differentdifferent resolution resolution bandwidth. bandwidth. In In particular, particular, at athigher higher frequencies, frequencies, the the resolution resolution bandwidth bandwidth is is larger,larger, resulting resulting in in time time records records with with shorter shorter duration, duration, while while at atlower lower and and lower lower frequencies, frequencies, the the

Electronics 2019, 8, 1197 10 of 14

Electronics 2019, 8, x FOR PEER REVIEW 10 of 14 records with shorter duration, while at lower and lower frequencies, the resolution bandwidth is proportionallyresolution bandwidth reduced. is All proportionally these virtual analyzersreduced. All operate these in virtual parallel, analyzers and this operate means, in with parallel, reference and tothis the means, calibration with approachreference to we the propose, calibration that approa the higherch we frequency propose, rangesthat the can higher be exploitedfrequency for ranges the calibrationcan be exploited of the gainfor the according calibration to the of the procedure gain according discussed to inthe the procedure previous discussed paragraph in while, the previous at the sameparagraph time, power while, spectra at the same estimation time, atpower the output spectraVO estimation1 can proceed at the with output the proper VO1 can (small) proceed∆f required with the toproper correctly (small) estimate f required power to spectra correctly in the estimate frequency power range spectra below in 1 the Hz. frequency range below 1 Hz. AsAs a a first first test test experiment, experiment, we we used used an an unbiased unbiased 50 50 k kΩresistor resistor as as a a DUT DUT at atthe theinput inputof of thethe system.system. 16 2 TheThe thermal thermal noise noise of of a a 50 50 k Ωkresistor resistor at at room room temperature temperature is aboutis about 29 29 nV nV//√Hz√Hz (8.3 (8.3 10× 10− −16V V/2Hz)/Hz) and and × itit is, is, therefore, therefore, much much larger larger than than the expectedthe expected equivalent equivalent input voltageinput voltage noise of noise the low of noise the low amplifier noise inamplifier the entire in frequency the entire range frequency of interest. range Hence, of interest. regardless Hence, of regardless the actual valueof the of actual the gain, value the of spectrum the gain, atthe the spectrum output V Oat1 thecan output be used V toO1 verifycan be the used shape to verify of the frequencythe shape response.of the frequency The result response. of the estimation The result ofof the the PSD estimation of the noise of the at PSD the of output the noiseVO1 atis the reported output in V FigureO1 is reported7 (black in curve Figure labeled 7 (blackS11 curve). As canlabeled be verified,S11). As thecan curvebe verified, is flat, the indicating curve is a flat, constant indicati frequencyng a constant response frequency from 200 response mHz up from to above 200 mHz 1 kHz, up afterto above which 1 kHz, the PSD after decreases. which the However,PSD decreases. this decrease Howeve canr, this be decrease traced back can tobe thetraced low back pass to filtering the low epassffect duefiltering to the effect capacitance due to the at thecapacitance input of theat the JFET input that, of inthe the JFET passband that, in and the inpassband the configuration and in the inconfiguration Figure4, is essentially in Figure the4, sumis essentially of the gate-to-source the sum of andthe ofgate-to-source the gate-to-drain and capacitancesof the gate-to-drain of the device,capacitances in the orderof the of device, 500 pF. in Indeed, the order an RCof 500 low pF. pass Indeed, filter with an RCR = low50 kpassΩ (the filter resistance with R = of 50 the k DUT(the atresistance the input) of and the DUTC = 500 at the pF input) would and result C = in 500 a pole pF would frequency result of in about a pole 6.4 frequency kHz, consistent of about with 6.4 kHz, the behaviorconsistent for withS11 observed the behavior in Figure for S7.11 All observed spectra in and Figure cross spectra7. All sp shownectra and in Figure cross7 spectraare obtained shown by in resortingFigure 7 toare QLSA. obtained The by sampling resorting frequency to QLSA. is The 51.2 sampling kHz. In thefrequency uppermost is 51.2 frequency kHz. In the range uppermost (above 1frequency kHz), the range resolution (above bandwidth 1 kHz), the∆ fresolutionis 100 Hz, bandwidth while in the f lowestis 100 Hz, frequency while in range, the lowest∆f = frequency25 mHz. Inrange, order  tof = obtain 25 mHz. very In smoothorder to spectraobtain very at very smooth low frequencies, spectra at very averaging low frequencies, for the spectra averaging in Figure for the7 wasspectra carried in Figure on for about7 was 2carried h. However, on for theabout correct 2 h. However, estimate of the the correct gain A estimateV1 was obtained of the gain just A afterV1 was a fewobtained seconds, just as after confirmed a few seconds, from the as plot confirmed of |AV1,M from| vs. the time plot in Figureof |AV81,a.M| vs. time in Figure 8a.

10-7 S 22 -8 |S | |S | S 10 23 12 11

10-9 /Hz) 2

S22 -10 (V 10 S11 VOUT S 10-11 R =50 k DUT |S12| R =100  DUT |S23| 10-12 10-1 100 101 102 103 104 frequency (Hz)

Figure 7. Cross spectra at the outputs VO1, VO2, and VO3 in Figure4. The black curves are obtained withFigureRDUT 7. Cross= 50 kspectraΩ; the grayat the curves outputs refer VO1 to, V theO2, and case VRO3DUT in Figure= 100 Ω4. .TheS11 blackis the curves power are spectral obtained density with (PSD)RDUT = at 50 the k output; the gray of the curves low noiserefer amplifierto the case (A R1DUT); S 22= 100is the . PSDS11 is at the the power output spectral of one density of the auxiliary (PSD) at amplifiersthe output (A of2); theS12 lowis the noise cross amplifier spectrum (A1 between); S22 is the the PSD low at noise the output amplifier of one (A1 )of and the oneauxiliary of the amplifiers auxiliary amplifiers(A2); S12 is ( Athe2); crossS23 is spectrum the cross between spectrum the between low noise the amplifier outputs of (A the1) and two one auxiliary of the amplifiers.auxiliary amplifiers (A2); S23 is the cross spectrum between the outputs of the two auxiliary amplifiers.

To verify the robustness of the approach we propose, the estimation of AV1,M was carried out over two frequency ranges. The circles in Figure 8a are relative to the estimation of the gain performed in the frequency range from 200 Hz to 1 kHz (with f = 6.25 Hz) where the recorded spectra appear to

Electronics 2019, 8, 1197 11 of 14

To verify the robustness of the approach we propose, the estimation of AV1,M was carried out over twoElectronics frequency 2019, 8 ranges., x FOR PEER The REVIEW circles in Figure8a are relative to the estimation of the gain performed11 of 14 in the frequency range from 200 Hz to 1 kHz (with ∆f = 6.25 Hz) where the recorded spectra appear tobe be flat, flat, while while the the squares squares are are relative relative to tothe the estimati estimationon of the of the gain gain in the in thefrequency frequency range range from from 2 to 215 tokHz 15 kHz(with (with f = 100∆f = Hz),100 where Hz), where SI, regardedSI, regarded as the noise as the at noise the input at the of input the gate of the of gatethe JFET, of the changes JFET, changeswith frequency with frequency because becauseof the filtering of the filteringeffect due eff toect the due input to the capacitance input capacitance of the amplifier of the amplifiercombined combinedwith the resistance with the resistance of the DUT, of the as previously DUT, as previously discussed. discussed. A careful A observation careful observation of Figure of 8a Figure indicates8a indicatesthat the thattime the required time required for the for correct the correct estimation estimation of AV of1 isA Vshorter,1 is shorter, as expected, as expected, in the in thecase case of ofthe theestimation estimation with with the the larger larger resolu resolutiontion bandwidth. bandwidth. In In any any case, case, afte afterr 2020 seconds,seconds, thethe two two estimates estimates essentiallyessentially coincide coincide ( A(AVV11 == 2160),2160), with with a adifference difference between between the the two two of ofless less than than 0.3%. 0.3%. This This result result also alsodemonstrates demonstrates that that the thegain gain from from the thegate gate of the of theJFET JFET toward toward the theoutput output is flat is flatup to up frequencies to frequencies well wellabove above 10 kHz. 10 kHz. Note Note that that the theability ability to obtain to obtain the thecorrect correct estimate estimate of the of thegain gain after after a few a few seconds, seconds, also alsoin the in thecase case of a ofrelatively a relatively small small f, is ∆duef, is to due the tofact the that fact the that noise the introduced noise introduced by the DUT by the is also DUT large is alsowith large respect with to respect the EIVN to the of EIVN the amplifiers of the amplifiers A2 andA 2Aand3 in Athe3 in frequency the frequency range range in which in which the thegain gainestimation estimation was wasperformed. performed. In order In order to test to our test a ourpproach approach in a much in a much more moredemanding demanding case, we case, used we a used100 a100 resistanceΩ resistance as a DUT. as a DUT. The thermal The thermal noise noise of a of100 a  100 resistorΩ resistor at room at room temperature temperature is about is about 1.29 1.29nV/ nV√Hz/√ Hz(≈ 1.66 ( 1.66 × 10−1810 V2/Hz).18 V2/ InHz terms). In terms of PSD, of PSD,this noise this noiselevel levelis about is about 30 times 30 times smaller smaller than thanthe EIVN the ≈ × − EIVNof A2 ofandA2 Aand3 forA f3 =for 1 kHz,f = 1 and kHz, it andis in it the is in same the sameorder order of magnitude of magnitude of the of BN the of BN the of best the bestlow noise low noiseamplifiers amplifiers for LFNM for LFNM applications. applications. The The spectra spectra ob obtainedtained in in this this case, case, in thethe samesame measurement measurement conditionsconditions as as before, before, are are reported reported in in Figure Figure7 (gray7 (gra curves).y curves). To To avoid avoid complicating complicating the the figure, figure, the the plot plot ofof the the cross cross spectra spectra is, is, in in this this case, case, shown shown only only in in the the highest highest frequency frequency range, range, that that is, is, the the one one used used V1 M forfor gain gain calibration. calibration. The The behavior behavior of ofA AV1,M, vs.vs. timetime isis shownshown inin FigureFigure8 b.8b. As As can can be be noted, noted, in in this this I 2 3 situation,situation, since sinceSI Sis veryis very small small compared compared to the to EIVNthe EIVN of A 2ofand A Aand3 even A even at higher at higher frequencies, frequencies, we need we toneed wait to a fewwait minutes a few minutes before reaching before reaching the situation the situation in which in the which estimated the estimated value no longervalue no changes longer withchanges time. with In any time. case, In any after case, 2 min, after the 2 estimatedmin, the esti valuemated sets value are withinsets are within2% of the±2% asymptotic of the asymptotic value. ± Therefore,value. Therefore, even in this even demanding in this demanding situation, we situatio obtainn, the we conclusion obtain the that conclusion a quite good that estimate a quite of good the amplifierestimate gainof the can amplifier be reached gain in can a fraction be reached of the in time a fraction required of forthe thetime measurement required for of the the measurement DUT noise. of the DUT noise. 2.5 2.5

V R =100  V 2.0 2.0 DUT A A f=100 Hz; R = 50 k | / | / | / DUT

1.5 M 1.5 2 kHz to15 kHz M 1, 1, V V f =100 Hz,  |A

|A 1.0 1.0 2 kHz to 15 kHz

0.5 f =6.25 Hz 0.5 200 Hz to 1kHz 0.0 0.0 0.1 1 10 100 1000 0.1 1 10 100 1000 meas. time (s) meas. time (s)

(a) (b)

FigureFigure 8. 8.Gain Gain of of the the low low noise noise ampliﬁer amplifier (A (VA1,MV1,M)) normalized normalized with with respect respect to to the the gain gainA AV Vof of the the auxiliary auxiliary ampliﬁersamplifiers vs. vs. measurement measurement time time according according to to Equation Equation (10). (10). The The plot plot on on the the left left (a ()a refers) refers to to the the case case RDUTRDUT= = 50 kΩ;the plot on the right (b) refers to the casecase RDUT == 100100 Ω. .

WithWith the the value value of of the the calibrated calibrated gain gain as as obtained obtained from from Figure Figure8, we 8, obtainwe obtain the equivalentthe equivalent noise noise at theat inputthe input of the of amplifier, the amplifier, as reported as reported in Figure in 9Figu. Notere that9. Note our approachthat our approach requires a requires DUT to generatea DUT to noisegenerate that isnoise present that atis thepresent input at of the the input amplifier of the to am calibrateplifier to the calibrate gain. In the order gain. to evaluateIn order to the evaluate BN of thethe low BN noiseof the amplifier, low noise that amplifier, is, the equivalentthat is, the inputequivalent noise input when noise the input when is the shorted, input is we shorted, used the we valueused ofthe the value gain of obtained the gain inobtained the case in in the which case thein which DUT wasthe DUT a 100 wasΩ resistance. a 100  resistance. To ensure To that ensure no appreciablethat no appreciable change in change the gain in could the gain occur, could the measurementoccur, the measurement with the shorted with inputthe shorted was performed input was performed immediately after the measurement with a 100  DUT was completed. As can be noted in Figure 9, the low noise amplifier is characterized by excellent noise performances, especially when we take into account its very simple structure. The EIVN at 200 mHz is about 6 nV/√Hz and becomes less than 1 nV/√Hz above 2 Hz.

Electronics 2019, 8, 1197 12 of 14 immediately after the measurement with a 100 Ω DUT was completed. As can be noted in Figure9, the low noise ampliﬁer is characterized by excellent noise performances, especially when we take into accountElectronics its 2019 very, 8,simple x FOR PEER structure. REVIEWThe EIVN at 200 mHz is about 6 nV/√Hz and becomes less than12 of 14 1 nV/√Hz above 2 Hz.

10-14

R =50 k 10-15 DUT

10-16 /Hz) 2

-17

(V 10 R =100  VIN DUT S 10-18 R = 0 DUT  10-19 10-1 100 101 102 103 104 frequency (Hz)

Figure 9. Referred noise obtained dividing the noise at the output VO1 by the calibrated gain squared. In theFigure case 9. of ReferredRDUT = 100,noise the obtained measured dividing noise is the above noise the at idealthe output one because VO1 by at the these calibrated low levels gain of squared. noise, theIn background the case of R noiseDUT = (BN)100, the is no measured longer negligible. noise is above the ideal one because at these low levels of noise, the background noise (BN) is no longer negligible. While the main purpose of this work was to demonstrate the effectiveness of the approach we propose,While it is worththe main noting purpose that the of performancesthis work was of to the demonstrate amplifier we the have effectiveness designed, of in termsthe approach of EIVN, we comparepropose, very it is well worth with noting other that designs the performances resorting to the of IF-3601the amplifier as the we active have device. designed, A comparison in terms of with EIVN, twocompare of such very designs well is with summarized other designs in Table resorting2. to the IF-3601 as the active device. A comparison with two of such designs is summarized in Table 2. Table 2. Comparison of the prototype in this work with other works. Table 2. Comparison of the prototype in this work with other works. Equivalent Input Voltage Noise (nV/√Hz) JFET Bias Offline ULNA 200 mHzEquivalent 1 Hz 10Input Hz Voltage 100 Hz Noise 1 kHz (nV/√ 10Hz) kHz Current Cal. * JFET Bias ULNA 200 1 10 Offline This work 6 1.7 0.7 0.6100 Hz 0.5 1 kHz 100.5 kHz 2.5Current mA NO Levinzon [22] 4mHz 1.4Hz 0.6Hz 0.6 0.5 0.5 5 mA YEScal. * CannatThisà et work al. [18 ] 6 6 1.4 1.7 1 0.7 0.9 0.6 0.8 0.5 1.1 0.5 4 2.5 mA mA NO NO Levinzon* Gain [22] calibration 4 must be performed 1.4 0.6 with a dedicate 0.6 measurement 0.5 with 0.5 a calibrated source. 5 mA YES Cannatà et al. NO The design in [22] relies6 on a single1.4 JFET1 stage and0.9 requires0.8 either1.1 dedicated 4 measurements mA for [18] gain calibration or acting on a variable component in the circuit for gain adjustment. As can be noted, * Gain calibration must be performed with a dedicate measurement with a calibrated source. noise performances are essentially comparable, notwithstanding the fact that the bias current for the JFET inThe our design isin smaller[22] relies than on the a single one used JFET in stag [22]e (theand equivalent requires either input dedicated noise for ameasurements JFET decreases for forgain increasing calibration bias current).or acting Theon a designvariable in component [18] relies on in a the feedback circuit configurationfor gain adjustment. employing As can an IFbe 3602noted, (a pairnoise of performances IF3601 in a single are essentially device) for comparable, the differential notwit inputhstanding stage. In the this fact case, that we the can bias observe current a lower for the levelJFET of noisein our in design the proposed is smaller design than forthe a one lower used bias in current. [22] (the It equivalent must be noted input that noise at higher for a JFET frequencies, decreases thefor EIVN increasing in [18] increasesbias current). as a The result design of the in eff [18]ect ofrelies the compensationon a feedback networkconfiguration employed employing to insure an IF stability.3602 (a Inpair any of case, IF3601 what in a we single believe device) sets ourfor the approach differential apart input from stage. others In is this its degree case, we of flexibility.can observe Thea lower fact that level we of do noise not resort in the to proposed a feedback design configuration for a lower greatly bias current. simplifies It themust design be noted of the that low at noise higher amplifier.frequencies, Different the EIVN JFETs in with [18] diincreasesfferent characteristics as a result of the of noiseeffect andof the input compensation capacitances network can be employed used to to insure stability. In any case, what we believe sets our approach apart from others is its degree of flexibility. The fact that we do not resort to a feedback configuration greatly simplifies the design of the low noise amplifier. Different JFETs with different characteristics of noise and input capacitances can be used to quickly and effectively design the low noise amplifier: the same measurement configuration, with the same auxiliary amplifiers can be used for obtaining gain calibration while performing measurements. In the same way, different OAs can be used for the auxiliary amplifiers

Electronics 2019, 8, 1197 13 of 14 quickly and effectively design the low noise amplifier: the same measurement configuration, with the same auxiliary amplifiers can be used for obtaining gain calibration while performing measurements. In the same way, different OAs can be used for the auxiliary amplifiers as long as they are characterized by low input noise current. While the auxiliary amplifiers are used at higher frequencies, resorting to JFET input OAs that have lower low frequency corners would allow, in many cases, a reduction in the time required for gain calibration by extending the frequency interval used for the estimation of the gain toward lower frequencies according to Equation (10). As a final remark, it must be noted that the gain calibration approach we propose works as long as the noise level introduced by the DUT in the frequency interval used for gain estimation is non-negligible. In the rare situations in which this condition cannot be satisfied, we would, however, not be worse off than in the case of the amplifier in [22], with the advantage that the presence of the auxiliary amplifiers would still allow the measurement of the gain of the low noise amplifier by connecting a simple resistor at the input of the system prior to the measurement on the actual DUT.

4. Conclusions We proposed an approach for the design of low noise voltage preamplifiers for application in the field of low frequency noise measurements on electron devices. We resorted to an open loop design for the amplifier front-end with the advantage of reducing the number of active components introducing noise and avoiding compensation networks for insuring stability. We have demonstrated remarkable noise performance with an extremely simple and compact design (only 9 components for the main amplifier, including the coupling networks). The problem of gain calibration, which is typical of an open loop design, has been successfully addressed by introducing two simple auxiliary OA-based voltage amplifiers with stable and known gain and resorting to cross correlation to extract the amplifier gain, even while actual measurements on a DUT are in progress.

Author Contributions: Conceptualization, G.S., and C.C.; methodology, G.S. and C.C.; software, C.C.; validation, G.G.; formal analysis, G.G. and C.C.; investigation, G.S.; data curation, C.C.; writing—original draft preparation, G.S. and C.C. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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