# Thermal Noise-Boosting Effects in Hot-Wire-Based Micro Sensors

Authors: Massimo Piotto; Alessandro Catania; Andrea Nannini; Paolo Bruschi
Journal: Journal of Sensors
 (2020)

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## Abstract

This article proposes an original approach aimed at modelling the noise density in sensors based on a single hot wire or pairs of thermally coupled wires. The model consists in an original combination of a previous electrothermal model of the wire with well-established assumptions on the thermal noise in conductors that carry moderate current densities. A simple method for estimating the model parameters with simple impedance spectroscopy is suggested. The predicted power spectral densities of the wire thermal noise differ from the result of previously presented analytical models, stimulating further experimental studies. The effects of the electrothermal feedback of both hot wires and hot-wire pairs on flicker noise is also intrinsically covered by the proposed approach.


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## Body

## 1. Introduction

Hot wires are used in a large variety of sensing devices and instruments. They consist of thin wires of an electrical conductor that, once biased with a sufficiently large current, reach a temperature that is significantly larger than ambient temperature. Hot wires are generally suspended at the extremities and immersed in a fluid. The category of hot-wire devices includes hot films, which consist of a thin or thick conducting stripe deposited over a thermally and electrically insulating substrate. The temperature difference between the wire and the fluid (overheating) is detected exploiting the dependence of the electrical resistance on temperature. Measuring the overheating, it is possible to detect several quantities of interest. Sensors based on hot wires gained significant importance since they can be easily fabricated using MEMS (Microelectromechanical Systems) technologies that allowed extreme miniaturization leading to reduction of the power consumption and response time of up to three orders of magnitude with respect to traditional macroscopic devices.Among the sensors that exploit this principle are vacuum sensors [1–3], gas concentration sensors [4], thermal conductivity probes [5], and anemometers [6–8]. An evolution of the single hot wire is represented by the pair of thermally coupled wires. These devices are formed by two hot wires placed at micrometric distances from each other, so that substantial thermal exchange occurs between them. Such an arrangement allows detection of both the magnitude and direction of airflows [9]. Recently, wire pairs with thermal mass as small as to allow temperature variations with frequencies up to several kHz have been used to detect the local fluid displacement induced by an acoustic wave [10–13]. In particular, this new class of sensors is capable of directly detecting the acoustic particle velocity (APV), enabling interesting applications that cannot be easily achieved with standard microphones [14]. In all the mentioned applications of hot wires, it is of primary importance to model the electrical noise produced by the wire, in order to estimate the actual resolution of the sensors. This aspect is very critical for the APV sensors, which are marked by relatively low sensitivities resulting in low signal-to-noise ratios even in the presence of large sound intensities.Noise in electrical conductors supplied with a dc current is due to two main phenomena, namely, thermal agitation of the charge carriers (Johnson-Nyquist noise or thermal noise) and resistance fluctuations, resulting in the well-known flicker noise. Thermal noise is universal and is related only to the wire resistance and temperature, while flicker noise is strongly material-dependent. Both types of noise cause fluctuations of the current and voltage of the wire, resulting in small fluctuations of the heating power.In hot wires, these power fluctuations have to be taken into account because they generate nonnegligible temperature fluctuations, due to the high thermal isolation of the conductor. The high TCR (temperature coefficient of resistance) of the wire material transforms the temperature fluctuations again into voltage and current variations, in a loop. The effect of this thermoelectrical feedback on the noise density has not been studied extensively so far, maybe because the thermal capacity of macroscopic hot wires allows development of temperature fluctuations only in the range of ultralow frequencies (typically sub-1 Hz). This is no more the case for MEMS sensors, where cut-off frequencies in the kHz range are common. This electrothermal feedback was described previously by Kohl et al. [15], highlighting its consequences on the sensitivity and noise of metal film resistance bolometers. The same phenomenon was observed earlier in bolometers based on superconductor materials [16].It should be recalled that the temperature of a body is subjected to temperature fluctuation even when the cited electrothermal cause is not present. This type of “natural” temperature fluctuations is due to the so-called phonon noise and has a total mean square value equal to<δT2>=kBT2/CTH, where T is the body temperature, kB the Boltzmann constant, and CTH the thermal capacity of the body. Obviously, even these temperature fluctuations produce resistance fluctuations, resulting in voltage noise when the wire is biased with an electrical current. Furthermore, resistance fluctuations modulate the heating power, so that, in thermally insulated wires, the electrothermal feedback affects also phonon noise. This work is focused on the changes produced by the electrothermal feedback on thermal noise and flicker noise, with particular emphasis on the former. Therefore, analysis of phonon noise is out of the scope of this paper, although its contribution can be dominant in the case of materials with particularly high TCRs [17].In Kohl et al.’s work [15], a model that implicitly takes into account the effects of the feedback on thermal noise was proposed. The limit of the approach proposed in [15] is, in our opinion, the use of an expression for the thermal noise voltage density that is valid only for electrical networks in perfect thermal equilibrium.In this paper, we present an alternative noise model that starts from well-established assumptions on the very basic phenomena that generate current fluctuations in conductors and then applies the electrothermal feedback in a direct and rigorous way. Different noise expressions are found for both single hot wires and pairs of thermally coupled wires. Depending on the sign of the TCR and the type of biasing used for the wire (e.g., constant voltage or constant current), the model predicts significant modifications of the noise spectra with respect to the case of standard conductors, where the low thermal insulation prevents the development of significant self-heating, disrupting the quoted feedback mechanism. The main differences with respect to the previously proposed approach [15] are highlighted. This manuscript does not include experimental results but, on the other hand, intends to be a stimulus for the execution of measurements that, if interpreted on the basis of the model prediction, could add useful information for the design of hot-wire sensors.


## 2. Electrothermal Model for Suspended Microwires

### 2.1. Noise in Wires out of Thermal Equilibrium

In thermal equilibrium, an electrical conductor will produce only thermal noise. In a Thevenin equivalent, a noiseless resistor is placed in series to a voltage noise source of PSD (Power Spectral Density) equal to4kBTRS, where kB is the Boltzmann constant, T the absolute temperature, and RS the conductor resistance. The Norton equivalent circuit consists of a noiseless resistor in parallel with a noise current source of PSD equal to 4kBTGS, where G=1/RS. The two circuits are recalled in Figures 1(a) and 1(b). The first theoretical derivation of these noise models, developed by H. Nyquist using thermodynamic arguments, dates back to 1928.Figure 1
Thevenin (a) and Norton (b) representations of the noise in a resistor.
(a)
(b)Use of the above model to represent the noise in hot wires is not rigorous for two reasons. First, the voltage vs. current dependence (V‐I curve) is not linear, due to self-heating and the relatively large TCR, so that defining the wire resistance is not straightforward. Second, a hot wire in operating conditions carries a nonzero electrical current and then is not in thermal equilibrium.Let us start from the nonlinearity problem and consider Figure2(a), where the V‐I characteristics of a wire subjected to self-heating is sketched for the case of positive TCR.Figure 2
Sketched characteristics of a thermally insulated wire with positive TCR: (a)V‐I plots; (b) magnitude of the small-signal impedance as a function of frequency.
(a)
(b)For any given operating point, marked by currentI and voltage V, we can define two resistances, namely, a large signal resistance, R, and a small-signal resistance, rDC:
(1)R=V/I,(2)rDC=dVdII,V.As the current increases, so does the overheating and, due to the assumption of positive TCR, alsoR increases, producing the nonlinear behaviour of the V‐I characteristic shown in Figure 2(a). On the other hand, rDC is the small-signal equivalent resistance. This resistance is applicable for variations around the operating point that are either constant (dc components) or slow (low frequencies) so as to allow the overheating to follow the heating power. At high frequencies, the thermal mass of the wire dampens the temperature variations and the equivalent small-signal resistance asymptotically tends to R. In between, the magnitude of the small-signal (complex) impedance, indicated with r, should reasonably follow a bode plot like that depicted in Figure 2(b). The exact expression of the small-signal impedance, which is in agreement with this intuitive behaviour, was derived in a previous study [15], which deals with thermistor-based bolometers.Since noise consists in small current and voltage variations around the operating point, the equivalent circuits of Figure1 are applicable to the wire once Rs is replaced with the small-signal impedance r. What is less obvious is the expression of the spectral density to be assigned to vn or in noise sources that appear in Figures 1(a) and 1(b), respectively, due to the mentioned nonequilibrium condition. It has been shown with either a semiclassical [18] or quantum mechanics [19] approach that the current fluctuation due to thermal agitation of the carriers has a spectral density that does not change when the conductor is subjected to a moderate dc current with respect to the case of thermal equilibrium. The reason is that the carrier drift velocity is much smaller than the thermal velocity in most practical cases of metallic conductors, so that the random walks of the carriers, from which the current fluctuation originates, are practically unaltered. These random walks depend on the same scattering mechanism from which the resistance R originates. Therefore, it is reasonable to consider that the current fluctuation has a spectral density equal to 4kBT/R. However, using this spectral density for the current source in the equivalent model of Figure 1(b) means neglecting the mentioned feedback effect that occurs in the hot wire. If the wire is biased with a constant voltage, a current change due to the random agitation of the carriers produces a variation of the heating power and, in turn, a change of the wire temperature and resistance R. The resulting resistance variation produces an additional current contribution that reinforces or diminishes the original current change depending on the sign of the TCR. Then, due to the wire self-heating, the actual current fluctuations seen from the wire terminals are different from those that would be predicted assigning the density 4kBT/R to the current source in the circuit of Figure 1(b).This electrothermal feedback effect was described in previous works on bolometers [15, 16], where the interest was mainly to model the effects on the device sensitivity. The feedback was also considered for its effect on thermal noise, and the authors simply propose to calculate the voltage noise power density as 4kBTRer, where “Rer” indicates the real part of impedance r. This approach was demonstrated to be applicable to RLC (resistance, inductance, and capacitance) networks in thermal equilibrium, but there is no physical justification for use of it for out-of-equilibrium systems in the presence of electrothermal feedback.The alternative approach proposed in this work starts from the following equation for the total current (I) through a conductor subjected to a voltage V, valid also in the presence of self-heating:
(3)I=VR−ie⟺V=IR+Rie,where ie is the thermal noise component, marked by the usual spectral density SIT=4kBT/R, and R is simply defined by the ratio V/Idrift, where Idrift is the current component due only to the electric field in the conductor. Notice that the temperature dependence of R is the cause of the electrothermal feedback. To complete the framework, we consider also that R is subjected also to random fluctuations that would be present also in the case of a perfectly constant temperature. These fluctuations, indicated with δRe, are the cause of flicker noise and can be boosted by the feedback as thermal noise. These are the premises that will be used in the next subsections to derive a model of the noise in single hot wires and in thermally coupled pairs of hot wires. All the models used in this work are of the lumped-element type. Therefore, quantities such as the wire temperature will represent averages calculated along the wire length.


### 2.2. Electrothermal Model of a Single Hot Wire

Figure3 schematically shows the elements of the single wire model.Figure 3
Elements of the electrothermal model for a single wire.In the electrical domain, the wire is represented by voltageV across its terminals and current I flowing through it. The two quantities are tied by the electrical resistance R, as shown by Equations (1) and (2). In the thermal domain, the wire is characterized by its absolute temperature T, by its thermal capacity CTH, and by thermal conductivity θ from the wire to the environment, which is considered to be at uniform temperature TA. Thermal conductivity θ includes (i) heat conduction along the solid suspension elements, (ii) conduction and convection through the surrounding fluid, and (iii) radiation. We will consider that both R and θ are temperature-dependent quantities, while CTH will be considered constant. Heat balance in the wire body will require that
(4)CTHdTdt=W−θT−TA,where W is the total heating power dissipated by the electrical current, simply given by
(5)W=V·I.In order to calculate the effects of fluctuationsie and δRe, it is convenient to use a small-signal analysis of the wire around a static operating point (dc components). Then, we can write
(6)V=V0+v,I=I0+i,T=T0+δT,W=W0+w,where V0, I0, T0, and W0=V0I0 define the operating point (OP) and v, i, δT, and w are the corresponding variations around the OP. We use lower-case symbols for v, i, and w since it is customary in small-circuit analysis of electrical circuits. For all other variations, we have used the prefix “δ.” Taking into account the temperature dependence of R and θ, their first-order approximations can be written as
(7)R=R0+αR0δT+δRe,θ=θ0+βθ0δTwithα=1RdRdT,β=1θdθdT,where we have included also possible temperature independent fluctuations (δRe) of R. Note that α is the TCR of the wire. Thermal equilibrium in the OP requires
(8)W0=I0V0=θ0T0−TA.With the above definitions, the following equation for variations in the Laplace domain can be derived from Equation (4):
(9)sCTHδT=w−βθ0T−TAδT−θ0δT,from which we find
(10)δT=γsw,(11)γs=γ01+s/ωg,(12)γ0=1θ01+βT−TA,(13)ωg=1CTHγ0.In terms of variations, Equations (3) and (5) become
(14)v=iR0+αR0δT+δReI0+R0ie,(15)w=iV0+vI0.Combining Equations (10)–(15), we can find with elementary algebraic passages
(16)v=iR01+αV0I0γs1−αR0I02γs+R01−αR0I02γsie+I01−αR0I02γsδRe,which can be synthetically written in the following way:
(17)v=i·r+ie·re+kRδRe,where coefficients r, re, and kR depend on the complex frequency s. This dependence can be made more explicit, by substituting the expression of γs from Equation (11) into Equation (16) obtaining
(18)r=R01+αV0I0γ01−αV0I0γ0·1+s/ωz1+s/ωp,(19)re=R011−αV0I0γ0·1+s/ωg1+s/ωp,(20)kR=I01−αV0I0γ0·1+s/ωg1+s/ωp,where the following angular frequencies have been introduced:
(21)ωz=ωG1+αW0γ0,(22)ωp=ωG1−αW0γ0.Equations (17)–(22) can be used to describe the small-signal behaviour of the wire. As far as the small-signal ac impedance of the wire is concerned, this is given by parameter r. It can be easily shown that for a positive TCR, the magnitude of r depends on the frequency as in Figure 2(b), with fp=ωp/2π and fz=ωz/2π. An equivalent expression was found in [15].Equation (17) can be used to find Thevenin and Norton representations of the wire noise. Referring to Figures 1(a) and 1(b), obviously, we have to replace resistor RS with the complex impedance r in both the Thevenin and Norton circuits, in order to correctly model the ac behaviour of the wire. In the case of the Thevenin equivalent circuit, the voltage of the equivalent source (Vn in Figure 1(a)) can be calculated by nulling the small-signal current through the wire terminals (i.e., we set i=0 in Equation (17)). This represents the case of a wire biased with a constant current (zero variations). Then, the PSD of the Thevenin equivalent noise source is given by
(23)Svn=4kBT0R0re2+SδRefkR2,where SδRe is the PSD of resistance fluctuations δRe, while the PSD of ie was assumed to be equal to 4kBT0/R0 as anticipated in the previous subsection. At high frequency, re tends to R0, so that the thermal noise voltage contribution is simply 4kBT0R0. Since also the complex impedance of the wire tends to R0, the high-frequency limit simply corresponds to calculating the noise PSD using the usual Johnson-Nyquist expression. At very low frequencies (f≪fp), the thermal component of the voltage PSD becomes
(24)Svn−th=4kBT0R011−αV0I0γ02.For a positive TCR (α>0), the denominator can approach zero when the static heating power W0=V0I0 is large enough. In that case, the voltage noise PSD can get much larger than the high-frequency limit. It can be easily verified that the low-frequency limit given by Equation (24) does not coincide with the value obtained by applying the Jonson-Nyquist expression with the low-frequency limit of the wire resistance (rDC) given by Equation (18) for s=0.A similar boosting effect can be observed for the flicker component, represented by the resistance fluctuation component. At high frequency,kR tends to I0, so that the voltage noise is simply given by the product of the resistance fluctuations and the bias current I0. At low frequency, this term is boosted by the same factor as the thermal noise.The noise voltage density of the Thevenin model represents the actual noise that should be expected when the wire is biased at constant current. It is interesting to observe that the boosting factor tends to infinity whenαV0I0γ0 tends to one. If αV0I0γ0 gets larger than one, the pole sp=‐ωp in the denominator of both re and kR becomes positive, denoting instability. In these conditions, a small variation due to noise triggers a catastrophic increase of the voltage, leading to failure. This phenomenon is the well-known thermal runaway. It is possible to find a notable expression for αV0I0γ0 when thermal conductance θ is assumed to be independent of temperature (β=0). In that case, γ0=1/θ0 and αV0I0γ0=αT0−TA. As a result, the noise-boosting factor 1−αV0I0γ0−1 tends to infinity when the product of the wire overheating T0−TA by the TCR tends to one. Thermal runaway occurs when this product is equal to or greater than one. For a negative TCR, Equation (24) predicts a noise reduction at low frequencies with respect to the 4kBT0R0 limit and no thermal runaway.The situation is reversed for the Norton equivalent model. The noise current source of the Norton model gives the current fluctuations when the wire is biased at constant voltage. Its value can be found settingv=0 in Equation (17) and solving for current variations i, finding
(25)in=−i=rerie+kRrδRe.Ratiosre/r and kR/r are now marked by a pole equal to −ωz. Instability, i.e., thermal runaway, occurs when the TCR is negative. This is a well-known difference between constant-current and constant-voltage biasing. Similarly, current noise boosting at low frequency occurs only for α<0. In particular, the thermal component of the noise current PSD in the low-frequency limit that can be found from Equation (25) is
(26)Sin−th=4kBT0R011+αV0I0γ02,where 4kBT0/R0 is the high-frequency limit which coincides with the PSD of the current fluctuation ie that appears in Equation (1).


### 2.3. Electrothermal Model of Two Thermally Coupled Wires

The elements of the model used for the pair of thermally coupled wires are illustrated in Figure4.Figure 4
Elements of the electrothermal model for a thermally coupled wire pair.The thermal and electrical quantities (V, I, R, T, and θ) are duplicated and an additional coupling thermal conductance θ12 is present. In this study, we assume that the system is symmetrical (the two wires are symmetrical) and are biased in such a way that they reach a symmetrical OP, which is defined by the following values for the quantities of interest:
(27)V1=V2=V0,I1=I2=I0,T1=T2=T0,R1=R2=R0,θ1=θ2=θ0.In the majority of applications, thermally coupled wires are biased with a symmetrical operating point. On the other hand, variations are different between the two wires, so that the total values (OP plus variations) of the key quantities will be different from one wire to the other.Equation (4), pertinent to a single wire, is replaced by the following set:
(28)CTHdT1dt=W1−θ1T1−TA−θ12T1−T2,CTHdT2dt=W2−θ2T2−TA+θ12T1−T2.In Equation (28), a heat exchange term due to conductance θ12 is clearly present. In terms of variations and Laplace transforms, we obtain
(29)sCTHδT1=w1−θ0+βθ0T0−TAδT1−θ12δT1−δT2,sCTHδT2=w2−θ0+βθ0T0−TAδT2+θ12δT1−δT2.Heat generation in the wires is described by the following set:(30)w1=v1I0+i1V0,w2=v2I0+i2V0,while Equation (14) (Ohm’s law plus Jonson-Nyquist current fluctuations) for the two wires is replaced by the following set:
(31)I1=V1R1−ie1,I2=V2R2−ie2,⟺V1=I1R1+R1ie1,V2=I2R2+R2ie2,which, in terms of variations, becomes
(32)v1=i1R0+αR0I0δT1+I0δRe1+R0ie1,v2=i2R0+αR0I0δT2+I0δRe2+R0ie2.At this point, it is convenient to separate all variations into a differential mode and common mode component. For example, we will replacev1 and v2 by vd=v1‐v2 and vc=v1+v2/2, respectively, where vd is the differential component and vc the common mode one. In the remainder of this document, differential mode and common mode quantities will be indicated with the “d” and “c” subscript, respectively. In this way, we can easily find decoupled equations for the differential mode and common mode variables. For the common mode components, the resulting equations are identical to those of the single wire. This is reasonable, since common mode components do not break the symmetry, and with no temperature difference, the two wires do not interact and behave as a single wire. Equations for the differential components are only slightly different, due to the presence of θ12, so that from the Equation set (29), we can derive the following differential mode equation:
(33)sCTHδTd=wd−θ0+βθ0T0−TAδTd−2θ12δTd,while from the Equation sets (30) and (32), we find differential mode equations that are identical to Equations (14) and (15), respectively:
(34)vd=idR0+αR0I0δTd+I0δRed+R0ied,wd=vdI0+idV0.From Equation (33), we can find the expression of δTd, analogous to Equation (10):
(35)δTd=γdswd,(36)γds=γ0d1+s/ωgd,(37)γ0d=1θ01+βT0−TA+2θ12,(38)ωgd=1CTHγ0d.Since Equations (34)–(38) are formally equivalent to the equations of the single wire, with the sole difference of the expression of γ0d (that replaces γ0), the solution has the same form as Equation (17):
(39)vd=id·rd+ied·red+kRdδRed,where
(40)rd=R01+αV0I0γ0d1−αV0I0γ0d·1+s/ωzd1+s/ωpd,red=R011−αV0I0γ0d·1+s/ωgd1+s/ωpd,kRd=I01−αV0I0γ0d·1+s/ωgd1+s/ωpd,ωzd=ωgd1+αW0γ0d,ωpd=ωgd1−αW0γ0d.In order to explain the way to use Equation (39) in a practical case, we can consider the very common situation in which the wires are biased with identical constant currents and the output signal is the voltage difference vd=v1‐v2. In the next subsection, we will analyse the more complex case of a full Wheatstone bridge. With constant-current bias, both i1 and i2 are forced to be zero and then also id is zero. The forcing terms, i.e., Jonson-Nyquist current fluctuations ie1 and ie2 and resistance fluctuations δRe1 and δRe2, are split into their common mode and differential mode components. Thanks to linearity of the small-signal equations and the fact that the equations are decoupled, we can calculate the effects of the two modes separately and then add them up. Common mode forcing terms produce only common mode variations, thus the effect on vout=vd is null. Therefore, we can focus only on the differential mode terms. From Equation (39) with id=0 and considering that ie1, ie2, δRe1, and δRe2 are independent stochastic processes, we can find the PSD of the output voltage:
(41)Svdf=2Siefred2+2SδRekRd2,where Sief=4kBT0/R0 and SδRef are the PSD of the Johnson-Nyquist (thermal) current fluctuations and resistance fluctuations, respectively, of each single wire. Again, as in the case of the single wire, a boosting effect of both thermal and flicker noise at low frequencies is predicted if the TCR is positive. For the same heating power (V0I0) and then the same overheating, we can expect a smaller noise increase with respect to the case of the single wire, due to the presence of the additional coupling term θ12 in γ0d, compared to γ0 of the single wire.


### 2.4. Noise in Wheatstone Bridges of Thermally Coupled Wire Pairs

Often, thermally coupled wire pairs are connected to form Wheatstone bridges as shown in Figure5(a), where the two wire pairs are represented by resistors pairs R1 and R2 and R3 and R4. Since the aim is generally to sense a physical quantity, connection should be made in such a way that the effect of the quantity of interest contributes to the output voltage in a constructive way. The bridge can be biased with different approaches, which are all equivalent in terms of the effect of noise on the output voltage. In the example of Figure 5, the bridge is biased by a dc voltage source VA (e.g., a battery) and a resistor RB, which forms a voltage divider with the bridge resistance setting the operating voltage of the bridge to VB.Figure 5
(a) Wheatstone bridge formed by two thermally coupled wire pairs; (b) equivalent circuit for differential mode variations.
(a)
(b)The equivalent circuit for differential mode variations is represented in Figure5(b), where the differential currents of the two wire pairs are indicated. Equation (39) for the two wire pairs becomes
(42)vd=id·rd+ied·red+kRdδRed,vd′=id′·rd+ied′·red+kRdδRed′,where variables with a prime character vd′, id′, ied′, and δRed′ belong to the upper wire pair of the bridge (R3, R4). By elementary analysis of the circuit in Figure 5(b), it is possible to find that vd′=vd and id′=‐id. Solving Equation set (42) with these relationships gives
(43)2vd=ied−ied′·red+kRdδRed−δRed′.Considering thatied, ied′, δRed, and δRed′ can be reasonably considered independent stochastic processes, and recalling the dependence of each one of these quantities with current fluctuations (ie) and resistance fluctuations (δRe) of the single wires of the bridge, we can finally find
(44)Svdf=Siefred2+SδRekRd2,where, again, Sief=4kBT/R0. Equation (44) indicates that the output noise PSD of a Wheatstone bridge is subjected to noise-boosting effects caused by overheating and that this effect is characterized by the differential mode parameters red and kRd.


### 2.5. Determination of the Main Parameters

In order to utilize the proposed model to predict the output noise of single hot wires or arrangements of thermally coupled wires, it is necessary to find the system parameters. Referring to Equation set (18), it is possible to estimate all required parameters of a single wire by measuring the small-signal impedance (r) as a function of frequency. Fitting the response by means of a bilinear (one pole, one zero) function, it is possible to find ωp and ωz. From these two singularities, we can find ωg and the αV0I0γ0 term. By these four quantities, it is possible to calculate coefficients r, re, and kR as a function of frequency (see Equation (18)). Measurements have to be performed by biasing the wire in the desired OP. To complete determination of the quantities present in Equation (17), it is also necessary to measure the spectrum of the resistance fluctuations (SδRe) at different temperatures, by placing the wire in an oven to set the desired temperature without resorting to self-heating in order to avoid the mentioned noise-boosting effect.The same parameters of a set of thermally coupled wires connected to form a Wheatstone bridge can be measured again from a single frequency sweep, using a current sourceip as in Figure 6(a).Figure 6
(a) Simulation of a Wheatstone bridge to calculate the differential mode parameters; (b) equivalent circuit for differential mode variations.
(a)
(b)Sourceip should have zero dc value and a sinusoidal ac component small enough to induce only a small displacement of the current and voltage of the bridge around the OP. Since we have demonstrated that only differential mode components contribute to the output PSD of the bridge, we can analyse only the differential mode equivalent circuit of Figure 6(b) where we can easily find that id+id′=ip. Solving Equation set (42) with this condition and neglecting the noise components, we get
(45)rd=2vdip.Sweeping the frequency of the sourceip and measuring voltage vd, it is possible to use Equation (45) to calculate the frequency response of rd from which the main parameters of the differential mode model can be determined.


## 2.1. Noise in Wires out of Thermal Equilibrium

In thermal equilibrium, an electrical conductor will produce only thermal noise. In a Thevenin equivalent, a noiseless resistor is placed in series to a voltage noise source of PSD (Power Spectral Density) equal to4kBTRS, where kB is the Boltzmann constant, T the absolute temperature, and RS the conductor resistance. The Norton equivalent circuit consists of a noiseless resistor in parallel with a noise current source of PSD equal to 4kBTGS, where G=1/RS. The two circuits are recalled in Figures 1(a) and 1(b). The first theoretical derivation of these noise models, developed by H. Nyquist using thermodynamic arguments, dates back to 1928.Figure 1
Thevenin (a) and Norton (b) representations of the noise in a resistor.
(a)
(b)Use of the above model to represent the noise in hot wires is not rigorous for two reasons. First, the voltage vs. current dependence (V‐I curve) is not linear, due to self-heating and the relatively large TCR, so that defining the wire resistance is not straightforward. Second, a hot wire in operating conditions carries a nonzero electrical current and then is not in thermal equilibrium.Let us start from the nonlinearity problem and consider Figure2(a), where the V‐I characteristics of a wire subjected to self-heating is sketched for the case of positive TCR.Figure 2
Sketched characteristics of a thermally insulated wire with positive TCR: (a)V‐I plots; (b) magnitude of the small-signal impedance as a function of frequency.
(a)
(b)For any given operating point, marked by currentI and voltage V, we can define two resistances, namely, a large signal resistance, R, and a small-signal resistance, rDC:
(1)R=V/I,(2)rDC=dVdII,V.As the current increases, so does the overheating and, due to the assumption of positive TCR, alsoR increases, producing the nonlinear behaviour of the V‐I characteristic shown in Figure 2(a). On the other hand, rDC is the small-signal equivalent resistance. This resistance is applicable for variations around the operating point that are either constant (dc components) or slow (low frequencies) so as to allow the overheating to follow the heating power. At high frequencies, the thermal mass of the wire dampens the temperature variations and the equivalent small-signal resistance asymptotically tends to R. In between, the magnitude of the small-signal (complex) impedance, indicated with r, should reasonably follow a bode plot like that depicted in Figure 2(b). The exact expression of the small-signal impedance, which is in agreement with this intuitive behaviour, was derived in a previous study [15], which deals with thermistor-based bolometers.Since noise consists in small current and voltage variations around the operating point, the equivalent circuits of Figure1 are applicable to the wire once Rs is replaced with the small-signal impedance r. What is less obvious is the expression of the spectral density to be assigned to vn or in noise sources that appear in Figures 1(a) and 1(b), respectively, due to the mentioned nonequilibrium condition. It has been shown with either a semiclassical [18] or quantum mechanics [19] approach that the current fluctuation due to thermal agitation of the carriers has a spectral density that does not change when the conductor is subjected to a moderate dc current with respect to the case of thermal equilibrium. The reason is that the carrier drift velocity is much smaller than the thermal velocity in most practical cases of metallic conductors, so that the random walks of the carriers, from which the current fluctuation originates, are practically unaltered. These random walks depend on the same scattering mechanism from which the resistance R originates. Therefore, it is reasonable to consider that the current fluctuation has a spectral density equal to 4kBT/R. However, using this spectral density for the current source in the equivalent model of Figure 1(b) means neglecting the mentioned feedback effect that occurs in the hot wire. If the wire is biased with a constant voltage, a current change due to the random agitation of the carriers produces a variation of the heating power and, in turn, a change of the wire temperature and resistance R. The resulting resistance variation produces an additional current contribution that reinforces or diminishes the original current change depending on the sign of the TCR. Then, due to the wire self-heating, the actual current fluctuations seen from the wire terminals are different from those that would be predicted assigning the density 4kBT/R to the current source in the circuit of Figure 1(b).This electrothermal feedback effect was described in previous works on bolometers [15, 16], where the interest was mainly to model the effects on the device sensitivity. The feedback was also considered for its effect on thermal noise, and the authors simply propose to calculate the voltage noise power density as 4kBTRer, where “Rer” indicates the real part of impedance r. This approach was demonstrated to be applicable to RLC (resistance, inductance, and capacitance) networks in thermal equilibrium, but there is no physical justification for use of it for out-of-equilibrium systems in the presence of electrothermal feedback.The alternative approach proposed in this work starts from the following equation for the total current (I) through a conductor subjected to a voltage V, valid also in the presence of self-heating:
(3)I=VR−ie⟺V=IR+Rie,where ie is the thermal noise component, marked by the usual spectral density SIT=4kBT/R, and R is simply defined by the ratio V/Idrift, where Idrift is the current component due only to the electric field in the conductor. Notice that the temperature dependence of R is the cause of the electrothermal feedback. To complete the framework, we consider also that R is subjected also to random fluctuations that would be present also in the case of a perfectly constant temperature. These fluctuations, indicated with δRe, are the cause of flicker noise and can be boosted by the feedback as thermal noise. These are the premises that will be used in the next subsections to derive a model of the noise in single hot wires and in thermally coupled pairs of hot wires. All the models used in this work are of the lumped-element type. Therefore, quantities such as the wire temperature will represent averages calculated along the wire length.


## 2.2. Electrothermal Model of a Single Hot Wire

Figure3 schematically shows the elements of the single wire model.Figure 3
Elements of the electrothermal model for a single wire.In the electrical domain, the wire is represented by voltageV across its terminals and current I flowing through it. The two quantities are tied by the electrical resistance R, as shown by Equations (1) and (2). In the thermal domain, the wire is characterized by its absolute temperature T, by its thermal capacity CTH, and by thermal conductivity θ from the wire to the environment, which is considered to be at uniform temperature TA. Thermal conductivity θ includes (i) heat conduction along the solid suspension elements, (ii) conduction and convection through the surrounding fluid, and (iii) radiation. We will consider that both R and θ are temperature-dependent quantities, while CTH will be considered constant. Heat balance in the wire body will require that
(4)CTHdTdt=W−θT−TA,where W is the total heating power dissipated by the electrical current, simply given by
(5)W=V·I.In order to calculate the effects of fluctuationsie and δRe, it is convenient to use a small-signal analysis of the wire around a static operating point (dc components). Then, we can write
(6)V=V0+v,I=I0+i,T=T0+δT,W=W0+w,where V0, I0, T0, and W0=V0I0 define the operating point (OP) and v, i, δT, and w are the corresponding variations around the OP. We use lower-case symbols for v, i, and w since it is customary in small-circuit analysis of electrical circuits. For all other variations, we have used the prefix “δ.” Taking into account the temperature dependence of R and θ, their first-order approximations can be written as
(7)R=R0+αR0δT+δRe,θ=θ0+βθ0δTwithα=1RdRdT,β=1θdθdT,where we have included also possible temperature independent fluctuations (δRe) of R. Note that α is the TCR of the wire. Thermal equilibrium in the OP requires
(8)W0=I0V0=θ0T0−TA.With the above definitions, the following equation for variations in the Laplace domain can be derived from Equation (4):
(9)sCTHδT=w−βθ0T−TAδT−θ0δT,from which we find
(10)δT=γsw,(11)γs=γ01+s/ωg,(12)γ0=1θ01+βT−TA,(13)ωg=1CTHγ0.In terms of variations, Equations (3) and (5) become
(14)v=iR0+αR0δT+δReI0+R0ie,(15)w=iV0+vI0.Combining Equations (10)–(15), we can find with elementary algebraic passages
(16)v=iR01+αV0I0γs1−αR0I02γs+R01−αR0I02γsie+I01−αR0I02γsδRe,which can be synthetically written in the following way:
(17)v=i·r+ie·re+kRδRe,where coefficients r, re, and kR depend on the complex frequency s. This dependence can be made more explicit, by substituting the expression of γs from Equation (11) into Equation (16) obtaining
(18)r=R01+αV0I0γ01−αV0I0γ0·1+s/ωz1+s/ωp,(19)re=R011−αV0I0γ0·1+s/ωg1+s/ωp,(20)kR=I01−αV0I0γ0·1+s/ωg1+s/ωp,where the following angular frequencies have been introduced:
(21)ωz=ωG1+αW0γ0,(22)ωp=ωG1−αW0γ0.Equations (17)–(22) can be used to describe the small-signal behaviour of the wire. As far as the small-signal ac impedance of the wire is concerned, this is given by parameter r. It can be easily shown that for a positive TCR, the magnitude of r depends on the frequency as in Figure 2(b), with fp=ωp/2π and fz=ωz/2π. An equivalent expression was found in [15].Equation (17) can be used to find Thevenin and Norton representations of the wire noise. Referring to Figures 1(a) and 1(b), obviously, we have to replace resistor RS with the complex impedance r in both the Thevenin and Norton circuits, in order to correctly model the ac behaviour of the wire. In the case of the Thevenin equivalent circuit, the voltage of the equivalent source (Vn in Figure 1(a)) can be calculated by nulling the small-signal current through the wire terminals (i.e., we set i=0 in Equation (17)). This represents the case of a wire biased with a constant current (zero variations). Then, the PSD of the Thevenin equivalent noise source is given by
(23)Svn=4kBT0R0re2+SδRefkR2,where SδRe is the PSD of resistance fluctuations δRe, while the PSD of ie was assumed to be equal to 4kBT0/R0 as anticipated in the previous subsection. At high frequency, re tends to R0, so that the thermal noise voltage contribution is simply 4kBT0R0. Since also the complex impedance of the wire tends to R0, the high-frequency limit simply corresponds to calculating the noise PSD using the usual Johnson-Nyquist expression. At very low frequencies (f≪fp), the thermal component of the voltage PSD becomes
(24)Svn−th=4kBT0R011−αV0I0γ02.For a positive TCR (α>0), the denominator can approach zero when the static heating power W0=V0I0 is large enough. In that case, the voltage noise PSD can get much larger than the high-frequency limit. It can be easily verified that the low-frequency limit given by Equation (24) does not coincide with the value obtained by applying the Jonson-Nyquist expression with the low-frequency limit of the wire resistance (rDC) given by Equation (18) for s=0.A similar boosting effect can be observed for the flicker component, represented by the resistance fluctuation component. At high frequency,kR tends to I0, so that the voltage noise is simply given by the product of the resistance fluctuations and the bias current I0. At low frequency, this term is boosted by the same factor as the thermal noise.The noise voltage density of the Thevenin model represents the actual noise that should be expected when the wire is biased at constant current. It is interesting to observe that the boosting factor tends to infinity whenαV0I0γ0 tends to one. If αV0I0γ0 gets larger than one, the pole sp=‐ωp in the denominator of both re and kR becomes positive, denoting instability. In these conditions, a small variation due to noise triggers a catastrophic increase of the voltage, leading to failure. This phenomenon is the well-known thermal runaway. It is possible to find a notable expression for αV0I0γ0 when thermal conductance θ is assumed to be independent of temperature (β=0). In that case, γ0=1/θ0 and αV0I0γ0=αT0−TA. As a result, the noise-boosting factor 1−αV0I0γ0−1 tends to infinity when the product of the wire overheating T0−TA by the TCR tends to one. Thermal runaway occurs when this product is equal to or greater than one. For a negative TCR, Equation (24) predicts a noise reduction at low frequencies with respect to the 4kBT0R0 limit and no thermal runaway.The situation is reversed for the Norton equivalent model. The noise current source of the Norton model gives the current fluctuations when the wire is biased at constant voltage. Its value can be found settingv=0 in Equation (17) and solving for current variations i, finding
(25)in=−i=rerie+kRrδRe.Ratiosre/r and kR/r are now marked by a pole equal to −ωz. Instability, i.e., thermal runaway, occurs when the TCR is negative. This is a well-known difference between constant-current and constant-voltage biasing. Similarly, current noise boosting at low frequency occurs only for α<0. In particular, the thermal component of the noise current PSD in the low-frequency limit that can be found from Equation (25) is
(26)Sin−th=4kBT0R011+αV0I0γ02,where 4kBT0/R0 is the high-frequency limit which coincides with the PSD of the current fluctuation ie that appears in Equation (1).


## 2.3. Electrothermal Model of Two Thermally Coupled Wires

The elements of the model used for the pair of thermally coupled wires are illustrated in Figure4.Figure 4
Elements of the electrothermal model for a thermally coupled wire pair.The thermal and electrical quantities (V, I, R, T, and θ) are duplicated and an additional coupling thermal conductance θ12 is present. In this study, we assume that the system is symmetrical (the two wires are symmetrical) and are biased in such a way that they reach a symmetrical OP, which is defined by the following values for the quantities of interest:
(27)V1=V2=V0,I1=I2=I0,T1=T2=T0,R1=R2=R0,θ1=θ2=θ0.In the majority of applications, thermally coupled wires are biased with a symmetrical operating point. On the other hand, variations are different between the two wires, so that the total values (OP plus variations) of the key quantities will be different from one wire to the other.Equation (4), pertinent to a single wire, is replaced by the following set:
(28)CTHdT1dt=W1−θ1T1−TA−θ12T1−T2,CTHdT2dt=W2−θ2T2−TA+θ12T1−T2.In Equation (28), a heat exchange term due to conductance θ12 is clearly present. In terms of variations and Laplace transforms, we obtain
(29)sCTHδT1=w1−θ0+βθ0T0−TAδT1−θ12δT1−δT2,sCTHδT2=w2−θ0+βθ0T0−TAδT2+θ12δT1−δT2.Heat generation in the wires is described by the following set:(30)w1=v1I0+i1V0,w2=v2I0+i2V0,while Equation (14) (Ohm’s law plus Jonson-Nyquist current fluctuations) for the two wires is replaced by the following set:
(31)I1=V1R1−ie1,I2=V2R2−ie2,⟺V1=I1R1+R1ie1,V2=I2R2+R2ie2,which, in terms of variations, becomes
(32)v1=i1R0+αR0I0δT1+I0δRe1+R0ie1,v2=i2R0+αR0I0δT2+I0δRe2+R0ie2.At this point, it is convenient to separate all variations into a differential mode and common mode component. For example, we will replacev1 and v2 by vd=v1‐v2 and vc=v1+v2/2, respectively, where vd is the differential component and vc the common mode one. In the remainder of this document, differential mode and common mode quantities will be indicated with the “d” and “c” subscript, respectively. In this way, we can easily find decoupled equations for the differential mode and common mode variables. For the common mode components, the resulting equations are identical to those of the single wire. This is reasonable, since common mode components do not break the symmetry, and with no temperature difference, the two wires do not interact and behave as a single wire. Equations for the differential components are only slightly different, due to the presence of θ12, so that from the Equation set (29), we can derive the following differential mode equation:
(33)sCTHδTd=wd−θ0+βθ0T0−TAδTd−2θ12δTd,while from the Equation sets (30) and (32), we find differential mode equations that are identical to Equations (14) and (15), respectively:
(34)vd=idR0+αR0I0δTd+I0δRed+R0ied,wd=vdI0+idV0.From Equation (33), we can find the expression of δTd, analogous to Equation (10):
(35)δTd=γdswd,(36)γds=γ0d1+s/ωgd,(37)γ0d=1θ01+βT0−TA+2θ12,(38)ωgd=1CTHγ0d.Since Equations (34)–(38) are formally equivalent to the equations of the single wire, with the sole difference of the expression of γ0d (that replaces γ0), the solution has the same form as Equation (17):
(39)vd=id·rd+ied·red+kRdδRed,where
(40)rd=R01+αV0I0γ0d1−αV0I0γ0d·1+s/ωzd1+s/ωpd,red=R011−αV0I0γ0d·1+s/ωgd1+s/ωpd,kRd=I01−αV0I0γ0d·1+s/ωgd1+s/ωpd,ωzd=ωgd1+αW0γ0d,ωpd=ωgd1−αW0γ0d.In order to explain the way to use Equation (39) in a practical case, we can consider the very common situation in which the wires are biased with identical constant currents and the output signal is the voltage difference vd=v1‐v2. In the next subsection, we will analyse the more complex case of a full Wheatstone bridge. With constant-current bias, both i1 and i2 are forced to be zero and then also id is zero. The forcing terms, i.e., Jonson-Nyquist current fluctuations ie1 and ie2 and resistance fluctuations δRe1 and δRe2, are split into their common mode and differential mode components. Thanks to linearity of the small-signal equations and the fact that the equations are decoupled, we can calculate the effects of the two modes separately and then add them up. Common mode forcing terms produce only common mode variations, thus the effect on vout=vd is null. Therefore, we can focus only on the differential mode terms. From Equation (39) with id=0 and considering that ie1, ie2, δRe1, and δRe2 are independent stochastic processes, we can find the PSD of the output voltage:
(41)Svdf=2Siefred2+2SδRekRd2,where Sief=4kBT0/R0 and SδRef are the PSD of the Johnson-Nyquist (thermal) current fluctuations and resistance fluctuations, respectively, of each single wire. Again, as in the case of the single wire, a boosting effect of both thermal and flicker noise at low frequencies is predicted if the TCR is positive. For the same heating power (V0I0) and then the same overheating, we can expect a smaller noise increase with respect to the case of the single wire, due to the presence of the additional coupling term θ12 in γ0d, compared to γ0 of the single wire.


## 2.4. Noise in Wheatstone Bridges of Thermally Coupled Wire Pairs

Often, thermally coupled wire pairs are connected to form Wheatstone bridges as shown in Figure5(a), where the two wire pairs are represented by resistors pairs R1 and R2 and R3 and R4. Since the aim is generally to sense a physical quantity, connection should be made in such a way that the effect of the quantity of interest contributes to the output voltage in a constructive way. The bridge can be biased with different approaches, which are all equivalent in terms of the effect of noise on the output voltage. In the example of Figure 5, the bridge is biased by a dc voltage source VA (e.g., a battery) and a resistor RB, which forms a voltage divider with the bridge resistance setting the operating voltage of the bridge to VB.Figure 5
(a) Wheatstone bridge formed by two thermally coupled wire pairs; (b) equivalent circuit for differential mode variations.
(a)
(b)The equivalent circuit for differential mode variations is represented in Figure5(b), where the differential currents of the two wire pairs are indicated. Equation (39) for the two wire pairs becomes
(42)vd=id·rd+ied·red+kRdδRed,vd′=id′·rd+ied′·red+kRdδRed′,where variables with a prime character vd′, id′, ied′, and δRed′ belong to the upper wire pair of the bridge (R3, R4). By elementary analysis of the circuit in Figure 5(b), it is possible to find that vd′=vd and id′=‐id. Solving Equation set (42) with these relationships gives
(43)2vd=ied−ied′·red+kRdδRed−δRed′.Considering thatied, ied′, δRed, and δRed′ can be reasonably considered independent stochastic processes, and recalling the dependence of each one of these quantities with current fluctuations (ie) and resistance fluctuations (δRe) of the single wires of the bridge, we can finally find
(44)Svdf=Siefred2+SδRekRd2,where, again, Sief=4kBT/R0. Equation (44) indicates that the output noise PSD of a Wheatstone bridge is subjected to noise-boosting effects caused by overheating and that this effect is characterized by the differential mode parameters red and kRd.


## 2.5. Determination of the Main Parameters

In order to utilize the proposed model to predict the output noise of single hot wires or arrangements of thermally coupled wires, it is necessary to find the system parameters. Referring to Equation set (18), it is possible to estimate all required parameters of a single wire by measuring the small-signal impedance (r) as a function of frequency. Fitting the response by means of a bilinear (one pole, one zero) function, it is possible to find ωp and ωz. From these two singularities, we can find ωg and the αV0I0γ0 term. By these four quantities, it is possible to calculate coefficients r, re, and kR as a function of frequency (see Equation (18)). Measurements have to be performed by biasing the wire in the desired OP. To complete determination of the quantities present in Equation (17), it is also necessary to measure the spectrum of the resistance fluctuations (SδRe) at different temperatures, by placing the wire in an oven to set the desired temperature without resorting to self-heating in order to avoid the mentioned noise-boosting effect.The same parameters of a set of thermally coupled wires connected to form a Wheatstone bridge can be measured again from a single frequency sweep, using a current sourceip as in Figure 6(a).Figure 6
(a) Simulation of a Wheatstone bridge to calculate the differential mode parameters; (b) equivalent circuit for differential mode variations.
(a)
(b)Sourceip should have zero dc value and a sinusoidal ac component small enough to induce only a small displacement of the current and voltage of the bridge around the OP. Since we have demonstrated that only differential mode components contribute to the output PSD of the bridge, we can analyse only the differential mode equivalent circuit of Figure 6(b) where we can easily find that id+id′=ip. Solving Equation set (42) with this condition and neglecting the noise components, we get
(45)rd=2vdip.Sweeping the frequency of the sourceip and measuring voltage vd, it is possible to use Equation (45) to calculate the frequency response of rd from which the main parameters of the differential mode model can be determined.


## 3. Results and Discussion

In this section, we compare the predictions of our model with respect to those that can be obtained from the approach proposed in [15]. The comparison will be limited to the boosting factor of the thermal noise, since no effects of the electrothermal feedback on the flicker noise were proposed in the previous work.Figure7 shows the noise voltage PSD of the thermal noise, normalized with respect to the high-frequency limit. Note that this limit is the same for the two models and is equal to 4kBT0R0, where R0 is the wire resistance (V/I) ratio at the operating temperature. Both curves have been calculated for a value of parameter αV0I0γ0 equal to 0.75. We recall here that in [15], the noise density is simply assumed to be 4kBT0Rer, while in the proposed model, the PSD is given by 4kBT0/R0re2 (see the thermal component in Equation (23)). Figure 7 clearly shows that the two models predict a similar frequency behaviour but with noise boosting occurring at low frequencies which is significantly larger for the proposed model.Figure 7
Calculated thermal noise PSD as a function of frequency, normalized to the high-frequency limit, for the proposed model and for the expression given in reference [15].The dependence of the noise-boosting factor,Sv0/Sv∞ on the parameter αV0I0γ0 is shown in Figure 8 for the two models. Negative values of αV0I0γ0 occur for negative values of the TCR. In both models, thermal noise is boosted at low frequencies for α>0 and reduced for α<0, but the extent of this effect is different in the two cases. The proposed model predicts more noise boosting for α>0 and less noise reduction for α<0 than the model in [15], for which the noise PSD tends to zero when αV0I0γ0 approaches -1.Figure 8
Ratio of the thermal noise PSD at 0 Hz (dc) over the high-frequency limit as a function of parameterαV0I0γ0, for the proposed model and for the model in reference [15].Figures7 and 8 demonstrate that the proposed model gives results that are significantly different from the previous model, and this enables discrimination between the two by means of noise measurements.


## 4. Conclusions

The analysis presented in this paper predicts in a quantitative way that the electrothermal feedback, resulting from the combination of self-heating with a nonzero TCR, changes the noise spectral density of a conducting wire. Both thermal noise and flicker noise are affected by a filtering effect that, depending on the sign of the TCR, boosts or dampens the noise density at low frequencies. This effect was already suggested in a previous work [15], where, however, an arbitrary assumption was made on the expression of the thermal noise PSD. The proposed model derives the thermal noise PSD with straightforward passages, starting from well-established properties of the thermal current fluctuations, and, in addition, is capable of predicting the effect also on the flicker noise spectrum. The approach is extended to pairs of thermally coupled wires, which constitute the core of thermal flow sensors and, more recently, have been proven capable of detecting acoustic particle velocity. In the proposed model, the filtering effect is characterized by parameters that can be easily determined by means of small-signal impedance measurements as a function of frequency.Calculations of the thermal PSD performed using the proposed model and the previous approach revealed that the two models yield significantly different predictions of the noise modification occurring at low frequency. This should facilitate discriminating the two models through noise measurements, considering also that in both cases, the model parameters can be easily determined by means of small-signal impedance measurements as a function of frequency. These experiments should contribute to widen the knowledge of noise in out-of-equilibrium electrical systems.



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*Source: 8523170-2020-01-07.xml*