Citation: Walter, D.; Bülau, A.; Zimmermann, A. Review on Excess Noise Measurements of Resistors. Sensors 2023, 23, 1107. https:// doi.org/10.3390/s23031107 Academic Editor: Pak Kwong Chan Received: 11 November 2022 Revised: 11 January 2023 Accepted: 16 January 2023 Published: 18 January 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). sensors Review Review on Excess Noise Measurements of Resistors Daniela Walter 1,* , André Bülau 1 and André Zimmermann 1,2 1 Hahn-Schickard, Allmandring 9b, 70569 Stuttgart, Germany 2 Institute for Micro Integration (IFM), University of Stuttgart, Allmandring 9b, 70569 Stuttgart, Germany * Correspondence: daniela.walter@hahn-schickard.de; Tel.: +49-711-685-84772 Abstract: Increasing demands for precision electronics require individual components such as resis- tors to be specified, as they can be the limiting factor within a circuit. To specify quality and long-term stability of resistors, noise measurements are a common method. This review briefly explains the theoretical background, introduces the noise index and provides an insight on how this index can be compared to other existing parameters. It then focuses on the different methods to measure excess noise in resistors. The respective advantages and disadvantages are pointed out in order to simplify the decision of which setup is suitable for a particular application. Each method is analyzed based on the integration of the device under test, components used, shielding considerations and signal processing. Furthermore, our results on the excess noise of resistors and resistor networks are presented using two different setups, one for very low noise measurements down to 20 µHz and one for broadband up to 100 kHz. The obtained data from these measurements are then compared to published data. Finally, first measurements on commercial strain gauges and inkjet-printed strain gauges are presented that show an additional 1/fα component compared to commercial resistors and resistor networks. Keywords: measurements; 1/f noise; excess noise; noise index; resistor; resistor network; strain gauge; sensor; inkjet; nanoparticles 1. Introduction When it comes to resistors, circuit designers have to choose out of a large variety of resistor technologies. Choosing the appropriate resistor depends on many parameters associated with it, such as resistance, tolerance, power rating, temperature coefficient and package, to mention only a few out of many. With increasing demands for precision electronics, such as those needed in sensor applications, high-resolution test equipment or high-stability references, the noise of resistors as an additional parameter plays an important role. While all resistors exhibit inevitable thermal noise, often referred to as Johnson noise, excess noise, also known as current noise, is highly dependent on the technology used to manufacture the resistor and the current flowing through it. To choose the best resistor technology for any given application, it is crucial to understand which technology contributes which amount of excess noise, a topic that is not solely of academic interest. Although a standard on how to measure excess noise in resistors was defined [1,2], improved resistor technology and decreased excess noise require improving test equipment for its characterization. While datasheet values given by manufacturers are often quite conservative or limited by the test equipment used, if specified at all, only a limited number of papers contain investigations on different resistors and resistor networks. Different setups have been used in different papers published to this date and addressing this topic. This makes it hard to decide which setup is suitable for which application. Furthermore, not every paper describes the used setup to a point that it is easy to reproduce. Very low excess noise has been found for a few specific resistor technologies. Common sense states that carbon resistors exhibit the largest amount of noise, followed by thick film, then thin film and finally metal foil and wirewound resistors [3], but large differences can Sensors 2023, 23, 1107. https://doi.org/10.3390/s23031107 https://www.mdpi.com/journal/sensors https://doi.org/10.3390/s23031107 https://doi.org/10.3390/s23031107 https://creativecommons.org/ https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ https://www.mdpi.com/journal/sensors https://www.mdpi.com https://orcid.org/0000-0002-4285-4907 https://orcid.org/0000-0002-9585-3422 https://orcid.org/0000-0003-1824-9376 https://doi.org/10.3390/s23031107 https://www.mdpi.com/journal/sensors https://www.mdpi.com/article/10.3390/s23031107?type=check_update&version=2 Sensors 2023, 23, 1107 2 of 29 be observed within thin film resistors depending on the resistive material and substrate being used [4]. Excess noise has been investigated [5–7] and there is an understanding for its causes [8,9]. Motivated by new technologies to create resistors for sensor applications, such as inkjet- and aerosol-jet-printed strain gauges and temperature sensors, and how they com- pare to existing resistor technologies, the theoretical background on noise in resistors is summarized. A review of the methods used in different papers to characterize excess noise is given, results of excess noise that have been provided so far are compared, a method to characterize very low excess noise is presented and the results of our own measurements performed on resistors and resistor networks to reproduce already published results are shown. Furthermore, measurement results from a commercial strain gauge in thin film technology as well as an inkjet-printed strain gauge made from silver ink are presented. 2. Theoretical Background The content of this paper starts with a summary of the necessary theoretical back- ground to understand the topic of noise measurements in resistors. In the following, the wide field of noise is briefly introduced, especially the noise types important in resistors. Afterwards, resistor technologies and the noise associated with them are presented. 2.1. Noise Every signal that is measured includes some kind of noise. According to [7], the voltage measured at a sensor output is v(t) = vsig(t) + vn(t), (1) where vsig is the actual voltage signal and vn is the voltage noise component. Figure 1a shows a measured signal and its arithmetic mean. The arithmetic mean is usually used to characterize the signal. The arithmetic mean of the noise is always vn(t) = 0; see Figure 1b. Hence, it is more suitable to take the root mean square values for noise characterization (see Figure 1c) with the root mean square value given as vrms = √ vn2(t) (2) which is usually used to quantify noise voltage [7]. Sensors 2023, 23, x FOR PEER REVIEW  2  of  31      Very low excess noise has been found for a few specific resistor technologies. Com‐ mon sense states  that carbon resistors exhibit  the  largest amount of noise,  followed by  thick film, then thin film and finally metal foil and wirewound resistors [3], but large dif‐ ferences can be observed within  thin  film resistors depending on  the resistive material  and substrate being used [4]. Excess noise has been investigated [5–7] and there is an un‐ derstanding for its causes [8,9].  Motivated by new  technologies  to create  resistors  for sensor applications, such as  inkjet‐ and aerosol‐jet‐printed strain gauges and temperature sensors, and how they com‐ pare to existing resistor technologies, the theoretical background on noise in resistors is  summarized. A  review of  the methods used  in different papers  to  characterize  excess  noise  is given,  results of excess noise  that have been provided  so  far are  compared, a  method  to  characterize very  low excess noise  is presented and  the  results of our own  measurements performed on resistors and resistor networks to reproduce already pub‐ lished  results are  shown. Furthermore, measurement  results  from a  commercial  strain  gauge in thin film technology as well as an inkjet‐printed strain gauge made from silver  ink are presented.  2. Theoretical Background  The content of this paper starts with a summary of the necessary theoretical back‐ ground to understand the topic of noise measurements in resistors. In the following, the  wide field of noise is briefly introduced, especially the noise types important in resistors.  Afterwards, resistor technologies and the noise associated with them are presented.  2.1. Noise  Every signal that is measured includes some kind of noise. According to [7], the volt‐ age measured at a sensor output is  𝑣 𝑡 𝑣 𝑡 𝑣 𝑡 ,    (1) where  𝑣   is the actual voltage signal and  𝑣   is the voltage noise component. Figure 1a  shows a measured signal and its arithmetic mean. The arithmetic mean is usually used to  characterize the signal. The arithmetic mean of the noise is always  𝑣 𝑡 0; see Figure  1b. Hence, it is more suitable to take the root mean square values for noise characterization  (see Figure 1c) with the root mean square value given as  𝑣 𝑣 𝑡   (2) which is usually used to quantify noise voltage [7].  (a)    𝑣 𝑡 𝑣   (b)  𝑣 𝑡 0  (c)  𝑣 𝑡   Figure 1. A common sensor signal can be divided into the actual sensor signal and the noise part.  The sensor signal can be described with its mean value (a), while the mean value of noise will al‐ ways be zero (b). Therefore, noise is characterized with its mean square value (c) (refer to [7]).  Figure 1. A common sensor signal can be divided into the actual sensor signal and the noise part. The sensor signal can be described with its mean value (a), while the mean value of noise will always be zero (b). Therefore, noise is characterized with its mean square value (c) (refer to [7]). Noise can be classified into extrinsic and intrinsic noise. Extrinsic noise describes all the noise affecting the system from the outside, e.g., from the environment. This might be Sensors 2023, 23, 1107 3 of 29 natural noise sources such as sky noise as well as manmade noise or noise from power lines or electric motors [6]. This noise couples into the circuit either conductively or inductively. Intrinsic noise is the noise generated inside the system. The origin of this noise is the discrete nature of charge carriers [6]. The most relevant noise types for intrinsic noise are thermal noise, shot noise, burst noise, generation–recombination noise, excess noise (1/f-noise) and 1/f2-noise [9]. While thermal noise and excess noise are described in detail in the following due to their importance in the context of resistor noise, detailed information about other noise sources can be found elsewhere [6–10]. Noise types in general have two main characteristics. The first one is related to the physical phenomena, which are produc- ing the noise. The second one is their frequency distribution [10]. Noise types are often associated with a color that corresponds to their distribution in the frequency spectrum. 2.1.1. Thermal Noise Thermal noise is the most prominent form of noise. It is also called Johnson or Johnson–Nyquist noise. Since its power spectral density is evenly spread over the whole frequency range, like the spectrum for white light, thermal noise is also referred to as white noise [9,11,12]. Another characteristic of thermal noise is its Gaussian amplitude distribu- tion [12]. At temperatures above absolute zero, free electrons are moving in conducting materials due to kinetic energy. This inherent kinetic energy is proportional to temperature and therefore to thermal energy. Thereby, fluctuating charge levels occur at the ends of every resistor. This time-dependent noise voltage can be measured as thermal noise [6,7,13]. Thermal noise does not depend on material or the configuration of an electrical circuit but only on constants [14], and the voltage noise density of thermal noise that is constant over frequency [15] can be given by the formula eth = √ 4 ∗ k ∗ T ∗ R, (3) where eth is the effective voltage of thermal noise in a given bandwidth of 1 Hz, k ≈ 1.38 ∗ 10−23 J K is the Boltzmann constant, T is absolute temperature in K and R is resistance in Ω. Similar to the definition of voltage noise density, the short-circuit noise density ith is given by [15] ith = √ 4 ∗ k ∗ T R (4) As Equation (3) indicates, there are only three possibilities to decrease voltage noise: 1. Decrease resistance, 2. Decrease bandwidth, 3. Decrease temperature. Figure 2 shows in a double logarithmic plot how noise voltage increases with resis- tance while current noise decreases with resistance. In fact, it is an increase and decrease, respectively, with 10 dB per decade. 2.1.2. Excess Noise Excess noise is also called 1/f-noise, flicker noise, current noise or pink noise. This type of noise cannot be described by using a single formula with some constants like thermal noise [1]. Hooge [5] shows in his review that covers the field of excess noise that many theories and models were proposed for excess noise. So far, no unifying principle can be identified that would explain this type of noise, though some sources have been determined [15]. By observing excess noise, some characteristics can be derived anyway. It is only present when current flows through the device [11,16]. As the name 1/f-noise al- ready indicates, the power spectrum of 1/f-noise is inversely proportional to frequency and follows 1 f α , where α ≈ 1 [9,11]. Some publications even distinguish between fundamental Sensors 2023, 23, 1107 4 of 29 1/f-noise and 1/fα-noise with α = 0.8− 1.2 [17]. According to [18], in a noise density plot, excess noise e f can be described by e f = K√ f , (5) where K is a constant representing the noise density value e f at a frequency of f = 1 Hz. Sensors 2023, 23, x FOR PEER REVIEW  4  of  31        Figure 2. Voltage noise of a resistor increases proportional to its resistance with 10 dB per decade  while noise current decreases with 10 dB per decade.  2.1.2. Excess Noise  Excess noise is also called 1/f‐noise, flicker noise, current noise or pink noise. This  type of noise cannot be described by using a single formula with some constants like ther‐ mal noise [1]. Hooge [5] shows in his review that covers the field of excess noise that many  theories and models were proposed for excess noise. So far, no unifying principle can be  identified that would explain this type of noise, though some sources have been deter‐ mined [15]. By observing excess noise, some characteristics can be derived anyway. It is  only present when current flows through the device [11,16]. As the name 1/f‐noise already  indicates, the power spectrum of 1/f‐noise is inversely proportional to frequency and fol‐ lows  , where  𝛼 1  [9,11]. Some publications even distinguish between fundamental  1/f‐noise and 1/fα‐noise with  𝛼 0.8 1.2  [17]. According to [18], in a noise density plot,  excess noise  𝑒   can be described by  𝑒 ,  (5) where 𝐾  is a constant representing the noise density value  𝑒   at a frequency of  𝑓 1 Hz.  How excess noise behaves at very high and very low frequencies is of high interest  but quite difficult to determine. At high frequencies, thermal noise always superimposes  1/f‐noise, which makes it difficult to spot the further trend. The measurement of 1/f‐noise  at very  low  frequencies  is very  time‐consuming, as𝑓 0 Hz  can never be reached  [19].  Hooge  [8] states that  the spectrum cannot be exactly    from  𝑓 0  to  𝑓 𝐼𝑛𝑓𝑖𝑛𝑖𝑡𝑦  be‐ cause of the mathematical rule that neither the integral of power density nor the Fourier  transformation are able to have infinite values. This gives rise to the assumption that ex‐ cess noise might not have a 1/f spectrum over the whole frequency range. However, in  [15], Horowitz and Hill argue that excess noise increases forever. Excess noise in contrary  to thermal noise has no Gaussian distribution in its power density function [6]. Addition‐ ally, excess noise is voltage‐dependent, more precisely, it is proportional to the applied  voltage [20] across a resistor. In general, excess noise is caused by a DC current flowing  through a discontinuous medium, the interaction between charge carriers and surface en‐ ergy, and imperfect contacts and crystal defects [6,7,9,17,21,22]. This means the magnitude  of the excess noise spectrum is dependent on inherent properties of resistors such as ma‐ terial composition, processing technology, size and shape [1,17]. The level of excess noise  Figure 2. Voltage noise of a resistor increases proportional to its resistance with 10 dB per decade while noise current decreases with 10 dB per decade. How excess noise behaves at very high and very low frequencies is of high interest but quite difficult to determine. At high frequencies, thermal noise always superimposes 1/f-noise, which makes it difficult to spot the further trend. The measurement of 1/f-noise at very low frequencies is very time-consuming, as f = 0 Hz can never be reached [19]. Hooge [8] states that the spectrum cannot be exactly 1 f from f = 0 to f = In f inity because of the mathematical rule that neither the integral of power density nor the Fourier transformation are able to have infinite values. This gives rise to the assumption that excess noise might not have a 1/f spectrum over the whole frequency range. However, in [15], Horowitz and Hill argue that excess noise increases forever. Excess noise in contrary to thermal noise has no Gaussian distribution in its power density function [6]. Additionally, excess noise is voltage-dependent, more precisely, it is proportional to the applied voltage [20] across a resistor. In general, excess noise is caused by a DC current flowing through a discontinuous medium, the interaction between charge carriers and surface energy, and imperfect contacts and crystal defects [6,7,9,17,21,22]. This means the magnitude of the excess noise spectrum is dependent on inherent properties of resistors such as material composition, processing technology, size and shape [1,17]. The level of excess noise is related to the quality of its lattice. Hence, high excess noise is an indication for poor material quality and low reliability [6]. To specify the level of excess noise inherent in a device, the noise index (NI) is used [21], which is explained in detail later. 2.1.3. Noise in Resistors This section introduces the combination of noise types occurring in a resistor. Figure 3 shows how the noise density of a resistor is distributed over frequency. The black curve shows the total noise in a resistor etot. It is a combination of thermal noise eth (red curve) and excess noise e f (blue curve). Since thermal noise and 1/f-noise are independent noise Sensors 2023, 23, 1107 5 of 29 sources and therefore uncorrelated, noise densities cannot be added linearly. Instead, the sum of the squares has to be used to obtain total noise [10,11] etot 2 = eth 2 + e f 2. (6) As can be derived from Figure 3, excess noise dominates in the lower frequency range while thermal noise dominates at higher frequencies. The frequency where thermal noise has exactly the same value as excess noise is called corner frequency fc. According to [15,18] it is also possible to describe the total noise by means of the corner frequency with etot = eth ∗ √ fc f + 1 (7) Thermal noise eth can be calculated by using Equation (3). In this example, a resistance of 1 kΩ and an ambient temperature of 300 K were assumed. Using (4), 1/f-noise can be described. The constant K of a certain resistor can only be derived from measurements. The type of these measurements and their performance will be analyzed in chapter 3. As stated earlier, excess noise is dependent on material composition, technology, size and shape and, therefore, is an indication for the material quality of the measured resistor [1,6,17]. The slope of excess noise in the spectral density plot increases with 10 dB/decade or with a factor of 10 in two decades towards lower frequencies, since the energy is the same in any bandwidth. The area under the curve of the total noise density in Figure 3 over any given bandwidth f1 to f2 is the root mean square noise voltage. This can mathematically be described by the integral of the square of Equation (6) in the frequency band f1 to f2 and will result in [18] vn = eth ∗ √ fc ∗ ln f2 f1 + f2 − f1 (8) The only way to decrease excess noise contribution for a given resistance is by reducing the current flowing through it. Sensors 2023, 23, x FOR PEER REVIEW  6  of  31        Figure 3. Noise in resistors is a combination of thermal noise and excess noise. At low frequencies  excess noise dominates while at high frequencies thermal noise dominates.  2.2. Measurement Units to Express Excess Noise  As has been shown in the previous section, excess noise is represented by a linear  slope in a double logarithmic noise density plot. From Section 2.1.2, it is already known  that excess noise is dependent on the current through the resistor, which results from a  supply voltage and the resistor being measured. Hence, it is necessary to find a measure‐ ment unit that describes the excess noise  in a device under test  independently from  its  supply voltage. For  this purpose, different measurement units have been proposed.  In  earlier publications, the expression microvolts per DC voltage was mostly used to quan‐ tify excess noise. In particular, this is the ratio of the mean rectified resistor noise in μV to  the DC voltage in volts applied to the resistor [23]. According to Conrad, Newman and  Stansbury [23], this index was used in lack of better options. However, the suggested in‐ dex is not comparable due to the often‐missing specification of the pass band.  2.2.1. Conversion Gain  In  1956, Conrad  suggested  a new  index  called  conversion gain  as a  reproducible  measurement unit that is independent of loading power, the used test equipment and test  procedures [24]. Thereby, conversion gain  𝐺   describes the noisiness of a resistor and the  efficiency of a resistor to convert applied DC voltage power  𝑃   to current‐noise power  𝑃 . The power ratio of those named powers is given in dB [24]. The corresponding equa‐ tion can be found in [23,24] as  𝐺 10 ∗ 𝑙𝑜𝑔 ,    (9) where  𝑃   is defined as  the current‐noise‐power spectral density  (𝑁𝑃𝑆𝐷)  in microvolts‐ squared at 1 kHz of a resistor with a resistance of  𝑅  and  𝑃 𝑁𝑃𝑆𝐷 ∗ 10 4 ∗ 𝑅   (10) Thereby, the power of DC voltage applied to the resistor is defined as  Figure 3. Noise in resistors is a combination of thermal noise and excess noise. At low frequencies excess noise dominates while at high frequencies thermal noise dominates. Sensors 2023, 23, 1107 6 of 29 2.2. Measurement Units to Express Excess Noise As has been shown in the previous section, excess noise is represented by a linear slope in a double logarithmic noise density plot. From Section 2.1.2, it is already known that excess noise is dependent on the current through the resistor, which results from a supply voltage and the resistor being measured. Hence, it is necessary to find a measurement unit that describes the excess noise in a device under test independently from its supply voltage. For this purpose, different measurement units have been proposed. In earlier publications, the expression microvolts per DC voltage was mostly used to quantify excess noise. In particular, this is the ratio of the mean rectified resistor noise in µV to the DC voltage in volts applied to the resistor [23]. According to Conrad, Newman and Stansbury [23], this index was used in lack of better options. However, the suggested index is not comparable due to the often-missing specification of the pass band. 2.2.1. Conversion Gain In 1956, Conrad suggested a new index called conversion gain as a reproducible measurement unit that is independent of loading power, the used test equipment and test procedures [24]. Thereby, conversion gain GC describes the noisiness of a resistor and the efficiency of a resistor to convert applied DC voltage power Pdc to current-noise power Pa. The power ratio of those named powers is given in dB [24]. The corresponding equation can be found in [23,24] as GC = 10 ∗ log ( Pa Pdc ) , (9) where Pa is defined as the current-noise-power spectral density (NPSD) in microvolts- squared at 1 kHz of a resistor with a resistance of R and Pa = NPSD ∗ 10−12 4 ∗ R (10) Thereby, the power of DC voltage applied to the resistor is defined as Pdc = V2 R , (11) where V is the applied DC voltage across the resistor. By combining Equations (9)–(11), the value of the resistance R does not have to be determined, which is convenient for measuring [24]. The drawback of the conversion gain is that there is no relation to the formerly used index µV-per-Volt and that values usually range from −140 dB to −200 dB, which is difficult for visualization [23]. 2.2.2. Noise Index To overcome these drawbacks, the noise index (NI) was proposed and picked up by Conrad, Newman and Stansbury [23]. This index has a simple relation to conversion gain and is also familiar to µV-per-Volts. It is defined as the ratio of the rms noise voltage vrms (in µV) in a pass band of one frequency decade to the applied DC voltage V (in V) and is expressed in dB [23]. The corresponding formula is NI = 20 ∗ log (vrms V ) . (12) In this expression, 1 µVrms of noise in a decade together with a supply DC voltage of 1 V would correspond to a NI value of 0 dB. vrms has to be determined from measurements [11]. Since the DC voltage supply is part of the equation, it does not have to be selected carefully, and NI values can be compared anyhow [23]. It is possible to transfer noise index into conversion gain by NI − GC = −159.6 dB. (13) Sensors 2023, 23, 1107 7 of 29 The exact derivation of this relation can be found elsewhere [23]. With Equation (12), NI can be calculated when vrms in a single decade is given or vice versa. Sometimes a certain value of noise density at a particular frequency might be of interest, e.g., to draw graphs. This eventually leads to Seifert’s work [22] where the mean-square noise voltage is given by v2 = vrms 2 = ∫ f2 f1 S( f )d f = ∫ f2 f1 e2( f ) ∆ f d f (14) Thereby, S( f ) is the power spectral density (PSD) of the resistor excess noise in the frequency band f1 to f2 and e2( f ) ∆ f is the mean-square noise spectral density (NSD) at frequency f . According to Section 2.1.2, spectral density of excess noise is proportional to 1/f. Due to this reason, the product of the power spectral density and frequency can assumed to be constant and according to [22] Equation (14) can be rearranged to vrms 2 = e2( f ) ∆ f ∗ f ∗ ∫ f2 f1 1 f d f (15) vrms 2 = e2( f ) ∆ f ∗ f ∗ ln f2 f1 (16) If one decade with the relation f2 = 10 ∗ f1 is considered as the frequency range, the part ln f2 f1 equals ln 10. Replacing this relation in Equation (16) and rearranging the equation in order to obtain the noise density at a certain frequency results in en( f ) = √ e2( f ) ∆ f = vrms√ f ∗ ln10 (17) 2.3. Resistor Types and Technologies Since excess noise in resistors is dependent on resistor-specific characteristics, different resistor types and technologies are described in the following section. 2.3.1. Resistor Types Resistors come in a variety of different resistance values, sizes and other characteristics. In many applications, it is not the absolute resistance that is important, but resistor ratios, such as gain setting resistors in an amplifier [4,25]. To reach much better performance than with single resistors, resistor networks can be used. Resistor networks are multiple resistors in one package that are created with the same processes simultaneously, often on the same substrate. This leads to good resistance matching between the individual elements as well as small tolerances, good tracking of temperature coefficients and thermal coupling of elements within the network [4]. A special type of resistors for sensing applications are strain gauges. In contrary to the already mentioned fixed resistor types, strain gauges are variable resistors [25]. Due to experienced strain, they alter their resistance. Strain gauges can be used as a single element or in a bridge configuration, e.g., in a Wheatstone bridge. 2.3.2. Resistor Technologies As mentioned in Section 2.1.2, excess noise is dependent on resistor technology. It is common sense that carbon composition resistors show the largest excess noise, followed by thick film and thin film resistors. Metal foil and wirewound resistors exhibit the smallest excess noise. In carbon composition resistors, the whole body acts as the resistive element [26]. They are produced by mixing carbon particles and a special binder and compressing them to a solid element. At their ends, termination wires are attached. The whole resistor is sintered Sensors 2023, 23, 1107 8 of 29 in a furnace [6,26]. Since no trimming is carried out, composition resistors have large tolerances in the range of ±10% and ±20%. Their advantages are a good high-frequency characteristic as well as the capability to be overloaded relative to their size. Composition resistors are often used in power supplies, welding controls or as “dummy loads” [26]. Thick film resistors are mostly produced as SMD devices [25]. They are fabricated by screen-printing resistive pastes on a ceramic substrate. The resistive paste contains powders (e.g., silver, chromium, palladium, glass). The printed paste is then sintered in a furnace at about 800 ◦C in order to evaporate the organic components and melt the glass particles to form a stable resistor layer [6,27]. With automated processes, thick film resistors can be produced in large quantities and their reproducibility is good [25]. Due to the junctions between metallic grains and glass particles, intrinsic defects in the conducting layer are present and thick film resistors exhibit large excess noise [17]. Thin film resistors come with many different materials for the resistive layer, e.g., nickel–chromium, carbon–boron, tantalum, tantalum–nitride, various oxides and other alloys are used [6,25]. The thin film layer can be deposited on the substrate using different technologies such as ion deposition, sputter deposition, chemical vapor deposition and evaporation [25]. The film thickness is around 0.5 µm and the film is usually patterned and laser-trimmed to increase and adjust the resistance value [6,25]. The substrate can be an alumina-based ceramic, sapphire or surface-oxidized silicon [25]. Within the class of thin film resistors, the electrical properties including excess noise can be different depending on the thin film and substrate materials used [4,26]. Metal film resistors as a category of thin film resistors are supposed to have the best noise properties but are still worse than bulk metal foil or wirewound technology due to occlusions, surface imperfections and non-uniform deposition [3]. Metal foil resistors are made of a pure metal or metal alloy foil on a carrier that is attached to a solid ceramic or glass substrate. As resistive material, nickel–chromium is often used. The resistive layer is patterned with a meander by using photolithography and etching [6,25]. To obtain the desired resistance value, connections in the pattern are cut with a laser beam. Each part of the pattern and its respective resistance is well known in order to trim the resistor according to the desired value. An algorithm shows the connections to be cut for a specific resistor value [25]. Metal foil resistors are available from the mΩ to kΩ range [26]. A great advantage of metal foil resistors is their outstanding temperature coefficient. There are two important effects. First, the resistance of the foil increases with increasing temperature. Second, the coefficient of thermal expansion of the substrate is smaller than the coefficient of thermal expansion of the foil. With increasing temperature, this leads to compressive stress of the foil and decreases resistance. Due to these opposite effects, the resulting change in resistance is almost zero when temperature changes [25]. Metal foil resistors are used for low-ohm currents called shunts and precision resistors for measurement applications [26]. Wirewound resistors consist of a wire made up of a metal alloy that is wound onto a bobbin that might be made up of plastic, glass or ceramic. The wire ends are soldered, crimped or welded to the leads and the whole body is coated with a protective glaze [6,25], coated with silicone and either molded or inserted into a plastic shell and potted. For applications with high demands, hermetic packages with glass feedthroughs are used. Due to their processing technology, they can be very stable also at high temperatures over 200 ◦C [25,26]. A drawback of wirewound resistors is their inductive behavior that also makes them a bad choice for high frequency applications [26]. Reactances do not generate noise in general, but if current noise is running through them, they develop voltage noise and associated parasitics [3]. To obtain non-inductive wirewound resistors, two identical wirings can be used that run in opposite direction [25]. Noise characteristics of wirewound resistors are comparable to that of metal foil, but wirewound resistors show much more inductive behavior [3]. Sensors 2023, 23, 1107 9 of 29 3. Measurement Techniques In the following section, different measurement techniques from the literature are presented. The standard method is presented first, and it is discussed why it is not recommended to measure modern components. The other methods are analyzed according to the following parameters: usage of a bridge setting, voltage supply, low-noise amplifier, shielding and measurement device. Block diagrams of selected papers are depicted in Figure 4. A summary of the parameters is provided in Table 1. 3.1. Standard Method The standard method is based on the paper of G. Conrad, Jr., N. Newman and A. Stansbury with the title “A Recommended Standard Resistor-Noise Test System” from 1960 [23]. The introduced method was transferred to the currently known standard methods MIL-STD-202-308 [1] and IEC60195 [2]. Method 308 describes a resistor test method to establish the noise quality characteristics and helps to choose a suitable resistor when current–noise requirements exist [1]. The setup consists of several parts that are depicted in the block diagram of Figure 4a. The first part is a variable DC power supply. The resistor under test is supplied through an isolation resistor. The latter one prevents noise appearing at the terminals of the resistor under test from being attenuated by the very low parallel impedance of the output terminals of the DC power supply. The isolation resistor is ideally free of current noise. Low noise wirewound resistors with values of 1 kΩ, 10 kΩ, 100 kΩ or 1 MΩ depending on the value of the resistor under test are used. Standard nominal values for DC voltage and voltage for the isolation resistor are given in a table in [1]. The setup also consists of a DC vacuum-tube voltmeter (VTVM). It measures the DC voltage across the resistor under test and its resulting noise (D). The voltage noise at the resistor under test is amplified and is shown by the AC indicating amplifier. The amplifier characteristics are high gain and low noise. The filter included has a 1 kHz pass band and is centered at 1 kHz. The setup is recommended to be shielded and placed at an ambient temperature of 25 ± 2 ◦C. The whole measurement consists of three steps. First, a calibration is needed. Then, the “open circuit” current–noise voltage of the resistor under test is measured. Afterwards, the system noise (S) is measured, followed by a simultaneous measurement of the DC voltage and the resulting total noise (T). By subtracting all components from the total noise, the noise of the resistor under test can be determined. The noise of the resistor under test can be expressed in the ‘microvolts-per-volt-in-a-decade’ index indexdB = T − f (T − S)− D (18) with f (T − S) = −10 ∗ log ( 1− 10− T−S 10 ) . (19) This noise quality index is expressed in dB. The essential drawback of this method is the use of an analog filter with a specified passband. The noise measured in this frequency band contains all sources of noise. Modern low excess noise resistors might be dominated especially by thermal noise in this pass band. This method cannot give an answer as to which noise is actually measured [11]. Additionally, a low-noise isolation resistor is used. However, it still contributes to the total noise, most notably if the resistor under test is also low-noise. Nevertheless, this method is still widely used by resistor manufacturers, although its sensitivity is poor, especially for modern low-power precision components [4]. This is also the reason why manufacturers only give an upper limit between −30 and −40 dB for excess noise in their datasheets. One exception is the LT5400, which is specified with a noise index of <−55 dB [4]. As shown in the current section, the standard method has some disadvantages and requirements on the noise of resistors have increased over time. Excess noise has greater and greater meaning for measurements. Today’s high-quality resistors cannot be measured with this test method due to several limitations. For this reason, other methods can be found in literature, such as bridge configurations. Important parameters that will be discussed in the following are the bridge setting of resistors, voltage supply, used amplifiers, shielding Sensors 2023, 23, 1107 10 of 29 of measurement setup and measurement equipment. Table 1 gives a summary of the most important papers on noise measurements. Other papers were found, but they had little information given on the measurement setup [17,28,29]. The papers are sorted by year of publication starting with the work of G. Conrad, Jr., N. Newman and A. Stansbury [23], which is the basis for the current standard method. There are other papers before 1960 where similar methods were used [30,31]. 3.2. Bridge Setting As shown in the previous section, the standard method uses a single resistor as the resistor under test in series with a wirewound resistor. This will work properly under the assumption that the wirewound resistor is noise-free or at least of lower noise than the resistor under test. To overcome this issue most papers since 1980 used a Wheatstone bridge configuration for the resistor under test, as indicated by the column ”bridge setting” in Table 1. Some of the papers used a bridge configuration with two sample resistors and two ballast resistors [10,32], but the prevalent method was to use four “identical” sample resistors [4,11,20,22,33–37]. The bridge as a differential configuration has the advantage to be able to suppress disturbances caused by the power supply and other common mode interferences [22,35]. In general, the bridge consists of four resistors. Bridge bias voltage supply is applied to one diagonal of the bridge; therefore, two dividers are fed by the same voltage [25]. Every half bridge experiences the full bias voltage of the Wheatstone bridge. Hence, the voltage across one resistor is half of the bridge bias voltage [22]. This is important when calculating the NI in Equation (12), where DC voltage V corresponds to the voltage drop across one resistor of the bridge. The voltage fed into the amplifier is picked up over the second diagonal of the bridge. If the bridge is totally balanced, this voltage would be exactly zero. In practice, the voltage measured across the second diagonal of the bridge for resistors with the same nominal value and a voltage noise of vrms,1, vrms,2, vrms,3 and vrms,4 is according to [37] ∆vrms,tot = √ vrms,1 2 + rrms,22 + vrms,32 + vrms,4 2 4 (20) Since this is the resulting voltage noise of all four resistors in the bridge, the factor√ 4 = 2 needs to be considered for voltage noise vrms of one resistor when calculating the NI value in Equation (12). Otherwise, as stated by Seifert in [22], ∆vrms,tot can be interpreted as the voltage noise of one resistor driven by the whole bias voltage of the bridge. In Figure 4, some block diagrams of setups published in the literature using a Wheat- stone bridge configuration are depicted. In Figure 4b, the setup used by Scofield [35] is shown, using AC excitation for thermal electromagnetic force (t.e.m.f.) cancellation, a commercial lock-in amplifier and a commercial FFT spectrum analyzer to measure and analyze the bridge voltage. The cross-correlation method with AC excitation is used by Stoll in [33] and displayed in Figure 4c. Besides a commercial FFT analyzer, it uses a fairly complex custom circuit. Seifert [22] uses a fairly simple setup with DC excitation and the bridge voltage measured using an instrumentation amplifier connected to an FFT analyzer as shown in Figure 4d. LaMacchia and Swanson [11] use a bipolar battery supply and a commercial high-performance audio analyzer for data acquisition, shown in Figure 4e. Finally, Beev [4] uses a bipolar polarity reversible DC excitation of the bridge, combined with cross-correlation using an 8.5 digit multimeter as a fast sampler, shown in Figure 4f. A field-programmable gate array (FPGA) is used to synchronize and control the switches of the bride supply, the amplifier input, the programmable amplifier and the multimeter. Sensors 2023, 23, 1107 11 of 29 Sensors 2023, 23, x FOR PEER REVIEW  11  of  31      second diagonal of the bridge  for resistors with the same nominal value and a voltage  noise of vrms,1, vrms,2, vrms,3 and vrms,4 is according to [37]  ∆𝑣 , 𝑣 , 𝑟 , 𝑣 , 𝑣 , 4   (20) Since this is the resulting voltage noise of all four resistors in the bridge, the factor  √4 = 2 needs to be considered for voltage noise vrms of one resistor when calculating the NI  value in Equation (12). Otherwise, as stated by Seifert in [22], ∆vrms,tot can be interpreted as  the voltage noise of one resistor driven by the whole bias voltage of the bridge.  In Figure 4, some block diagrams of setups published in the literature using a Wheat‐ stone bridge configuration are depicted. In Figure 4b, the setup used by Scofield [35] is  shown, using AC  excitation  for  thermal  electromagnetic  force  (t.e.m.f.)  cancellation,  a  commercial lock‐in amplifier and a commercial FFT spectrum analyzer to measure and  analyze the bridge voltage. The cross‐correlation method with AC excitation is used by  Stoll in [33] and displayed in Figure 4c. Besides a commercial FFT analyzer, it uses a fairly  complex custom circuit. Seifert [22] uses a fairly simple setup with DC excitation and the  bridge voltage measured using an instrumentation amplifier connected to an FFT analyzer  as shown in Figure 4d. LaMacchia and Swanson [11] use a bipolar battery supply and a  commercial high‐performance audio analyzer for data acquisition, shown  in Figure 4e.  Finally, Beev [4] uses a bipolar polarity reversible DC excitation of the bridge, combined  with cross‐correlation using an 8.5 digit multimeter as a fast sampler, shown in Figure 4f.  A field‐programmable gate array (FPGA) is used to synchronize and control the switches  of the bride supply, the amplifier input, the programmable amplifier and the multimeter.      (a)  (b)    (c)  Sensors 2023, 23, x FOR PEER REVIEW  12  of  31          (d)  (e)  (f)  Figure 4. Block diagrams of different setups from the literature: (a) proposed standard method in  [23], (b) setup in [35], (c) setup in [33], (d) setup in [22], (e) setup in [11] and (f) setup in [4].  Figure 4. Block diagrams of different setups from the literature: (a) proposed standard method in [23], (b) setup in [35], (c) setup in [33], (d) setup in [22], (e) setup in [11] and (f) setup in [4]. Sensors 2023, 23, 1107 12 of 29 Table 1. Summary of different measuring methods and their characteristics. Paper Year Bridge Setting Power Supply Amplifier Measurement Range Analyzed Resistors Measurement Device Shielding Techniques to Improve Measurement Conrad et al. [23] 1960 5 DC power (DUT) + ac power (ampl.) AC band-pass amplifier (1 kHz) At 1 kHz 100 Ω–22 MΩ VTVM Shielded enclosure for DUT - Hawkins and Bloodworth [38] 1971 5 - AC coupled amplifier 10 Hz–5 kHz Thick film resistor Homodyne spectrum analyzer - - Digital technique 1 × 10−4 Hz–4 Hz - Stoll [33] 1980 X AC current Ampl. (not specified) - - 2-channel Fourier analyzer - Cross-correlation method Demolder et al. [34] 1980 X DC current, batteries PAR 113 0.1 Hz–1 kHz 1 Ω–1 kΩ - Twisted cables, metal shield around setup - Cross-correlation method Scofield [35] 1987 X AC current PAR 124A lock-in amplifier 0.1 Hz–100 Hz Thin continuous metal films Spectrum analyzer Bridge in aluminum box surrounded by Styrofoam - Verbruggen et al. [36] 1988 X AC current with 45◦ phase shift PAR 113 0.3 Hz–60 Hz Thin film Al-sample 2-channel Fourier analyzer - - Moon et al. [32] 1992 X AC current Stanford SR560/PAR 116 0.1 Hz–100 Hz 480 Ω Digital signal processor + PC - Digital mixing of supply current with 0◦ or 90◦ (or ±45◦) Leon and Hebard [10] 1999 X AC voltage Lock-in amplifier 0.1 Hz–1 Hz 1 kΩ carbon composite Spectrum analyzer - - Crupi et al. [39] 2006 5 - OP27 1 Hz–1 kHz 1 kΩ Spectrum analyzer - Cross-correlation method, different amplifier configuration measurements Seifert [22] 2009 X DC voltage, battery INA103 1 Hz–30 kHz 100 Ω, 1 kΩ FFT analyzer Aluminum box - AD620 10 kΩ Maerki [20] 2013 X DC voltage, battery AD8676 + INA103 0.1 Hz–1 MHz 470 Ω, 1 kΩ, 10 kΩ, 50 kΩ, 100 kΩ, 200 kΩ - - Trimmer to zero bridge offset INA103 Maerki [40] 2016 Half bridge DC voltage Differential amplifier 2015 dc - 1 MΩ–50 GΩ - - - LaMacchia and Swanson [11] 2018 X Bipolar DC voltage, battery - 5 Hz–40 kHz 2 kΩ AP515 Cookie tin Amplifier can be added for parts with low noise Miyaoka and Kurosawa [37] 2019 X Bipolar DC voltage, battery AD620 + AD797 1 Hz–100 kHz 10 kΩ (different technologies) Oscilloscope - - Beev [4] 2022 X Bipolar battery-based DC voltage INA163 + PGA 0.001 Hz–10 Hz Resistor net-works < 10 kΩ HP 3458A digital voltmeter Die-cast aluminum box, steel shielding box for carrier board and instrumentation amplifier Correlated double sampling method with opposite bridge bias polarities, temperature stabilizationLT1167 + PGA Resistor net-works ≥ 10 kΩ Sensors 2023, 23, 1107 13 of 29 3.3. Voltage Supply The resistor under test for a single resistor as well as in a bridge configuration needs to be driven by a voltage to generate excess noise. Different possibilities are used in the papers mentioned in Table 1. It is conspicuous that most papers before the year 2000 used an AC voltage supply [10,32,33,35,36] in their setup while after 2000 all papers used a DC voltage supply [4,11,20,22,37,40]. The advantage of using AC supply is that resistance fluctuations are shifted to frequencies near the carrier frequency of the AC signal and therefore away from DC, where the excess noise of the amplifier is much greater than at higher frequencies [32,33]. Using two oscillating signals with a phase shift of 0◦ and 90◦ (or ± 45◦) and performing measurements in a two-channel device helps to further decrease background noise [36]. Alternating supply voltages, on the other hand, have the advantage of eliminating contributions of thermal electromagnetic forces. The advantage of AC voltage supply at the same time is the disadvantage of DC voltage supply as amplifiers are noisier with DC [10]. Nevertheless, DC technique is predominant in newer papers since amplifiers improved over the years and exhibit low noise today [22]. The usage of batteries for DC voltage supply is prevalent [4,11,20,34,37] since they add the least additional disturbances to the signal. Some papers use an unipolar supply voltage [20,22] while others use a bipolar supply voltage [4,11,37] with a symmetrical arrangement to keep the bridge output near zero and further decrease common mode signals [4]. As investigated by Seifert [22], excess noise is dependent on the supply voltage. When specifying the NI value with Equation (12), supply voltage is already considered and NI values can be compared between setups of different supply voltage. It gets more challenging when comparing diagrams with measured noise densities since they have different absolute values when supplied with different voltages. Furthermore, mostly no limits for specified NI values are depicted in the diagrams, which makes it even harder to compare. To give a demonstrative example: Beev [4] used 10 V per element while Seifert [22] used 5 V per element. If the NI is calculated, they can be directly compared. The diagrams in the case of excess noise are different by a factor of two. This needs to be considered when comparing the results. 3.4. Low-Noise Amplifier Measuring noise is associated with the measurement of small signals. This makes the usage of an amplifier mandatory for most setups. A subsequent requirement to the amplifier in a noise measurement setup is that the amplifier contributes as little noise as possible. Table 1 specifies the amplifier used in the respective papers. One exception is [11], where no amplifier is used. However, it is stated that for more sensitive measurements, an amplifier is a necessary improvement for the setup. While in the 1980s and 1990s, low-noise preamplifiers [32,34,36] and lock-in amplifiers [10,35] were used, more recent papers use low-noise operational [39] or instrumentation amplifiers [4,20,22,37] that come as integrated circuits. Instrumentation amplifiers have differential inputs and a single-ended output. Classically, an instrumentation amplifier consists of three operational amplifiers. They have several desirable characteristics such as very high input impedance (10 MΩ–10 GΩ) and a wide gain range (G = 1–1000). Additionally, they have a very high common mode rejection ratio (CMRR), especially at higher gains [15,19]. A disadvantage in using instrumentation amplifiers is the maximum allowed offset voltage at its input, resulting in a maximum allowed tolerance of the resistors in the bridge to not drive the output of the amplifier into saturation [22]. Operational amplifiers suffer from voltage and current noise, hence, for a given bridge resistance the adequate amplifier has to be chosen to account for both noise sources [21]. As can be seen in [4,22], different instrumentation amplifiers were used for different ranges of resistance values. This is because at low resistance values, voltage noise has the greatest impact on the overall noise of the amplifier, while at high resistance values, the current noise of the amplifier dominates the overall noise [19]. Therefore, for low resistance values, an Sensors 2023, 23, 1107 14 of 29 instrumentation amplifier with low voltage noise might be chosen and for high resistance values, an instrumentation amplifier with low current noise might be required. 3.5. Shielding The standard method only mentions that adequate shielding is necessary [1]. Shielding is inevitable since excess noise measurements are challenging. The aim of shielding is to keep away unwanted electromagnetic radiation from the outside. Otherwise, unwanted signals can appear in the measured signal. Machinery, power lines or computers are known sources for such radiations [13]. The purpose of an electrical shield is to protect the important parts from disturbances and to couple into ground [19]. A simple remedy can be found by isolating the measurement setup with a Faraday cage and using batteries for the power supply [13]. Grounded shields around wires or twisted wires can help to reduce capacitive or inductive coupling, but grounding of different parts has to be performed carefully in order to not create ground loops [19]. As shown in Table 1, there are measurement setups that use batteries such as in [4,11,20,34,37] for exactly this purpose. The column shielding reflects what the respective papers did to protect their measurement setup from unwanted signals. There can be found twisted cables [34] as well as aluminum boxes. The cheapest way is to use a tin can to create a Faraday cage [4,11,22,34,35]. 3.6. Measurement Equipment There are different possibilities for recording the data. Some papers use a voltmeter [1,4] or an oscilloscope [37], but most papers use fast Fourier transformation (FFT), digital signals or spectrum analyzers [10,22,33,35,36,38,39]. In some rare cases, modern audio cards are used [11]. For voltmeters and similar acquisition devices, the FFT has to be performed subsequently on a computer. One drawback of the standard method is that only the noise at 1 kHz is measured. By finding the FFT of a signal, information on noise over a wide range of frequencies can be obtained. This provides more insight into the behavior of noise [11]. In any case, the noise floor of the measurement device has to be taken into consideration. To improve measurement resolution, two techniques were identified from the papers introduced in Table 1. The first method uses a correlation method. With the cross-correlation technique used in [32–34,39], uncorrelated noise as noise from the amplifier can be sup- pressed. Beev [4] introduced the correlated double sampling (CDS) method for resistor noise measurements, which suppresses parasitic components near DC, amplifier drift and excess noise of the amplifier itself. Additionally, Beev [4] used a PT1000 sensor and a thermoelectric cooler to thermally stabilize the measured resistors in order to suppress fluctuations due to self-heating of the sample or temperature variations of the environment. Depending on the measurement setup, the measured noise density spectrum is not necessarily the noise density of the measured resistor. This is only true if no further noise is added by the measurement setup, e.g., the noise of the amplifier is orders of magnitudes lower. In case the noise of the amplifier is in the order of the device under test, voltage noise density and current noise density of the amplifier have to be determined and subtracted from the measured signal. In the case of uncorrelated noise sources, their powers are added [3,6]. 4. Measurement Results 4.1. Most Important Measurement Results from Previous Publications Many previous papers published measured data of one or more resistors. Table 1 gives an overview of measured resistance values and resistor types in the respective papers. Often, only the technology of the analyzed resistors and their resistance value are given without the exact part number, like is done in [37]. These results give a raw impression in which range the noise of a certain technology can be expected, but do not reflect the noise of a specific resistor series. Furthermore, in most cases, only the graphical results are given, not the noise indices. This makes it hard to compare the results to other measurements and even more if the supply voltage is not specified or the setup is completely different. The Sensors 2023, 23, 1107 15 of 29 papers of Seifert [22] and Beev [4] solely show comprehensive results on specific resistor series. Table 2 summarizes a few simplified results from both papers in terms of a range for the NI, similar to how Beev presented his results of resistor networks. Additionally, he distinguished between different resistive materials and different substrates. In contrary, Seifert only presented diagrams in his paper with resistor values of 100 Ω, 1 kΩ and 10 kΩ. Some of their results were selected to be presented in Table 2. Table 2. Some results from the papers of Seifert [22] and Beev [4]. NI [dB] Resistors From [22] Resistor Networks From [4] ≥−40 MIRA Electronic 1206, 1%, 100 ppm, 100 Ω Philips RC01, 5%, 200 ppm, 100 Ω NOMCA Vitrohm RGU 526-0, 2%, 100 ppm, 100 Ω Phoenix PR01, 5%, 250 ppm, 100 Ω AORN Mira 0805, 1%, 100 ppm, 100 Ω Philips RC01, 5%, 200 ppm, 10 kΩ T914 Panasonic ERJ-8ENF, 1%, 100 ppm, 10 kΩ Mira 0805, 1%, 100 ppm, 10 kΩ ACAS −40 to −60 Microtech CMF0805, 0.1%, 25 ppm, 100 Ω Vishay SMM0204-MS1, 1%, 50 ppm, 100 Ω PRA Yageo MF0207, 1%, 100 ppm, 1 kΩ Arcol MRA 0207, 0.1%, 15 ppm, 100 Ω DFN Vishay Dale CMF55, 0.1%, 25 ppm, 1 kΩ Vishay MMA0204, 0.1%, 15 ppm, 10 kΩ DIP-1999 Vitrohm ZC0204, 1%, 50 ppm, 10 kΩ VSOR Tyco RN73, 0.1%, 10 ppm, 10 kΩ MAX549x ≤−60 Vishay Beyschlag MMA0204, 1%, 50 ppm, 100 Ω Phycomp TFx13 series, 0.1%, 25 ppm, 100 Ω NOMC PRND Ohmite, 5%, 100 Ω Vishay Beyschlag MBB0207, 1%, 50 ppm, 100 Ω MORN RIA Welwyn RC55Y, 0.1%, 15 ppm, 1 kΩ Yageo PO 593-0, 5%, 200 ppm, 100 Ω OSOP ORN Vishay Beyschlag MMA0204, 1%, 50 ppm, 100 Ω Phycomp TFx13 series, 0.1%, 25 ppm, 100 Ω TDP HTRN Ohmite, 5%, 100 Ω Vishay Beyschlag MBB0207, 1%, 50 ppm, 100 Ω DIV23 MPM Welwyn RC55Y, 0.1%, 15 ppm, 1 kΩ Yageo PO 593-0, 5%, 200 ppm, 100 Ω SMN/SMNZ LT5400 4.2. Our Measurement Results Motivated by new technologies to create resistors for sensor applications, such as inkjet- and aerosol-jet-printed strain gauges and temperature sensors, and the question of how they compare to existing resistor technologies, we performed our own measurements on commercial resistors. Therefore, noise measurements of commercial resistors given in the papers mentioned above were reproduced to verify the measurement equipment. Two different setups were used, the first one based on the setup of the Seifert paper [3], while the second setup used a nanovoltmeter. In the following, the analyzed resistors and their characteristics are given, the different setups are described and the results presented. 4.2.1. Analyzed Resistors The measurements were performed on different resistor types. The first category was single resistors with values of 100 Ω, 350 Ω, 1 kΩ and 10 kΩ. Table 3 lists all the analyzed single resistors with parameters such as manufacturer part number, resistance value, tolerance, power and resistor technology, if specified by their datasheets. In case the resistors of the same series with different resistance values were available, they are grouped in the table. All these resistors were measured with the first setup except for the 10 kΩ metal film resistor, which was analyzed with the second setup. The measurement setups are introduced in the following section. Table 4 lists the analyzed resistor networks. In addition to the parameters from Table 3, there is an additional column with the material used for the resistor network since most datasheets contain this information. Again, resistor networks of the same series with different resistance values are grouped in the table. The resistor network with 100 Ω was measured with setup 1, the 1 kΩ networks with both setups and the 10 kΩ networks with setup 2 only. Sensors 2023, 23, 1107 16 of 29 Table 3. The analyzed single resistors and their parameters from datasheets. Manufacturer Part Number Abbreviation Value [Ω] Tolerance [%] Power [W] Technology AlphaElectronics MC Y 000100 T AE 100R 100 ±0.01 0.3 (at 125 ◦C) - AlphaElectronics MC Y 1K0000 T AE 1k 1000 ±0.01 0.3 (at 125 ◦C) - AlphaElectronics MC Y 000350 T AE 350R 350 ±0.01 0.3 (at 125 ◦C) - Arcol MRA 0207 Arcol 1k 100 ±0.1 0.25 Metal film Vishay Beyschlag MBB 0207 MBB 100R 100 ±1 0.6 (at 70 ◦C) Thin film Vishay Beyschlag MMA 0204 MMA 1k 1000 ±1 0.4 (at 70 ◦C) Thin film Vishay Bccomponents PR01 PR01 100R 100 ±5 1 (at 70 ◦C) Metal film Vishay Bccomponents PR02 PR02 100R 100 ±5 2 (at 70 ◦C) Metal film Vishay Dale PTF-56-1K0000 PTF 1k 1000 ±0.1 0.125 (at 85 ◦C) Metal film Vishay Dale PTF-56-350R00 PTF 350R 350 ±0.1 0.125 (at 85 ◦C) Metal film Neohm UPW25 UPW25 1k 1000 ±0.1 0.25 (at 125 ◦C) Wirewound Vishay Foil Resistors S Series (S102J) VFR 100R 100 ±0.01 0.6 (at 70 ◦C) Bulk Metal® Foil Vishay Foil Resistors S Series (S102J) VFR 1k 1000 ±0.01 0.6 (at 70 ◦C) Bulk Metal® Foil Vishay Foil Resistors S Series (S102J) VFR 350R 350 ±0.01 0.6 (at 70 ◦C) Bulk Metal® Foil YAGEO 10.0K 0207 MetalFilm 10,000 ±1 0.6 (at 70 ◦C) Metal film Table 4. Analyzed resistor networks and their parameters from datasheets. Manufacturer Part Number Abbreviation Value [Ω] Tolerance Absolute [%] Power/Resistor [W] Technology Material LT5400-1 LT5400-1 10,000 ±15 0.8 - Chromium silicide on silicone substrate [4] LT5400-4 LT5400-4 1000 ±15 0.8 - Chromium silicide on silicone substrate [4] Vishay Dale NOMCA-1603-1002 NOMCA16031002 10,000 ±1 0.1 (at 70 ◦C) Thin film Tantalum nitride on alumina substrate Vishay Dale NOMCA-1603-1001 NOMCA16031001 1000 ±1 0.1 (at 70 ◦C) Thin film Tantalum nitride on alumina substrate Vishay Thin Film TDP-1603-1002 TDP16031002 10,000 ±0.1 0.8 (at 70 ◦C) Thin film Passivated nichrome on silicone/alumina substrate Vishay Thin Film TDP-1603-1001 TDP16031001 1000 ±0.1 0.8 (at 70 ◦C) Thin film Passivated nichrome on silicone/alumina substrate Vishay Dale TOMC-1603-1002 TOMC16031002 10,000 ±1 0.1 (at 70 ◦C) Thin film Passivated nichrome Vishay Dale TOMC-1603-1001 TOMC16031001 1000 ±1 0.1 (at 70 ◦C) Thin film Passivated nichrome Vishay Dale TOMC-1603-1000 TOMC16031000 100 ±1 0.1 (at 70 ◦C) Thin film Passivated nichrome Furthermore, strain gauges were analyzed in this paper. Table 5 lists the three strain gauges and their parameters. The first strain gauges were commercial 1-LY11-6/350 from Hottinger Brüel & Kjaer GmbH (HBK) with a nominal resistance value of 350 Ω and were free of strain. Additionally, strain gauges of type 1-XY33-6/350 that were attached to an aluminum substrate were measured. The third strain gauges were inkjet-printed with silver ink (SicrysTM I30EG-1) on a polyimide substrate and were sintered at 200 ◦C for 120 min. For the bridge configuration, the strain gauges with the four best matching resistances were selected. The average resistance value of the four selected strain gauges was 468 Ω. The strain gauges were measured with both measurement setups. Table 5. Analyzed strain gauges and their parameters. Manufacturer Part Number Abbreviation Value [Ω] Tolerance [%] Technology Material HBK 1-LY11-6/350 Comm. strain gauge 350 ±0.35 - Constantan on polyimide HBK 1-XY33-6/350 Comm. strain gauge 350 ±0.35 - Constantan on polyimide Printed Strain Gauge Printed strain gauge 468 - Inkjet printing Silver ink on polyimide Sensors 2023, 23, 1107 17 of 29 4.2.2. First Setup: Measurement of Resistor Noise between 0.1 Hz and 100 kHz For the first setup, the resistors under test were arranged in a Wheatstone bridge. The setup was a variant of the setup used in [22], but with a different instrumentation amplifier and an oscilloscope instead of an FFT analyzer. To easily swap the resistors under test, they were mounted on pin headers, as shown in [7]. The bridge was powered by a 10 V supply voltage at the first diagonal. This means that every resistor in the bridge was biased with 5 V. The voltage was battery-based with two 6 V-cells in series, regulated to 10 V with a low-drop-out regulator (LDO) type LT3045. The resulting voltage across the second diagonal of the bridge was connected to the inputs of an AD8429 low-noise amplifier (LNA). The amplifier itself was battery-powered with ±12 V by two 6 V batteries in series for each rail. The gain was set to 60 dB. The whole setup was placed inside a tin can for electromagnetic shielding. The amplified signal was forwarded to a 12 bit Teledyne LeCroy Oscilloscope HDO6054-MS. Figure 5a shows a block diagram of the described setup. The signal was measured over a time period of 10 s. The oscilloscope has the ability to perform fast Fourier transformation (FFT) and rescaling to calculate the power spectral density as shown in [8]. Afterwards, the square root was calculated to obtain the noise density in V/ √ (Hz). An average of 100 measurements was taken. Hence, the measured result is a combination of resistor noise, amplifier voltage noise and amplifier current noise. To determine the noise contribution of the amplifier, the inputs of the amplifier were shorted for voltage noise measurement. The current noise of the amplifier was determined with a LT5400 100 kΩ resistor network, which is known for having very low excess noise. The measured voltage and current noise of AD8429 are depicted in Figure 6a. Figure 6b shows the contribution of voltage and current noise of the amplifier for different resistor values. Sensors 2023, 23, x FOR PEER REVIEW  18  of  31      Table 5. Analyzed strain gauges and their parameters.  Manufacturer Part Number  Abbreviation  Value [Ω]  Tolerance [%]  Technology  Material  HBK 1‐LY11‐6/350  Comm. strain gauge  350  ±0.35  ‐  Constantan on polyimide  HBK 1‐XY33‐6/350  Comm. strain gauge  350  ±0.35  ‐  Constantan on polyimide  Printed Strain Gauge  Printed strain gauge  468  ‐  Inkjet printing  Silver ink on polyimide  4.2.2. First Setup: Measurement of Resistor Noise between 0.1 Hz and 100 kHz  For the first setup, the resistors under test were arranged in a Wheatstone bridge. The  setup was a variant of the setup used in [22], but with a different instrumentation ampli‐ fier and an oscilloscope instead of an FFT analyzer. To easily swap the resistors under test,  they were mounted on pin headers, as shown in [7]. The bridge was powered by a 10 V  supply voltage at the first diagonal. This means that every resistor in the bridge was bi‐ ased with 5 V. The voltage was battery‐based with two 6 V‐cells in series, regulated to  10 V with a low‐drop‐out regulator (LDO) type LT3045. The resulting voltage across the  second diagonal of the bridge was connected to the inputs of an AD8429 low‐noise ampli‐ fier (LNA). The amplifier itself was battery‐powered with ±12 V by two 6 V batteries in  series for each rail. The gain was set to 60 dB. The whole setup was placed inside a tin can  for electromagnetic shielding. The amplified signal was forwarded to a 12 bit Teledyne  LeCroy Oscilloscope HDO6054‐MS. Figure 5a  shows a block diagram of  the described  setup. The signal was measured over a time period of 10 s. The oscilloscope has the ability  to perform fast Fourier transformation (FFT) and rescaling to calculate the power spectral  density as shown  in [8]. Afterwards, the square root was calculated to obtain the noise  density  in V/√(Hz). An average of 100 measurements was  taken. Hence,  the measured  result  is a  combination of  resistor noise, amplifier voltage noise and amplifier  current  noise. To determine  the noise contribution of  the amplifier,  the  inputs of  the amplifier  were shorted for voltage noise measurement. The current noise of the amplifier was de‐ termined with a LT5400 100 kΩ resistor network, which  is known for having very  low  excess noise. The measured voltage and current noise of AD8429 are depicted in Figure  6a. Figure 6b shows the contribution of voltage and current noise of the amplifier for dif‐ ferent resistor values.  (a)    (b)  Figure 5. Block diagram of measurement setups used for measurement of noise in resistors. (a) cor‐ responds  to setup 1 with an AD8429 as  instrumentation amplifier and an oscilloscope  to  record  measurement values. (b) corresponds to setup 2 with a nanovoltmeter for capturing data.  Figure 5. Block diagram of measurement setups used for measurement of noise in resistors. (a) corresponds to setup 1 with an AD8429 as instrumentation amplifier and an oscilloscope to record measurement values. (b) corresponds to setup 2 with a nanovoltmeter for capturing data. This contribution had to be subtracted from the measurement results. This subtraction was performed with GNU Octave on a computer after each measurement. The setup could measure noise of resistors with values between 100 Ω and 1 kΩ. The frequency span used for the investigation was 0.1 Hz to 100 kHz. Sensors 2023, 23, 1107 18 of 29 Sensors 2023, 23, x FOR PEER REVIEW  19  of  31          (a)  (b)  Figure 6. (a) Voltage noise and current noise of AD8429 and (b) total noise from the amplifier for  different resistor values.  This contribution had to be subtracted from the measurement results. This subtrac‐ tion was performed with GNU Octave on a computer after each measurement. The setup  could measure noise of resistors with values between 100 Ω and 1 kΩ. The frequency span  used for the investigation was 0.1 Hz to 100 kHz.  4.2.3. Second Setup: Measurement of Very‐Low‐Frequency Resistor Noise  For very‐low‐noise resistors and measurements down to the mHz and μHz regime,  a different setup was used. For 10 kΩ resistors, the bridge was fed by a battery‐powered  10 V, low noise voltage reference based on a temperature‐compensated, oven‐controlled  LTZ1000CH Zener that could provide up to 10 mA. For resistors ≤1 kΩ, the bridge was  fed by 10 V from a low‐noise linear dropout regulator type LT3045, which itself was pow‐ ered by a 12 V battery  (Figure 5b). The bridge voltage was measured using a Keithley  2182A nanovoltmeter set  to  the 10 mV  input  range.  It  featured an  input  impedance of  >10 GΩ, hence, it was not loading the bridge. On the other hand, its own noise equaled a  1 kΩ resistor. To suppress the influence of line voltage, the sample rate was set to multi‐ ples of power line cycles (NPLC), with a maximum possible sample rate of 25 NPLC. Fur‐ thermore, all  filters on  the nanovoltmeter were  turned off. To prevent  the  influence of  temperature fluctuations, the measurements were performed in a temperature‐controlled  environment at 23±1 °C and direct air drafts avoided.  The time series data for each measurement were captured over multiple hours and  logged to a file. Since the sample rate slightly varied with the line frequency, the measure‐ ment was corrected for equally spaced samples using interpolation. The equidistant sam‐ ples were then processed using the Welch algorithm, resulting  in the power spectrum.  The square root of the power spectrum then represented the noise in nV/√(Hz) format.  4.2.4. Results  In the following section, the measured results are presented. First, the results from  setup 1 are given, followed by the ones of setup 2. Afterwards, the results from both setups  are combined to show the noise over a larger frequency range.      Figure 6. (a) Voltage noise and current noise of AD8429 and (b) total noise from the amplifier for different resistor values. 4.2.3. Second Setup: Measurement of Very-Low-Frequency Resistor Noise For very-low-noise resistors and measurements down to the mHz and µHz regime, a different setup was used. For 10 kΩ resistors, the bridge was fed by a battery-powered 10 V, low noise voltage reference based on a temperature-compensated, oven-controlled LTZ1000CH Zener that could provide up to 10 mA. For resistors ≤1 kΩ, the bridge was fed by 10 V from a low-noise linear dropout regulator type LT3045, which itself was powered by a 12 V battery (Figure 5b). The bridge voltage was measured using a Keithley 2182A nanovoltmeter set to the 10 mV input range. It featured an input impedance of >10 GΩ, hence, it was not loading the bridge. On the other hand, its own noise equaled a 1 kΩ resistor. To suppress the influence of line voltage, the sample rate was set to multiples of power line cycles (NPLC), with a maximum possible sample rate of 25 NPLC. Furthermore, all filters on the nanovoltmeter were turned off. To prevent the influence of temperature fluctuations, the measurements were performed in a temperature-controlled environment at 23 ± 1 ◦C and direct air drafts avoided. The time series data for each measurement were captured over multiple hours and logged to a file. Since the sample rate slightly varied with the line frequency, the mea- surement was corrected for equally spaced samples using interpolation. The equidistant samples were then processed using the Welch algorithm, resulting in the power spectrum. The square root of the power spectrum then represented the noise in nV/ √ (Hz) format. 4.2.4. Results In the following section, the measured results are presented. First, the results from setup 1 are given, followed by the ones of setup 2. Afterwards, the results from both setups are combined to show the noise over a larger frequency range. Results from Setup 1 Figures 7–9 show the measured results of excess noise with setup 1 on single resistors arranged in a bridge with resistance values of 100 Ω, 350 Ω and 1 kΩ, respectively. Table 6 recaps the calculated NI values for every resistor. For the 100 Ω resistors in Figure 7, AE 100R and VFR 100R show the smallest excess noise, with NI smaller than −60 dB. Most excess noise can be found in PR01 100R and PR02 100R, with NI of about −45 dB. It can be seen that, especially between 0.1 Hz and 1 Hz, the resistors are superimposed by 1/fα-noise with α > 1. Figure 8 shows that for AE 350R and VFR 350R, the setup from setup 1 reaches its limitation. The NI is better than −60 dB, but no certain number can be given. In Figure 7 AE 1k, VFR 1k and UPW25 1k reached the limitation of the setup too. The largest excess Sensors 2023, 23, 1107 19 of 29 noise was measured for the PTF 1k resistor. Within one resistor series, all NI for different resistor values were comparable. Sensors 2023, 23, x FOR PEER REVIEW  20  of  31      Results from Setup 1  Figures 7–9 show the measured results of excess noise with setup 1 on single resistors  arranged in a bridge with resistance values of 100 Ω, 350 Ω and 1 kΩ, respectively. Table  6 recaps the calculated NI values for every resistor. For the 100 Ω resistors in Figure 7, AE  100R and VFR 100R show the smallest excess noise, with NI smaller than −60 dB. Most  excess noise can be found in PR01 100R and PR02 100R, with NI of about −45 dB. It can be  seen that, especially between 0.1 Hz and 1 Hz, the resistors are superimposed by 1/fα‐noise  with α > 1. Figure 8 shows that for AE 350R and VFR 350R, the setup from setup 1 reaches  its limitation. The NI is better than −60 dB, but no certain number can be given. In Figure  7 AE 1k, VFR 1k and UPW25 1k reached the limitation of the setup too. The largest excess  noise was measured for the PTF 1k resistor. Within one resistor series, all NI for different  resistor values were comparable.    Figure 7. Noise spectral density for some 100 Ω resistors biased with 5 V per element.  Figure 7. Noise spectral density for some 100 Ω resistors biased with 5 V per element. Sensors 2023, 23, x FOR PEER REVIEW  21  of  31        Figure 8. Noise density for some 350 Ω resistors biased with 5 V per element.    Figure 9. Noise spectral density for some 1 kΩ resistors biased with 5 V per element.      Figure 8. Noise density for some 350 Ω resistors biased with 5 V per element. Sensors 2023, 23, 1107 20 of 29 Sensors 2023, 23, x FOR PEER REVIEW  21  of  31        Figure 8. Noise density for some 350 Ω resistors biased with 5 V per element.    Figure 9. Noise spectral density for some 1 kΩ resistors biased with 5 V per element.      Figure 9. Noise spectral density for some 1 kΩ resistors biased with 5 V per element. Table 6. Noise indices for 100 Ω, 350 Ω and 1 kΩ resistors measured with setup 1. Resistor Noise Index [dB] AE 100R −64 VFR 100R −63 Arcol 100R −59 MBB 100R −55 PR01 100R −45 PR02 100R −46 AE 350R <−60 PTF 350R −47 VFR 350R <−60 AE 1k <−60 VFR 1k <−60 PTF 1k −51 MMA 1k −57 UPW25 1k <−60 Figures 10 and 11 show the measurement results of resistor networks analyzed with measurement setup 1 and their respective NI values given in Table 7. Figure 10 shows the TOMC16031000 resistor network with NI of about −59 dB. As for the 100 Ω single resistors, the spectrum is superimposed with 1/fα-noise with α > 1. Measurement results for 1 kΩ resistor networks are shown in Figure 11. TDP16031001 shows the least excess noise and cannot be determined quantitatively due to reaching the limit of the setup. NOMCA16031001 shows the largest excess noise with NI of about −30 dB, which equals the value given in the datasheet. Sensors 2023, 23, 1107 21 of 29 Sensors 2023, 23, x FOR PEER REVIEW  22  of  31      Table 6. Noise indices for 100 Ω, 350 Ω and 1 kΩ resistors measured with setup 1.  Resistor  Noise Index [dB]  AE 100R  −64  VFR 100R  −63  Arcol 100R  −59  MBB 100R  −55  PR01 100R  −45  PR02 100R  −46  AE 350R  <−60  PTF 350R  −47  VFR 350R  <−60  AE 1k  <−60  VFR 1k  <−60  PTF 1k  −51  MMA 1k  −57  UPW25 1k  <−60  Figures 10 and 11 show the measurement results of resistor networks analyzed with  measurement setup 1 and their respective NI values given in Table 7. Figure 10 shows the  TOMC16031000 resistor network with NI of about −59 dB. As for the 100 Ω single resistors,  the spectrum is superimposed with 1/fα‐noise with α > 1. Measurement results for 1 kΩ  resistor networks are shown in Figure 11. TDP16031001 shows the least excess noise and  cannot  be  determined  quantitatively  due  to  reaching  the  limit  of  the  setup.  NOMCA16031001 shows the largest excess noise with NI of about −30 dB, which equals  the value given in the datasheet.    Figure 10. Noise spectral density for a 100 Ω resistor network biased with 5 V per element.  Figure 10. Noise spectral density for a 100 Ω resistor network biased with 5 V per element. Sensors 2023, 23, x FOR PEER REVIEW  23  of  31        Figure 11. Noise spectral density for some 1 kΩ resistor networks biased with 5 V per element.  Table 7. Noise indices for 100 Ω and 1 kΩ resistor networks measured with setup 1.  Resistor Network  Noise Index [dB]  TOMC16031000  −59  LT5400‐4  −53  NOMCA16031001  −31  TDP16031001  <−60  TOMC16031001  −57  Figure 12a shows the result of a commercial strain gauge 1‐LY11‐6/350. The strain  gauges in the bridge were free of strain. Down to 1 Hz, the density spectrum follows 1/f.  Between 0.1 Hz and 1 Hz, the spectrum is superimposed by 1/fα ‐noise with α > 1. A second  bridge with 1‐XY33‐6/350 strain gauges attached to an aluminum substrate was measured  and shown in Figure 12b. They show a similar behavior in this frequency range. NI values  for commercial strain gauges are shown in Table 8.    The printed strain gauge could not be measured with setup 1 since the output voltage  of AD8429 went into its limits. It is probable that the single resistors of the bridge config‐ uration differ too much, which created a large offset voltage at the input of the LNA and  drove the output into saturation.  Figure 11. Noise spectral density for some 1 kΩ resistor networks biased with 5 V per element. Table 7. Noise indices for 100 Ω and 1 kΩ resistor networks measured with setup 1. Resistor Network Noise Index [dB] TOMC16031000 −59 LT5400-4 −53 NOMCA16031001 −31 TDP16031001 <−60 TOMC16031001 −57 Sensors 2023, 23, 1107 22 of 29 Figure 12a shows the result of a commercial strain gauge 1-LY11-6/350. The strain gauges in the bridge were free of strain. Down to 1 Hz, the density spectrum follows 1/f. Between 0.1 Hz and 1 Hz, the spectrum is superimposed by 1/fα -noise with α > 1. A second bridge with 1-XY33-6/350 strain gauges attached to an aluminum substrate was measured and shown in Figure 12b. They show a similar behavior in this frequency range. NI values for commercial strain gauges are shown in Table 8. Sensors 2023, 23, x FOR PEER REVIEW  24  of  31          (a)  (b)  Figure 12. Noise spectral density for a commercial 350 Ω strain gauge (a) 1‐LY11‐6/350 free of  strain and (b) 1‐XY33‐6/350 attached to an aluminum substrate.  Table 8. Noise index for 350 Ω commercial strain gauge.  Strain Gauge  Noise Index [dB]  Comm. strain gauge 1‐LY11‐6/350  −60  Comm. strain gauge 1‐XY33‐6/350  −61  Results from Setup 2  Figure 13 shows the noise spectral density for some 1 kΩ resistors and resistor net‐ works. The lowest noise was found for the TDP16031001 network, directly followed by  LT5400‐4, both in the order of <−60 dB. As shown in previous papers, NOMCA16031001  exhibited  the  largest noise with  −30 dB, matching  the datasheet  specification. A wire‐ wound resistor type UPW50 was measured and excess noise was found.    Figure 13. Noise spectral density for some 1 kΩ resistors and resistor networks biased with 5 V per  element.  Figure 12. Noise spectral density for a commercial 350 Ω strain gauge (a) 1-LY11-6/350 free of strain and (b) 1-XY33-6/350 attached to an aluminum substrate. Table 8. Noise index for 350 Ω commercial strain gauge. Strain Gauge Noise Index [dB] Comm. strain gauge 1-LY11-6/350 −60 Comm. strain gauge 1-XY33-6/350 −61 The printed strain gauge could not be measured with setup 1 since the output voltage of AD8429 went into its limits. It is probable that the single resistors of the bridge configu- ration differ too much, which created a large offset voltage at the input of the LNA and drove the output into saturation. Results from Setup 2 Figure 13 shows the noise spectral density for some 1 kΩ resistors and resistor net- works. The lowest noise was found for the TDP16031001 network, directly followed by LT5400-4, both in the order of <−60 dB. As shown in previous papers, NOMCA16031001 exhibited the largest noise with−30 dB, matching the datasheet specification. A wirewound resistor type UPW50 was measured and excess noise was found. Figure 14 shows the noise spectral density plot for some 10 kΩ resistors and resistor networks. Here, TDP16031002 and LT5400-1 are on par with around −60 dB, which differs from the results given in [4], indicating variations between different samples and measurements. A wirewound resistor type UPW50 was measured and here the excess noise is on par with TDP16031002 and LT5400-1. In Figure 15a, the noise spectral density for a resistor bridge formed by four commercial single-strain gauges type HBK 1-LY11-6/350 intended for measurements on steel is shown. During the measurement, the resistive elements were not attached to a substrate and, thus, were free of strain. It is worth noting that besides 1/f-noise, the noise spectral density is superimposed by a 1/fα component with α > 1. The noise index is in the order of −30 dB and thus comparable to the noise index of a NOMCA16031001 resistor network. On the other hand, the results of a similar strain gauge attached to a substrate with the proper Sensors 2023, 23, 1107 23 of 29 adhesive are shown in Figure 15b. Here, the 1/f component almost vanished and the spectrum is dominated by 1/fα-noise with α > 1. Sensors 2023, 23, x FOR PEER REVIEW  24  of  31          (a)  (b)  Figure 12. Noise spectral density for a commercial 350 Ω strain gauge (a) 1‐LY11‐6/350 free of  strain and (b) 1‐XY33‐6/350 attached to an aluminum substrate.  Table 8. Noise index for 350 Ω commercial strain gauge.  Strain Gauge  Noise Index [dB]  Comm. strain gauge 1‐LY11‐6/350  −60  Comm. strain gauge 1‐XY33‐6/350  −61  Results from Setup 2  Figure 13 shows the noise spectral density for some 1 kΩ resistors and resistor net‐ works. The lowest noise was found for the TDP16031001 network, directly followed by  LT5400‐4, both in the order of <−60 dB. As shown in previous papers, NOMCA16031001  exhibited  the  largest noise with  −30 dB, matching  the datasheet  specification. A wire‐ wound resistor type UPW50 was measured and excess noise was found.    Figure 13. Noise spectral density for some 1 kΩ resistors and resistor networks biased with 5 V per  element.  Figure 13. Noise spectral density for some 1 kΩ resistors and resistor networks biased with 5 V per element. Sensors 2023, 23, x FOR PEER REVIEW  25  of  31      Figure 14 shows the noise spectral density plot for some 10 kΩ resistors and resistor  networks. Here, TDP16031002 and LT5400‐1 are on par with around −60 dB, which differs  from the results given in [4], indicating variations between different samples and meas‐ urements. A wirewound resistor type UPW50 was measured and here the excess noise is  on par with TDP16031002 and LT5400‐1.    Figure 14. Noise spectral density for some 10 kΩ resistor networks biased with 5 V per element.  In Figure 15a, the noise spectral density for a resistor bridge formed by four commer‐ cial single‐strain gauges type HBK 1‐LY11‐6/350 intended for measurements on steel  is  shown. During the measurement, the resistive elements were not attached to a substrate  and, thus, were free of strain. It is worth noting that besides 1/f‐noise, the noise spectral  density is superimposed by a 1/fα component with α > 1. The noise index is in the order of  −30 dB and thus comparable to the noise index of a NOMCA16031001 resistor network.  On the other hand, the results of a similar strain gauge attached to a substrate with the  proper adhesive are shown in Figure 15b. Here, the 1/f component almost vanished and  the spectrum is dominated by 1/fα‐noise with α > 1.      (a)  (b)  Figure 15. (a) Noise spectral density of a resistor bridge formed by four single‐strain gauges biased  with 5 V per element, (b) noise spectral density of a resistor bridge formed by four single‐strain  gauges that are attached to an aluminum substrate biased with 5 V per element.  Figure 14. Noise spectral density for some 10 kΩ resistor networks biased with 5 V per element. A similar measurement was performed for the aforementioned inkjet-printed single- element strain gauge sensors. The noise spectral density is shown in Figure 16. Although limited by the upper frequency, a combination of 1/f-noise and 1/fα noise with α>1 can be observed. The noise index is in the order of 10 dB and, thus, exceptionally large compared to other resistor technologies. This is probably due to the nature of resistive elements created by sintered silver nanoparticles. Sensors 2023, 23, 1107 24 of 29 Sensors 2023, 23, x FOR PEER REVIEW  25  of  31      Figure 14 shows the noise spectral density plot for some 10 kΩ resistors and resistor  networks. Here, TDP16031002 and LT5400‐1 are on par with around −60 dB, which differs  from the results given in [4], indicating variations between different samples and meas‐ urements. A wirewound resistor type UPW50 was measured and here the excess noise is  on par with TDP16031002 and LT5400‐1.    Figure 14. Noise spectral density for some 10 kΩ resistor networks biased with 5 V per element.  In Figure 15a, the noise spectral density for a resistor bridge formed by four commer‐ cial single‐strain gauges type HBK 1‐LY11‐6/350 intended for measurements on steel  is  shown. During the measurement, the resistive elements were not attached to a substrate  and, thus, were free of strain. It is worth noting that besides 1/f‐noise, the noise spectral  density is superimposed by a 1/fα component with α > 1. The noise index is in the order of  −30 dB and thus comparable to the noise index of a NOMCA16031001 resistor network.  On the other hand, the results of a similar strain gauge attached to a substrate with the  proper adhesive are shown in Figure 15b. Here, the 1/f component almost vanished and  the spectrum is dominated by 1/fα‐noise with α > 1.      (a)  (b)  Figure 15. (a) Noise spectral density of a resistor bridge formed by four single‐strain gauges biased  with 5 V per element, (b) noise spectral density of a resistor bridge formed by four single‐strain  gauges that are attached to an aluminum substrate biased with 5 V per element.  Figure 15. (a) Noise spectral density of a resistor bridge formed by four single-strain gauges biased with 5 V per element, (b) noise spectral density of a resistor bridge formed by four single-strain gauges that are attached to an aluminum substrate biased with 5 V per element. Sensors 2023, 23, x FOR PEER REVIEW  26  of  31      A similar measurement was performed for the aforementioned inkjet‐printed single‐ element strain gauge sensors. The noise spectral density is shown in Figure 16. Although  limited by the upper frequency, a combination of 1/f‐noise and 1/fα noise with α>1 can be  observed. The noise index is in the order of 10 dB and, thus, exceptionally large compared  to other resistor technologies. This is probably due to the nature of resistive elements cre‐ ated by sintered silver nanoparticles.  In Figure 17, the measured results of 1 kΩ resistor networks of both setups are com‐ bined in one diagram. As can be seen, the noise spectral densities for NOMCA16031001,  TDP16031001, TOMC16031001 and LT5400‐4 are in good agreement, indicating that both  setups lead to the same results.    Figure 16. Noise spectral density of a resistor bridge formed by four single inkjet‐printed strain  gauges biased with 5 V per element.    Figure 17. Combined noise spectral densities of some 1 kΩ resistor networks measured with setup  1 and 2.  Figure 16. Noise spectral density of a resistor bridge formed by four single inkjet-printed strain gauges biased with 5 V per element. In Figure 17, the measured results of 1 kΩ resistor networks of both setups are com- bined in one diagram. As can be seen, the noise spectral densities for NOMCA16031001, TDP16031001, TOMC16031001 and LT5400-4 are in good agreement, indicating that both setups lead to the same results. Sensors 2023, 23, 1107 25 of 29 Sensors 2023, 23, x FOR PEER REVIEW  26  of  31      A similar measurement was performed for the aforementioned inkjet‐printed single‐ element strain gauge sensors. The noise spectral density is shown in Figure 16. Although  limited by the upper frequency, a combination of 1/f‐noise and 1/fα noise with α>1 can be  observed. The noise index is in the order of 10 dB and, thus, exceptionally large compared  to other resistor technologies. This is probably due to the nature of resistive elements cre‐ ated by sintered silver nanoparticles.  In Figure 17, the measured results of 1 kΩ resistor networks of both setups are com‐ bined in one diagram. As can be seen, the noise spectral densities for NOMCA16031001,  TDP16031001, TOMC16031001 and LT5400‐4 are in good agreement, indicating that both  setups lead to the same results.    Figure 16. Noise spectral density of a resistor bridge formed by four single inkjet‐printed strain  gauges biased with 5 V per element.    Figure 17. Combined noise spectral densities of some 1 kΩ resistor networks measured with setup  1 and 2.  Figure 17. Combined noise spectral densities of some 1 kΩ resistor networks measured with setup 1 and 2. 5. Comparison with the Literature Results Comprehensive results of the individual parts can be found in [4,22,41]. Seifert gives sufficient information about the setup to estimate or read the NI of the single components and Beev [4] gives a range for NI values for individual components. Results from [22,41] were digitized and compared to our own. Arcol 100 Ω and MBB 100 Ω were used in Seifert’s paper. As shown in Figure 18a, the respective curves do not match our own results. Arcol 100 Ω was measured noisier in Seifert’s work while MBB 100 Ω was measured noisier here. For resistor networks, shown in Figure 18b, our own results on LT5400-1, NOMCA16031002 and TOMC16031002 are in fair agreement with [41], while good matching is given for LT5400-1. For NOMCA16031002, the results are not far apart; our measurement shows slightly more excess noise. For TOMC16031002, it is the opposite way and our results are slightly lower in noise. These small differences could probably be explained by variations from sample to sample. Sensors 2023, 23, x FOR PEER REVIEW  27  of  31      5. Comparison with the Literature Results  Comprehensive results of the individual parts can be found in [4,22,41]. Seifert gives  sufficient information about the setup to estimate or read the NI of the single components  and Beev [4] gives a range for NI values for individual components. Results from [22] and  [41] were digitized and compared to our own. Arcol 100 Ω and MBB 100 Ω were used in  Seifert’s paper. As shown in Figure 18a, the respective curves do not match our own re‐ sults. Arcol 100 Ω was measured noisier in Seifert’s work while MBB 100 Ω was measured  noisier here. For resistor networks, shown  in Figure 18b, our own results on LT5400‐1,  NOMCA16031002 and TOMC16031002 are in fair agreement with [41], while good match‐ ing is given for LT5400‐1. For NOMCA16031002, the results are not far apart; our meas‐ urement shows slightly more excess noise. For TOMC16031002, it is the opposite way and  our results are slightly lower in noise. These small differences could probably be explained  by variations from sample to sample.      (a)  (b)  Figure 18. (a) Comparison of Seifert’s results for Arcol 100 Ω and MBB 100 Ω results with the re‐ sults of the authors and (b) comparison of results from [41] for 10 kΩ resistor networks LT5400‐1,  NOMCA16031002 and TOMC16031002.  6. Discussion  Different papers on noise measurements of resistors were reviewed and analyzed  based on criteria such as bridge setting, supply voltage and others. This work gives an  overview of existing methods as well as an insight in the theory behind such measure‐ ments. The standard method is limited in its ability to measure noise in modern resistors.  On the other hand, improved setups have been proposed, resulting in the ability to meas‐ ure much  lower noise  in resistors based on bridge configurations.  Improvements were  achieved via bipolar DC or AC bridge excitation and/or cross‐correlation methods. The  setup proposed by [4] is superior over the others. It has the largest flexibility with respect  to the resistor values to be measured and uses cross‐correlation for improved resolution.  Temperature stabilization allows for very long measurements and the setup can also be  used for very‐low‐frequency noise measurements. However, it is a custom setup, while  most other setups use commercially available equipment. Nevertheless, it is hard to make  any recommendations on which setup should be used for a certain type or technology of  resistor, since for most setups the noise floor is unknown and only a few measurement  results are available.      Figure 18. (a) Comparison of Seifert’s results for Arcol 100 Ω and MBB 100 Ω results with the results of the authors and (b) comparison of results from [41] for 10 kΩ resistor networks LT5400-1, NOMCA16031002 and TOMC16031002. Sensors 2023, 23, 1107 26 of 29 6. Discussion Different papers on noise measurements of resistors were reviewed and analyzed based on criteria such as bridge setting, supply voltage and others. This work gives an overview of existing methods as well as an insight in the theory behind such measurements. The standard method is limited in its ability to measure noise in modern resistors. On the other hand, improved setups have been proposed, resulting in the ability to measure much lower noise in resistors based on bridge configurations. Improvements were achieved via bipolar DC or AC bridge excitation and/or cross-correlation methods. The setup proposed by [4] is superior over the others. It has the largest flexibility with respect to the resistor values to be measured and uses cross-correlation for improved resolution. Temperature stabilization allows for very long measurements and the setup can also be used for very-low-frequency noise measurements. However, it is a custom setup, while most other setups use commercially available equipment. Nevertheless, it is hard to make any recommendations on which setup should be used for a certain type or technology of resistor, since for most setups the noise floor is unknown and only a few measurement results are available. 6.1. Reproduction of Measurement Results In the present work, measurements on resistors and resistor networks were performed with two different setups to reproduce results from previous publications. A first setup, a variant of [22], used DC bridge excitation, a low-noise amplifier and the FFT functionality of a digital oscilloscope to measure the bridge voltage, while a second setup for measuring very low excess noise used a commercial nanovoltmeter. The advantage of setup 2 was the ability to measure bridges with higher imbalance, which led to larger offset voltages and saturated the LNA of setup 1. While setup 1 is good for components showing significant noise for frequencies up to 100 kHz, setup 2 is feasible for very-low-noise resistors, making it necessary to measure down to very low frequencies. However, it is worth noting that setup 1 can be used for very-low-frequency measurements too, but is limited by the 1/f noise of the LNA. A differential, very-low-noise chopper amplifier with JFET input stage could overcome the 1/f noise limit. Several different approaches for such chopper-based LNAs have already been proposed in the literature. The results obtained from both setups on the same resistors show conclusive and matching results. The noise indices for commercial resistors and resistor networks calcu- lated from these measurements agree with previous publications. This indicates that both setups are adequate to characterize excess noise in resistors. However, slight differences, probably due to sample variation, were observed. While setup 1 requires an LNA, setup 2 uses commercial test equipment only. However, with the NPLCs used in the measurements of setup 2, the maximum frequency is limited to 0.5 Hz. To overcome this limit, the NPLC could be increased up to 0.1 NPLC on a Keithley 2182A or by using another nanovoltmeter such as an Agilent 34420A with sampling rates up to 0.02 NPLC, with the disadvantage of a higher noise floor. In general, there are only a few NI values in the literature given at all. This makes it hard to estimate how big the sample-to-sample variation is as well as by how much the setup or maybe other unknown parameters affect the results. The effect of sample variations for resistors in general and the reproducibility of noise measurements on the same component could be investigated. 6.2. Measurements on Commercial and Inkjet Printed Strain Gauges Additionally, a few commercial strain gauges were measured to understand how they compare to different existing resistor technologies. Thin film strain gauges made of constantan (typically Cu55Ni44Mn1) on polyimide demonstrated a noise index similar to resistor networks made of tantalum nitride resistor films on high-purity alumina substrates, such as NOMCA1603. It is conspicuous that besides excess noise, the strain-free strain gauges demonstrated 1/fα noise with α > 1. The reason for that should be investigated Sensors 2023, 23, 1107 27 of 29 further to extract the nature of that extra noise contribution. This could be done by measuring the strain gauges prior and post-attaching them to a substrate. A first inkjet-printed strain gauge made of sintered silver nanoparticles on a polyimide foil was measured and exhibited a 100 times larger noise index compared to the commercial strain gauge. Besides that, it also exhibited the very same 1/fα noise with an α > 1 component, with the root cause for that being yet unknown, but obviously representative for strain gauges. Hence, a systematic investigation on inkjet-printed strain gauges should be performed to separate influencing parameters from printing