Twin‐Wire Sensor Networks

A fundamental limit to high‐density sensing is that adding sensors increases the number of wires, pads, and interconnections. This problem is even worse for low‐values resistive or impedance sensors which, for high accuracy, require 4‐wire measurements. Here, twin‐wire sensor networks are described which enable 4‐wire measurements with significantly fewer wires, pads, and interconnections. The effects of the resistor noise can be minimized by minimum‐resistance sensing paths. A single chopper switch and straightforward digital operations can reject the equivalent input offset and low‐frequency noise voltages of the instrumentation amplifier, thus requiring no hardware changes. The inclusion in the network of a reference resistor can compensate the errors of both the biasing current and the instrumentation amplifier voltage gain. For validation, an extremely compact, robust, and easy‐to‐connect flexible‐PCB twin‐wire 29‐temperature‐sensors network requiring only 32 pads is demonstrated. This device is placed on an anthropomorphic head phantom for detecting the temperature and location of a touching object or on the hand of a volunteer for monitoring skin temperature during mental and physical stimulations. A twin‐wire 29‐photoresistors network is also presented. The strategies reported here can be applied to any high‐density array of resistive or impedance sensors (temperature, strain, blood flow, light, etc.) and may find wide application in robotics and wearable or epidermal devices.


Introduction
Arrays of minuscule sensing elements can enable the evaluation, with unprecedented spatial resolution, of relevant properties such as temperature, [1] strain, [2] macrovascular and microvascular blood flow, [3] multimodal mechanical stimuli, [4] force, [5] DOI: 10.1002/adsr.202300031bioimpedance, [6] tissue oxygenation, [7] pressure, [8] light, [9] and more.The individual sensing element in the array is often a simple resistor [1][2][3][4][5] or an impedance [6] or, more in general, a bipole [7][8][9] whose current-voltage characteristics somehow depend on the property of interest.When measuring small resistances or impedances or, more in general, when series parasitic elements can introduce significant error voltages, 4-wire (or 4-terminal) sensing, [10] introduced by Lord Kelvin in 1861, is the standard technique for accurate measurements.However, with this approach, the characterization of N R elements requires that the number of pads, N P , and of external wires, N W , are both equal to 4N R or, if the ground terminal can be shared by all the devices, with possible tradeoffs between compactness and parasitic resistances, 3N R + 1.This fundamental constraint is a key issue for highdensity 4-wire measurements and may be complicated by other limits associated with parasitic resistances, lithographic resolution, lay-out and errors originated by the electronic interface (e.g., the uncertainty of biasing current or errors of the instrumentation amplifier such the input offset voltage or gain errors).
Recently, twin-wire networks [11] have been proposed for enabling zero interconnect and high-density 4-wire material characterizations (e.g., silver-nanoparticle conductive inks printed on polyimide, paper, or photo paper [11] ), which are more efficient than conventional 4-wire systems in terms of the numbers of pads and wires.The twin-wire network strategy can be generalized to other 4-wire measurements, but networks of resistive sensors do not necessarily inherit the zero-interconnect property.In some cases, the sensors can be made of the same conductive material used for connecting the sensors to the electronic interface.For instance, an array of twin-wire resistors printed by silver nanoparticles conductive ink, [11] once the temperature coefficients of the resistors are calibrated, can be seen as an array of resistive temperature sensors and, if the entire network is printed by the same silver nanoparticles conductive ink without adding any other conductive material, the resulting twin-wire sensor network has the zero-interconnect property. [11]However, in practical systems, discrete components are often employed as sensors or, in microsystems, the sensing elements may be made of a different material in comparison with the connections or must be smaller than the area required for connections, so parasitic resis- tances must be taken into account.Additionally, non-idealities of both the sensors and the electronic interface translate into measurement errors, so that accurate measurements may be hindered by the uncertainties on the biasing current and the nonidealities of the instrumentation amplifier, with special reference to the gain error and to the input offset and low-frequency noise voltages.
Here, we extend the twin-wire network strategy to twin-wire networks of resistive sensors and systematically discuss how to minimize the main sources of errors, including the parasitic resistances introduced by the interconnects, resistor noise, errors of the biasing current and of the instrumentation amplifier (i.e., gain and the input offset and low-frequency noise voltages).Finally, as an example, we demonstrate a very robust, compact, and easy-to-connect flexible PCB twin-wire 29 temperaturesensors network requiring only 32 pads.For validation, this device has been placed on an anthropomorphic head phantom as a temperature-sensitive robotic skin or placed on the hand of a healthy volunteer during both mental and physical stimuli.As another example, a twin-wire network of 29 photo-resistors has also been shown.The strategies reported here can be generalized to any resistive or impedance sensor and can provide crucial advantages for high-density 4-terminal measurements.

Results and Discussion
Figure 1a shows how standard 4-wire measurements allow to measure the sensing resistance R S independently, with excellent approximation, on the parasitic resistances R P1 , R P2 , R P3 , and R P4 . [10]In fact, ideally, since the instrumentation amplifier has zero input currents, the current I 0 entirely flows through R S and, therefore, the voltage across R S is exactly equal to R S I 0 .Moreover, ideally, the currents through the parasitic resistors R P3 and R P4 , which are in series with the input terminals of the instrumentation amplifiers, are also zero, so that there is no voltage (Ohm law) across R P3 and R P4 .As a consequence, the voltage between the two sense terminals ("Sense +" and "Sense −") or, equivalently, the voltage between the input terminals of the instrumentation amplifier is exactly equal to the voltage across R S (or, equivalently, to R S I 0 ) and shows no dependence on the parasitic resistors R P1 , R P2 , R P3 , and R P4 .Though this analysis is only approximate because of the non-zero input bias and offset currents of real instrumentation amplifiers, in several cases such input currents are extremely small and therefore accurate measurements are possible.The twin-wire network strategy, which has been recently proposed for enabling zero interconnect, high-density 4wire electrical characterizations of materials, can also be applied to networks of resistive or impedance sensors.The general results [11] for network generation, iterations, numbers of pads/wires/resistors and multiply-by-M expansions (i.e., bifurcation for M = 2, trifurcation for M = 3,…) apply, but twin-wire sensing networks do not necessarily have the zero-interconnect property.As an example, Figure 1b schematically shows a twin-wire network of 29 resistive sensors which may be formally generated by applying three bifurcation steps [11] to an initial resistor R A .Each sensing resistor in the network can be measured with the standard 4-wire method.In fact, if each pad is connected by two distinct wires (a couple of wires connected to a certain pad are referred to as "twin-wires") to switches (properly driven by a microcontroller), one or more configurations of the switches can connect any resistor of interest in the network to external instrumentation as required for performing 4-wire measurements.For instance, the sensing resistor R J can be measured by using the pads P N and P V as positive and negative force terminals (Figure 1a), respectively, and the pads P X and P W as positive and negative sense terminals (Figure 1a), respectively.Clearly, all the switches, except the ones required for connecting to these 4 pads, during this measurement, must be kept as open.In such a configuration, the resistors R D , R A , R B , R F , and R N would be essential (as interconnects) for performing the measurements, but would not affect the 4-wire measurement as they are in series with the positive force terminals (because of the open circuit state of all the switches except 4 switches connecting with the interface circuit the 4 pads P N , P V , P X , and P W ) and therefore have the same role as R P1 in Figure 1a or, equivalently, are excluded from the 4-wire measurement.Similarly, in this configuration of the switches, R V (in series with the negative force terminal) would have the same role as R P2 in Figure 1a and would therefore not affect the 4-wire measurement.In the same way, R K and R X (in series with the positive sense terminal) would have the same role as R P3 in Figure 1a and R W (in series with the negative sense terminal) would have the same role as R P4 in Figure 1a so that all these resistances would also not affect the 4-wire measurement.Similar observations can be applied to other configurations (e.g., the sensing resistor R J can also be measured by using the pads P Q and P W as positive and negative force terminals, respectively, and the pads P Y and P V as positive and negative sense terminals, respectively) and to all the other sensors in the twin-wire network.The presence of twin-wires, for each pad, is necessary for performing 4-wire measurements on most peripheral resistors (i.e., the resistors directly in contact with the pads).More details on the twin-wire strategy can be found in literature. [11]For each sensor, among all possible connecting options, in order to minimize the effects of thermal noise, the total resistance along the sense path (i.e., the sum of the resistances of the network resistors connecting the positive and the negative input sense terminals) should be minimized.As to 1/f noise, [12] the voltage noise generated by the resistors in series with the sense terminals (i.e., with almost zero currents) will generally be negligible and only the 1/f noise generated by the resistor under test is important.As schematically shown in Figure 1c for a representative sensor in the twin-wire network, R J , each ter-minal of the sensing resistor is connected by series resistors to its adjacent nodes.In some cases, the entire sensor network can be made of a single conductive material, so that these series resistors can, if there are no different constraints on the sensing area, be seen as part of the sensor itself and, therefore, do not introduce parasitic resistances or, equivalently, the network inherits the zero-interconnect property. [11]However, in general, each sensor is 4-wire self-connected by the network itself but also requires two wires for connecting to its adjacent nodes.In fact, sensors are often implemented by discrete components on PCBs or, in microsystems, can be made by a material different from connections or must be smaller than the area required for connections.Clearly, in such cases, the parasitic resistances of the two wires connecting the sensor to its adjacent nodes must be taken into account.The two series parasitic resistors corresponding to a certain sensing resistor can be distinguished as the superior and the inferior parasitic resistances (e.g., R P,UP,J and R P,DOWN,J for the sensor R J ).From the external point of view, both the superior and inferior parasitic interconnects may not be distinguished from their sensing resistor and, therefore, introduce a resistance error ΔR equal to.
where W and t are the width and the thickness of the interconnect tracks, respectively,  INTERCONNECTS and R □,INTERCONNECTS are the resistivity () and the sheet resistance (i.e., the ratio between resistivity and thickness, /t) of the interconnect material, respectively, and the ordered pairs (L P,UP , L P , DOWN ), (N □,UP , N □,DOWN ) and (R C,UP , R C,DOWN ) are the lengths, numbers of squares (L/W) and contact parasitic resistances (e.g., due to the junctions or solder joint between the interconnects and the sensing resistive material) of the superior and inferior parasitics, respectively.In several cases, the contact parasitic resistances can be ignored (e.g., for conventional PCBs, the product of the solder joint resistance and the solder joint area is typically in the order of 1 mΩ mm 2 or less, [13][14][15][16] so that for typical solder joint areas in the order of 1 mm 2 , these resistances are in the mΩ range, Note S1, Supporting Information).As a consequence, the resistance error ΔR can be made extremely small by adopting good electrical conductors (i.e., low resistivity) and by minimizing the number of squares of their parasitic resistances or, equivalently, by designing interconnects as short and wide as possible.In order to illustrate the advantages of the proposed strategies, as shown in Figure 1d, we have designed a twin-wire network of 29 Pt100 resistive temperature sensors on a standard, flexible, single-metal layer PCB using 35 μm thick copper interconnects (i.e., sheet resistance at room temperature around 0.5 mΩ □ −1 ).The device enables 4-wire measurements of 29 sensors, requires only 32 pads and is extremely compact, robust, and easy-to-connect by standard, low-cost connectors.Figure 1e shows a few Pt100 temperature sensors and illustrates the compactness of the device, which is a consequence of the small number of interconnects and the absence of metal traces directly connecting each sensor to external pads (such traces would be required for conventional arrays of 4-wire resistive sensors, see Figures S1 and S2, Supporting Information for details on the dimensions and for a comparison with a correspondent PCB comprising 29 resistors interfaced with conventional 4-wire strategies).The solder joint area for our standard Pt100 sensors was about 2.6 mm 2 so that the solder joint resistances can be neglected.The copper traces which are extremely short or whose resistances are rejected by 4-wire measurements have the smallest width (100 μm) allowed by the technology, whereas copper traces whose resistances affect 4-wire measurements have a larger width (300 μm) so that both the parasitic resistances (≤ 0.1 Ω) and their equivalent temperature errors (0 °C ≤ ΔT ≤ 0.25 °C) (Notes S2 and S3, Table S1, and Figure S3, Supporting Information) would not excessively increase the temperature error of Pt100 sensors (around ± 0.4 °C at room temperature for our Pt100 sensors) even in absence of any compensation.However, clearly, the nominal values of the parasitic resistances (Table S1, Supporting Information) can be easily taken into account, so that only their spread and drift (e.g., variations with temperature, strain, or aging) may not be compensated, thus reducing the final temperature errors to much smaller values, which, for a specific application, can be estimated by taking into account spread (e.g., the spread of the PCB-copper sheet resistance and of its temperature coefficient) and drift (e.g., the temperature range for the PCB during normal operations and aging).
Though the 4-wire measurements can be performed by highaccuracy instrumentation [11] (e.g., conventional multimeters, Figure S4, Supporting Information), in practical applications the electronic interface, including the basic circuit for 4-wire measurements (Figure 1a) and switches, [11] must be integrated in a single chip or assembled on a PCB.In these cases, a dominant source of error is typically constituted by the equivalent input offset and low-frequency noise voltages of the instrumentation amplifier.For instance, in case of CMOS electronic interfaces, [10] the input offset voltage of CMOS op amps can be as large as a few mV.As schematically shown in Figure 2a, even in such circumstances, all the sensors in the twin-wire network can still be accurately measured by taking advantage of two chopper switches [10,17,18] which enable the straight connections in one phase and the cross-connections in the second phase, provided that the output is somehow low-pass filtered.In fact, since each chopper switch can be seen as a modulator (i.e., its output voltage is equal to the input voltage multiplied by a square wave with levels equal to 1 and −1), the voltage across the sensing resistor is modulated by the first chopper switch, amplified, and then demodulated back to the base-band by the second chopper switch. [10,17,18]By contrast, the equivalent input offset and low-frequency noise voltages are amplified by the instrumentation amplifier and then modulated to high frequency by the second chopper switch, so that the low-pass filter can easily filter out their contributions to the output voltage. [10,17,18]In practice, the schematic diagram shown in Figure 2a can often be greatly simplified as the fully differential instrumentation amplifier can be replaced by a conventional single-ended instrumentation amplifier followed by an analog-to-digital converter so that the functions of both the second chopper switch and the low pass filter can be performed in the digital domain.With this observation, since switches are already necessary for sequentially connecting to each sensor, the chopper strategy can be applied without any hardware modification and at the only (often negligible) cost of halving the measurement speed by performing, instead of a single measurement for each sensing resistor, two measurements with interchanged positive and negative sense terminals.For validation, we considered a test PCB (Figure S5, Supporting Information) and artificially created a disturbing voltage in the order of 5 mV by connecting a thermocouple (with a hot plate for generating a significant temperature difference between the hot and cold junctions) in series with the positive sense terminal of the instrumentation amplifier (Figure S6, Supporting Information).In order to quantitatively demonstrate the effectiveness of chopper compensation we performed three distinct sessions of measurements.In the first session, the accurate values of all 29 resistors are measured with errors which, as a first approximation, can be neglected because of high accuracy of the 6½digit high-performance digital multimeter (Agilent 34410A).In the second session, we purposely introduced the input offset voltage and recorded the values of the measurements (affected by the input offset voltage) for all the 29 resistors; this second session allowed us to verify that the effects of the input offset voltage, in absence of compensation, were very large.In the third session, the input offset voltage is still present, but the values of the measurements for all 29 resistors are determined after applying for chopper compensation.After the end of the measurements, for each resistor, for each session, its resistance was computed as the mean of all the measured values (i.e., an average of 16 values for each sensor, see Statistical Analysis in the Experimental Section; see Figure S7 and Table S2, Supporting Information for the mean and the standard deviation of the measured values for all the sensors).After determining the values of the resistances (average of 16 values for each sensor) for all the sensors, for each session, we estimated the relative resistance errors, defined, for each resistor, for each session, as the ratio between the resistance error (i.e., the difference between the measured value and the resistor value initially measured by the multimeter in absence of offset) and the resistor value initially measured by the multimeter, in agreement with the general definition for relative errors (i.e., the ratio between the error and the ideal value).Figure 2b,c show the relative resistance errors before (Figure 2b) and after (Figure 2c) chopper compensation, thus confirming that the chopper strategy can reduce the relative errors by orders of magnitude for all the 29 sensors in the network.
Besides chopper compensation, errors in the biasing current and in the voltage gain of the instrumentation amplifier can also be important.In fact, similar to the strategies discussed for twinwire networks, [11] a microcontroller can drive arrays of switches for sequentially connecting each sensor to a conventional 4-wire interface (Figure 1a) comprising a current source with nominal current equal to I 0 and an instrumentation amplifier with nominal gain equal to A. The biasing current error ΔI and the voltage gain error ΔA may typically not be ignored and can degrade the measurement accuracy.For instance, in integrated circuits, it is generally possible to design bandgap voltage references [19][20][21][22][23][24][25] which, even in processes that do not allow the integration of good quality resistors, [26,27] can be reasonably accurate (curvature correction [20,21,[23][24][25]28] may be required for improved accuracy). By cntrast, it is generally complex to design accurate current references because of the typically large tolerance of integrated resistors, thus resulting in significant ΔI errors.As another example, the voltage gain of instrumentation amplifiers is often set by a discrete resistor which is not integrated on the same chip as the rest of the instrumentation amplifier (in particular, not integrated on the same chip as the other resistors of the instrumentation amplifier), so that the voltage gain is typically not very accurate (ratios between integrated and discrete resistors are generally not accurate).However, as schematically shown in Figure 3a, both these types of errors (ΔI and ΔA) may be easily compensated if a resistive sensor in the network is replaced by a reference resistor.In fact, the output voltage during phase 1 (switches 1 closed, switches 2 open), V OUT,1 , and during phase 2 (switches 1 open, switches 2 closed), V OUT,2 , will be.
so that, once the reference resistor is accurately known and the output voltages have been digitalized, the sensor resistance can be accurately calculated as In integrated circuits, this ratiometric technique would also allow us to reject low-frequency errors of the voltage reference used for analog-to-digital conversion.
For validation, we have fabricated a PCB with a twin-wire network comprising 28 resistors and 1 reference resistor (Figure S8, Supporting Information).In practice, we used a first multimeter used for generating the biasing current, an AD620 instrumentation amplifier with a voltage gain controlled by a variable discrete resistor, and a second multimeter for measuring the output voltage of the instrumentation amplifier (Figure S9, Supporting Information).Similarly to the case of chopper, for ratiometric compensation, we also performed three distinct experiments for measuring the values of all the 29 resistors with negligible errors (first session) and with errors due to purposely introduced variations of the current I 0 and of the amplifier gain before (second session) and after (third session) both ratiometric and chopper compensation (chopper compensation is necessary as, otherwise, the input offset voltage of the AD620 may easily introduce excessive errors).After the end of the measurements, for each resistor, for each session, its resistance was computed as the mean of all the measured values (i.e., an average of 13 values for each sensor, see the Statistical Analysis in the Experimental Section; see Figure S10 and Table S3, Supporting Information for the mean and the standard deviation of the measured values for all the sensors).After determining the values of the resistances (average of 13 values for each sensor) for all the sensors, for each session, we estimated the relative resistance errors.Figure 3b shows that with this arrangement, by simultaneously taking advantage of both the chopper (for compensating the input offset voltage of the AD620) and the ratiometric compensation, even extreme variations of both the biasing current (0.1 mA instead of a nominal value equal to 1 mA) and the instrumentation amplifier voltage gain (2.3 instead of a nominal value equal to 11.5) can be effectively compensated.
As an application example, we placed the flexible PCB with the 29-Pt100 sensor network with only 32 pads on an anthropomorphic head phantom for detecting the temperature and location of a touching object.As shown in Figure 4, after a finger touches one of the temperature sensors of the network (Figures 4a,b), the temperature of the touched sensor and, to a reduced extent, the temperatures of adjacent temperature sensors increase (Figure 4c).In our experiment, after about 30 s, the sensor R D is touched by a finger for about 30 s (Figure 4d).Similarly, after 80 s the sensor R G is touched for about 30 s (Figure 4d).As expected, when a sensor is heated by contact with the skin, the other sensors are also heated, to a smaller extent in order of proximity (see Figure 1D and Figure S1, Supporting Information for the positions of the different resistors in the network).
In another experiment, we used twin-wire networks for real-time monitoring of the skin temperature distribution.As shown in Figure 5a, the flexible PCB comprising the 29 Pt100 sensors was placed on the hand of a healthy volunteer who, after an initial relaxation period, was asked to perform mental calculations.Since for accurate temperature measurements, it is necessary to guarantee good thermal contact between skin and the temperature sensors, the flexible PCB was placed with the temperature sensors in contact with the skin and was firmly attached to the hand by a biocompatible, sweat-resistant wig glue smeared on the sensorized side of the PCB, but not on the Pt100 resistors.Since the PCB is larger than the hand, only the central temperature sensors are in contact with the skin.Though the stratum corneum, which is the outermost layer of the skin, is electrically insulating, sweat would easily introduce significant leakage currents and, therefore, large errors.However, the PCB can be easily passivated by applying an electrically insulating layer (e.g., based on nitrocellulose, resin, and plasticizers, for instance, nail polish).As a test, taking advantage of the simplicity of the quasi-simultaneous characterization of all the sensors, we verified that such a passivation procedure, with just two layers, was sufficient for reducing leakage currents to negligible levels (Figure S11, Supporting Information).For comparison, the hand temperature was simultaneously monitored by the twin-wire network and by a thermal camera.Since the PCB is not transparent, the thermal camera may not measure the temperature in the same points as the sensors, but the large holes in the PCB allow monitoring by the thermal camera at points of the skin which are very close to the sensors.In order to easily locate sensors in the thermal images, we attached small rectangles made of an adhesive copper tape (Figure 5b) on the outer side of the PCB, each rectangle in correspondence with an underlying temperature sensor (the sensors are in contact with the skin, that is, on the inner side of the PCB).The volunteer was a 34-year-old male mathematical engineer who sat in a reclined office chair, with his left forearm resting on the table and a soft pillow under his hand to ensure a comfortable posture during the entire test.The thermal camera was held about 20 cm from the hand with the lens pointed toward the palm.The test consists of an initial relaxation phase (25 minutes) followed by mental arithmetic calculations (15 minutes).During the 25 minutes relaxation phase, the volunteer did not focus on any task and was asked to rest (e.g., close the eyes and rest, as when trying to take a nap).During the mental stimulus, lasting 15 minutes, the subject was asked to correctly answer several arithmetic calculations (multiplications and divisions).The difficulty of calculations increased during the performance so an increasing concentration was required.All measurements were conducted at nominally constant room temperature in order to annihilate the effect of changes in room temperature on skin temperature.Figure 5c shows a representa- tive thermal image of the PCB on the hand of the volunteer (the scale bar for temperature is omitted as the emissivity of the skin is not accurately known).The copper rectangles are easily recognizable (dark blue spots) in the thermal image because of their lower emissivity.After the on-body temperature measurements the data are pre-processed (Note S4, Supporting Information) and then the temperatures evaluated by the different sensors can be directly visualized as measured (in contrast with experiments for validating chopper and ratiometric compensation, in order to not reduce the sampling rate, we did not compute mean values).Figure 5c also shows the elliptical skin areas (each elliptical area adjacent to a copper rectangle, i.e., adjacent to a sensor) used for computing the average temperature to be compared with the temperature of the correspondent sensor, with a representative sensor and the correspondent elliptical area enclosed by a dashed green rectangle.Finally, Figure 5d shows the nominal temperature measured during the test by the representative Pt100 sensor (red trace) with its uncertainty (pink area) and the temperatures estimated by the thermal camera in the correspondent adjacent elliptical area (blue traces) when the emissivity of skin is considered equal to different values between 0.95 and 0.98, which include the default value for opaque bodies (0.95, added for comparison) and the typical range (between 0.96 and 0.98) for the emissivity of skin. [29]If we consider the uncertainty of the skin emissivity and the errors of the twin-wire network system, which are dominated by the intrinsic errors of the Pt100 sensors (pink area), there is excellent agreement between measurements taken by the twin-wire network and the thermal camera (Figure S12, Supporting Information).These measurements are also consistent with similar experiments with epidermal devices for measuring skin temperature. [1]s an additional experiment, we have also monitored the skin temperature during a physical stimulus test (Figure S13, Supporting Information), consisting of a healthy volunteer, a 25-yearold male, who sat in a reclined office chair, with his left forearm resting on the table.The volunteer, after an initial 40 minutes relaxation phase, performed 5 minutes of biceps curl with the right arm (i.e., opposite arm as monitoring the temperature on a moving hand with both the thermal camera and the PCB would be more complex) followed by a 25 minutes relaxation phase.The sensors were kept in contact with the palm of the left hand by a sweat-resistant glue smeared on the sensorized side of the flexible PCB.The thermal camera was held about 20 cm from the hand with the lens pointed toward the palm.The palm temperature distribution was measured with both the twin-wire network of resistive temperature sensors and the thermal camera.As expected, [1] both the twin-wire sensor network and the thermal camera monitored a decrease in the hand skin temperature compared with the resting condition because of the peripheral vasoconstriction evoked by the physical stimulus.Moreover, similarly to the mental stimulus test, the measurements taken with the thermal camera and the twin-wire sensor network are in very good agreement, with discrepancies likely due to the uncertainties of the skin emissivity, the imperfect thermal contact between the non-epidermal device and skin, the residual error introduced by the parasitic resistances after compensation with their nominal values (computed by considering the nominal copper resistivity at a temperature of interest and the lengths, widths, and thickness of the copper tracks) and the Pt100 errors (around ± 0.4 °C at room temperature, Note S3, Supporting Information).
In principle, the proposed twin-wire sensor network strategy is suitable for any type of resistive sensor, with the only condition that there is not an exaggerated spread (e.g., orders of magnitude) on the resistances of the sensors in the twin-wire network or, equivalently, that all the resistances in the network have comparable values and no resistance becomes extremely high.In fact, intuitively, as an extreme case, when measuring a certain resistive sensor (under test), other resistive sensors are used as interconnects between the sensor under test and the four terminals of the interface (i.e., the two force terminals and the two sense terminals) and, clearly, such interconnects must not degenerate into open circuits.Apart from the theoretical case of open circuits, the other resistive sensors used as interconnects between the sensor under test and the force terminals must also not become excessively high (i.e., orders of magnitude higher than other resistors) as, otherwise, the injection of a current which could guarantee a satisfactory signal-to-noise ratio for the sensing voltage, would require very large voltages (as the injection current must flow through high resistances).By duality, if the resistors used as interconnects between the sensor under test and the sense terminals become excessively high, their noise voltages will add to the sensing voltage and may deteriorate the signal-to-noise ratio.This is, however, not an issue for most arrays of resistive sensors; for instance, platinum resistive temperature sensors roughly only double when the temperature goes from −50 to 150 °C.In conclusion, with the only exception of arrays comprising sensors that may have very different resistances (e.g., orders of magnitude) at the same time, the twin-wire network strategy is suitable for every type of resistive sensor.As another illustrative example, besides resistive temperature sensors, we also tested a twin-wire network of 29 light-dependent resistors or photoresistors.In practice, we used 29 commercial CdS photoconductive resistors (Luna Optoelectronics NSL-19M51).With these photoconductors, photons of visible light (with energy greater than the bandgap) result in the generation of electrons-holes pairs which increase the conductivity.For instance, we placed the network (Figure S14, Supporting Information) at a basal illumination of soft light inside a black camera box and the resistance of each photoresistor was in the kΩ range.Afterward, we sequentially illuminated two sensors within the network with the beam of a laser pointer, thus reducing their resistances about five times, but, despite such a large variation, all the sensors in the network could still be accurately measured (Figure S14, Supporting Information), which would become problematic in case of differences between the simultaneous values of different sensors of many orders of magnitude (e.g., the dark resistance of the photoconductors can be as high as MΩ, whereas the resistance under extreme illuminations, such as the beam of our laser point, could be around 200 Ω).

Conclusion
In summary, we have pesented twin-wire sensor networks for alleviating a fundamental limit for high-density 4-wire measure-ments.Each sensor is 4-wire self-connected by the rest of the network and has two connecting wires, each joining one terminal of the sensor to its adjacent node.Similar to zero-interconnect twinwire networks, [11] the twin-wire strategy results in advantages in terms of the numbers of wires, pads, and interconnections, which can be crucial advantages for high-density sensing (e.g., compare Figures S1 and S2, Supporting Information).Unlike twin-wire networks for zero interconnect 4-wire electrical characterizations of materials, [11] twin-wire sensor networks do not necessarily have the zero-interconnect property because of, for each sensor, two connecting wires having their parasitic resistances.However, not only such parasitic resistances can be easily evaluated and minimized by proper design (e.g., see the thicker copper traces in Figure S1, Supporting Information), but their nominal values can be easily taken into account so that only the spread and drift of these parasitics will introduce an error.Though different sensing paths are possible for connecting the sensor to the input terminals of the instrumentation amplifier, in order to minimize the noise, minimum-resistance sensing paths should be used.Since switches are necessary for sequentially connecting to each sensor, if the second chopper and low-pass filtering are implemented in the digital domain, a conventional singleended instrumentation amplifier is sufficient and the chopper does not require any hardware modification.If a reference resistor replaces a sensor in the network, both the biasing current and the instrumentation amplifier voltage gain errors can be easily compensated.For validation, we designed an extremely simple, robust, and easy-to-connect flexible PCB comprising a twin-wire 29-temperature-sensors network with only 32 pads.This device has been used both as a robotic skin and for monitoring skin temperature distribution.As another example, we have also demonstrated a twin-wire network of 29 photoresistors.The proposed strategies can provide crucial advantages for high-density 4-wire interfacing of arrays of resistive or impedance sensors, independently on materials (conventional or nano-cracked metals, [2,[30][31][32] liquid metals, [33] etc.), signal to be measured (temperature, strain, blood flow, light, etc.), technology and fabrication procedures (PCBs, system-on-chip, system-in-package, etc.).

Experimental Section
The twin-wire sensor network was designed with the KiCad Schematic Editor.The flexible PCB was manufactured by PCBway and the 29 Pt100 sensors (experiments shown in Figures 2, 4, and 5) or the 28 Pt100 sensors and the reference resistor (experiments shown in Figure 3 to illustrate ratiometric compensation) have been directly soldered onto the pads of the flexible PCB.4-wire measurements were taken with a Digital Multimeter Agilent 34410A.The twin-wire network has been connected to the multimeter through a pair of multiplexers that enabled the sequential and individual measurement of each resistive sensor (Figure S4, Supporting Information).The multimeter was set to the Ω4W (4-wire measurement) function, and the sample count, sampling interval, and integration time were set using an appropriate Java program to control the Agilent digital multimeter.The sample count (i.e., the number of resistance values stored in the multimeter) was set according to the duration of the experiment and the sampling rate.After the experiment, the resistance values stored in the multimeter were exported as a text (.txt) file and imported into MATLAB for pre-processing.
Additional information on the configuration, firmware, and software for the basic measurement system can be found at https://data.mendeley.com/datasets/pywhr745ns/1 which was previously used for verifying the general twin-wire strategy with resistors printed by a standard silver nanoparticles conductive ink rather than with an array of resistive sensors.The thermal image was analyzed with R (GNU language and environment for statistical computing and graphics) with the Thermimage package and with FLIR Tools software.The data analysis on the resistances of the Pt100 temperature sensors measured by the multimeter was performed with MATLAB (R2022b).Cubic-spline interpolation of the temperatures was measured within the same elliptical area between consecutive thermal images.
Statistical Analysis: The resistive sensors within the twin-wire network were sequentially and cyclically measured and all the measurements were stored inside the RAM of the Agilent digital multimeter, with the introduction of marker values for simplifying the identification of the different cycles in the data vector and, therefore, the association of the measured values to the correct sensors.The pre-processing consisted of the following four steps: a) separation of the "raw" data vector into 29 vectors, one vector for each sensor resistance; b) subtraction, for each sensor, of the value of the parasitic resistance; c) determination, for each sensor, of temperature; d) exclusion of eventual outliers.
For the experimental validation of chopper compensation (Figure 2), the sample sizes were 960 (raw vector for the 29 resistors and the equivalent resistor measured as marker), 32 (values for each sensor before the chopper compensation, 32 = 960/30), and 16 (values for each sensor after the chopper compensation, which required two values for producing a compensated value, 16 = 32/2).
For the experimental validation of the ratiometric strategy (Figure 3) which also took advantage of chopper compensation, the sample sizes were 780 (raw vector for the 29 resistors and the equivalent resistor measured as marker), 26 (values for each sensor before the chopper compensation, 32 = 960/30), and 13 (values for each sensor after the chopper compensation, which required two values for producing a compensated value, 13 = 26/2).
In both cases (chopper and ratiometric compensations) the data were presented as mean values (in the main text) and SD (in the Supporting Information), as discussed in the correspondent figure captions.MATLAB R2022b had been used for all the data processing and analysis.More information on the statistical analysis is given in Note S4, Supporting Information.

Figure 1 .
Figure 1.Twin-wire network of resistive sensors.a) 4-wire measurements (fundamental principle).b) Twin-wire network of 29 resistive sensors with the illustration of the parasitic resistances (red wires) in series with the resistive sensors (black).c) Equivalent electric circuit for the resistive sensor R J including its parasitic resistances R P,UP,J and R P,DOWN,J .d) Flexible PCB with a twin-wire network of 29 resistive Pt100 sensors and e) zoom showing the Pt100 sensors in the middle of the PCB.

Figure 2 .
Figure 2. Chopper compensation.a) Schematic diagram.b,c) Relative resistance error maps with an input equivalent offset voltage around 5 mV, b) without chopper compensation, and c) after chopper compensation.The relative resistance errors are calculated after averaging 16 resistance values for each sensor (obtained from a raw vector of 960 values).

Figure 3 .
Figure 3. Reference resistor for compensating the gain error ΔA and the current error ΔI. a) Circuit for compensating the gain error and the current error with a reference resistor (basic principle).b) Relative resistance error map with a gain equal to 2.3 (nominal gain A equal to 11.5) and a 0.1 mA current (nominal current I 0 equal to 1 mA) after ratiometric reference-resistor compensation.The relative resistance errors are calculated after averaging 13 resistance values for each sensor (obtained from a raw vector of 780 values).

Figure 4 .
Figure 4. Twin-wire temperature sensor network for robotic skin.a,b) Flexible PCB with a twin-wire network of 29 Pt100 sensors placed on a) an anthropomorphic head phantom and b) zoom showing a finger approaching the Pt100 sensor R D .c) Resistances measured by the 29 Pt100 sensors when a finger consecutively touches the sensor R D and the sensor R G .d,e) Temperatures measured by the sensors d) R D and e) R G during the experiment.

Figure 5 .
Figure 5. Twin-wire sensor network for skin temperature monitoring.a) Flexible PCB with a twin-wire network of 29 Pt100 sensors placed on the hand of a healthy volunteer during an experiment consisting of an initial relaxation period followed by mental calculations.b) Zoom of the flexible PCB showing the copper rectangles for simplifying the identification of the sensors in the thermal images.c) Thermal image of the hand with the illustration of the elliptical areas adjacent to the sensors, with a representative sensor and the correspondent elliptical area enclosed by a dashed green rectangle.d) Temperatures measured during the experiment by the representative Pt100 sensor (red trace, with pink area showing the error of the Pt100 sensor) and by the thermal camera in the correspondent adjacent elliptical area (blue traces for different values of the skin thermal emissivity).