Two-point Resistance Research Articles

Two-point resistances in symmetric bracelet resistor networks: accurate estimates and exact expressions

Resistor networks, used to model new types of natural or artificial matter, also provide generic examples for practising the methods of physics for obtaining estimates, revealing the main properties of a system and deriving exact expressions. Symmetric bracelet resistor networks are constructed by connecting n identical resistors in a circle, and then connecting two such circles by another set of n identical resistors. First, using van Steenwijk’s method, we establish that the equivalent resistance or two-point resistance (TPR) between any two nodes is derived when the layer-to-layer resistance R 0n is known. We then determine R 0n by an elementary recurrence relation which converges rapidly to its large n limit. Using this reference value of R 0n , accurate estimates of other TPRs follow for all values of n, characterised by a leading 1/n variation. In addition, exact explicit expressions of the TPRs can be calculated for any value of n. These networks, prototypes of three-dimensional networks considered in research, can be used to illustrate the diversity of the physical approach, the power of elementary methods, and to learn to be comfortable with approximations. Easy to make and use for experimental tests, they can support hands-on activities and conceptual changes.

Automated superconducting qubit characterization platform based on a modified 3D printer

Josephson Junctions are important components in superconducting qubits. It introduces anharmonicity to the energy level spacings of the qubit which allow us to identify two unique quantum energy states for computing. It is difficult to fabricate multiple junctions within the same desired parameter range. Characterisation of the junctions is, therefore, a necessary step after fabrication. In particular, the critical current of the junctions is determined by measuring their normal state resistance. This is done via two-point or four-point resistance measurement at a manual probe station which is a time-consuming process, especially for wafer-scale fabrication. This bottleneck can be circumvented by automation with object detection. The base of the automated probe station is a 3D printer modified with multiple Arduino Uno microcontrollers and motorized linear stages. The automation process is achieved via auto-alignment of the probes and an automatic measurement procedure. As a result, the fully automated process will take about 27–29[Formula: see text]s to measure the resistance of one junction which saves 28%–51% of the time compared to the manual probe station and can be unsupervised. Due to the reuse of a commercial 3D printer, the cost of this system is 800 SGD which is much less than comparable commercial solutions.

Effect of MXene fillers on the electrical conductivity of Ag-coated Cu conductive adhesives

The conductive adhesives were made using Ag-coated Cu and V2CTx + Ag + MWCNTs + Ag composites as fillers. The effects of V2CTx + Ag + MWCNTs + Ag composites content on the conductivity and microstructural properties of the conductive adhesives were investigated using two-point resistance measurements, scanning electron microscopy and XRD, respectively. The results showed that when the Ag package Cu content was 70%, the resistivity increased inversely with the V2CTx + Ag + MWCNTs + Ag (0.3%∼0.6%) content. Two-dimensional V2CTx + Ag + MWCNTs + Ag composites were obtained by solution mixing, offering the possibility to build up a large number of conductive networks. The synergistic effect of the Ag-claded Cu and V2CTx + Ag + MWCNTs + Ag composites significantly reduces the percolation threshold and conductivity of the conductive glue. In particular, the resistivity of the conductive glue is only 3.52 × 10−6 Ω m at a content of 70% Ag-coated Cu and 0.3% V2CTx + Ag + MWCNTs + Ag composites. Inside the epoxy resin Ag-coated Cu is connected to each other to form a conductive pathway and V2CTx + Ag + MWCNTs + Ag composites is interspersed between the conductive pathways to form a conductive network for electron transport. The synergistic effect of the Ag-coated Cu and V2CTx + Ag + MWCNTs + Ag composites makes the epoxy resin conductive adhesive with good conductive properties.

A type of generalized mesh-star transformation and applications in electrical networks

The calculation of two-point resistance and the number of spanning trees of a resistor network is a classic problem in electric circuit theory, which has been studied extensively by many mathematicians and physicists for many years. Ge (2021) obtained the formulae for the resistance distances and the number of spanning trees in a complete bipartite graph plus a matching. In this paper, we give a so-called complete multipartite graph-blossomed star transformation which is similar to the mesh-star transformation in electrical networks. As applications, we obtain the close formulae of resistance distances and the number of spanning trees in a complete multipartite graph adding a matching. We also derive the formulae of the resistance distances in the almost complete graph, which is obtained from a complete graph by deleting a forest in which each component is a star.

Silver-coated copper particles as a new raw material for manufacturing electrical contacts

Electrical contacts were prepared from copper, silver, and silver-coated copper (Cu@Ag) powders by cold-press and sintering. Then, they were used in an electrical circuit and evaluated their performance under AC (7 A, 6000 switchings) and DC (10 A, 20 A, 200 switchings) currents. The microstructure, phase structure, thermal stability, and electrical resistance of the contacts were studied using Scanning Electron Microscope (SEM), X-Ray Diffraction (XRD), Thermogravimetric Analysis (TGA) and Differential Scanning Calorimetry (DSC), and two-point electrical resistance measurement techniques. Results showed that the electrical resistance of copper contacts after 6000 switchings under AC current reaches from the initial value of 0.1 to 0.7 Ω. However, the electrical resistance of the silver and Cu@Ag contacts remained unchanged close to 0.1 Ω under the same conditions. The electrical resistance of copper contacts under 10 A-DC current increased from 0.11 Ω to 18 kΩ after 200 switchings, while the electrical resistance of the air and argon-sintered Cu@Ag contactsafter 200 switchings in the same conditions increased from 0.11 to 0.14 and 0.18 Ω, respectively. Increasing the DC current from 10 to 20 A increased the final electrical resistance of the copper contact from 18 kΩ to 35 kΩ and increased the electrical resistance of the Cu@Ag contact (sintered in air) from 0.18 Ω to 0.93 Ω after 200 switchings. Increasing the DC current from 10 A to 20 A did not affect the electrical resistance of the Cu@Ag contact sintered in argon. The study of microstructure of the contacts showed that surface of the copper contact oxidizes and cavities and micro-cracks appear on it during switchings. The Cu@Ag contacts sintered in argon were more resistant against oxidation than other contacts under switching.

Soft Pressure Sensor for Underwater Sea Lamprey Detection

In this paper, an economical and effective soft pressure sensor for underwater sea lamprey detection is proposed, which consists of an array of piezoresistive elements between two layers of perpendicular copper tape electrodes, forming a passive resistor network. With multiplexers, the apparent resistance corresponding to each pixel of the sensing matrix can be measured directly, where the pixel is identified with the row and the column of the respective electrodes. However, this measured two-point resistance is not equal to the actual cell resistance for that pixel due to the crosstalk effect in the resistor network. Since the cell resistance reflects directly the pressure applied on each pixel, the relationship between the cell resistance and the measured two-point resistance is analyzed for a passive matrix of any size. More importantly, several regularized least-squares algorithms are proposed to reconstruct the cell resistance profile from the two-point resistance measurements, with enhanced robustness of the reconstruction in the presence of measurement noises and modeling errors. The proposed pressure sensor is applied to detect the suction attachment of sea lampreys, a devastating invasive species in the Great Lakes region. Experimental results demonstrate that the pressure sensor can successfully capture the rim profile of the lamprey’s sucking mouth. Moreover, the performance and computational complexity of the reconstruction algorithms with different regularization functions are compared.

Two-point resistances in Archimedean resistor networks

The rise of novel materials such as graphene compounds or carbon nanotubes recently revived the interest on the role of symmetries upon electrical properties. Among possible solids, symmetrical polytopes, widely realized in natural and artificial matter, deserve particular attention. When equal resistors are attached between neighbouring nodes of Perfect (Platonic) solids (PS) along the edges, the two-point resistance between any pair of nodes was established in the 1990s by van Steenwijk using an elegant and efficient method. Using van Steenwijk’s method, Moddy and Aravind subsequently derived the resistance values for two Archimedean and two Catalan solids. Here, with the same method, we derive the exact expression for the two-point resistance between any two nodes for resistor networks made of equal resistors placed along the edges of the thirteen Archimedean solids (AS). While the calculation remains elementary for the simplest AS, a dedicated method using computer assistance was developed for the Snub Cube, the Snub Dodecahedron and the three AS with three different rotation symmetries (the Great Rhombicuboctahedron, the Small Rhombicosidodecahedron and the Great Rhombicosidodecahedron). The largest resistance value (191/90) is obtained between the two opposite nodes of the Truncated Dodecahedron, and the second largest (42815/21114) is obtained between the two opposite nodes of the Great Rhombicosidodecahedron. The smallest resistance value (14137/38016) is obtained between some of the neighbouring nodes of the Snub Cube, whereas the resistance between neighbouring nodes of the Icosahedron (11/30) is slightly smaller. Some general symmetry relations between two-point resistances in AS networks are also derived, as well as some relations with two-point resistances in PS networks. The complete set of exact two-point resistance values for PS and AS networks can be used to check numerical codes or to evaluate the capacity of regular solids to provide appropriate models for given experimental situations.

Microstructure, resistivity, and shear strength of electrically conductive adhesives made of silver-coated copper powder

Electrically conductive adhesives were made using silver-coated copper powders (filler) and epoxy. Resistivity, microstructure and shear strength of prepared adhesives were studied using two-point resistance measurements, Scanning Electron Microscope (SEM) and universal tensile tests, respectively. Effect of filler concentration (70–85 wt%), silver concentration (10–50 wt%), particles morphology (flake or spherical) and addition of graphite (2–15 wt%) were investigated on prepared adhesives properties. Results showed that by increasing the filler percentage from 70 to 85 wt%, the electrical resistivity decreases from 6.2 × 10−3 to 3 × 10−3 Ω·cm. Furthermore, the electrical resistivity of adhesives is proportional to the silver content of the filler particles. Regarding the morphological effect of filler particles, it was found that by replacing 40 wt% of the flake with spherical particles in the adhesive with a filler content of 75 wt%, the electrical resistivity of the adhesive reduces from 5.6 × 10−3 to 4.5 × 10−3 Ω·cm, while the electrical resistivity of the adhesive containing 50 wt% of the spherical particles reached to 1.1 × 10−2 Ω·cm. Graphite addition to adhesive formula in concentrations less than 6 wt%, slightly lowered the resistivity of the adhesives. Increasing the graphite addition from 6 to 15 wt% enhanced the electrical resistivity from 2.8 × 10−3 to 1.7 × 10−2 Ω·cm. The shear strength of the adhesives is inversely proportional to the filler percentage in the adhesives.

Recurrence relations in m × 3 scaffolding and globe resistor networks

Resistor networks are popular because they offer solvable models of transport between connected discrete points and can represent natural or artificial systems such as mycorhizzal networks or carbon composite chains. When the connectivity pattern is repeated, two-point resistances can be expressed by recurrence relations. Here, we illustrate this approach in the case of three-dimensional m × 3 scaffolding and globe networks, characterized by a repeated pattern along a three-fold invariant axis. We show that a first set of recurrence relations follows from three-fold invariance and Kennelly’s Y-Δ transform, providing the two-point resistance between any pair of neighbouring nodes, including the case of infinite networks. Using van Steenwijk’s method, a second set of recurrence relations is obtained between non-neighbouring nodes. Numerous explicit expressions are thus derived using elementary methods, which can be compared with the results of numerical codes or novel integration methods such as Tan’s recursion-transform. Having at hand general properties of these networks is useful to evaluate their capacity of representing natural or industrial systems.

TWO-POINT RESISTANCES IN SAILBOAT FRACTAL NETWORKS

The two-point resistance of fractal network has been studied extensively by mathematicians and physicists. In this paper, for a class of self-similar networks named sailboat networks, we obtain a recursive algorithm for computing resistance between any two nodes, using elimination principle, substitution principle and local sum rules on effective resistance.

Aerosol Jet Printing of Ti3C2 Mxene Aqueous Ink

An emerging class of two-dimensional (2D) transition metal carbides, nitrides, and carbonitrides, known as MXenes have garnered significant interest in the scientific community due to their unique electrical, optical, and chemical properties [1]. Studies have reported high electrical conductivity and response, surface hydrophilicity, compositional control, and mechanical strength [1]. This unique combination of properties makes MXenes a contender for a wide variety of applications including batteries, super-capacitors, electronics, sensors, and others [2]. However, for MXenes to become industrially relevant, their compatibility with large scale synthesis and manufacturing techniques needs further investigation. In this study, we report the first aerosol jet printing (AJP) of water based titanium carbide (Ti3C2) MXene ink on a flexible polyimide substrate (Fig. 1-a,b). AJP is a high resolution additive manufacturing technique expected to shorten production times, provide greater control over size, design and variability, and a demonstrated capability of conformal printing and flexible substrate processing [3]. Previous studies have reported nano-material inks composed of various 2D materials including molybdenum disulfide, black phosphorous and hexagonal boron nitride [4]. However, only graphene-based inks have been shown to be AJP compatible [5]. Compared to graphene, MXenes offer greater compositional control. About 30 different kinds of MXenes have been synthesized, while many more are theoretically studied [1]. 2D Ti3C2 was the first to be discovered and has been studied most extensively [6]. To develop our AJP MXene ink, multi-layer MXenes were obtained by in situ HF etching of the pre-cursor Ti3AlC2 MAX Phase [7]. The obtained colloidal solution, containing 2D MXene flakes was sonicated with deionized water for 3 hours at 55 W. The resultant ink had a concentration of 1 mg mL-1 and an average flake size of 100 nm +/- 10 nm, as confirmed by dynamic light scattering (DLS) measurements. Contact angle measurements of the ink on polyimide substrate revealed an angle of 58.13o (Fig 1-c), showing good wettability. Our aqueous ink was printed with Optomec AJ-200 printer using the ultrasonic atomizer and printer parameters such as nozzle diameter, sheath gas flow, atomizer gas flow, atomizer current and platen temperature were optimized to get even dispersion and uniform printing. The MXene solutions were then printed in 0.5 mm x 1 cm linear structures with varying pass numbers (N), N = 5 to N = 25 in 5 pass increments. To investigate the quality of the printed structures, optical microscopy, height measurements, scanning electron microscopy (SEM) imaging and two-point probe resistance measurements were carried out. As revealed by optical images, the lines were even and didn’t show overspray, “coffee ring” effects, or runs typically associated with a solvent rich print (Fig. 1-d). SEM also showed even dispersion and consistent printing (Fig 1-e). However, some microcracks could be seen within the lines; which could be due to the absence of an annealing treatment. Stylus profilometer was used to determine the height of our printed MXene lines which varied from ~ 620 nm to 4.80 µm (Fig. 1-f). Our most uniform prints were N = 15 and N = 20 which had a height of 2.50 µm +/- 0.10 µm and 3.50 µm +/- 0.10 µm, respectively (Fig. 1-f). Electrical resistance (R) varied from R = 568 to R = 60 kΩ for N = 5 and N = 25 respectively (Fig. 1-g). In conclusion, we report the first AJP of aqueous Ti3C2 MXene inks. We find that the dimensions and electrical resistance of AJP MXene devices scale with increasing pass number, as expected. Microscopy analysis suggests additional work remains to be done to optimize our ink rheology and post-processing. However, our work highlights the important potential of additive manufacturing of MXene based devices which could include microelectrodes, battery electrodes, thermoelectrics, micro-capacitors and flexible sensors. [1] B. Anasori, M. R. Lukatskaya, and Y. Gogotsi, Nature Reviews Materials, 2 (2017). [2] K. Hantanasirisakul and Y. Gogotsi, Advanced Materials, 1804779 (2018). [3] A. A. Gupta, A. Bolduc, S. G. Cloutier, and R. Izquierdo, 2016 IEEE International Symposium on Circuits and Systems (ISCAS) (2016). [4] G. Hu et al., Chemical Society Reviews, 47, 3265–3300 (2018). [5] T. S. Tran, N. K. Dutta, and N. R. Choudhury, Advances in Colloid and Interface Science, 261, 41–61 (2018). [6] M. Naguib et al., ChemInform, 42 (2011). [7] M. Alhabeb et al., Chemistry of Materials, 29, 7633–7644 (2017). Figure 1

Two-point resistances and random walks on stellated regular graphs

A formula for computing the resistance between any two vertices in the stellated graph of a regular graph is obtained. It turns out that the two-point resistance of the stellated graph can be expressed in terms of the two-point resistance of the original graph. As a consequence, the Kirchhoff index (i.e. the sum of the effective resistances between all pairs of vertices) for the stellated graph is obtained, which extends the previously known result. The correspondence between random walks and electric networks is then used to obtain the mean first passage time and mean commute time for random walks on stellated graphs.

The two-point resistance on the diamond cubic lattice

The problem of calculating the equivalent resistance between any two lattice sites in several lattice structures of resistors has recently received much attention. In this paper, we study the problem of an infinite diamond cubic lattice structure formed from identical resistors using the lattice Green’s function technique. Some numerical results for the resistance are presented. The asymptotic values of the resistance are calculated numerically as the distance between lattice sites goes to infinity.

Two-Point Resistance on the Centered-Triangular Lattice

The resistance between any two lattice points in an infinite, centered-triangular lattice of equal resistors is determined using the lattice Green function method. It is shown that the two-point resistance on the centeredtriangular lattice is expressed in terms of the resistance of a triangular lattice. Some exact values for the resistance near the origin of the lattice are presented. For large separation between lattice points the asymptotic forms of the resistance are calculated.

The role of hydrogen and oxygen in the persistent photoconductivity of strontium titanate

Annealed strontium titanate (SrTiO3 or STO) single crystals exhibit persistent photoconductivity (PPC) at room temperature. Illumination with sub-gap light reduces the resistance by three orders of magnitude, which persists for weeks or longer. The defects responsible for this remarkable phenomenon have not been identified. In this work, we report on the importance of hydrogen and oxygen during the annealing process that is used to induce PPC. The results from IR spectroscopy and two-point resistance measurements indicate that water vapor at 1200 °C yields hydrogen and oxygen-vacancy populations that result in large PPC. Deuterium substitution experiments show evidence for a two-hydrogen center that forms after exposure to light. The results suggest that light causes substitutional hydrogen to leave the oxygen site, forming metastable O-H bonds. This process liberates electrons and causes PPC.

Using electrical resistance to evaluate the chip seal curing process

Chip seals are among the most cost-effective surface treatments available for pavement preventive maintenance. However, frequent issues associated with asphalt emulsion early mechanical strength development, resulting in premature surface treatment failure, have led to the need to improve the characterisation of the chip seal curing process. As such, the use of an electrical resistance measurement has been studied to develop a sound construction methodology that prevents common failures that occur soon after construction. This paper presents a novel approach, based on electrical resistance measurements, to determine when a chip seal has developed enough binder adhesive strength to bond to the existing pavement while keeping the aggregate chips in place. The electrical resistance measurements provide a rapid, non-destructive indication of the amount of curing that has occurred. By implementing this methodology the user can determine when a chip seal has gained sufficient mechanical strength to allow for brooming or opening to unrestricted traffic without an undue loss of cover aggregate. Laboratory and full-scale field trials were conducted using a variety of materials. The electrical properties of the fresh seal coats were quantified by employing a handheld electrical device with a two-point probe resistance measurement. The experimental results suggest that chip seal systems have gained significant mechanical strength when the initial electrical resistance measurement increases by a factor of 10. As a result, this study establishes that electrical resistance measurements can be used to determine when a fresh chip seal has sufficiently cured to withstand the shear forces of brooms and uncontrolled traffic. The implementation of the technique could potentially impact chip seal construction quality, as well as service life performance.

Using persistent photoconductivity to write a low-resistance path in SrTiO3

Materials with persistent photoconductivity (PPC) experience an increase in conductivity upon exposure to light that persists after the light is turned off. Although researchers have shown that this phenomenon could be exploited for novel memory storage devices, low temperatures (below 180 K) were required. In the present work, two-point resistance measurements were performed on annealed strontium titanate (SrTiO3, or STO) single crystals at room temperature. After illumination with sub-gap light, the resistance decreased by three orders of magnitude. This markedly enhanced conductivity persisted for several days in the dark. Results from IR spectroscopy, electrical measurements, and exposure to a 405 nm laser suggest that contact resistance plays an important role. The laser was then used as an “optical pen” to write a low-resistance path between two contacts, demonstrating the feasibility of optically defined, transparent electronics.

Detection of current induced spin polarization in epitaxial Bi2Te3 thin film

We electrically detect charge current induced spin polarization on the surface of a molecular beam epitaxy grown Bi2Te3 thin film in a two-terminal device with a ferromagnetic MgO/Fe contact and a nonmagnetic Ti/Au contact. The two-point resistance, measured in an applied magnetic field, shows a hysteresis tracking the magnetization of Fe. A theoretical estimate is obtained for the change in resistance on reversing the magnetization direction of Fe from coupled spin-charge transport equations based on the quantum kinetic theory. The order of magnitude and the sign of the hysteresis are consistent with the spin-polarized surface state of Bi2Te3.

Two-Point Resistance of a Non-Regular Cylindrical Network with a Zero Resistor Axis and Two Arbitrary Boundaries* *Supported by the Natural Science Foundation of Jiangsu Province under Grant No. BK20161278

We study a problem of two-point resistance in a non-regular m × n cylindrical network with a zero resistor axis and two arbitrary boundaries by means of the Recursion-Transform method. This is a new problem never solved before, the Green’s function technique and the Laplacian matrix approach are invalid in this case. A disordered network with arbitrary boundaries is a basic model in many physical systems or real world systems, however looking for the exact calculation of the resistance of a binary resistor network is important but difficult in the case of the arbitrary boundaries, the boundary is like a wall or trap which affects the behavior of finite network. In this paper we obtain a general resistance formula of a non-regular m × n cylindrical network, which is composed of a single summation. Further, the current distribution is given explicitly as a byproduct of the method. As applications, several interesting results are derived by making special cases from the general formula.

Calculation of the equivalent resistance and impedance of the cylindrical network based on recursion-transform method

A classic problem in circuit theory first studied by German physicist Kirchhoff more than 170 years ago is the computation of resistances in resistor networks. Nowadays, resistor network has been an important model in the fields of natural science and engineering technology, but it is very difficult to calculate the equivalent resistance between two arbitrary nodes in an arbitrary resistor network. In 2004, Wu F Y formulated a Laplacian matrix method and derived expressions for the two-point resistance in arbitrary finite and infinite lattices in terms of the eigenvalues and eigenvectors of the Laplacian matrix, and the resistance results obtained by Laplacian matrix method is composed of double sums. The weakness of the Laplacian matrix approach is that it depends on the two matrices along two orthogonal directions. In 2011, Tan Z Z created the recursion-transform (RT) method, which can resolve the resistor network with arbitrary boundary. Using the RT method to compute the equivalent resistance relies on just one matrix along one direction, and the resistance is expressed by single summation. In the present paper, we investigate the equivalent resistance and complex impedance of an arbitrary mn cylindrical network by the RT method. Firstly, based on the network analysis, a recursion relation between the current distributions on three successive vertical lines is established through a matrix equation. In order to obtain the eigenvalues and eigenvectors of the matrix, and the general solution of the matrix equation, we then perform a diagonalizing transformation on the driving matrix.Secondly, we derive a recursion relation between the current distributions on the boundary, and construct some particular solutions of the matrix equation. Finally by using the matrix equation of inverse transformation, we obtain the analytical solution of the branch current, and gain the equivalent resistance formula along the axis of the arbitrary mn cylindrical network, which consists of the characteristic root and expressed by only single summation. As applications, several new formulae of equivalent resistances in the semi-infinite and infinite cases are given. These formulae are compared with those in other literature, meanwhile an interesting new identity of trigonometric function is discovered. At the end of the article, the equivalent impedance of the mn cylindrical RLC network is also treated, where the equivalent impedance formula is also given.