Abstract
We investigate a dc electric concentrator for steady current fields theoretically and experimentally. Based on the transformation electrostatics, we show that the dc concentrator can focus electric currents into the central concentrated region and enhance the electric field and current density. Outside the concentrator, the current lines are distributed as the same as those in a homogeneous conducting material. Hence, such a dc electric concentrator has no impact on other external devices. Using the analogy between electrically conducting materials and resistor networks, we design, fabricate, and test a dc concentrator using the circuit theory. The measured results agree very well with the theoretical predictions and numerical simulations, demonstrating the perfect concentrating performance.
Introduction
In the past a few years, transformation optics (TO) devices have caused much attention^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}, such as electromagnetic (EM) invisibility cloaks. Most TO devices were considered and designed in the timevarying EM fields^{1,2,3,4,5,6,7,8,9,10}, and a few devices have also been dealt with for static fields^{11,12,13,14,15,16,17,18,19}. In fact, the static fields play a significant role in several applications, e.g. they are involved in photocopy machines, electrostatic spraying systems, and the electric impedance tomography. They can also be used to detect landmines and torpedoes.
Early in 2003, Greenleaf et al. studied the coordinate transformation in static potential^{11}, in which they constructed anisotropic conductivities that give rise to the same voltage and current measurements on the boundary as those of a homogeneous and isotropic conductivity. Then in 2007, Wood and Pendry proposed a dc metamaterial design, which is based on superconducting materials, and the cloaking for static magnetic field was suggested^{12}. Later, dc metamaterials have been further investigated theoretically and verified experimentally by different groups^{13,14,15,16,17,18,19}. The EM concentrators for timevarying fields were first proposed by Rahm et al.^{9}, and developed in^{10}. However, the EM concentrators have never been demonstrated experimentally since they demand highly anisotropic material parameters with extreme values, which cannot be realized using the current technologies.
Results
In this paper, we theoretically and experimentally investigate the concentrating device for the steady current fields and realize a dc electric concentrator. Using the transformation electrostatic theory and exploiting the connection between conductivities in conducting materials and resistors in the circuit theory, we fabricate such the dc concentrator using the resistor network. The proposed dc concentrator could be useful in concentrating or enhancing current densities, and other staticfield manipulations.
The principle of dc concentrator is shown in Fig. 1, in which Panel a illustrates a bundle of electric currents flowing in a homogeneous and isotropic conducting material, while Panel b demonstrates the current flows when we put a dc electric concentrator inside the material, which can smoothly guide the electric currents into the central region, but does not make perturbations outside. As has been proved, the Laplace Equation maintains the form invariance under the coordinate transformation, and hence the TO theory applies well to dc fields^{11,20}. Using the TO theory, the dc concentrator can be easily designed by concentrating a sphere into the center of the sphere while diluting a spherical shell into the remaining region. For the twodimensional (2D) case, we adopt a linear transformation between the physical and virtual spaces: in which, k_{1} = b/a, k_{2} = (c−a)/(c−b), and k_{3} = (b−a)/(c−b) are constant coefficients under the cylindrical coordinate system. The electric concentrator contains three regions. In the central circular core with radius ρ'≤ a, the current density is enhanced by k_{1} times; in the middle layer (a ≤ ρ’≤ b), the current density is concentrated into the central core; in the outer layer (b≤ ρ’≤ c), the current density is “diluted” by k_{2} times. Then the transformed conductivity for the electric concentrator is then derived using the transformation electrostatics as in which the primed variables belong to the physical space, the unprimed ones belong to the virtual space, and . It is interesting to notice that, implied in equation (2), the central “concentrated” region is isotropic and the conductivity is the same as that of the background material for 2D case. In the “diluted” region, the conductivity is anisotropic, whose ρ component is a monotonously decreasing function on radius, while the ϕ component is a constant.
We, firstly, take an example to illustrate the focusing effect of the dc electric concentrator based on numerical simulations, in which we choose a = 1 cm, b = 10 cm, c = 11 cm, and background material is homogeneous with conductivity σ_{0} = 1 S/m. In a homogeneous material, the potential is governed by the Laplace equation in the absence of source At the position of x = −0.25 m, there is a planar metallic plate with the potential of U_{0} = 5V, and the metallic plate at x = 0.25 m is connected to the ground. If the potential is invariant in the y and z directions, then it will vary linearly in the x direction as U = −10x+2.5, as shown in Fig. 2a. Figure 2c illustrates the electric potential distributions inside and outside the electric concentrator. We can see that, outside the concentrator, the electric potential distribution is the same as that in the homogeneous material. Compared with the virtual region shown in Fig. 2b, the electric potential distribution inside the concentrated region (U’ = −100x+2.5) is “focused” but not enhanced, as shown in Fig. 2d. In the other word, the electric potential in the region ρ≤b in Fig. 2b is concentrated into the central region ρ’≤ c, as illustrated in Fig. 2d. In the diluted region a≤ ρ’≤ c displayed in Fig. 2c, the electric potential can be transformed from the region b≤ ρ≤ c by the mapping expressed in Eq. (1).
Fig. 3a demonstrates the electric field intensity E inside and outside the concentrator. From the Maxwell's equation, we know that In the homogeneous material, E_{x} = 10 V/m, which is the same as that in the region outside concentrator. In the concentrated region, predicted by the transformation electrostatic theory, the electric field will be enhanced by k_{1} = b/a = 10 times, i.e., 100 V/m. This has been verified by the numerical simulations, as shown in Fig. 3a. In the “diluted” region, the minimum electric field has been reduced by k_{2} = (c−a)/(c−b) = 10 times, i.e., 1 V/m. Fig. 3b illustrates the total current densities inside and outside the concentrator, which are given by We observe from Fig. 3b that, outside the concentrator, the current densities are the same as those in the homogeneous materials. As predicted, in the central concentrated region, the current densities are enhanced by k_{1} = 10 times. In the “diluted” region, due to the anisotropic properties of the conductivity, the distribution of current densities is different from that of the electric field. From Eq. (2), we know that the conductivity tensor in the “diluted” region is orthometric and the determinant equals 1, so it is a rotational matrix. Combining with Eq. (5), we can obtain the current density by rotating the distribution of electric fields, as shown in Fig. 3b. It is apparent that the current density is strongly enhanced in the concentrated region.
Eq. (2) shows that the realization of dc concentrator requires anisotropic and inhomogeneous conductivities. Fig. 4a demonstrates the required radial and tangential components of the conductivity tensor for the dc concentrator with inner and outer radii of 5 cm and 15 cm, respectively. Clearly, the required conductivities are difficult to be realized in nature, but they can be easily emulated using the circuit theory. Suppose that a continuous conducting material plate with the conductivity σ and thickness h, which may extend to infinity in the radial direction. The material may be inhomogeneous and anisotropic. To make an equivalence of the material to a resistor network, the continuous material is discretized using the polar grids. According to Ohm's law, each elementary cell in the grid can be implemented by two resistors where Δρ and Δϕ are step lengths in the radial and tangential directions, respectively. Thus the anisotropic conductivity tensor can be implemented easily using different resistors in different directions. To make simulations and measurements, the infinitelylarge material should be tailored to have a suitable size. Like the perfect matching layers in the timevarying problems, matching resistors are added in the outer ring to emulate an infinite material. Using the theorem of uniqueness, the matching resistors can be easily obtained as^{19} in which r_{0} is the distance between the ground and the source point and the definitions of other geometrical parameters are presented in Fig. 4c. Hence the required resistors for the dc concentrator are illustrated in Fig. 4b. It is clearly shown that all resistors have moderate values and can be commercially obtained.
To verify the correctness and effectiveness of the design, we first simulate the homogeneous background material based on the resistor network. The simulation results of potential distributions under the excitation of a point source are shown in Fig. 5a. As predicted from electrostatic theory, the equipotential lines are concentric circles. Figs. 5b and 5c demonstrate the simulation and measurement results of a dc concentrator with a = 5 cm, b = 10 cm and c = 15 cm based on the resistor network. The fabricated dc concentrator is illustrated in Fig. 6, and will be discussed in details in Methods. In experiments, the potential is much easier to measure than the current density, and hence we simulate and measure potential distributions. All simulations are performed using the commercial software, the Agilent Advanced Design System (ADS). The simulated potential distributions of the dc concentrator are shown in Fig. 5b. We clearly observe that the potential distributions outside the concentrator keep the original equipotential lines as those in the homogeneous material, which indicates that the concentrator has no effect on any external devices. The measurement results of the dc concentrator are presented in Fig. 5c, demonstrating excellent concentrating performance. To verify such a design for a dc concentrator with large “concentrated” factor, we fabricated another device. Figs. 5d and 5e demonstrate simulation and measurement results of a dc concentrator with a = 1 cm, b = 10 cm and c = 11 cm. We ploted the potential distributions in the simulation and experiment along a line y = 0 cm in Fig. 5f. We observed that the tested result is agreed very well with the simulation one.
Discussion
A careful comparison between Figs. 5b and 5c shows excellent agreements between the experiments and simulations, both inside and outside the concentrator. Although the current densities in the resistors cannot be measured directly, we can compare the current densities in the central region of the dc concentrator and in the homogeneous material because the resistors are constants in such two cases. Hence, we can calculate the current density enhancement factor from the voltage difference between the corresponding resistors. In the excitation of point source, the potential distribution in the isotropic and homogeneous circular material can be described as where A_{1} and A_{2} are coefficients determined by the positions of the source and ground. In the ADS simulations and experiments, we observe that the current density in the concentrated region has been enhanced by 1.7 times for the case of a = 5 cm, b = 10 cm and c = 15 cm, and by 5.6 times for the case of a = 1 cm, b = 10 cm and c = 11 cm, compared to the same region in the homogeneous material due to potential distribution in the logarithmic law. We remark that the enhancement factors are a little bit smaller than the theoretical calculations due to the errors of practical resistors. Therefore, in the dc concentrator, the electric potential is concentrated, while the electric field and current density are enhanced.
In conclusion, we have proposed and made the first experimental verification of the dc electric concentrator in the steady current fields. Such a transformation optics device can concentrate the current into a designed central region and enhance the current density inside. With the help of modern integrated circuit technologies, it is possible to extend such a device to the nano scale. Hence the proposed device has potential applications in the electric impedance tomography technology, graphene, and other integrated circuits.
Methods
In the experimental setup of the first concentrator sample, the background material has the conductivity of 1 S/m, which is cut into a circular region with radius 25 cm. In our design, the geometrical parameters are chosen as a = 5 cm and c = 15 cm. To layout the 4mmlong patch resistors along the tangential direction appropriately in the sample, we put on a circular metallic plate with radius 5 cm in the central region, which is equivalent to a conducting point with infinitesimal radius due to its perfectconductivity nature in the steady currents. Hence, in the actual operation to construct the resistor circuit, the space between two adjacent layers is 0.8 cm instead of the theoretical interval of 1 cm, as shown in Fig. 5. However, thoughout the paper, we design the dc concentrators and plot the results according to the geometry size shown in Eq. (6) and (7). We adjust the spacing between adjacent resistors in the radial redirections to layout the innerlayer resistors in the actual circuits. Following the abovementioned design procedure, the background material together with the dc concentrator is divided into 25×36 cells using the polar grids. The resistor network contains 25 concentric layers in the radial direction, and 36 nodes in the tangential direction, which is built on a printed circuit board (PCB) with thickness of 2 mm. The concentrated region of the dc concentrator occupies an area of 5 layers, the diluted region employs another 10 layers, and the remaining 10 layers are background material. The fabricated sample is shown in Fig. 5.
For the second sample, the central concentrated region has onelayer resistors with radius a = 1 cm and the diluted region has 10 layers of resistors, respectively. The remaining 9 layers are background material. In both samples, all resistors are commercially available chipfixed resistors with a high accuracy of 0.1%. An Agilent voltage source with 5V magnitude is connected to the circuit network at the 24th layer. The voltage at each node is measured by using a 4.5digit multimeter.
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Acknowledgements
This work was supported in part from the National Science Foundation of China under Grant Nos. 60990320, 60990321, 60990324, 61171024, 61171026, 60901011, and 60921063, in part from the National High Tech (863) Projects under Grant Nos. 2011AA010202 and 2012AA030702, in part from the 111 Project under Grant No. 111205, in part from the Talent Project of Southeast University, and in part by the Joint Research Center on Terahertz Science.
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Author notes
 Wei Xiang Jiang
 & Chen Yang Luo
These authors contributed equally to this work.
Affiliations
State Key Laboratory of Millimeter Waves, Department of Radio Engineering, Southeast University, Nanjing 210096, P. R. China
 Wei Xiang Jiang
 , Hui Feng Ma
 & Tie Jun Cui
School of Electronic Science and Engineering, Southeast University, Nanjing 210096, P. R. China
 Chen Yang Luo
School of Information Science and Engineering, Lanzhou University, Lanzhou 730000, P. R. China
 Zhong Lei Mei
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Contributions
W.X.J. and T.J.C. proposed and designed the theory of dc concentrators and performed initial verification, W.X.J. and C.Y.L. conceived the experiments and fabricated the devices. H.F.M. and Z.L.M. performed numerical simulations. T.J.C. supervised the design and experiments, and wrote the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Tie Jun Cui.
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