Inverses of k-Toeplitz matrices with applications to resonator arrays with multiple receivers

https://doi.org/10.1016/j.amc.2020.125185Get rights and content

Abstract

We find closed-form algebraic formulas for the elements of the inverses of tridiagonal 2- and 3-Toeplitz matrices which are symmetric and have constant upper and lower diagonals. These matrices appear, respectively, as the impedance matrices of resonator arrays in which a receiver is placed over every 2 or 3 resonators. Consequently, our formulas allow to compute the currents of a wireless power transfer system in closed form, allowing for a simple, exact and symbolic analysis thereof. Small numbers are chosen for illustrative purposes, but the elementary linear algebra techniques used can be extended to k-Toeplitz matrices of this special form with k arbitrary, hence resonator arrays with a receiver placed over every k resonators can be analysed in the same way.

Introduction

This work concerns the theory and applications of some special tridiagonal matrices, known in the literature as tridiagonal k-Toeplitz matrices. Those are tridiagonal matrices of order n, say, where the entries along the main diagonal and its adjacent diagonals are periodic sequences of period k, so that they have the form(a1b1c1akbkcka1b1c1akbkcka1b1c1), where the entries are real or complex numbers, with bjcj ≠ 0 for j=1,,k. These matrices have proved to be a very useful tool in many contexts of pure and applied mathematics, e.g., in partial differential equations (appearing in the discretization of elliptic or parabolic partial differential equations by finite difference methods), in chain models of quantum physics ([7]), and in sound propagation theory ([9], [10]). Gover in [13] solved the eigenproblem associated with such matrices for the special case k=2. Gover’s results were recasted by Marcellán and Petronilho in [14] using tools from the theory of orthogonal polynomials. Later, in [15] these authors solved the associated eigenproblem for the case k=3, using again tools from orthogonal polynomials theory and polynomial mappings (see also the work [7] by Álvarez-Nodarse et al.). The eigenproblem of a general tridiagonal k-Toeplitz matrix was solved by da Fonseca and Petronilho ([12]), motivated by the need of finding explicit formulas for the entries of the inverses of such matrices (whenever they are nonsingular), the special case k=3 having been considered previously by these same authors in [11]. Recently these explicit formulas for the entries of the inverses have proved to be very useful in real world problems involving circuit models (e.g. [6]). Such formulas for the entries were obtained in [11], [12] as expressions involving polynomial mappings and Chebyshev polynomials of the second kind, a fact that (despite the beauty of such formulas) may be regarded as an additional difficulty in their applications, especially for those which are not so familiar with the theory of orthogonal polynomials.

Our aim in this contribution is twofold. On the one hand we will determine, without using the theory of orthogonal polynomials, explicit algebraic expressions for the entries of the inverses of symmetric 2- and 3-Toeplitz matrices which have constant and equal upper and lower diagonals (b1==bk=c1==ck). To do so we will only resort to elementary linear algebra: we will compute the determinants of such matrices by linear recurrence relations, and then apply those determinants to compute the minors appearing in the cofactor matrix, which directly relate to the elements of the inverse. It is clear that the methods found here can be applied to k-Toeplitz matrices with constant and equal upper and lower diagonals, for arbitrary k.

On the other hand, we will apply these results to achieve closed formulas for wireless power transfer (WPT) systems using resonator arrays with multiple receivers. WPT systems have been going through intensive research lately, as they allow one to avoid electrical contact and transfer power in rough environments with water, dust or dirt. Nowadays they are being used in several applications as electrical vehicle charging ([1]), mobile devices charging ([18]) and powering biomedical devices ([22]). However, they have the drawback that, in case of misalignment or distance from the transmitter to the receiver, the efficiency and power transmitted can drop abruptly. So, in order to overcome this inconvenience, arrays of resonators can be used to transfer power over longer distances ([16], [17], [23]). In these arrays the first resonator is usually connected to a power source and transmits power through magnetic coupling to the other resonators of the array, which are arranged in a plane with parallel axes, and a receiver is placed over the array to absorb the power transmitted ([2], [3], [4], [5], [17], [20]). In the literature, these arrays have been examined mostly using magnetoinductive wave theory ([17], [20]) or through the circuit analysis of the array ([3], [23]), in which the array is represented by an impedance matrix which contains the impedance of each resonator and the mutual inductances between pairs of resonators ([2], [17], [19], [23], [24]). In [5], [6] the inversion of the impedance matrix is performed using generic tridiagonal matrices. In this way, it is possible to determine closed-form expressions for equivalent impedance, the power transmitted and the efficiency of these systems. However all these works consider only one receiver placed over the array. Instead, the array could possibly transmit power to several receivers at the same time. In this paper we study and give closed-form algebraic formulas for the currents, power transmission and efficiency in an array powering multiple receivers placed over every two or three resonators. These small numbers have been chosen for the sake of simplicity of the exposition, but the same methods work equally well for arrays with receivers placed every k resonators, with k arbitrary.

Section snippets

Description of the circuit

In this paper we consider an array with N identical resonators and some identical receivers placed over them. If the lth resonator has a receiver over it (Fig. 1(A,B)) then an impedance Z^d is added, which is the impedance of the receiver as seen from the resonator ([17], [20]). The last resonator (Nth) is connected to a termination impedance Z^T. The first resonator is connected to a voltage source V^s, which we consider to generate an ideal sinusoidal voltage, as has been done in other WPT

2-Toeplitz matrix

Recall that the (i, j)th cofactor of the matrix AMn(C) isCij=(1)i+jdet(Aij),where AijMn1(C) is the submatrix of A formed by removing the ith row and the jth column. Then the cofactor matrix of A is the matrix C(A)=(Cij)Mn(C). The inverse of a regular matrix A can be computed asA1=adj(A)detA,where the adjugate matrix of A is adj(A)=C(A)T, the transpose of its cofactor matrix.

Denote Mn(a1,a2,b)=(a1b00ba2b000ba1b0000ba2b00b000bα)Mn(C), where α is a1 when n

3-Toeplitz matrix

Let Mn(a1,a2,a3,b)=(a1b00ba2b000ba3b0000bα)Mn(C), where α={a1,n1(mod3)a2,n2(mod3)a3,n3(mod3). Note that the impedance matrix in Section 2.2 is of the special form Mn(a1, a1, a2, b), but the cases Mn(a1, a2, a1, b) and Mn(a2, a1, a1, b) will be needed when computing its inverse. We compute the inverse of Mn(a1, a2, a3, b) via its adjugate matrix, which is again its cofactor matrix.

Application of the mathematical results

In this section we will use the generic expressions obtained for the elements of the inverse of the tridiagonal matrix to determine the expressions for the currents, power transmitted and efficiency of the resonator array. Subsequently we will use said expressions to illustrate the mathematical results and understand how the behaviour of the system changes with the variation of its characteristics and parameters. In particular, we will analyse the behaviour of the system for different values of

Conclusions

In this paper, the inversion of special 2- and 3-Toeplitz matrices is used to analyse and assess the power transfer capability of a resonator array with multiple receivers. By replacing the generic parameters with the parameters of the circuit, it is possible to obtain closed-form expressions for the currents in the resonators, power transfer and efficiency of the system. Using these expressions, some examples were made in order to illustrate the mathematical results obtained and show their

References (24)

  • R. Álvarez-Nodarse et al.

    On some tridiagonal k-Toeplitz matrices: algebraic and analytical aspects. Applications

    J. Computat. Appl. Math.

    (2005)
  • C.M. da Fonseca et al.

    Explicit inverses of some tridiagonal matrices

    Linear Algebra Appl.

    (2001)
  • M.J.C. Gover

    The eigenproblem of a tridiagonal 2-Toeplitz matrix

    Linear Algebra Appl.

    (1994)
  • F. Marcellán et al.

    Eigenproblems for tridiagonal 2-Toeplitz matrices and quadratic polynomial mappings

    Linear Algebra Appl.

    (1997)
  • A. Ahmad et al.

    A comprehensive review of wireless charging technologies for electric vehicles

    IEEE Trans. Transport. Electrific.

    (2018)
  • J. Alberto et al.

    Experimental study on the termination impedance effects of a resonator array for inductive power transfer in the hundred kHz range

    Proc. 2015 IEEE Wireless Power Transfer Conf. (WPTC 2015), Boulder, CO, USA

    (2015)
  • J. Alberto et al.

    Circuit model of a resonator array for a WPT system by means of a continued fraction

    Proc. 2016 IEEE 2nd Int. Forum on Research and Technol. for Society and Ind. (RTSI), Bologna, Italy

    (2016)
  • J. Alberto et al.

    Magnetic near field from an inductive power transfer system using an array of coupled resonators

    (2016)
  • J. Alberto et al.

    Fast calculation and analysis of the equivalent impedance of a wireless power transfer system using an array of magnetically coupled resonators

    PIER B

    (2018)
  • J. Alberto et al.

    Accurate calculation of the power transfer and efficiency in resonator arrays for inductive power transfer

    PIER B

    (2019)
  • J.T. Boys et al.

    An appropriate magnetic coupling co-efficient for the design and comparison of ICPT pickups

    IEEE Trans. Power Electron.

    (2007)
  • S.N. Chandler-Wilde et al.

    On the application of a generalization of Toeplitz matrices to the numerical solution of integral equations with weakly singular convolution kernels

    IMA J. Numer. Anal.

    (1989)
  • This work was partially supported by the Centre for Mathematics of the University of Coimbra – UID/MAT/00324/2019, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020. The second author was supported by the Portuguese Government through the Fundação para a Ciência e a Tecnologia, from Portugal grant SFRH/BPD/118665/2016.

    View full text