2017 Progress In Electromagnetics Research Symposium — Fall (PIERS — FALL), Singapore, 19–22 November Matlab Simulation of Nonlinear Electrical Networks via Volterra Series Expansion and Multidimensional NILT Lubom´ ır Branˇ c´ ık, Nawfal Al-Zubaidi R-Smith, and Filip Z´ aplata Department of Radio Electronic, Brno University of Technology Technicka 12, Brno 616 00, Czech Republic Abstract— The paper deals with a simulation of nonlinear networks based on a classical ap- proach of Volterra series expansion. It is known that a multidimensional Laplace transform (MLT) of a time-domain nonlinear impulse response results in the respective Laplace-domain transfer function which helps in finding Volterra kernels, for example via a harmonic input method. After solving the system in the Laplace domain, a final step is to transfer the solution back into the time domain. For this purpose proper multidimensional numerical inverse Laplace transforms (MNILT) are applied with advantages avoiding the usage of rather impractical associate vari- ables method required to receive a single-variable Laplace image. To ensure good convergence and stability of the method the networks are limited to be rather weakly nonlinear when usually the kernels into the third order already yield reasonable results. That is why, methods for up to the third-dimensional NILT (3D NILT) are discussed in the paper, both the FFT-based one with a quotient-difference algorithm and a hyperbolic one with the Euler transformation. All the discussed methods are programmed and tested in Matlab language while considering a proper model of a nonlinear electrical network. 1. INTRODUCTION There are a number of methods to analyze nonlinear electrical circuits, mostly based on proper numerical techniques for the solution of nonlinear differential equations as their mathematical models [1]. When limiting the circuits to be rather weakly nonlinear, a classical approach based on Volterra series expansion is still worth considering [2]. Its main practical limitation lays on difficult way for finding Volterra kernels up to orders guaranteed acceptable accuracy, and possible problems with convergence of the solution. Considering just weakly nonlinear circuits helps to resolve the above difficulties, when usually the kernels into the third order already yield reasonable results. The approach is further suppported by the fact that the Volterra kernels are obtainable experimentally, by measuring the X -parameters through the use of a nonlinear vector network analyser (NVNA) and after it by properly transforming the results [3]. In this way the comparison between simulations and practical measurements is enabled in case of hardware realization of the circuit. It is well-known that a multidimensional Laplace transform (MLT) of a time-domain nonlinear impulse response leads to the respective Laplace domain transfer function which can help in finding required Volterra kernels [4], often through a harmonic input method [5]. After solving the system in the Laplace domain, a final step is to transfer the solution back into the time domain. An often used method, an association of variables, enables to achieve a resultant single-variable Laplace image, however until after an impractically repetitive usage of residue theorems [6]. From a com- putational viewpoint, therefore, a direct application of proper multidimensional numerical inverse Laplace transforms seems to be more efficient [7, 8]. For this purpose, methods for multidimensional NILTs based on complex Fourier series approximations and FFT/IFFT with a quotient-difference algorithm are shortly discussed [9, 10]. All the methods are programmed and tested in the Mat- lab language while considering a model of simple nonlinear electrical network as an example, and verifying the results based on the Matlab built-in ODE function. 2. THEORETICAL FOUNDATIONS Hereafter some principles of Volterra series expansion and solution via multidimensional Laplace transforms technique will be recapitulated. Let us consider a response y(t) to a stimulus x(t), then 2822