Continuous-time Analog Computing Circuits for Solving The Electromagnetic Wave Equation

Two continuous-time mathematical computing methods are proposed for solving the multidimensional wave equation leading to realizable analog computing circuits. The proposed analog computing processors will potentially be able to solve a certain special classes of computational problems involving partial differential equations, which are defined from continuous-time systems. The new analog computing methods are first derived and physically implemented for the first-time using low-frequency operational amplifier circuits in order to experimentally verify the correctness of the proposed methods. Both algorithms approximate the spatial domain partial derivatives using discrete finite differences. The first method performs a direct Laplace transform (with respect to the time variable) on the resulting expression. The second method applies the finite difference along the time dimension and then replaces the discrete time difference with a continuous-time delay operator, which in turn, can be realized as an analog all-pass filter. Analog circuit architectures are introduced for different boundary conditions relevant to common electromagnetic simulation problems. A low frequency prototype of the analog wave equation solver (based on method 1) has been designed, realized and tested using board-level operational amplifier circuits. Test results and measurements are provided to demonstrate the wave propagation in the space-time domain.

[1]  Ladis D. Kov Ach,et al.  The analog computer as a teaching aid in diffential equations , 1961 .

[2]  Leonid Belostotski,et al.  A Steerable DC-1 GHz all-pass filter-Sum RF space-time 2-D beam filter in 65 nm CMOS , 2013, 2013 IEEE International Symposium on Circuits and Systems (ISCAS2013).

[3]  Leonard T. Bruton,et al.  Network Transfer Functions Using the Concept of Frequency-Dependent Negative Resistance , 1969 .

[4]  Simha Sethumadhavan,et al.  Energy-Efficient Hybrid Analog/Digital Approximate Computation in Continuous Time , 2016, IEEE Journal of Solid-State Circuits.

[5]  Simha Sethumadhavan,et al.  Continuous-time hybrid computation with programmable nonlinearities , 2015, ESSCIRC Conference 2015 - 41st European Solid-State Circuits Conference (ESSCIRC).

[6]  Simha Sethumadhavan,et al.  Hybrid Analog-Digital Solution of Nonlinear Partial Differential Equations , 2017, 2017 50th Annual IEEE/ACM International Symposium on Microarchitecture (MICRO).

[7]  Ahmed S. Elwakil,et al.  A New Second-Order All-Pass Filter in 130-nm CMOS , 2016, IEEE Transactions on Circuits and Systems II: Express Briefs.

[8]  G.E.R. Cowan,et al.  A VLSI analog computer/math co-processor for a digital computer , 2005, ISSCC. 2005 IEEE International Digest of Technical Papers. Solid-State Circuits Conference, 2005..

[9]  G. Mur.  Absorbing Boundary Conditions for the Finite-Difference Approximation of the Time-Domain Electromagnetic-Field Equations , 1981, IEEE Transactions on Electromagnetic Compatibility.

[10]  Simha Sethumadhavan,et al.  Evaluation of an Analog Accelerator for Linear Algebra , 2016, ISCA.

[11]  Carretera de Valencia,et al.  The finite element method in electromagnetics , 2000 .

[12]  Magdy F. Iskander.  Electromagnetic Fields and Waves , 2000 .

[13]  Arjuna Madanayake,et al.  Continuous-Time Analog Two-Dimensional IIR Beam Filters , 2012, IEEE Transactions on Circuits and Systems II: Express Briefs.

[14]  Pierre-Antoine Absil,et al.  Continuous-time Systems that Solve Computational Problems , 2006, Int. J. Unconv. Comput..

[15]  Matthew N. O. Sadiku,et al.  Elements of Electromagnetics , 1989 .

[16]  Umran S. Inan,et al.  Numerical Electromagnetics: The FDTD Method , 2011 .

[17]  W.C. Chew.  150 Years of Maxwell's equations: A reflection , 2014, 2014 USNC-URSI Radio Science Meeting (Joint with AP-S Symposium).

[18]  C. Papas.  Theory of electromagnetic wave propagation , 1965 .

[19]  Jacob K. White,et al.  Electromagnetic Wave Theory and Applications , 1987 .

[20]  Stanley Fifer.  Analogue computation : Theory, techniques and applications , 1961 .

[21]  Arjuna Madanayake,et al.  Continuous-time algorithms for solving the electromagnetic wave equation in analog ICs , 2017, 2017 IEEE 60th International Midwest Symposium on Circuits and Systems (MWSCAS).

[22]  J. Kong.  Electromagnetic Wave Theory , 1986 .

[23]  Jian-Ming Jin,et al.  Fast and Efficient Algorithms in Computational Electromagnetics , 2001 .

[24]  Olivier Bournez,et al.  A Survey on Continuous Time Computations , 2009, ArXiv.

[25]  R. Sallen,et al.  A practical method of designing RC active filters , 1955, IRE Transactions on Circuit Theory.

[26]  W. D. Little,et al.  On the analog computer solution of first-order partial differential equations , 1965 .