Continuous-time Algorithms and Analog Integrated Circuits for Solving Partial Differential Equations

Analog computing (AC) was the predominant form of computing up to the end of World War II. The invention of digital computers (DCs) followed by developments in transistors and thereafter integrated circuits (IC), has led to exponential growth in DCs over the last few decades, making ACs a largely forgotten concept. However, as described by the impending slow-down of Moore’s law, the performance of DCs is no longer improving exponentially, as DCs are approaching clock speed, power dissipation, and transistor density limits. This research explores the possibility of employing AC concepts, albeit using modern IC technologies at radio frequency (RF) bandwidths, to obtain additional performance from existing IC platforms. Combining analog circuits with modern digital processors to perform arithmetic operations would make the computation potentially faster and more energy-efficient. Two AC techniques are explored for computing the approximate solutions of linear and nonlinear partial differential equations (PDEs), and they were verified by designing ACs for solving Maxwell\u27s and wave equations. The designs were simulated in Cadence Spectre for different boundary conditions. The accuracies of the ACs were compared with finite-deference time-domain (FDTD) reference techniques. The objective of this dissertation is to design software-defined ACs with complementary digital logic to perform approximate computations at speeds that are several orders of magnitude greater than competing methods. ACs trade accuracy of the computation for reduced power and increased throughput. Recent examples of ACs are accurate but have less than 25 kHz of analog bandwidth (Fcompute) for continuous-time (CT) operations. In this dissertation, a special-purpose AC, which has Fcompute = 30 MHz (an equivalent update rate of 625 MHz) at a power consumption of 200 mW, is presented. The proposed AC employes 180 nm CMOS technology and evaluates the approximate CT solution of the 1-D wave equation in space and time. The AC is 100x, 26x, 2.8x faster when compared to the MATLAB- and C-based FDTD solvers running on a computer, and systolic digital implementation of FDTD on a Xilinx RF-SoC ZCU1275 at 900 mW (x15 improvement in power-normalized performance compared to RF-SoC), respectively