Adaptive steady-state analysis of circuits using wavelets

Abstract

This thesis presents research into utilizing the sparse representations of waveforms that are possible in the wavelet domain to increase the computational efficiency of the steady-state analysis of electric circuits. The system of non-linear equations that represent the circuits are formulated in the wavelet domain and solved using Newton-Raphson method. Factoring the Jacobian matrix each iteration is a major contributor to the computational time required for solving the circuit equations with NewtonRaphson method. This research aims to reduce the computational time of factoring the Jacobian matrix and has led to the following contributions: 1. A study on the effect of wavelet selection on the sparsity of the Jacobian matrix and nodal variable vectors: Results show that there is no one wavelet that provides the sparsest Jacobian matrices in every case but the Haar wavelet tends to be a good choice if Jacobian matrix sparsity is a concern. However, the time domain provides sparser Jacobian matrices than all of the wavelets tested. Selection of a wavelet to provide the sparsest nodal variable vectors is much more difficult and no one wavelet stood out as providing sparser vectors than the others. 2. A method for increasing the sparsity of the Jacobian matrix via removal of low amplitude entries: The threshold to determine which elements to remove is adaptively controlled during the simulation. Results show that there can be a significant decrease in Jacobian matrix density with adaptive thresholding but the Haar wavelet tends to provide the sparsest matrices with the test cases. The results show that adaptive Jacobian matrix thresholding can lead to a speedup over the non-thresholded wavelet domain steady-state analysis. In some cases, this speedup was enough to lead to a speedup over the time domain when the non-thresholded simulations ran slower than the time domain. 3. Two new methods that reduce the problem size by taking advantage of the sparse representations that are possible with the nodal variable vectors in the wavelet domain: A unique feature of one of these methods is that it allows for the automatic selection of a wavelet for each nodal variable. Results show a speedup over wavelet domain steady-state analysis for some test cases. There were some test cases where there was a slowdown compared to wavelet domain steady-state analysis which was caused by the computational overhead associated with these methods. With one circuit, the number of columns in the Jacobian matrix was not reduced for most iterations. More work is required to determine if this is due to the method used to select columns from the Jacobian matrix, the method used to control the error introduced into the update vectors by the column reduction method, or if there are some problems that cannot benefit from the column reduction method.