This paper develops a theoretical analysis of harmonic balance method, based on the cubic spline wavelet and Daubechies wavelet, for steady state analysis of nonlinear circuits under periodic excitation. The properties of the resulting Jacobian matrix for harmonic balance are analyzed. Numerical experiments illustrate the theoretical analysis.

The rapid growth in integrated circuits has placed new demands on the simulation tools. Many quantities properties of circuits are of interest to circuits designer. Especially, the steady-state analysis of nonlinear circuits represents one of the most computationally challenging problems in microwave design.

Harmonic balance (HB) [

The remainder of this paper is organized as follows. In Section

The harmonic balance (HB) method is a powerful technique for the analysis of high-frequency nonlinear circuits such as mixers, power amplifiers, and oscillators. The basic idea of HB is to expand the unknown state variable

Let us consider the general approach of HB which assumes obtaining the solution

Hence, the sparsity of this Jacobian matrix

Given the base

Two functions

The function

We observe that once the filter

In HB method the wavelets on the interval

This linear transform can be represented by the

The periodized Daubechies wavelet HB formulation has been formulated in [

Consider the cubic spline wavelets as the expansion base in HB technique. The cubic spline wavelets are constructed in [

Due to the periodic condition

Denote by

We obtain the derivative matrix

For the whole nonlinear equation system

Using HB method to solve nonlinear ODEs, the Newton iterative form is obtained. Here we want to analyze the sparsity of derivative matrix

Now we analyze the sparsity of the matrixes

The sparsity of the derivative matrix

By the formulation in Section

In this section, we will give the sparsity figures of the transform matrix and the derivative matrix based on two kinds of wavelets. For simplicity, assume the matrices

(a): Sparsity pattern of the periodized transform matrix

In Figure

(a): Sparsity pattern of the derivative matrix

In this paper, we formulate the comparison analysis of harmonic balance method based on the cubic spline wavelets and periodic Daubechies wavelets. It is shown that the cubic spline wavelet HB method has the special structure for Jacobian matrix compared to the Daubechies wavelet HB method to solve steady-state analysis of nonlinear circuits.

The author declares that there is no conflict of interests regarding the publication of this paper.

The work was supported by the Natural Science Foundation of China (NSFC) (Grant nos. 11201370 and 11171270) and the Fundamental Research Funds for the Central Universities.