A robust and efficient oscillator analysis technique using harmonic balance

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Abstract

In this paper, we present a new technique, based on a continuation method, for oscillator analysis using harmonic balance. With the use of Krylov subspace iterative linear solvers, harmonic balance has become a very powerful method for the analysis of general nonlinear circuits in the frequency domain. However, application of the harmonic balance method to the oscillator problem has been difficult. The strong dependence of the oscillator behavior on frequency, and the dependence of the frequency of oscillation on the circuit waveforms result in a very small region of convergence for harmonic balance. The main contribution of this paper is a robust and efficient continuation method to obtain global convergence.

Introduction

Oscillators are essential components found in almost any electronic system, from wrist watches to sophisticated instrumentation and communication systems, and computers. Oscillators generate the signals that are used as the time or phase references. In communication systems oscillators generate the signals transmitted by antennas or by a wire line system. They also generate the local oscillator signals used in processing receiver signals. There are as many different kinds of oscillators as there are applications.

The design of an oscillator, a nonlinear circuit, requires a large signal steady state analysis. Using traditional time domain integration techniques for simulating these circuits can be very difficult, especially for high-Q crystal oscillators. Without special intervention, such simulations settle to the trivial DC solution. If oscillations do start, due to the high-Q, many cycles have to be simulated before the steady state is obtained. In addition, the truncation error tolerances need to be extremely stringent. The accuracy requirements and the high-Q mean that, in some cases, the simulation can proceed for a few days. Even with more accurate integration schemes [1], time domain simulation of oscillators is inefficient.

Since many periodic waveforms typically consist of a fundamental frequency and a few harmonics, it is more natural to represent the waveforms with a truncated Fourier series. Contrast this with use of piece–wise polynomials used in time domain simulations. Using the Fourier representation converts the differential–alegbraic equations of the circuit equations to a set of nonlinear algebraic equations which connect the Fourier coefficients of the signals. The harmonic balance method achieves the simultaneous solution of these equations employing Newton–Raphson based methods. A very important advantage of harmonic balance is that the truncation error inherent in time domain analysis is absent when the waveforms are band limited. This makes harmonic balance the method of choice for a large class of important circuits in the wireless communications area. Also, the transient response of the circuit is completely ignored by the method, greatly speeding up the analysis of the steady state response.

Harmonic balance has been applied to oscillator simulation by adding the unknown frequency of oscillation to the set of circuit state variables. However, oscillator simulation using harmonic balance has proven to be difficult due to the small region of convergence and the existence of the degenerate solution (the DC solution is a valid steady state). This is due to nature of the oscillator circuit itself, often requiring an extremely good guess for the frequency of oscillation and the oscillator waveforms to aid convergence. There have been many methods to improve the convergence of harmonic balance when applied to oscillator circuits [2], [3], [4]. However, given the wide range of circuit techniques used to realize an oscillator, these techniques are not always suitable. In this paper we propose a new continuation method that significantly improves our ability to simulate oscillators using harmonic balance.

We first present an overview of the theory of harmonic balance, including recent advances in the application of Krylov methods. We then discuss the oscillator problem and present a new continuation method. We conclude the paper with an example that demonstrates the robustness and efficiency of the new method.

Section snippets

State of the art of harmonic balance

Harmonic balance is well established as a simulation technique for nonlinear circuits driven by one or more periodic inputs [5], [6], [7]. Harmonic balance exploits the fact that the periodic or quasi-periodic waveforms in many circuits are most compactly described in terms of their Fourier coefficients.

If the periodic inputs driving the circuit are harmonically related, all the circuit waveforms are periodic and can be expanded using the Fourier series. If the period T of a waveform x contains

The oscillator problem

The periodic steady state of an oscillator is determined not by external sources, but by the circuit itself, so, the period is an additional unknown. With a small extension, the harmonic balance method discussed in the preceding sections can be applied to the analysis of oscillators. One approach is to add the unknown frequency to the harmonic balance equations of the circuit [15]. An additional equation fixing the phase of the fundamental component of one waveform is added to choose one of

An example

One advantage of the harmonic balance technique is that the oscillation frequency can be determined very accurately. This is particularly important in high Q circuits. It is impossible to attain this level of accuracy using time domain techniques. In this section, we consider the simulation of a simple single transistor crystal oscillator, called the Pierce oscillator, shown in Fig. 8. The element values for the Pierce oscillator are: R1=100 KΩ, R2=2.2 KΩ, C1=100 pF, C2=100 pF, CP= 24.8679559

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