Traditional models of high-frequency oscillators have guided design engineers for many years. Recently, design approaches have focused on transmission analysis, with many benefits compared to negative-resistance approaches. By adopting a "virtual-ground" design concept, it has been possible to reconfigure traditional oscillator topologies to simplify analysis,¹ and over time this approach has been supplemented with numerous reference works.^{2,3, and 6} By building on fundamentals,⁷ however, it may be possible to redefine the traditional models of high-frequency oscillators.

Most oscillator theory begins with a simple sine function describing an ideal waveform. In order to represent real-world noise and modulation, however, noise components must be added to the simple expression. So, the nominal amplitude, V₀, is augmented by means of amplitude noise as well as phase noise. The expression,

[V₀+ε(t)]sin[ω₀t+ω(t)]

appears at the beginning of most fundamental oscillator papers. It has been a starting point because of its twofold definition of oscillator impurities. Because of the dominance of phase noise, amplitude noise was generally treated as unmeasurable and neglected, or treated as frequency stability and modeled in terms of modulation theory. But what if an analysis of oscillators did not include a noise definition? The basic expression above describes a modulation process, and when modulation is examined there should be three factors: there is something which operates on one signal and modulates it by another to produce a resulting waveform. In most analysis, the modulated noise is well treated. But in the case of a pure sine wave, there is still a modulation function that must be understood.

A different way to consider an oscillator model is by describing the generation of a signal. An oscillator's noise spectrum, when shown on a logarithmic plot, reveals that the power spectral density can be characterized by a power-law model. The sidebands can be depicted as descending slopes of f⁻⁴ close to the carrier, followed by f⁻³, f⁻², f⁻¹, and the flat f⁰ noise background further from the carrier. Yet, the f⁻⁴ slope may be more theoretical than real, and most measurements of normal, simple oscillators do not expect such a slope, leaving the f⁻², and f⁻¹ slopes of practical interest for analysis.

Figure 1 may help to enlighten the meaning of these three generic slopes. It shows there to be only two possible and practically measurable basic oscillator spectra. These spectra arise because of two essential factors: 1/f noise (10 log/decade) and the common f⁻² (20 log/decade) oscillator transfer function. The 1/f noise is practically characterized by its corner frequency, f_c, where it rises 3 dB above the base noise level. Similarly, the limit of the oscillator transfer function's influence can be marked as f_g on an asymptotic curve, with no regard to its origin. The two kinds of spectra result simply from the f_c to f_g relationship. It should be noted that the 1/f noise is usually characterized as low-frequency noise with reference to zero frequency (DC), while the other curves in Fig. 1 are referenced to the generated frequency, f₀, with the sideband (offset) frequency (f_s) shown along the horizontal axis. In normal oscillator modeling, upconversion of the 1/f noise is normally assumed. The influence of the 1/f noise on the oscillator noise spectra, although very important, can be regarded as a secondary effect. To simplify oscillator behavior, it is necessary to disregard 1/f noise for a while, adding it to the model as a real-world noise feature. With this assumption, any basic oscillator would generate an extremely simple signal with the close-to-carrier spectrum of 20 log/decade slope, resulting from the flat, "white noise" and f⁻² oscillator transfer function. These conclusions are based on decades worth of noise spectra measurements, and observations rather than modeling. An accurate model should explain the common f⁻² transmittance on a physical basis, and the widespread popularity of modern oscillator models seems to arise from this fact.⁷

Engineers should not be impressed with close agreement between measurements and oscillator models, since such measurements are difficult and the parameters needed for simulation are often loosely estimated. As ref. 7 suggests, the model parameters should be chosen from a range expected to fit the measurements.