Oscillator designers have relied on certain assumptions based on modulation theory. But by abandoning traditional beliefs, it is possible to formulate new models for analyzing oscillators. In Part 1 of this article last month, some of these long-standing conventions were shown to be less than ideal for noise analysis, leading to the development of a parameter called time inertia to explain the behavior of a

resonator within an electromagnetic (EM) field (Fig. 6). The discussion continued with the derivation of Eqs. 6 and 7 to define quality factor, Q, and a variable, T0, that relates to Q and the resonant frequency.

The variable T0, expressed by τi, is directly applicable for such different oscillator types like LF ring oscillators with multiple inverters in the feedback loop or Gunn diode oscillators, where the transmission time through the inverter chain or the charge transit time through the semiconductor body directly determine the sideband noise level. Equation 6 makes it possible to calculate the equivalent Q for such an oscillator, where an amplitude-selective resonator does not exist. For the 100-MHz example examined earlier, the equivalent Q can be calculated to be close to 1, which is quite low. For example, well-designed wideband VCOs have an equivalent Q near 10 and, according to Eq. 7, have about 20 dB or lower sideband noise levels. Applying the same methodology to oscillators at 10 times higher frequency (1 GHz), Eq. 7 would predict noise levels about 20 dB higher for typical VCOs (at an offset frequency of 100 kHz), around −110 dBc/Hz.

The measured noise seems to be "white" in nature, i.e., having a flat, thermal kT density. Only at very low frequencies (acoustic range) does its behavior essentially change. As the measurement frequency decreases, the noise level increases according to the 1/f rate of noise power (10log/decade). The phenomenon appears not only with DC current, but also with any AC current, always exhibiting 1/f slope sidebands. For any generated frequency having f−2 sidebands according to the very oscillator action, as was described earlier, it adds its own impact, resulting in the f−3 (−30 dB/decade) sideband noise slope, close to the carrier f0. Despite experimental results, this phenomenon still remains a mystery. A possible explanation is that any current flowing through any physical material excites 1/f power sidebands. So, the spectrum of any real current may not be abrupt, but must essentially have a smooth, 1/f slope. Any dislocations or impurities within the material will create traps for charge flow.

Defining sideband resistance, Rs, as the resistance presented for an instantaneous sideband frequency, fs, it is evident that its value will be proportional to the trapping or sideband frequency. If so, than an instantaneous sideband frequency current Is will be inversely proportional to the sideband frequency. Accordingly, the instantaneous sideband power Ps given as Is2Rs must have a 1/f sideband frequency dependence, which is often observed in practice. Such an explanation applies to both DC as well AC current sidebands. Contrary to the standard literature in which 1/f noise in oscillators is described as the effect of upconversion of baseband 1/f noise, nonlinear action is not required for 1/f oscillator noise (AGCs have been known to raise sideband noise in semilinear oscillators). As confirmed by measurements, any current flowing through a device, whether linear or nonlinear, excites 1/f sidebands, with sideband power density proportional to the current value.

It should be possible to describe the 1/f noise level for a device at given conditions in practical terms, for example at a 1-Hz sideband frequency. For bipolar transistors, this level is typically about −120 dBc/Hz. Another (less meaningful but more convenient) way to describe this noise is to identify the 1/f noise corner frequency, fc, where it increases 3 dB above the noise floor. Typical values of fc are near 5 kHz for bipolar transistors and even megahertz frequencies for gallium-arsenide field-effect transistors (GaAs FETs). The values are directly related to the amplifying mechanism for each device type, especially to its sensitivity for surface actions where device current is dispersed due to surface irregularities, contamination, and dislocations in the semiconductor crystal lattice structure. Generally, 1/f noise is device and technology dependent, and devices designed for low-noise oscillators should list this quantity in product sheets. For predicting overall oscillator sideband noise, it is practicable to establish the fc value, and then include the 1/f noise influence, by the additional (fc/fs + 1) term which modifies the noise floor. Applying that component to Eq. 4, including noise plateau, and taking into account the generalized τi parameter, leads to the complete sideband noise form:

SEE EQ. 8

Its equivalent form with Q instead of τi is:

SEE EQ. 9

Page Title

Modulation theory has long been accepted as the cornerstone of oscillator spectra considerations, although this is a fundamental misconception. Oscillators have their own particular operating mechanism, in which modulation explains little. Parasitic modulation can appear at any time, but should be negligible in a properly designed oscillator. If anything, an oscillator can be considered as a special noise amplifier. More specifically, it is a "phase-selective," infinite-gain, noise amplifier that produces a very narrow and symmetric "burst" of noise. Phase-selective means that the oscillator loop phase zero-crossing point determines the oscillation frequency, while its quality is determined by the phase characteristic slope at this point. In general, the amplitude characteristic is irrelevant for oscillation analysis, although amplitude selectivity can be very desirable for secondary reasons. The natural coexistence of both phase and amplitude responses of resonance circuits makes it possible to confuse their roles. Proper distinction is possible when general cases with transmission-line oscillators, as well as inverted amplitude responses, are considered.

The close-to-the-carrier spectrum (discerned from harmonics) should always be considered as one entity, without separation of the carrier from the sidebands. Only in subsequent system stages it is justifiable to approximate an oscillator's spectrum by a single spectral line. An oscillator's signal sidebands have an inherent f−2 slope, while their level (signal quality) is determined by the instantaneous loop phase derivative. The natural measure of generated signal quality is sideband noise density (SND). This parameter is commonly in use, although it has been called phase noise according to improperly understood theory.

Traditional Visualization
Traditional visualization of generalized signal sidebands with several descending power-law sections on log-log diagrams leads to another common misunderstanding. As it was shown earlier, the oscillator action yields only a f−2 sideband slope. However, any currents flowing through a physical structure excite sidebands of 1/f power density slopes which add to the sideband slope, resulting in a common f−3slope at the lowest offsets. There are two possible transitions down to the flat noise floor. The traditional explanation of oscillator 1/f sideband noise, as a result of upconversion from baseband, cannot be justified. DC as well as AC currents excite inherent 1/f sidebands, because of microstructural irregularities, thus causing elementary charge trapping, especially within semiconductor devices with significant surface action.

Oscillator quality has traditionally been categorized as short-term and long-term stability, without any transition between the two. The meaning of long-term stability or instability is easily understood with such influences as temperature, humidity, aging, vibration, etc. Short-term stability is erroneously based on modulation theory to explain oscillation behavior. But, as has been shown, it is not correct to apply the terms stability or instability in regard to oscillator sidebands. It is more accurate to make the connection to the selectivity of the oscillator as an active noise filter. The term "low-noise oscillator" should be used instead of saying an oscillator with good short-term stability.

The parameter for the phase-response derivative provides a more precise concept than group delay in the design an analysis of oscillators. It provides a more general conceptualization of oscillator behavior than a mere time delay, reflecting the EM inertia of a circuit. In understanding this parameter in this way, problems are eliminated in terms of possible negative values as well as with determining it at an instantaneous frequency points.

FOR FURTHER READING
  1. S. Alechno, "Analysis Method Characterizes Microwave Oscillators (four parts)," Microwaves & RF, November 1997-February 1998.
  2. S. Alechno, "The Virtual Ground in Oscillator Design—A Practical Example," Applied Microwave & Wireless, July 1999.
  3. S. Alechno, "Microwave Switched VCO with Microstrip Branch Resonator—The Virtual Ground in Practice," Eagleware Design Contest Winning Paper, October 1999.
  4. S. Alechno, "Analysis of the Cascode Crystal Oscillator," RF Design, May 1997.
  5. S. Alechno, "Oscillator Loaded-Q Measurement," RF Design, November 1995.
  6. S. Alechno, "Advancing The Analysis Of Microwave Oscillators," Microwaves & RF, June 2000.
  7. D.B. Leeson, "A Simple Model of Feedback Oscillator Noise Spectrum," Proceedings of the IEEE, February 1966.
  8. D.B. Leeson, "Short Term Stable Microwave Sources," The Microwave Journal, June 1970.
  9. L.S. Cutler, C.L. Searle, "Some Aspects of the Theory and Measurements of Frequency Fluctuations in Frequency Standards," Proceedings of the IEEE, February 1966.
  10. R.A. Baugh, L.S. Cutler, "Precision Frequency Sources," The Microwave Journal, June 1970.
  11. J.A. Barnes et al., "Characterization of Frequency Stability," IEEE Transactions on Instrumentation and Measurement, May 1971.
  12. J. Rutman, "Characterization of Phase and Frequency Instabilities in Precision Frequency Sources: Fifteen Years of Progress," Proceedings of the IEEE, September 1978.
  13. D. Scherer, "Today's Lesson—Learn About Low-Noise Design," MicroWaves, April 1979.
  14. Chronos Group, Frequency Measurement and Control, Chapman & Hall, London, 1994.