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 τ_i, 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, R_s, as the resistance presented for an instantaneous sideband frequency, f_s, it is evident that its value will be proportional to the trapping or sideband frequency. If so, than an instantaneous sideband frequency current I_s will be inversely proportional to the sideband frequency. Accordingly, the instantaneous sideband power P_s given as I_s²R_s 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, f_c, where it increases 3 dB above the noise floor. Typical values of f_c 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 f_c value, and then include the 1/f noise influence, by the additional (f_c/f_s + 1) term which modifies the noise floor. Applying that component to Eq. 4, including noise plateau, and taking into account the generalized τ_i parameter, leads to the complete sideband noise form:

SEE EQ. 8

Its equivalent form with Q instead of τ_i is:

SEE EQ. 9