# Oscillators: A New Look At An Old Model

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, V_{0}, is augmented by means of amplitude noise as well as phase noise. The expression,

0+ε(t)>sin0t+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 f^{0} 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, f_{0}, 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.

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The main idea of the traditional oscillator noise spectrum model, as originated in ref. 7, lies in the explanation of the f^{−2} noise slope character. For this, an oscillator loop consisting of an amplifier and simple (ideal) resonator is assumed. The slope of the simple resonator's asymptotic characteristic is 20log/decade, while its action in a feedback loop leads to just such overall transmittance, as shown in the second curve in Fig.1. From this reasoning, the f^{−2} slope starts below the resonator's half-bandwidth at f_{0}/2Q. Based on this, many simulation were performed and verified, with good agreement to measurements.

What happens to the traditional model if the resonator is not ideal? Let the oscillation point be at its slope rather than at the top (and such dislocation is normal with phase balance condition usually set off of the optimum peak). Let the resonator be extremely asymmetric, and let it be more complex with steeper slopes. The signal close to the carrier will still be ideally symmetrical with ideal 20log/decade slope (excluding 1/f noise). But if the resonator is made less and less ideal, until its slope is flat, it would lead the theory to predict something like infinite noise. But measurements will show a clear spectrum line, not ideal, but symmetric and with an inherent 20log/decade slope.

If a microwave approach is taken to the oscillator model, then a piece of transmission line would be included in the connection from the output to the input, since every connection (at higher frequencies) must have distributed inductance as well as capacitance (Fig. 2). In analytical terms, the circuit consists of a simple feedback loop with closed-loop transfer function like:

For analysis purposes, the amplifier is considered to have plain voltage transmittance, to be flat in frequency with constant phase shift, φ, and described by Ae^{jφ}. The transmission line is assumed to be ideal, described by the delay element β(jω = e^{−jωτ}). Oscillation will occur at the frequency ω_{0} (for simplicity, the angular frequency will simply be called the frequency), where the wave transmitted through the overall loop coincides in phase, with overall phase shift of zero, described by φ − ω_{0}τ = 0. It is useful to describe frequency Ω as the sum of Ω_{0} + ω_{s}, where Ω_{s} is the sideband frequency. The main product in Eq. 1 can then be written as:

The amplitude or loop gain, A, is needed to excite oscillation. But the real world restricts a generated waveform to a certain level, determined by the power supply and by the limitations of active devices. In a settled state, A drops to unity. The initial value of A is essential for transient analysis, while Eq. 1 relates to the steady state (this analysis does not include nonlinear effects) with its denominator going to zero at ω_{0}, implying infinite closed-loop gain.

This can be understood as a possibility for any high settled signal level circulating in the loop, despite an arbitrarily low excitation, indicating that a value of A=1 should be assumed for further calculations. So far, the voltage transmittance was developed with complex functions but the transmittance of interest is the power oscillator transmittance, denoted as T_{0}. For this, the above designations will be set in Eq. 1, with a transition from an exponential complex form to a trigonometric form, according to the known equivalence that e^{−jωτ} = cosωτ − jsinωτ. Next, simple transformations are needed to get the absolute value of the complex expression, and further, after squaring it, the desired overall power transmittance is derived as:

Can this trigonometric-like function describe complex oscillators and their resultant spectrum line shapes? Plotting the function provides graphs such as Fig. 3(a) which shows generated spectral lines with oscillations theoretically possible at every frequency where the loop phase condition n(2π) exists. But all real oscillators seem to have a transfer function with 20log/decade slope close to carrier, and the expression above does not appear to yield such a response. Nevertheless, if the simple trigonometric function is plotted on a log-log scale, Fig. 3(b) results, with a 20log/decade slope. This shows that the basic oscillator behavior was represented without any amplitude selectivity. The only selectivity is for phase, for phase selectivity of noise (not to be confused with traditional phase noise).

Although Fig. 3 predicts many possible oscillation frequencies, the primary phase condition will dominate all others and produce a single spectrum line (neglecting harmonics). As such, it should be possible to simplify Eq. 2 and using a trigonometric identity and the efficiency approximation of sinx ≈ x, a simple expression can be found:

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Such a simple form clearly expresses the fundamental oscillator behavior resulting in f^{−2} close to the carrier spectrum slope (assuming no 1/f noise) with the only required parameter being delay in the loop. Based on this form, it should be possible to build a practical circuit (Fig. 4). The amplifier is simple, comprised of one or two transistors, having about 10 dB gain and phase shift of about 150 deg. typical of RF transistors. Amplifier feedback should be avoided to maintain high backward isolation and simplicity of the oscillator loop. Input/output matching should be to 50 Ω, using attenuator pads if required. A simple −6-dB resistive power divider composed of three 15-Ω resistors will maintain the loop at 50 Ω without draining excessive output power. A 0.5-m connecting cable completes the design. This simple design provides a fairly clean output spectrum, with sideband noise levels low enough to be measured only with a good spectrum analyzer, such as the HP8564E from Agilent Technologies (Santa Rosa, CA), and only at a fairly distant offset, such as 100 kHz (Fig. 5).

The noise of −110 dBc/Hz at a 100-kHz offset is not much worse than performance levels of −120 to −130 dBc/Hz typical of commercial voltage-controlled oscillators (VCOs) at this frequency. The 30 dB/decade region extends to a few kHz, and for higher offset frequencies, ideally a 20 dB/decade slope exists. This is in agreement with typical bipolar-junction-transistor (BJT) devices having a corner frequency, F_{c} (1/f noise), on the order of 5 kHz. For comparison, the 100-kHz offset frequency lies well inside the 20log/decade region that is free of 1/f noise influence. To verify the measured results, the required expression for sideband noise, N_{s}, must be will determined as a function of the oscillator noise floor, N_{f}, multiplied by the oscillator transmittance, T_{0}, while referenced to generated power, P:

where:

k = Boltzmann's constant (1.38 × 10^{−23}J/K),

T = temperature (in K, at 290 K),

kT= −174 dBm/Hz,

F = amplifier thermal noise factor,

G = amplifier gain, and

P = generated power.

This expression is valid only well within the f^{−2} region.

Accounting for G in Eq. 4 requires some justification. Although G is assumed to be unity under basic oscillation conditions, it should not be treated as an ideal power follower through the band of operation. Amplifier compression at f_{0} does not change the influence of G on the noise floor, expressed in terms of rising noise according to F as well as G.

Measurements of the example oscillator include risetime (τ) of approximately 3 ns, noise factor (F) of about 5 dB, gain (G) of about 10 dB, and output power (P) of about +7 dBm. At f_{s} = 100 kHz, these values give a result that is close to the measured value of −110 dBc/Hz. Although some of these parameters were (coarsely) estimated, the agreement between the prediction and the measurement is satisfactory.

Although not an ideal solution, the primitive oscillator circuit of Fig. 4 may be suitable for a student's laboratory. According to ref. 4, lower sideband noise will result with increased delay in the oscillator loop. A transmission line can add delay, but will add size and temperature instability. A better source of delay would include a circuit with a few LC components to provide delay independent of phase shift. The amplitude characteristics of such a circuit may not be ideally flat, although this may not be such a bad thing since this resonator circuit can help attenuate signals beyond the desired passband.

Although the traditional oscillator noise model was based upon a resonator's amplitude characteristics, neither the resonator or its amplitude selectivity is needed for a general oscillator description. Still, a variety of resonators are available for phase management, with the deepness of resonance corresponding to the amount of delay as well as the selectivity. A resonator also makes possible impedance transformations in the oscillator loop. The most prevalent resonator structure is the shunt-C coupled series type (Fig. 5b).^{1} It exists in many different configurations, although only reconfigurations according to the virtual-ground concept^{1} allow proper visualization of the true resonator structure.

But is the delay added by the resonator the same as group delay? Measurements of the simple circuit of Fig. 6(c) will yield negative values of group delay, although negative time has no meaning. By examining the three generic responses of Fig. 6, the first response shows a strict relationship between the phase derivative and the signal-transmission time according to the definition of group delay. For the bandpass responses (Fig. 6b), this relationship can provide good approximations, but only for circuits that are not very selective, since any signal with a reasonable bandwidth and containing information will be distorted after transmission. For example, a burst carrier will have smoothed slopes at its output, so the group-delay definition cannot be easily related to this case. For circuits such as Fig. 6(c), this becomes more evident. The definition gives a function determined at an instantaneous frequency while any physical signal involves some range of frequencies, transmitted differently through a selective network. This alone suggests the lack of a direct relationship to the overall time delay. Thus, naming the phase derivative as group delay, envelope delay, or signal delay seems inadequate.

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What is really needed for proper oscillator operation: the pure time delay offered by a transmission line or a high rate of phase change with respect to frequency near the oscillation point? Because an oscillator is not dealing with distributed signals (like a filter), but rather almost an instantaneous frequency, it is the second case that is desirable. Thus, the phase derivative is an important parameter in oscillator theory, and there is need for closer examination of its physical meaning. The stronger that an electromagnetic (EM) field can be made within the resonator, the more inert it will appear to an incoming wave—and this behavior is directly determined by the transmittance phase slope. A parameter called time inertia, τ_{i}, in units of seconds, might be useful here:

An oscillator's resonator can be compared to the flywheel of a mechanical engine. In both cases, inertia ensures stability. Equation 5 can be negative or positive, denoting rotation in two directions. Note that the defined time constant of EM inertia in Eq. 4 is in the second power, so its sign has no significance for generated spectra. Is there a primary direction of rotation for oscillators, as there is for motor engines? Examining one of the resonators from Fig.5(c), an obvious problem appears: near resonance, where reasonable τ_{i} can be obtained, there is also high attenuation. Of course, an amplifier can be added to overcome the loss. Assuming a 100-MHz oscillator in this analysis, compare resonators with good Q within 50-Ω ports. The first, which is a series resonator inserted in parallel, behaves better than the second configuration. Note that for a typical inductor Q of about 40 (at a reactance of 100 Ω), an optimum τ_{i} of about −50 ns results, with tolerable attenuation of 18 dB and phase angle of −40 deg. Practical resonator component values are 18 pF and 150 nH. Using the appropriate amplifier, it is possible to use a shorter cable section to ensure zero phase balance at the proper point on the resonator response curve.

This oscillator circuit seems to operate within a few megahertz of the expected frequency. But the resonator's phase response must be added to the model, to the phase of the transmission line, resulting in a higher oscillator frequency (shifted by the amount of the amplifier phase shift).

The resultant phase response shows three zero-crossing points, with one lower and one higher than the expected point because the gain margin is much higher at these points. Oscillation builds more quickly at these points because of the lower τ_{i}.

Some additional notations may be useful for oscillator close-to-the-carrier spectrum analysis (still ignoring harmonics and nonlinear behavior). In Fig. 1, frequency f_{g} indicated the point where oscillator transmittance begins to rise above the noise floor. According to Eq. 3, f_{g} will be defined by f_{s} when T_{0} = 1 to Ω_{g} = 1/τ_{i} (f and Ω = 2πf) are treated without distinction here). The sideband bandwidth, B, is defined as 2f_{g}, which is 1/πτ_{i}. With these parameters, the oscillator quality factor can be defined as:

This is directly related to the well-known quality factor (Q). For a resonant tank, Eq. 6 can also be found by deriving τ_{i} as the phase derivative and setting Q as the quotient of the reactance to the resistance. one also obtains the result of Eq. 6. The term T_{0}, in terms of τ_{i}, appears more general, but an equivalent form in terms of Q may be more practical:

Parameter Q is of great significance, indicating circuit quality: the reactance level in reference to resistance level. Here Q refers to loaded Q (often denoted Q_{L}. For the same design approach and different bands, with scaling in frequency, Q remains constant while τ_{i }changes proportionately as well as generated sideband bandwidth and relevant sideband noise levels.

*Editor's Note:* The conclusion of this article, including Fig. 6 and the references, will be published in the January 2003 issue of *Microwaves & RF*. That installment will offer an explanation for the generation of 1/f noise in oscillators as a function of current flow through electronic materials.