Analysis of high-order sub-harmonically injection-locked oscillators

High-order subharmonically injection-locked oscillators have recently been proposed for low phase-noise frequency generation, with carrier-selection capabilities. This work focuses on the analysis of this frequency-synthesis procedure, with emphasis on the oscillator capability to discriminate the input tones, under the variation of a tuning voltage. The oscillator is described with an analytical envelope-domain formulation, leading to an expression for the oscillator phase shift with respect to the input source. The average of this phase shift is shown to evolve in a continuous manner in the distinct synchronization bands obtained versus the tuning voltage. Then, the subsynchronized operation at a high frequency ratio (N = 30) is considered, showing the applicability of envelope-domain simulations under various Fourier decompositions and sampling rates. The synchronization bands are obtained through the phase averaging technique. The analysis has been applied to a prototype at 2.7 GHz that has been manufactured and measured.


I. INTRODUCTION
The recent works [1]- [3] present a frequency-synthesis technique based on the subharmonic injection locking of an oscillator at a frequency fo by an independent source at a much lower frequency. The high ratio N between the two frequencies (in the order of N = 30) is enabled by the periodic switching of the higher frequency oscillator by a low frequency input, which gives rise to a high number of spectral lines at the oscillator output. This output signal is introduced into a second oscillator, which gets locked to one or another spectral line, depending on its tuning voltage [1]- [3]. Thus, the combination of the two oscillators enables a programmable frequency selection, with frequency spacing fo/N. The whole system has the advantage of a low phase noise in comparison with standard phase-locking methodologies [2]. However, the realistic simulation of this complex system is extremely demanding and, to our knowledge, has not been carried out in any previous work. In [1]- [3] the injection-locked oscillator (ILO) is analyzed in time domain, using simplified models of the Van der Pol type. Then, the synchronization bandwidths are detected through a comparison of the oscillation amplitude and phase values at the beginning and end of each period of the switching signal.
Here the analysis will be carried out in the envelope domain [4]- [5], better suited at microwave frequencies. Two distinct investigations will be performed: the oscillator frequency selectivity and its subsynchronized operation under a high frequency ratio. Initially the ILO response in the presence of an arbitrary multitone input signal is analyzed, in terms of bandwidth and spectral content. An analytical formulation is used to derive an expression for the oscillator phase shift with respect to the locking tone. As will be shown, the averaging of this phase shift enables a reliable detection of the distinct synchronization bands obtained versus a tuning parameter. In a second stage, a high subharmonic order ILO is considered. Its envelope-domain analysis is based on a judicious choice of the Fourier frequency basis and integration time step, combined with the phase averaging technique to detect the synchronization bands. The analysis has been applied to a prototype at 2.7 GHz that has been manufactured and measured.
II. FREQUENCY SELECTIVITY Let an oscillator with an input signal composed of multiple tones, at the frequencies ω1, ω2,…,ωK, be considered. For a good insight, closely-spaced tones about the free-running frequency, ωo, will be initially assumed. The oscillator will be modeled in terms of its current-to-voltage ratio Y(V,ω,η) at a particular observation node [6]- [7], where V is the voltage amplitude at ω and η is a tuning parameter. In the absence of the input signals, one has Y(Vo, ωo, ηo) = 0, where Vo and ωo, are the free-running amplitude and frequency at ηo. The oscillator will be formulated at the particular input frequency ωq, expressing the node voltage as ( Applying the implicit function theorem [6]- [7] to the circuitlevel envelope-transient system, one obtains the following equation: where s is a complex frequency increment and Ik is the complex equivalent current of the each input signal. Under small input power and close spacing of the input tones, the function Y can be expressed in a first-order Taylor series expansion about Vo, ωo and ηo [6]- [7]. Taking into account that s acts as a time differentiator, one obtains: where q q o ω ω ω Δ = − and YV, Yη and Yω are the derivatives of Y with respect to V, η and ω, calculated at Vo, ωo, ηo, as shown in where |Ik| and φk are the magnitude and phase of the input tones, αvω = ang(Yω)-ang(YV), αvη = ang(Yη)-ang(YV) and αv = ang(YV). The phase ϕ(t) can be expressed as ,..., has zero average. Averaging equation (3): From inspection of (4), Δη depends on φo, which varies between 0 to 2π, and the averaged term on the right side. As in the case of an oscillator locked to a single tone, only about one half of the total phase φo range (from 0 to 2π) will correspond to stable solutions. As derived from (2), the voltage magnitude at ωq is Vo+ΔV(t), so the power at ωq is enhanced by the self-oscillation.
The increment ΔV(t) is: where αω = ang(Yω), αηω = ang(Yω)-ang(Yη). The first two terms in (5) are not frequency dependent and the first one increases with the magnitude |Iq|.  The above analysis has been applied to the voltage-controlled oscillator at fo = 2.7 GHz in Fig. 1, tuned with the varactor SMV1235. The tones, generated with the Agilent E4438C ESG Vector Signal Generator, are equally spaced by default. In order to generate as many tones as possible, the spacing Δf = 16 MHz is chosen, enabling six input tones, between 2.7 GHz and 2.78 GHz. Initially, synchronization at ωq /(2π) = 2.7 GHz is assumed. Fig. 2 validates the spectrum obtained through the integration of (2) and with circuit-level envelope transient simulations [4]- [5], using ωq as the only fundamental frequency, as in (2) is not centered about φo = 0º because αv is different from zero.
The three bands in Fig. 3(b) correspond to synchronization at f1 = 2.7 GHz, f2 = 2.716 GHz and f3 = 2.732 GHz. The circuit is synchronized where the frequencies are constant. In Fig. 3(d) the output power at the frequencies of the input tones is evaluated versus VD. The boundaries of the synchronization bands are easily detected by the discontinuity of the curves. The bandwidths slowly decrease when moving away from the oscillator free-running frequency (2.7 GHz). Fig.  4 presents the measurement results. Fig. 4(a) shows the six input tones, with Δf = 16 MHz. Fig. 4(b) presents an unlocked spectrum for VD = 0.28 V. Fig. 4(c) and (d) present the spectrum for VD = 0.8 V (synchronization to f2), and VD = 2.85 V (to f5).
There is a deviation in the tuning voltages, attributed to the inaccuracies in the device models.
III. SUB-HARMONIC INJECTION LOCKED OPERATION The analysis of the subharmonically synchronized oscillator is more demanding than the one in Section II, since the subsynchronization is a nonlinear phenomenon and the relevant spectral lines, resulting from the input signal, plus frequency multiplication/mixing effects, cover the entire frequency bandwidth from dc to the oscillation frequency (and all the frequency intervals between the harmonic terms). The harmonic balance (HB) analysis of a subsynchronized solution at a high order, such as N = 30 considered here, is    ϕp(t) = ϕo,p + Фmix,p(t), will exhibit a slower time variation for a higher P. In the limit situation P = N, the phasors p V  will be constant in the synchronization band. The oscillator in Fig. 1 has been injection-locked by a rectangular signal with frequency 90 MHz, amplitude 1 Vpp, and duty-cycle of 25%. To facilitate the locking, the input dc block has been suppressed. Although the oscillation frequency of this prototype (2.7 GHz) is lower than the one in [1]- [3], the frequency ratio is the same N = 30, so the analysis complexity must be similar. The circuit has been analysed with circuit-level envelope transient, following the criteria in (6), and using an auxiliary generator (AG) at fo, connected to the circuit at the initial time only [6], to initialize the oscillation. Fig. 5(a) presents the spectrum at 30 90 MHz out f = × for P = 1,4,6, which predicts a synchronized behaviour. Only the central spectral lines can agree since for P > 1 part of the spectrum corresponds to terms P-1 and P+1. For P = 1 the integration time step is 0.05 ns, whereas for P = 6 it is 0.3 ns. With P = 6, the time variation is more regular and the synchronization bands can be determined with higher accuracy. The analysis has been validated through an independent HB analysis (superimposed). This costly HB analysis has been carried out providing the amplitude and phase of the central spectral line, obtained with envelope transient, to an AG at 30 90 MHz out f = × . Through AG optimization [7], it was possible to trace the synchronization curve versus VD [ Fig. 5 Note that envelope transient only provides the upper (stable) section of the synchronization curve. There is an excellent agreement with the HB predictions, though the computation time is four times shorter. Moreover, it was not possible to obtain any HB convergence when changing VD in order to select a different spectral line, for instance N+1 and N+2. This could be easily done with the envelope-transient method, as shown in Fig. 6(a), where three different spectral lines, corresponding to N = 30, N+1 and N+2, are selected by changing VD. Finally, the output of the subsynchronized oscillator has been connected to an analogous voltage-controlled oscillator to increase the frequency selectivity. To initialize the two individual oscillations an AG is connected to each circuit at the initial time only. The simulated spectra are shown in Fig. 6(b). The ratio between the selected spectral line and the highest-power neighbouring line is in the order of 30 dB. Fig. 6(c) presents the experimental results. In a manner similar to Fig. 4, there is a deviation in the tuning voltages, attributed to the device models.
IV. CONCLUSION An envelope-domain analysis of high-order subharmonically injection-locked oscillators has been presented. The locking bands can be efficiently detected by averaging the phase of the oscillation spectral line. Application of this technique to a highorder subharmonic synchronization requires a suitable selection of the Fourier decomposition and time sampling. The procedure has been illustrated with a two-oscillator system at the subharmonic order N = 30. To our knowledge, it is the first realistic analysis of a system of this high complexity.