Frequency-Domain Modeling and Analysis of Injection-Locked Oscillators

A Charge-Sharing Locking Technique With a General Phase Noise Theory of Injection Locking

“…Due to the nature of CSL/IL, the lack of any significant delay in the feedforward path makes it impossible to directly downsample the final t out [k] by N to obtain t out,down [n] = t out [n N] for the purpose of feeding it back to compare it with t ref [n]. Since t out [k] has already been corrected at k = n N (i.e., t = nT ref ), doing so anyway would cause an incomputable "delay-free loop" in the discrete-time model, such as in [26].…”

“…The CSL jitter-tracking bandwidth (describing how much information in the output is influenced by the reference injection, i.e., "loop" bandwidth), f BW,ref , can be derived by forcing the first term (excluding N 2 L ref ) of ( 22) to equal 0.5 (i.e., 3-dB bandwidth). It can also be used to estimate the "one-sided" lock range f LR of CSL (or IL) [24], [26], [27]. When β 1, it is simplified as…”

“…Failing to address this, the output jitter in ILOs might not be any better than the reference jitter [16], [23]. Surprisingly, a general PN theory in IL is still missing [24]- [26]. The conventional s-or z-domain (at reference rate) analyzing techniques [16], [19], [26], [27] model ILOs in the same manner as in the conventional type-I PLLs with a single integrating pole [28], but they fail to predict the ILO's special PN phenomena, such as the extremely strong suppressing ability of the oscillator's PN and the PN folding effect resulting in the ILO's PN being 3 dB higher outside of the "loop" bandwidth than when the oscillator is free-running [22], [23], [29]- [31].…”

“…2(b)]. Assuming a perfect lock, if the oscillator's free-running frequency f osc suddenly gets a little lower than N× the reference frequency (i.e., N f ref ), then, at the following reference event, the sharing charge 2 It can be calculated analytically by the slope of "phase domain response (PDR)" [27], including the methods of "impulse sensitivity function (ISF)" [25], [26], "perturbation projection vector (PPV)" [32], [33] (i.e., a more rigorous version of ISF, considering the phase shift of ISF itself when perturbation happens [34], which enables the ISF to predict some IL phenomena [35]), and others [16], [27]. 3 For a fractional-N operation, the DAC would set an aliased [4] version of the sinusoidal waveform of V osc .…”

“…[k] and t osc [k] & t ref[n], we start with an equation based on t corr [n] in Fig.6asβ(t ref [n] − t out,down,uncorr [n]) = t corr [n] − t corr [n − 1]. (26)Substituting (20) into(26), we obtain its z-transform asβ( T ref (z ref ) − z −1 ref T corr (z ref ) − T osc,down (z ref )) = T corr (z ref ) − z −1 ref T corr (z ref ) (27)and thenT corr (z ref ) = β T ref (z ref ) − T osc,down (z ref ) 1 − (1 − β)z −1 refOn the other hand,t out [k] = t corr,up-zoh [k] + t osc [k](29)and its z-transform isT out (z osc ) = H zoh (z osc ) • T corr,up (z osc ) + T osc (z osc ) = H zoh (z osc ) • T corr (z ref ) + T osc (z osc ) (30)…”