IEEE Circuits and Systems Magazine - Q3 2020 - 51

-R

C

R

L

1 4kT
2π R
ω0 - 3∆ω

ω0 - 2∆ω

ω0 - ∆ω

ω0

ω0 + ∆ω

ω0 + 2∆ω

ω0 + 3∆ω

ω

Figure 6. Partitioning of white noise current into small segments of length D~ [119].

where K ij = K p 2 0 , if the edge e ij ! Er ; otherwise,
K ij = K n 1 0 , if the edge e ij ! E . Kmax is the maximum degree of all the nodes in the complement graph1. By the
means of adaptively adjusting Kn and Kp, the clustering
phenomenon can be observed, and the cluster diameter,
defined as the maximum phase difference of oscillators
in a cluster, can be estimated at the same time, which has
been applied into coloring a graph [107]. Moreover, the
cluster synchronization appears in the Kuramoto-oscillator networks with external equitable partition (EEP)commensurate intrinsic frequencies, and bipolar cluster
synchronization emerges in signed Kuramoto-oscillator
networks [109], respectively. Nevertheless, determining
the number of clusters is not an easy task. Moreover, the
model of two coupled groups of phase oscillators is used
to investigate the global phase synchronization by taking
the effects of asymmetric distributions, finite-sized systems, and time delays into consideration in [110].
Besides the cluster phase synchronization, the frequency clustering appears among Kuramoto-oscillator
networks [111]. In addition, the model of Kuramoto-oscillator networks can also exhibit other types of collective phenomena, such as resonances [112] and frequency locking [113].
3. Selected Application Examples
3.1. Locking of Circuit Oscillators
The relationship between the Kuramoto model and
the circuit oscillator ha s been seldom discussed

1

Given an original graph G(V, E), V and E are the sets of vertices and
r (V, Er ) is defined as follows
edges, respectively, its complement graph G
r (V, Er ) has the same set of vertices
[108]: (1) The complement graph G
as G(V, E), Er is the edge set of complement graph; (2) For any a pair of
vertices, if there exists an edge between them in G(V, E), then there is
r (V, Er ); or vice versa.
no edge between them in G
THIRD QUARTER 2020 		

so far. Let's go back to Adler's orig ina l differentia l
equation [114]:
	

di = ~ + ~ 0 r I inj sin (i - i), (6)
0
inj
dt
2Q 4 I

where i is the phase, ~ 0 signifies the resonant frequency, Q is the quality factor, I represents the tail
current, and I inj is the injection current with its phase
i inj. Eq. (6) is a special case that the coupling strength
K = r~ 0 I inj 4QI and the network size N = 2 in the classical Kuramoto model (1). This original equation has been
using for presenting the nonlinear analysis of injectionlocked frequency dividers [115], [116], exploring the
-behavior of two mutually coupled phase-locked (or freerunning) oscillators with nearby frequencies [117], and
studying quadrature voltage-controlled oscillators with
bidirectional current injection [118]. If there exist multiple injection currents in white noise circuits (power
spectral density of 4KT/R enters the tank), the Adler's
equation can be modified as [119]:
	

+3
di = ~ + ~ 0 i n / sin (~ t + z - i), (7)
0
0, m
0, m
dt
2Q I s m = - 3

where in is the injection amplitude, Is is the peak of
the fundamental component of the tank current, m
is an integer ranging from - 3 to + 3 , ~ 0 is the freerunning frequency, ~ 0,m is the center of the interval
[~ 0 + mD~, ~ 0 + (m + 1) D~] with the width D~ (see
Fig. 6), and z 0,m is a random phase that is uniformly distributed in [- r, r]. Eq. (7) is a more general case for the
classical Kuramoto model (1) with K = ~ 0 i n 2QI s and
~ 0, m t + z 0, m = i m . The duple injection has been reflected
by the use of triple push oscillators in a typical millimeter-wave frequency synthesizer [120].
IEEE CIRCUITS AND SYSTEMS MAGAZINE	

51
IEEE Circuits and Systems Magazine - Q3 2020

Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q3 2020
