IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 4, APRIL 2010
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Analysis of Strong Coupling in Coupled Oscillator Arrays
Venkatesh Seetharam and L. Wilson Pearson, Fellow, IEEE
Abstract—Significant improvement in the mutual injection locking range (MILR) and pha noi can be obtained by strengthening the coupling between the oscillator elements in a coupled oscillator array (COA). An analysis of the array properties in the strongly coupled regime is provided in this paper. Previous analys of COAs have employed a so-called broadband condition that substantially simplifies analysis. The broadband condition is obrved to break down in the strongly coupled regime, a feature that is central in the understanding of the behavior of strongly coupled arrays. The obrved improvement in the pha noi performance can be attributed to the highly resonant nature of the coupling network in the strongly coupled regime. The theory is verified using a five-element linear COA operating at 3.75 GHz. Significant improvements in the MILR and pha noi is reported. The beam steering capabilities of strongly coupled arrays are also prented. Index Terms—Beam steering, pha noi, phad arrays, radiation patterns, voltage controlled oscillators.
I. INTRODUCTION T is well known that nsitivity to component variation from cell-to-cell in a coupled oscillator array (COA) is problematic in array performance. This nsitivity is minimized if one employs wide-locking bandwidth oscillators. It has been shown that oscillator design can be optimized for wide-locking bandof the oscillators [1]. Operating on width by lowering the their own, low Q oscillators lead to poor pha noi performance in the array. However, this pha noi can be controlled by injection locking the array to an external source that exhibits pha stability commensurate with the eventual application of the array. The mutual injection locking range (MILR) between two oscillators in an array is directly proportional to the coupling strength [2]. In the weakly coupled regime, the pha noi performance of the array improves with increa in the coupling strength. Thus strong coupling can possibly be utilized to widen the mutual injection locking range of the oscillators and improve the pha noi performance intrinsic to a given COA
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Manuscript received October 09, 2008; revid September 28, 2009; accepted October 13, 2009. Date of publication January 22, 2010; date of current version April 07, 2010. V. Seetharam is with Ansoft LLC, San Jo, CA 95132 USA (e-mail: ; venki_vs_). L. W. Pearson is with the Holcombe Department of Electrical and Com
puter Engineering, Clemson University, Clemson, SC 29634 USA (e-mail: pearson@ces.clemson.edu). Color versions of one or more of the figures in this paper are available online at ieeexplore.ieee. Digital Object Identifier 10.1109/TAP.2010.2041141
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(prior to external injection locking). However, strong coupling between the elements complicates analysis of array interaction. The amplitude variation along the array is negligible for the ca of weak or loo coupling between the oscillator elements. This variation becomes significant for strongly coupled arrays. Reducing the value of the coupling resistors to achieve strong coupling between the oscillator elements results in increasing the Q of the coupling network thereby making the network highly resonant at higher coupling strengths. For the reasons the analysis of strongly coupled oscillator arrays is not straightforward. Strongly coupled arrays have been investigated by Nogi et al. [3]. They analyzed the issue of multiple modes in such arrays and showed that all but one of the resulting modes has amplitude variation across the array. They also prent a technique to suppress the undesired modes. Lynch and York [4] investigated the effects of narrowband coupling networks on the synchronization properties of an array. They employed resistive coupling networks and performed analysis on weakly and strongly coupled oscillators for broadband and narrowband coupling. They concluded that narrowband coupling decreas the probability for the oscillators in an
array to lock with each other. Georgiadis et al. [5] performed a stability analysis on one-dimensional coupled oscillator array systems under weak and strong coupling conditions to understand the factors that limit the achievable values of constant inter-oscillator pha difference. In this paper, an analysis of strongly coupled oscillator arrays is performed. The analysis focus on obtaining a detailed understanding of the amplitude and pha distribution along the array for strongly coupled oscillators. Section II provides a brief introduction to COA theory. The oscillator array dynamics in the strongly coupled regime is prented. Section III provides computed and the measured amplitude and pha distributions. The measured mutual injection locking ranges are also prented in this ction. The array amplitude variations result in undesired radiation patterns. Section IV shows a means of reducing the side lobe levels by suitably adjusting the oscillator amplitudes. The beam steering capabilities of strongly coupled arrays are also evaluated. Section V discuss the pha noi characteristics of strongly coupled arrays. The pha noi theory developed by Chang et al. [6], [7] assuming weak coupling, is ud to understand the effect of incread coupling strength on the pha noi performance of a COA. Then, the improvement in the pha noi performance of strongly coupled arrays is explained by computing the quality factor of the coupling network for various coupling strengths. The measured array free-running and injection locked pha noi characteristics are prented.
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 4, APRIL 2010
arrays. The amplitude and pha dynamic equations can be derived from (1) and are given as
(5a) and
Fig. 1. An N-element coupled oscillator array.
斑斓什么意思(5b) II. AMPLITUDE AND PHASE DYNAMICS A. Array Dynamics Assuming Coupling Network is Broadband Previous analysis of coupled oscillator arrays like that in Fig. 1 have demonstrated that the amplitude and pha of the individual oscillators can be expresd as [2], [8]: In (2), (4) and (5) wherever sign is ud, the upper sign applies to ries resonant oscillators while the lower sign applies to parallel resonant oscillators. Equation (5) is valid provided the oscillator and the coupling network satisfy the condition [8] (6) Applying this to a typical bilateral, nearest neighbor cou, where pling network (e Fig. 1) and tting can be expresd as [8]
(1) where , and frequency of the are the amplitude, pha and free-running oscillator respectively, and
(2) and compri the elements of the vectors The variables [A] and . The parameters in (2) are , the amplitude satuoscillator; ration factor; , the uncoupled amplitude of the , oscillator quality factor; , the load conductance; is and oscillators and , the coupling pha between the the admittance of the coupling network from port to port . as the coupling coefficient, (2) can Defining be simplified when the frequency dependence of the coupling network is much weaker than that of the oscillator, or (3)
(7) is the magnitude of the coupling coefficient and is where termed the coupling strength. The coupling strength can be varied by changing the value of . The mutual injection locking range, which is defined as the maximum range by which the free-running frequencies of the oscillators can deviate collectively and still maintain a mutually locked state, is expresd as (8) results in a wider mutual injection locking From (8), a large caus an undesirable change in the range. However, a large array amplitudes [9] and prents a significant load to the oscilbetween the oscillator lator [10]. Strong coupling elements is desirable as long as other coupling features are not compromid. For in-pha synchronization, ries resonant oscillators re, where n is odd and parallel quire a coupling pha, , where n is even [11]. The resonant oscillators require amplitude and pha dynamic equatio
ns (5) would specialize to
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(4) Condition (3) is the broadband condition defined in [8]. The left member of (3) appears in the denominator in (2), and neglecting it greatly simplifies the analysis of coupled oscillator
(9a) and (9b)
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Equation (9) is traditionally ud to ascertain the amplitude and pha dynamics of COAs. The equations break down with large coupling factor, consistent with exceeding the limitations of the broadband condition. B. Array Dynamics Assuming Coupling Network is Not Broadband When the broadband condition (3) does not hold, a COA must be modeled with (2) including the resonant denominator term that the broadband condition obviates. This term bring the resonances of the coupling networks into play. Viz: assuming
VARIATION OF
Q
AND THE
TABLE I BROADBAND FACTOR WITH
otherwi (10) is the resonant frequency of the coupling network. where is a uful The quality factor of the coupling network parameter through which to view the networks’ influence. can be derived by applying the analogy of a resonant cavity [10]. Alternatively, can also be obtained from the denominator terms of (2) as
Fig. 2. Photograph of the five-element COA. The top ports are the outputs, and the lower ports are injection points.
(11) For in-pha synchronization of the oscillators, using (10), (11) simplifies to
The resonant nature of the coupling network complicates the interaction of strongly coupled oscillators. Therefore, (9) does not prent an accurate reprentation of the amplitude and pha of the oscillator elements when the broadband condition is invalid. An accurate estimate of the behavior of the oscillator array under strong coupling conditions can be obtained by deriving the amplitude and pha dynamics equations without using (3). We u (7) and (12) in (2) and note that for in-pha synchro, where is even for parallel resonant oscilnization lators and odd for ries resonant oscillators, the amplitude and pha dynamics equations result from (1) as in (13a) and (13b) shown on the following page. III. EXPERIMENTAL VERIFICATION The theoretical amplitude and pha distributions are verified on a five-element coupled oscillator array, which is built bad on the oscillator cell design prented in [12]. The oscillators are coupled with 50- transmission lines resistively loaded with chip resistors (e Fig. 1) and a coupling pha of . The array is fabricated on 0.635-mm thick Rogers TMM 10 board. A photograph of the array is shown in Fig. 2. Injection ports introduced at each element are utilized to measure the inter-element pha difference. Output signals are obtained by using a vector network analyzer configured for S21 measurement. The given injection port is taken as “port 1” in the measurement and supplied with the signal from the network analyzer. An oscillator output port is taken as “port 2.” All other output ports are terminated with matched loads. Each output is measured in
(12) Clearly, is directly proportional to the coupling strength. and the broadband factor Table I shows the variation of with for and . The is to introduce loss main purpo of the coupling resistor, . Decread increas the and and thereby reduce thereby the inequality (3). The coupling resistors also act as a mode killer and suppress the undesired modes that ari in a strongly coupled array [3]. From Table I one obrves that as the value of resistor is reduced to increa the coupling strength, also increas. The lowered loading caus the coupling network to become more highly resonant resulting in the breakdown of the broadband condition. When the broadband factor value is of order 1 and larger, (3) does not hold. From Table I, and is one obrves that (3) starts to break down around . clearly violated for
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 4, APRIL 2010
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succession by changing the network analyzer connection and the terminations. The amplitude distribution across the array is measured for 10 , 20 and 30 inter-element pha differences using a network analyzer. Fig. 3 shows the computed and for measured array amplitude variation for various values of 20 inter-element pha difference. The oscillator amplitudes are normalized with respect to
the measured amplifor each tude of the center element. Equation (13a) is ud to compute the array amplitude variation. The computed and measured amplitudes are within a dB of each other. No amplifiers or attenuators were connected to the oscillators for the measurements. The mutual injection locking range exhibited by two nearest neighboring elements for various coupling strengths is determined using the procedure detailed in [12]. Table II prents the
measured MILR exhibited by the oscillators for various coupling strengths. One obrves that the MILR increas proportionately with the coupling strength. For , the MILR exhibited by oscillator #5 is about 750 MHz or 10% of 3.75 GHz. To the authors’ knowledge, this is the largest MILR reported in literature to date. IV. RADIATION PATTERNS AND BEAM STEERING The amplitude variations exhibited in Fig. 3 lead to poor radiation patterns. Since the end oscillators have greater amplitudes than the interior oscillators, when such an array is employed in a phad array system, the side lobe level is undesirably large. A better radiation pattern can be obtained by introducing an amplitude taper wherein the end oscillator elements have the lowest
(13a)
(13b)
SEETHARAM AND PEARSON: ANALYSIS OF STRONG COUPLING IN COUPLED OSCILLATOR ARRAYShigh five
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TABLE II MUTUAL INJECTION LOCKING RANGE EXHIBITED BY THE OSCILLATORS FOR VARIOUS COUPLING STRENGTHS
祝贺 英文Fig. 3. Amplitude variation with (a) : , (b) , (c) : , , (e) : . The solid and dashed lines reprent measured and (d) computed data respectively.
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amplitude in the array. In our experimental work, this is accomplished by amplifying each VCO output with a Mini-Circuits ERA-1SM+ buffer amplifier and coaxial attenuators were ud to trim the output power. Pha relationships were maintained by using equal numbers of attenuators in all paths. In practice, one might employ amplifiers with trimmable gain so that efficiency is maintained. The trimmed power levels are delivered to
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a patch antenna, 17.8 mm wide and 12.5 mm long, which employed a quarter wave transformer matching ction to prent a 50- load to the oscillator. The antenna spacing is 29.5 mm at the oscillation frequency. The buffer which is about 0.4 amplifier and patch elements are realized on a parate Rogers TMM 10 board. Fig. 4 shows the photograph of the oscillator array connected to the patch antennas. Figs. 5–9 show the measured array broadside and steered rawith and without the amplitude diation patterns for various taper. The buffer amplifiers are driven by the oscillator outputs (e Fig. 4) and the radiative coupling from the antennas disturbs the symmetry that is prent in Fig. 3. Table III displays the radiated amplitudes with and without the taper. Table IV shows the maximum beam steering from broadside exhibited by the COA for different coupling strengths. The beam steering is limited by the mutual injection locking range. It is noteworthy that the MILR limit was not encountered. The tuning for
