Mode change analysis of a two element oscillator-type antenna array

Mode change analysis of a two element oscillator-type antenna array

D.E.J.Humphrey V.F. Fusco

Indexing terms: Antenna system, Array behaviour

Abstract: An array of two radiation coupled oscillator-type antennas is investigated to confirm the behaviour of the antenna system as a function of the strength of mutual coupling between elements. Array behaviour prediction requires correct array mode identification; thus, a simple method whereby the stability of these modes can be identified is presented. An analysis is performed which assumes radiation coupling between identical active antenna oscillators. Explicit equations are given for the locked frequency and individual amplitudes of the array elements for each mode of operation. Theoretical predictions are compared with experimental results to confirm amplitude, frequency and phase variation features in the vicinity of the element separation where mode change occurs. The results compare well with those observed in practice where a mode change is observed as the element separation is increased by half wavelength intervals. In addition, the available modes in a two element array, the in-phase and antiphase conditions, are examined at full-wavelength and half-wavelength separations. The in-phase mode is seen to be stable for full-wavelength separations while the antiphase mode is stable at half- wavelength separations.

1 Introduction

At present, a number of advanced communication products are being introduced into the commercial and consumer marketplace, including broadband wireless, automotive cruise control and small aperture terminal systems for direct satellite access [l]. These systems, operating across the microwave and millimetre frequency spectrum, have one common problem, namely, that they all require portable high DC to RF efficient power sources, having low mass, volume and cost and which are reliable and have high bandwidth capability PI. 0 IEE, 1997 IEE Proceedings online no. 19971340 Paper first received 20th August 1996 and in final revised form 29th April 1997 The authors are with the High Frequency Electronics Laboratory, Department of Electrical and Electronic Engineering, The Queen’s University of Belfast, Ashby Buildings, Stranmills Road, Belfast BT9 5AH, Northern Ireland, UK

To comply with rising demand for inexpensive high performance transmitter circuits, spatially distributed power-combining active antenna arrays potentially appear to offer a solution. Since this class of circuit can operate in a multimode fashion [3], these modes have to be predictable and ultimately controllable.

Modern methods for power-combining microwave devices often involve the use of quasi-optical tech- niques where the power combining takes place in free space in the near-field region of the array. With this approach devices can be integrated to form an active antenna array and thus losses normally associated with the array feed network are avoided. Such arrays can power combine by means of interinjection locking or mutual (radiation-type) coupling of the individual elements. It has been shown previously that when active elements are mutually coupled, they exhibit different operating modes depending on element separation [3]. Thus, it is necessary to know exactly how the operating mode of the active array depends on element separation. Without this knowledge the array may exhibit characteristics different from those required for its application.

In this paper equations are developed to enable modal prediction of an array of radiation coupled oscillator-type active antennas. The two element array considered is a building block for more complex studies and lends itself to detailed experimental investigation. Here there are two modes of operation that result in quite different far-field radiation patterns. The two modes are the ‘in-phase mode’ and the ‘antiphase mode’ [3--61. The behaviour of the coupled two element array with weak mutual coupling between elements is analysed as a function of element separation and the predictions made compared with experimental observations.

mutual (weak) coupling

4 12 12 exp(-,> t

1 I

d12

active patch 1 active patch 2

Fig. 1 tors

Equivalent circuit for the two-element array of’ radiating oscilla-

IEE ProcMicrow. Antennus Piopag., Vol. 144, No. 6, December 1997 483

2 Experimental observations

Fig. 1 shows an electrical equivalent circuit model for the two element active antenna array [6]. This model is based on van der Pol equivalent circuits for each oscillator [6, 71.

The plots in Figs. 2-5 showing phase, frequency and R F amplitude deviation, were obtained using a Hewlett Packard HP 70820A microwave transition analyser operated in vector voltmeter mode. Here the individual elements were embedded on an adjustable perspex fix- ture to provide the necessary collinear element separations. Measurements can only be made at separations of greater than about one wavelength. Frequency and output power variations were observed by measuring the RF leakage through the 'bias-tee' required for element biasing, and thus absolute measurements of amplitude are not presented. Fig. 4 presents relative amplitude deviations from the free running level. The features of Figs. 2-5 are in keeping with effects previously observed [8]; an offset phase shift of 64" is present due to reactive near field energy effects.

s 1 7 - C 0

0 >

.- 1

.-

2 0- W U Y - .- a

-1.7-

$ 150

2 100

e, 50

10 656.-

3 10.65L- (3

s c

U

L

U

10 6 5 2 -

L

2 10.650- 2 . _ a c W

10.6L8

x 3 A / 2 2x

element seperation

-

*

., I

5 W 2

Fig.2 Relative phase between antennas in a two-element active antenna

10 6L6- h 3A/2

I

I 5A/2

element separation Fig. 3 array

Measured frequency results for a two-element active antenna

The theoretical analysis presented hereafter confirms the following: (i) A phase or mode change occurs (indicated by the vertical solid lines in Fig. 2) when the element separation is increased by half wavelength amounts.

484

(ii) As a mode change occurs the synchronous frequency of the array tends to its highest value, Fig. 3. (dashed lines indicate loss of oscillator synchronism). (iii) At these points the output amplitude of the array tends towards its lowest value, Fig. 3. In addition, Fig. 4 shows that the ratio of oscillator power outputs is within 0.5 dB for most element separations (1.2dB worst case) with a larger jump being observed during the mode change condition.

We shall now consider the theoretical aspects involved at critical points where mode change is observed, in other words, at intervals of A/2 in element separation.

3 L r

-3.L L A

*

I 3A/2

+

i. - A

2A

element separation Fig. 4 array

Measured peak amplitude results for a two-element active antenna

Fi an

3

-8 A

p.sa

I

I 2A

A

I 5A/2

element separation 5 Measured relative oscillator power in a two-element active nu array

Theoretical consideration

Previously interinjection locked active antenna type oscillators in a two element mutually coupled array were analysed using a simple van der Pol model [7, 91, Fig. 1. The oscillator equations are

fil = E1 (1 - PI (U1 + P12v2)2)fi1w1

+ Uf(.l + P l 2 V 2 ) = 0 i;2 = € 2 ( 1 - P2(v2 + P21v1)2)fi2w2

+ W22(V2 + P21V1) = 0

(14

( I b ) where pi (= 4/Ajree) is the factor controlling the free running amplitude of the oscillator. Afree is the free running peak output R F amplitude of the ith oscillator.

IEE Proc -Microw. Antennas Propag., Vol. 144, No. 6, December 1997

The oscillator dissipation factor (required to define oscillation limiting) is ii and ai is the free-running radian frequency of the oscillator. For identical oscillators, p, = p2 = p, = E and w1 = % = w. Mutual radiative coupling, as a function of element separation dI2, between the two oscillators [lo] is represented by

where qb12 = 27cdI2l;1 is the assumed phase retardation due to separation between elements, (Note:- p12 = p21)

To investigate the stability of a two element array (i.e. can an entrained frequency solution be found), we consider the stability of the limit cycle [l 11 by using the perturbation technique in [12]. The analysis assumes a van der Pol model as in eqn. 1, but now the assumed solution for vl and v2 must allow for instantaneous changes in amplitude and phase. The assumed solution becomes

=

Pl2 = P12 exp(-j412) (2)

VI = A1 cos(wLt + a i )

V2 = A2 COS(WLt + a2)

(34 (3b)

Assuming slow variation means that we can ignore second order derivative terms in the amplitudes and phases A , , A2, a, and a, so that the second order derivative terms in v become

i;1 = - A l ( w ~ + &1)2 cos(wLt + a i )

- 2A1(wL + til) sin(wLt + ~ ( 1 )

- 2 A 2 ( w L + c i 2 ) sin(wLt + a2)

(4a)

(4b) Since phase variation is allowed in both oscillators the coupling terms become

(5a)

( 5 b )

U2 = - A ~ ( w L + cos(wLt + ~ ( 2 )

,~12va = pA2 cos (w~t + az + 4) ~ 2 1 ~ 1 = PAI c o s ( w ~ t + ai + 4)

Eqns. 3, 4 and 5 are substituted into eqn. 1, and higher frequency order terms are ignored since sinusoidal operation is assumed due to antenna filtering. After equating sine and cosine terms, linearising by adding the small perturbations, AA,, AA2, Aa,, and Aa2 to the amplitudes and phases, and after noting small angles and slow variation approximations, eqn. 6 results

AA1 - W: + w l + ~ E ~ w ~ w L P ~ A ~ A ~ sin4

3 1 + - E ~ w ~ P ~ w L ~ ~ A ; sin24 - E ~ w ~ D + -E~LJ~PIA:D

4 4 1 3

+-p2~1P1~1A;D + ~ P I w ' P A I A ~ cos 4 D 2

[ It is noted that DAl can be ignored if the variation of A , is slow enough.

An equivalent set of equations for the second element also exists. The stability of the system at certain points is examined to establish mode information at separations where a mode change is observed to have occurred.

4 Mode change conditions

Consideration of eqn. 6 when the separation between the two identical active antennae is a full wavelength enables further simplification in that qb = 2nn. With this condition enforced, the array equations reduce to

w; = w2(1+ p ) 4

A2 = p(1+ PI2

It is noted that since p > 1, for this scenario > w and A < Afyee, where Afre, is the free running peak amplitude of both oscillators (since AI = A2 = A) . Figs. 2 and 3 show that this result is true once oscillator syn- chronisation has been restored. This represents the minimum energy condition for the array [14]. By inserting eqn. 8 into eqn. 6, a further simplification is

485 IEE Proc.-Microw. Antennas Propag., Vol. 144, No. 6, December 1997

( ~ 5 + ~ 6 )

For a stable limit cycle the characteristic equation described in eqn. 11 must be stable since it is a repre- sentation of the responses to a small perturbation from the stable limit cycle. Since terms in D3 and D2 are always positive, if the term in D is positive, eqn. 11 will represent a stable system. The common factor (a5 + a6) will always have a negative real root and will not effect the system. Hence, consideration of the term in D shows

(a1 - a 2 ) a3 a4 (a5 -a6 ) a7 -a7 (10) ( a 2 - a l ) a4 a3

which can be seen to have a minimum value for p of

(13) -(1 - 2 2 ) + J(l - 2 ~ ~ ) ~ - 8~~

2 Pmzn =

which gives a negative value for pmln for all values of E. Since p is always positive by definition this ensures that the term in D is positive. Thus, for antenna separations of an integral number of wavelengths, the in-phase mode will be stable since mutual reinforcement is occurring.

When the separation distance between two identical in-phase active antennas is exactly an odd number of half wavelengths, @ = (272 - 1)n, eqn. 6 reduces to

W ; = w2(1 - p )

Here, it is seen that o, < w and A > Afrec. Figs. 2 and 3 disagree with this result showing rather a lower peak amplitude and higher frequency after loss of synchroni-

486

sation has been accounted for. With 4 = (2n - l)n, after manipulating the charactenstic equation the system is found to be stable between the limits:

However, it is possible that these limits may be complex over a certain range of E and hence forming an unstable system. Therefore, for stability

( 2 ~ ~ - 1)2 > 8~~ 4 E 4 - 4e2 + 1 > 8~~ 4E4 - 12&2 + 1 > 0 (16)

Since E cannot be negative it is seen that E > 1.7071 or E 0.2929. However, if E > 1.7071 the solutions of eqn. 15 give a negative value for pmln which is not possible and an unstable condition results. But if 0 < E < 0.2929, both solutions of eqn. 15 are positive and yield a stable limit cycle provided the value for p falls within the given range.

Cunningham [15] related E to the ‘quality factor’ of the van der Pol circuit showing that E is equal to the reciprocal of the quality factor. Thus, the quality factor of the active antennas must be greater than 110.2929 or the array will be unstable at all odd number of half wavelength separations. From eqn. 15, as E decreases the range of values where p will produce a stable limit cycle increases, since a higher Q-factor results in a more stable oscillator which is less inclined to mode change.

In this work [16] E was approximated to be about 0.148, setting pllmlts as 0.096 < p < 1.0 since the condition set by D3 shows that p < 1.0. Hence a very large value of p is needed to produce stability. Our experi- ments and those of others, (e.g. York [lo]) have shown that a typically large value for p is 0.0021. Hence, this range will not usually be achieved in a weak coupling situation and so an unstable solution is predicted. Thus the in-phase mode will be unstable for an odd number of half wavelength separations.

At full wavelength separations (@ = 2nn), the antiphase mode condition is identical to that described by the in-phase mode at a half wavelength separation. Therefore, it can be concluded that the antiphase mode is unstable at full wavelength antenna separations.

For a separation of one half wavelength and with the antiphase mode, an equivalent result as for eqns. 8 and 11 are returned which provide for a stable limit cycle. Figs. 3 and 4 illustrate this. Thus, the antiphase mode is shown to be siable at odd numbers of half wavelength separations. These results are in agreement with York’s predictions [ 171.

array frequency

Fig. 6 - in-phase mode - preferred frequency of active array ~ _ _ ~ antiphase mode

Modal stability in a two-element array

IEE Proc.-Microw. Antennas Propag , Vol. 144, No. 6, December 1997

The stable mode is shown as a function of mode, frequency and separation in Fig. 6 which shows that from (n - 1)d to [(2n - 1)/2]A a mode change occurs from the in-phase mode to the antiphase mode. In the range [(2n - 1)/2]d to nil the antiphase mode changes to the in- phase condition once near field effects have been taken into account. This is consistent with the results shown in Figs. 2-5 and with the experimental results previously reported [8].

In summary, it can be seen from Figs. 2-5 that as the mode change approaches, for each stable case, the frequency of the array changes from a low value to a high value and the peak amplitude fluctuation for the array tends to the mode with the lowest peak amplitude condition in accordance with eqn. 8.

5 Conclusions

Analysis has shown that for a two element active antenna array used for quasi-optical power-combining, the array mode changes from the in-phase mode to the antiphase mode between separations of (n - l ) A and [(2n - l)A]/2, (the antiphase mode swaps between separations of [(2n - l)A]/2 and nA). It has also been shown that at the separations where mode change occurs, twice per wavelength separation distance, the mode running with the highest frequency at this separation is the stable mode. At these points the output amplitude of each active antenna oscillator tends to a minimum value. Simple closed formulae have been presented to allow these predictions to be made. The experimental evidence presented in this paper confirms all of the features of the theoretical predictions.

The procedure has demonstrated that the higher the Q-factor of the elements, the more stable the running mode. It also shows that if injection-locking by mutual coupling can occur, then modal stability may be further enhanced by introducing stronger coupling mecha- nisms.

6 Acknowledgments

The authors would like to thank the Department of Education for Northern Ireland, and the Engineering

and Physical Science Research Council for funding this work.

7

1

2

3

4

5

6

7

8

9

References

GEPPERT, L.: ‘Industrial R & D: the new priorities’, ZEEE Spectrum, Sept. 1994, 31, (9), pp. 30-36 MACDONALD, P.A., and MATLOUBIAN, M.: ‘Millimeter wave technology for commercial applications’. Microw. RF, Lon- don, Oct. 1996, pp. 114-117 CHEW, S.T., and ITOH, T.: ‘A 2 x 2 beam-switching active antenna array’. IEEE MTT-S Digest, Orlando, June 1995, pp. 925-928 YORK, R.A., LIAO, P., and LYNCH, J.J.: ‘Oscillator array dynamics with broadband N-port coupling networks’, ZEEE Trans. Microw. Theory Tech., Nov. 1994, 42, (Il), pp. 2040-2045 MINEGISHI, M., LIN, J., ITOH, T., and KAWASAKI, S.: ‘Control of mode-switching in an active antenna using MESFET’, ZEEE Trans. Microw. Theory Tech., Aug. 1995, 43, (8), pp. 1869- 1874 HUMPHREY, D.E.J., FUSCO, V.F., and DREW, S.: ‘Active antenna array behaviour’, IEEE Trans. Microw. Theory Techn.,

VAN DER POL, B.: ‘The nonlinear theory of electrical oscillators’, Proc. IRE, Sept. 1934, 22, (9), pp. 1051-1086 STEPHAN, K.D., and YOUNG, S.L.: ‘Mode stability of radiation-coupled interinjection-locked oscillators for integrated phased arrays’, IEEE Trans. Microw. Theory Techn., May 1988, 36, (5), pp. 921-924 STEPHAN, K.D., and MORGAN, W.A.: ‘Analysis of interinjection-locked oscillators for integrated phased arrays’. IEEE

Aug. 1995, 43, (8), pp. 1819-1825

. . T;ans. Antennas Propug., July 1987,35, pp.-771-781

10 YORK, R.A., and COMPTON, R.C.: ‘Measurement and model- ling of radiative coupling in oscillator arrays’, IEEE Trans. Microw. Theory Tech., March 1993, 41, (3), pp. 438-444

11 JORDAN, D.W., and SMITH, P.: ‘Nonlinear ordinary differen- tial equations’ (Clarendon Press, Oxford, 1990)

12 LINKENS, D.A.: ‘Stability of entrainment conditions for a par- ticular form of mutually coupled van der Pol oscillators’, ZEEE Trans. Circuits Syst., Feb. 1976, 23, (2), pp. 113-121

13 HUMPHREY, D.E.J., and FUSCO, V.F.: ‘Nonlinear two element active antenna array stability analysis’, Electron. Lett., Apr. 1996, 32, (9), pp. 788-789

14 KURAMITSU, M., and TAKASE, F.: ‘Analytical method for multimode oscillators using the averaged potential’, Electron. Commun. Jpn., 1983, 6 4 (4), pp. 10-19

15 CUNNINGHAM, W.J.: ‘Introduction to nonlinear analysis’ (McGraw-Hill, New York,1958)

16 HUMPHREY, D.E.J., FUSCO, V.F., and DREW, S.A.: ‘Active antenna frequency and voltage characteristics’. Proc. Microwaves 94 Conference, London, Oct. 1994, pp. 212-215

17 YORK, R.A.: ‘Nonlinear analvsis of rJhase relationships in auasi- optical oscillator arrays’, ZEEE Trans.‘ Microw. Theorf Tech.,‘ Oct. 1993, 41, (lo), pp. 1799-1809

IEE Proc.-Microw. Antennas Propug., Vol. 144, No. 6 , December 1997 487

Documents

Mode change analysis of a two element oscillator-type antenna array