pacitively coupled oscillator array ehaviour

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

Abstract: A simple lumped element coupled occillator theory which permits in-phase distributed power combining to occur has been developed for planar distributed oscillator combinations. Individual oscillators are mutually locked together through a coupling capacitance network. The amplitude dnd frequency characterisiics of ihe system are found to be relatively insensitive to capacitance variations. Expcrimcntal arid time domain simulation results are presented for two and three coupled oscillator caseo, these results show excellent correlation with the analytical theory presented.

1 introduction

For phased array radar applicaiions it would be useful to have reduced complexity in the networks used io fccd thc antcnna ciements. One way of achieving this is to distribute ihe power generating devices to positions in the array which are local to groups of radiating ele- ments. Current practice is to provide a single power source and complex feed network. In configuring this new arrangement a. multiple coupled oscillator problem ariscs. Coupling between individual elements can be of a distributed [ I , 21 or a lumped nature [3]. The work by Endo and Mori [3] indicated that oscillators could be stably coupled using lumped elements.

In this paper a simple lumped element coupled oscil- lator theory which pcrmits frequency and amplitude bchaviour io be established as the coupling element value is varied is presented. The resulting network of oscillators can be considered a distributed multiple out- put in-phase power source. Full time domain simula- tions based on coupled Van der Pol [4] equivalent circuits are augmented with experimcntal results for a microwave frequency oscillator operating at 1 GHz in order to dcmonstrate the validity of the analytical the- ory presented

2 Lumped capacitive coupling

For each of ihe Van der Pol [4] oscillator elements in the oscillator array given in Fig. I , it is possible to for-

0 IEE. 1996 IEE Procc.cx/ing,r online no. I0960268 Paper rcccivcd 7th August 1995 The authors arc with thc High Frequency Electronics Laboratory, Department of Elcclricai and Electronic Enginccring, The Queen's University of Belfast, Ashby Buildings, Stranmills Road. Belfast BT9 SAH. UK

mulate a circuit equation which shows the performance of the oystem I n Fig 1 the 1,ubscript k refers to any element in the chain array Uvng Kirchhoffs law<, the conditions for Z,< and Zk

d d t

are

(1)

( 2 )

I , = C,-(V, ~ ' J k + l )

where CO IS the coupling capdcitnnce

Fig. 1 Coupled oscillritoi cii cult mmem lutlri e

Setting the current through the lzth nonlinear device i,< as

i.e. the simplest form necessary to form a stable limit cycle, we obtain

2k = -< lvk + 1!77/: ( 3 )

By substituting eqns. lL3 into eqn. 4 and by differenti- ating we find an equation of the form

( ( k + 2 ) & -t&(l-Bkz:)n'kwA +d&Jk -?kp1 -?k+l = 0 (5)

m

(6)

where: n 1 3b & = -

c CLl/k U

W k = -- F k = ~

Using the assumption that tlhe solution is sinusoidal and of the form

Ck = ci,

xk, = Ai, cos(wl t + Nk) and that there are no 3rd-harmonic terms, it is possible to formulate two equations for each individual oscilla- tor in the array:

( ~ A k d ? COS ( t k - 2Aku~; ( : O S i l k + C ~ E ~ W ~ A ~ L ~ J L sill (kk

(8)

I67

The solution of these equations for the array yields val- ues for the iiidividual peak dinplitudes and phabes. C I A (relativc to the first oscillator) of the elements and the overall entrained frequency

- /

777

Fig. 2 Distributed feed array

16 8mm

-~~ __ 17 pF

3 b

5 OV RF out to load Fig.3 Orcrllutor ebnient

3 Oscillator characterisation

A 1 GHz oscillator was designed using coniniercial MMIC amplifiers [Note 11 with added distributed 5OQ transmission line parallel feedback constructed on a PTFE softboard substrate (Figs. 2 aiid 3). By measure- ment of the injection lockiiig characteristics of each oscillator and by applying Adler’s equation [j]; the oscillator quality factors were found, from which the Van der Pol damping factor E~ = l / Q was determined. Measurement of the free-running peak amplitude pro- vides an estimate for the coefficient P k since from eqns. 7 and 8, when there is only one oscillator present cou- pled via Go12 to earth,

4 oi, = - A? (9)

where A/ , is the peak amplitude of the oscillator Hence p,, can be estimated

To find the Van der Pol capacitance C. the individual peak aniplitudes and the entrained frequency of an array of wveral elements are recorded Eqns 7 and 8 are then solved for foi each element Knowing the coupling Capacitance CO, the Van der Pol capacitance C can be calculated From the free-running frequencq the inductance requiied to complete the Van der Pol model can be calculated

Table 1

Oscillator 1 Oscillator 2 Free-running frequency Free-running frequency = 1.0094GHz = 1.0051 GHz

a 0.58666 0.69354

0 37.24830 26.42156

L, pH 38.1299 36.9821

C, pF 652.010 678.003

Notc 1 MINI-CIRCUITS, P O Box 350166 Brooklyn, New Yolk, 11235-0003, U S A

4 Experimental verification

4.7 Two-oscillator case Using the technique described above the two oscillators to be coupled were characterised with experimentally derived coefficients of the Van der Pol type in Fig. 1 (Table 1).

Eqns. 7 and 8 were solved using a quasi-Newton least-squares numerical optimising package [6].

0 2 5 -

< 0 15- aJ a

0 10-

, # I # , # / / , , , / , , , , , ,

12 40 13.12 13.84 14.56 15.28 16.00 coupling capacitance, pF

Fig. 4 * measured: ~

T1i.o coupled oscillators, amplitude tuning behaviour analytical theory: A time domain simulation

1000 ,

992-

9 9 0 ’ , 1 I , I I , -3 I , I , , , , I I

12-40 1312 13.84 14.56 15.28 16.00 coupling capacitance, pF

Fig. 5 - measured: ~ ~ ~ analytical theory; A time domain simulation Tiro coupled oscillators, fieyuency tuning behaviour

For different values of coupling capacitance the results obtained are compared to the measured values. The Van der Pol model was also implemented using a time domain transient circuit analysis program [Note 21 and peak amplitude and frequency sensitivity to the tuning capacitance calculated. In Figs. 4 and 5 two coupled oscillators are examined. As can be seen, both simulations approximate the frequency and the peak amplitude as the coupling capacitance is varied. The results of the analytical technique presented in this work agree with the full time domain transient simula- tion, the experimental results in each case being com- mensurate with the theory. Near identical results for all sets of results occur for a coupling capacitance of 16pF which was the value of coupling capacitance that was

Note 2: Hewlett-Packard, ‘Microwave and RF design system’. Hewlett- Packard Company, Santa Rosa Systems Division, 1400 Fountaingrove Parkway, Santa Rosa, ‘2.495403, USA

I bX 1EE Pvoc.-Circuits Devices Syst., Vol. 143. No. 3, June 1996

used to calibrate the oscillators. It can be seen from Fig. 4 that the variation in amplitude o f each coupled oscillator tracks in-phase in a relatively insensitive way with tuning capacitance. Fig. 5 indicates that the fre- quency shift of the coupled oscillator pair due to cou- pling capacitance variation is also relatively small. Due to the coupling network a small overall circuit detiuning of about 12MHz from the nominal oscillator free-run- ning frequency is observed.

Using a method similar to Mortazawi [7] and imple- menting Kurokawa's multiple device theory for esti- mating the eigenvectors [SI for the passive coupling circuit connection in Fig. 1, confirms that both oscilla- tors running in-phase is an acceptable mode of opera- tion for the coupling configuration used in this work.

In the Appendix a linearisation-based approach to achieve an approximate stability analysis for the two identical oscillator systems is employed. It can be seen that the equation required for stability gives a value for cmin = 1.0. It is also noted here that the stability analy- sis presented in the Appendix is appropriate provided a stable limit cycle can be initially formed and is then only slightly perturbed. From Table 1 the coupled oscillators have a mean Van der Pol equivalent circuit capacitance of 665 pF. Hence the maximum predicted coupling capacitance should be of the order of 665pF. In practice the values used in this work are typically two orders of magnitude less than this.

4.2 Three-oscillator case Next, three oscillators were experimentally character- ised using the methodology described above to Van der Pol types with the coefficients given in Table 2.

Table 2

Oscillator 1 Oscillator 2" Oscillator 3 (1.002841 GI+) (1.004041 GHz) (1.002941 GHz)

a 0.53624 0.35890 0.61220

b 3.27785 1.83720 3.06178

L, pH 41.9879 62.6527 36.7655

C, pF 599.8628 40 1.0494 684.9343

* It is noted that oscillator 2 was selected f rom a different batch of MMlC ampli l iers

Again eqns. 7 and 8 and the full transient analysis indicate good correlation with the experimental per- formance of the system (Figs. 6 and 7). Again it is found that over the tuning range each of the coupled oscillators operates in-phase with a different free- running amplitude which varies in a relatively insensi- tive way to tuning capacitance.

5 Conclusions

It has been shown that a simple lumped element cou- pled oscillator theory, when compared to experimental and full time domain simulation, can predict the ampli- tude and frequency behaviour of a network of multiple coupled oscillators. The calibration and accuracy of a simple Van der Pol model for the system has been dem- onstrated and validated for two and three elements. The results show that it is possible to entrain in-phase by lumped capacitive coupling several oscillators run- ning at similar frequencies to run at an identical fre- quency and to thereby produce a power source whose output is distributed over several oscillators.

IEE Pr~c.-C,~irczrifs Devices Sy~sf., Vol. 143, No. 3, June 1996

0.60 -

0.401,?-~,-,~,..-,-~.~-~-~~.-~~ v j - 7.-

35-48 36.20 36.92 37.6.4 coupiing capacitance& pF

Fig. 6 Three coupled oscillators, umplitude tuning behaviour + measurrd; - analytical theory; 6 Lime domain simulation

L ' c

977 1

975 L-T.-7---- ,.--,.-. T ~ - - - - . . ~ . - ~ . . ~ 7--

35.48 36.20 36-92 37.6h coupling capacitance, pF

Fig. 7 Three coupled oscillutors, jreqwwcy tuning behaviour -t measured: - - - analytical theory: A time domain siiaulation

6

1

2

3

4

5

6

n

8

9

7

References

NOGI, S., LIN, J.; and JTOH, T.: 'Model analysis and stabilisa- tion of a spatial power combining array with strongly coupled oscillators', IEEE Trans. Microw. Theory Tech., 1993, 41, (IO), pp. 1827T1831 LIN; J., and ITOH, T.: 'TWO - dimensional quasi - optical power _. combining array using strongly coupled oscillators', IEEE Trans. Microw. Theory Tech., 1994, 42, (4), pp. 734-741 ENDO, T., and MORI, S . : 'Mode analysis of a multimode ladder oscillator', ZEEE Traris. circuit.^ !$st.. 1976. CAS--23, pp 100.- 113 VAN DER POL, B,: 'The nonlinear theory of electrical oscilla- tors', Proc. IRE, 1934, 22, ( O ) , pp. 1051L1086 ADI,ER, R.: 'A study of locking phenomena in oscillators', P m c . IRE, 1946, 34, pp. 351--357 MCKEOWN; J.J.: 'An interactive approach to solving non-linear least squares problems', IMACS 91, Proceedings of the 13th IMACS World Congress on Con?yulntion and applied mathiwut- ics, Trinity College, Dublin, July 199 I ~ pp. 1 4 4 146 MORTAZAWI, A., FOLTZ, H.D.. and ITOH, T.: 'A periodic second harmonic spatial power combining oscillator', IEEE Truns. Microw. Theory Tech., 40, ( 5 ) , PI?. 851 -856 KUROKAWA, K.: 'Analysis of Rucker's multidevice symmetri- cal oscillator', IEEE Truus. Microw. T/iecwy Tech., 1970: MTT- 18, pp. 961 -969 LINKENS, D.A.: 'Stability of entrainment conditions for a par- ticular form of mutually coupled Van der Pol oscillators', JEEE Trans. Circuit.r Syst., 1976, CAS-23, pp. 113- 121

Appendix

The stability study introduced here is limited to two icou- pled oscillators capacitively coiinected to ground at the end o f the linear chain (Fig. 1). To investigate the stabil-

169

ity of the limit cycles analytically it is necessary to con- sider the response of the network to small perturbations from an already established limit cycle; thus the analysis assumes that a stable mode of operation is already formed. Hence the assumed solutions for both .Y] and .I-? must allow for instantaneous changes in amplitude and phase. Differentiating eqn. 6 for both the first and sec- ond oscillators with respect to time and with the assumption of slow variations, meaning we can ignore 2nd derivative tenns in A i , A2, al and a?, cf. [9], yields: kl = - A ~ ( L ~ ) L +- r ~ 1 ) sin(iC)Lt + 01) + =i1 c o s ( ~ : L t + 01)

2, = -.41 ( W L + &l)2 COS(WLt + 011) - 2 A I ( u L + til) s i r i (WLt + 01)

x 2 = -Aa(dJ, + & 2 ) 2 cos(wi,t + a%)

(10)

i2 -A2(uI, + iY2) sin(dLt + 01%) + A? cos(d.12 + a?)

- 2 A 2 ( u L + sin(dLt + a2) (11) By substituting eqns. 10 and 11 into eqn. 7. equating the coefficients in coswLf and sino,t and allowing small perturbations in the amplitudes and phases. AAl . AA2. A a l , Aa, of the resultant equations we can linearise the system. Hence by making the substitutions A , = A2 +

such that, for any angle p,.sin (@ + AB) 1 AD, cos (p + AD) = 1, and specifically A, = A2 = Ccl = ci2 = 0. eqns. 12 and 13 result:

AAi, A, = A2 + AA,, a, = a1 + Act,, and ~ ( 2 = + ACX,

AA,[-<LJ: - 2 4 , + J?C - t C ~ v 1 D + $ F <

+ AA2 [U?,]

+ A 0 1 l J F Q J 1 A l u L - ~ ~ < ~ L L ) ~ W L A : - ~ < - o ' L A ~ D - 4d,5AL D ]

+ A o ~ [ ~ A ~ ~ L D ] = 0 (12)

AA1[-4d[,D + E < ~ J ~ W L - f E < , ? L u ' l d ~ ~ 4 f - 2<;.'~ Dl + AA2 [2!JJT,D]

+ A c x ~ [ < A ~ I J ~ + 2 A l 4 - "?<A1 + E < ~ L A ~ D - ~FCIIJLJ~AYD]

+ A012[-~iAa] = 0 (13)

where D is the differential operator. From eqns. 7 and 8 for the case of identical oscillators expressions for both the entrained frequency and peak amplitude when a, = CL? = 0 are

and

These are the nominal unperturbed values of amplitude and frequency and can replace A , , A2 and oL in eqns. 12 and 13.

In addition, symmetry means that similar equations will result for the second oscillator. Therefore a matrix of four equations with perturbed amplitudes and phases can be formed. The resulting matrix determi- nant leads to a characteristic equation in the differen- tial operator D , which can be factorised to leave

This characteristic equation gives a representation of the response to small perturbations from the normal stable limit cycle. For stability, each coefficient should have the same sign, which will always be positive since the constant term is always positive. Hence, by ensur- ing that all the terms in eqn. 16 are all positive, stabil- ity is ensured. We see that the term in D3 will always be positive provided [ > 1.0. This constraint also ena- bles us to say that the terms in D2 and D will always be positive. Hence this minimum value for stability to occur is that (,,,,,, = 1.0. Thus the maximum value of coupling capacitance will be equal to the Van der Pol oscillator capacitance.

I70 IEE Proc -Czriuit.s Devices Syst , Vol 143, No. 3, June 1996