Experimental characterization of mutually synchronized ... - J-Stage

LETTER

IEICE Electronics Express, Vol.14, No.6, 1–8

Experimental characterization of mutually synchronized voltage edges in point-coupled tunnel diode transmission lines Koichi Naraharaa) Department of Electrical and Electronic Engineering, Kanagawa Institute of Technology, 1030 Shimoogino, Atsugi, Kanagawa 243–0292, Japan a) [email protected]

Abstract: We investigate experimentally the mutual synchronization of oscillating pulse edges developed in dissipatively point-coupled transmission lines periodically loaded with tunnel diodes (TDs). Based on the phasereduction scheme, the mutually synchronized edges are shown to pass simultaneously the connected points. It then becomes possible to design the phase difference between synchronized edges by the arrangement of connected points. Using test TD lines fabricated on breadboards, the fundamental properties of mutually synchronized edges are successfully validated. Keywords: mutual synchronization, multiphase oscillation, phase-reduction method, tunnel diodes Classification: Electron devices, circuits and modules References

© IEICE 2017 DOI: 10.1587/elex.14.20170054 Received January 18, 2017 Accepted February 27, 2017 Publicized March 13, 2017 Copyedited March 25, 2017

[1] K. Narahara: “Mutual synchronization of oscillating pulse edges in pointcoupled transmission lines with regularly spaced tunnel diodes,” Commun. Nonlinear Sci. Numer. Simulation 42 (2017) 236 (DOI: 10.1016/j.cnsns.2016. 04.029). [2] W. C. Black and D. A. Hodges: “Time interleaved converter arrays,” IEEE J. Solid-State Circuits 15 (1980) 1022 (DOI: 10.1109/JSSC.1980.1051512). [3] P. J. A. Harpe, et al.: “Analog calibration of channel mismatches in timeinterleaved ADCs,” Int. J. Circuit Theory Appl. 37 (2009) 301 (DOI: 10.1002/ cta.545). [4] G. Souliotis and C. Psychalinos: “Electronically controlled multiphase sinusoidal oscillators using current amplifiers,” Int. J. Circuit Theory Appl. 37 (2009) 43 (DOI: 10.1002/cta.486). [5] A. Mazzanti, et al.: “Analysis and design of injection-locked LC dividers for quadrature generation,” IEEE J. Solid-State Circuits 39 (2004) 1425 (DOI: 10. 1109/JSSC.2004.831596). [6] K. Narahara: “Multiphase oscillator using dissipatively coupled transmission lines with regularly spaced tunnel diodes,” Int. J. Circ. Theor. App. (2016) (DOI: 10.1002/cta.2284). [7] P. Maffezzoni, et al.: “Analysis and design of weakly coupled LC oscillator array based on phase-domain macromodel,” IEEE Trans. Comput.-Aided Des.

1

IEICE Electronics Express, Vol.14, No.6, 1–8

Integr. Circuits Syst. 34 (2015) 77 (DOI: 10.1109/TCAD.2014.2365360). [8] Y. Kuramoto: Chemical Oscillations, Waves, and Turbulence (Springer-Verlag, New York, 1984). [9] E. M. Izhikevich: “Phase equations for relaxation oscillators,” SIAM J. Appl. Math. 60 (2000) 1789 (DOI: 10.1137/S0036139999351001). [10] Ö. Suvak and A. Demir: “Quadratic approximations for the isochrons of oscillators: a general theory, advanced numerical methods, and accurate phase computations,” IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 29 (2010) 1215 (DOI: 10.1109/TCAD.2010.2049056).

1

Introduction

It has been observed that a voltage edge repeatedly turns around halfway on a tunnel diode (TD) line with an appropriate boundary condition. Here, the edge oscillation is shown to be a limit cycle that exhibits several synchronization phenomena. Recently, we investigated the dynamics of the oscillating edges on point-coupled TD lines based on the phase-reduction scheme [1] and found that the mutual synchronization is established as the leading (trailing) edge simultaneously passes the edge on the other line at the connected cell. It results in the synchronization properties such that the phase difference and frequency shifts depend on the cell at which the TD lines couple. Using this property, we proposed a scheme to develop multiphase oscillators [2, 3, 4, 5], which contribute to several cost-effective electronic systems including parallelism and effective encoding, using several coupled TD lines. [6] As mentioned above these previous studies have been carried out with the aid of numerical methods. In the case of time-domain calculations it requires more than several hundred cycles for the calculated system to reach the synchronized state, so that we cannot discard the possibility that the accumulated numerical errors contribute to the development of mutual synchronization constructively. It then becomes very important to examine how much the results obtained by numerical methods can be applied to practical devices. In this study, we fabricate breadboarded circuit using TDs and measure it in both the frequency and time domains. Fundamental properties of mutually synchronized edges, including the simultaneous passage of connected points, phase multistability, and possible design of phase difference between edges, are examined. In particular, it is established that the phase difference between two edges can increase by the sequential assignment of connected cells. This fact is essential to guarantee the operation principle of multiphase oscillators. In this paper, we describe experimentally obtained results for the mutually synchronized edges after briefly reviewing their fundamental properties. 2

© IEICE 2017 DOI: 10.1587/elex.14.20170054 Received January 18, 2017 Accepted February 27, 2017 Publicized March 13, 2017 Copyedited March 25, 2017

Fundamental properties of point-coupled TD lines

Fig. 1(a) shows the point-coupled TD lines that we investigated. For brevity, the upper and lower TD lines are called lines 1 and 2, respectively. Given this, the M1 th cell of line 1 is coupled to the M2 th cell of line 2 through Rc . The total cell number for lines 1 and 2 is commonly set to N. Initially, all line voltages are set to zero.

2

IEICE Electronics Express, Vol.14, No.6, 1–8

Then, the input end is connected to a voltage source with negligible internal resistance that outputs the DC voltage of Ai , with the other end short-circuited for line i (i ¼ 1; 2). The nth voltage and current on lines i (i ¼ 1; 2) are denoted as Ii;n and Vi;n , respectively. Moreover, we define 2N þ 1 variables Xi;n (n ¼ 1; . . . ; 2N þ 1) as Xi;n ¼ Ii;n for n ¼ 1; . . . ; N þ 1 and Xi;Nþ1þn ¼ Vi;n for n ¼ 1; . . . ; N. Additionally, Li;n (i ¼ 1; 2; n ¼ 1 . . . N þ 1) and Ci;n (i ¼ 1; 2; n ¼ 1 . . . N ) represent the line inductance and capacitance, respectively, at the nth cell of line i to incorporate the variations of reactance values. Then, the transmission equations of the point-coupled TD lines are given by dXi;1 Ai  Xi;Nþ2  RXi;1 ; ¼ dt Li;1 dXi;n Xi;Nþn  Xi;Nþ1þn  RXi;n ; ¼ dt Li;n

ð1Þ ðn ¼ 2; . . . N Þ;

dXi;Nþ1 Xi;2Nþ1  RXi;Nþ1 ; ¼ dt Li;Nþ1 dX1;Nþ1þn Xi;n  Xi;nþ1  ID ðXi;Nþ1þn Þ ¼ Ci;n dt X1;Nþ1þn  X2;Nþ1þM2  n;M1 ; C1;n Rc dX2;Nþ1þn X2;n  X2;nþ1  ID ðX2;Nþ1þn Þ ¼ C2;n dt X2;Nþ1þn  X1;Nþ1þM1  n;M2 ; C2;n Rc

ð2Þ ð3Þ

ðn ¼ 1; . . . N Þ;

ð4Þ

ðn ¼ 1; . . . N Þ;

ð5Þ

for i ¼ 1; 2. Here, k;l is the Kronecker’s delta. In addition, R represents series resistance and a TD is modeled by its current ID . By the application of a DC voltage at the first cells, the voltage edge develops and moves to the far end. Because of losses and leakage, the edge is gradually attenuated; it nearly disappears. At this stage, a stable traveling front develops and starts propagating back to the near end. Once the voltage edge returns to the input end, it is reflected again to propagate from the input end. The voltage edge repeats this process, thus oscillating on the line. To discriminate the position-dependent portion from line inductance, we denote Li;n as L þ Li;n , where L is the average inductance of line i (i ¼ 1; 2) and Li;n shows the deviation at the nth cell. Similarly, for capacitance, we set Ci;n to C þ Ci;n . We also set A1 equal to A2 ð A0 Þ. Then, edge oscillations on both lines become identical without the L and C variations, with oscillation angular frequency denoted by ½. Assuming sufficiently small reactance variations, the above-mentioned transmission equations of point-coupled TD lines are conveniently written as

© IEICE 2017 DOI: 10.1587/elex.14.20170054 Received January 18, 2017 Accepted February 27, 2017 Publicized March 13, 2017 Copyedited March 25, 2017

dXi ¼ Fi ðXi Þ þ Fi ðXi Þ þ gi ðXj Þ; ð6Þ dt where j ¼ i mod 2 þ 1; further, Fi, Fi , and gi are (2N þ 1)-vectors representing the contributions of the reactance average, reactance variations, and coupling, respectively (See [1] for explicit expressions of g, F, and F). According to the phasereduction scheme described in [7, 8], the identity of the edge oscillation is represented by 2-periodic (2N þ 1)-vector function ZðÞ ¼ rX , i.e., the phase 3

IEICE Electronics Express, Vol.14, No.6, 1–8

sensitivity. In actuality, the temporal evolution of the limit-cycle phase of line i i (i ¼ 1; 2) is given by di ¼ !i þ i ði  j Þ; dt

ð7Þ

where Z 1 2 !i ¼ ! þ d ZðÞT Fi ðÞ; 2 0 Z 1 2 d ZðÞT gi ð; i  j Þ; i ði  j Þ ¼ 2 0 with j ¼ i mod 2 þ 1. Phase difference

ð8Þ ð9Þ

 1  2 is then shown to satisfy

d ¼ ! þ ð Þ; ð10Þ dt where !  !1  !2 and ð Þ  G1 ð Þ  G2 ð Þ. When the right-hand side of Eq. (10) becomes zero for a certain ¼ and dð Þ=d < 0, ¼ becomes timeinvariant for a sufficiently large period of time. Therefore, edge oscillation is mutually phase locked. The phase sensitivity is then numerically obtained by solving the adjoint of the linearized transmission equations with the initial condition that the eigenvector corresponds to eigenvalue 1 of the transpose of the statetransition matrix of the edge oscillation [9]. We searched all ¼ values corresponding to the phase-locked state to vary the position of connected cells while keeping M1 and M2 coincident. Fig. 1(b) and (c) show our results for A0 ¼ 0:6 V, L ¼ 1:0 µH, C ¼ 470 pF, R ¼ 1:0 Ω, and Rc ¼ 50 Ω, wherein the phase differences obtained for five different reactance variations are superimposed. The reactance variations were given by the Box-Muller method with the standard deviation of 5% and N was set to 30. In Fig. 1(b), the horizontal axis measures M1 ¼ M2 ð MÞ while the phase differences that allow mutual synchronization are shown vertically as red circles. The turning point occurred approximately at the 25th cell, such that no synchronized states were observed for M > 25. At M  3, the edges were only allowed to contribute to in-phase synchronization. Another synchronized state appeared at M > 3, having nonzero phase difference, becomes approximately ³ at M ¼ 9. At other M values up to 18, we observe three synchronized states whose respective phase difference centers at 0, 0 , and 2  0 with some 0 2 ð0; Þ. In contrast, we observe many synchronized states at 18 < M < 25. Based on this observation, the mutual synchronization is established when the edge on each line passes the connected cell at the same time. On the other hand, frequency shift by synchronization is obtained by substituting the phase-locked values of ¼ into 1  2 in the right-hand side of Eq. (7). Fig. 1(c) shows the properties of this frequency shift, with its dependence on the phase difference. The frequency shift tends to increase for the phase difference lying in ð0; =2Þ, otherwise the frequency shifts only slightly. 3 © IEICE 2017 DOI: 10.1587/elex.14.20170054 Received January 18, 2017 Accepted February 27, 2017 Publicized March 13, 2017 Copyedited March 25, 2017

Experimental observations

We experimentally characterized a test circuit built on breadboards. We employed NEC 1S1763 Esaki diodes as TDs. Series inductance, resistance, and shunt

4

IEICE Electronics Express, Vol.14, No.6, 1–8

capacitance were implemented using 1.0-µH inductors, 1.0-Ω resistors, and 470-pF capacitors, respectively. N was set to 30. In addition, Rc was implemented by the 50-Ω resistors. By setting the input DC voltage value to 0.6 V lines 1 and 2 exhibit the fundamental oscillation frequency of 1.07 and 1.17 MHz, respectively. We first varied the position of connected cells while keeping M1 and M2 coincident (M1 ¼ M2 ð MÞ). In Fig. 2(a), the dependence of the frequency shift on M is shown using the phase-reduction scheme (Simple rearrangement of Fig. 1(b) and (c)). On the other hand, Fig. 2(b) shows the measured result. The spectrum was detected with little disturbance to the oscillatory dynamics by using a high-impedance probe. The spectrum of the lines connected at M is shown, such that the top and bottom correspond to M ¼ 1 and 20, respectively and M increases in ascending order to the direction from the top to bottom. At M ¼ 1, 2, 19, and 20, the Q value of the spectral peaks exhibits no improvement over the uncoupled case; therefore, no synchronized oscillation is considered to be detected at these values of M. In contrast, for 3  M  13, red circles correspond to the highest spectral peaks, whereas blue squares shows them for 14  M  17. In addition, we can see two fundamental frequencies coexist for 12  M  14. The similarity between Figs. 2(a) and (b) is sufficiently good, such that the red-circle and blue-square peaks in Fig. 2(b) correspond to sequences A (in-phase synchronization) and B

Fig. 1. © IEICE 2017 DOI: 10.1587/elex.14.20170054 Received January 18, 2017 Accepted February 27, 2017 Publicized March 13, 2017 Copyedited March 25, 2017

Fundamental properties of dissipatively point-coupled TD lines. (a) The structure of the investigated lines, (b) the dependence of the phase difference between two edge oscillations on the connected cell, and (c) the relationship between the frequency and phase difference for synchronized edges.

5

IEICE Electronics Express, Vol.14, No.6, 1–8

(out-of-phase synchronization) in Fig. 2(a), respectively. In order to establish this correspondence, we detected the voltage on each cell in the time domain. Fig. 3(a) and (b) show the resulting spatiotemporal profiles of the oscillating edges for M ¼ 4 and 14, respectively. The detection was carried out using active highimpedance probe and the trigger signal was detected at the 18th cell of line 2. The 64-times average was taken, by which the intensity is subject to an apparent degradation owing to phase noises (more eminent in Fig. 3(a)). As expected, the inphase synchronization is confirmed for M ¼ 4, whereas a finite phase difference exists for M ¼ 14. Consistently, we can see that the oscillating frequency is larger for M ¼ 14 than that for M ¼ 4. Finally, Fig. 4 shows the properties of synchronized edges for M1 ≠ M2 . We fixed M1 to 5 and varied M2 from 1 to 20. Fig. 4(a) shows the temporal waveforms detected at the 7th cell of lines 1 and 2. The trigger signal is detected at the 23rd cell of line 1, so that the waveform detected at the 7th cell of line 1, which is typically shown in the top of Fig. 4(a), has only little dependence on M2 including the timing of the leading edge. In the same figure the line-2 waveforms detected for varied M2 are shown in ascending order from top to bottom (The number on the waveform represents the corresponding M2 ). The larger M2 becomes, the more the timing of the leading edge in line 2 precedes that in line 1. Assuming the frequency of the uncoupled oscillation to be 1.07 MHz, the measured phase difference is shown in Fig. 4(b) with blue triangles. In the same figure, red circles show the dependence of phase difference on M2 for M1 ¼ 5 obtained by the phase-reduction scheme; here, M2 varies from 1 to 20 and the phase differences for five different reactance variations are superimposed. Good similarity is confirmed. Note that the phase difference does not fluctuate too much for varied reactances; therefore, the phasereduction scheme can be regarded as a powerful tool to design the phase difference

Fig. 2. © IEICE 2017 DOI: 10.1587/elex.14.20170054 Received January 18, 2017 Accepted February 27, 2017 Publicized March 13, 2017 Copyedited March 25, 2017

Frequency response of edge oscillations in TD lines connected with the cells having a common address. (a) The frequency shift by synchronization predicted by the phase-reduction scheme and (b) the measured frequency responses.

6

IEICE Electronics Express, Vol.14, No.6, 1–8

Fig. 3.

Measured spatiotemporal profiles of synchronized edge oscillations. The upper and lower half shows the voltage profile detected on lines 1 and 2, respectively. M was set to 4 and 14 to obtain Figs. 3(a) and (b), respectively.

in the present system. In contrast, the frequency shift by synchronization fluctuates more than the phase difference (red circles in Fig. 4(c)). The measured frequency shift (blue triangles in Fig. 4(c)) are quantitatively well predicted by the phasereduction method. However, its M2 dependence loses sufficient correspondence. At present, the discrepancy observed in M2 dependence is supposed to be caused by the phase noise [10], which also depends on M2 .

© IEICE 2017 DOI: 10.1587/elex.14.20170054 Received January 18, 2017 Accepted February 27, 2017 Publicized March 13, 2017 Copyedited March 25, 2017

Fig. 4.

The synchronization for M2 ≠ M1 . (a) The measured waveforms, (b) the phase difference, and (c) the frequency shift.

7

IEICE Electronics Express, Vol.14, No.6, 1–8

4

Conclusion

In this paper, the mutually synchronized edges developed in point-coupled TD lines are experimentally studied and analyzed on the basis of the phase-reduction scheme. The phase difference increases almost linearly as the connected cells separate sequentially, so that the coupled TD lines can contribute to developing high-frequency multiphase oscillators. Characterizing phase noises in the framework of the phase-reduction scheme may give refined frequency shift induced by synchronization. Acknowledgment This work was supported by JSPS KAKENHI Grant Number 26420296.

© IEICE 2017 DOI: 10.1587/elex.14.20170054 Received January 18, 2017 Accepted February 27, 2017 Publicized March 13, 2017 Copyedited March 25, 2017

8