Dynamics of high-power coupled nonlinear oscillator arrays for electronic beam steering technique: case of the triode based van der Pol oscillator

Abstract

This paper presents the dynamics of coupled nonlinear oscillator arrays (CNOAs) as phase shifter for electronic beam steering technique. The proposed phase shifter uses oscillators of van der Pol type with a fifth-order nonlinear resistance and is built on the two architectures based injection-locked oscillator array (ILOAs) and coupled oscillator arrays (COAs). The dynamics equations of the phase shifter are proposed in the frequency and time domain. With the help of mathematical tools such as bifurcation diagrams, maximum Lyapunov exponent and Hilbert transform, important cartography of different oscillating states is deduced over large range of system parameters. Different behavior of the CNOAs including chaos, 2D-torus, multistability, locked and unlocked states are depicted. The phase shift between output voltages of each pair of coupled oscillators is presented for both architectures. A detailed analysis of locked states shows that the required phase shift is synthesized by slightly varying the frequency of the injected signal, or simply by detuning the free-running frequencies of the edge oscillators in equal but opposite directions. The proposed circuit of the phased shifter is built using the original van der Pol oscillator based on a vacuum triode. The maximal phase shift reaches 63° and 75° respectively for ILOAs and COAs. With the presence of triode in the circuit, the output voltage of each oscillating element in the array reaches 100V around 60KHz and is boosted by increasing the value of the triode bias potential to more than 420V. This, therefore, increases the total power dissipation of a single oscillator in the array to more than 47dB ( 50.12W ). This circuit is a candidate for high-power electronics beam steering at ultrasonic frequency.