Autodyne signal features of frequency-locked microwave oscillators

. The research results on the autodyne signal formation in microwave oscillators, which are exposed under the impact of the proper reflected emission and, at the same time, of the external frequency-locked signal are presented. The basic relations are obtained for signal analysis under the condition of the quasistatic target movements. The theoretical research results are confirmed by experimental data obtained on the example of an oscillator made on the basis of the 8mm-range Gunn diode. 2, d . 4 Conclusion Thus, a mathematical model of an autodyne transceiver stabilized in frequency by a signal from an additional oscillator has been developed. The basic relationships for calculating its signal characteristics are obtained. It


Introduction
Autodyne transceivers (or just autodynes -AD) are widely used in various Short Range Radar Systems, in process parameter sensors and in the measuring systems in industry and in the transport as well as for various scientific applications, military systems and medicine devices [1][2][3][4][5]. The correct signal processing, which are received by AD, makes it possible to determine the electrophysical and kinematic characteristics of the radar objects. The structural simplicity and the low cost of microwave modules as well as their high sensitivity facilitate to the AD widespread application.
In order to search for new AD features and to improve its parameters, a number of publications were prepared concerning the research results of AD synchronised from the additional (external) microwave oscillator [5]. However, in the literature known to the authors, the features of the generation of synchronised AD (SAD) signals, which should be taken into account at its utilization, have not been sufficiently studied [6]. In this connection, the aim of this paper is to fill up this gap.

The mathematical model of the synchronised autodyne
An equivalent SAD circuit reduced to the plane of the active element (AE) is shown in Fig.  1. In this diagram, OS Y displays the conductivity of an oscillatory system (OS). In the case of the single-circuit OS, the expression for this conductivity (taking into account the AD load) has the following form: is the loaded Q-factor and the natural frequency of OS;  is the current oscillator frequency. Average AE oscillation conductivity over the oscillation period with the voltage-current characteristic of the N-type,  The equivalent sources of instantaneous current j ex (t) and j s (t, τ) on the diagram in Fig. 1 represent the effect on AD of an external synchronising signal and a signal on the oscillator load caused by the reflected proper emission delayed by time τ, relatively. Here  are the amplitude and the frequency of the 'external' signal. Provided that the balance of amplitudes and phases in the circuit under the influence of AE current, the undamped oscillations () ut will be generated. Considering usually that we have the fairly high value of the Q-factor L Q of OS, we believe that the oscillations on AE are quasi-harmonic: are amplitude and phase, which are slowly-changing over a period of oscillations. Then, the oscillations of the equivalent source s ( , ) jt are also quasi-harmonic: t  is the reflected signal phase incursion.
According to Kirchhoff's laws for the circuit shown in Fig. 1, we have: To obtain the abbreviated equations of the perturbed oscillator, in expression (2), after substituting (1) into it, we first replace  by the expression  We further assume that the relative level of reflected emission 1   . In this case, we assume that the synchronising effect is also low ex 1 k  but it can cause significant deviations of the steady-state mode of the stand-alone oscillator in comparison with the reflected emission effects, ex k   . Therefore, to describe the AD behaviour, we further linearized equations (3), (4) with respect to the steady-stat mode of synchronous oscillations.
The reflected emission radiation, which produces an effect on the oscillator OS, as noted above, causes changes in the amplitude where a / K    is the autodyne gain coefficient; a  ,   are time constants of changes (relaxation) in the amplitude and phase, respectively: are non-isochronous and non-isodromic coefficients of synchronised AD respectively; are parameters characterising the strength of the limit cycle, non-isodromity and non-isochronity of the oscillator, respectively; 3 The resulting system of linearized equations (5) and (6), taking into account the wellknown duality principle, has sufficient generality for the SAD analysis with any type of AE.

Calculation of the parameters of the autodyne signal of SAD
To obtain analytical solutions of differential equations describing the behaviour of selfoscillating systems, the so-called quasi-static method is widely used [8]. This method makes it relatively easy to calculate the AD characteristics that determine the generation of their output signals. To find these characteristics, we set the derivatives in (5) and (6) to be zero and take into account that in the synchronous mode ex    there is also a phase where H are transfer coefficients due to the internal parameters of the oscillator and conditions for its synchronisation: As can be seen from (10), (11), the first terms of the right-hand sides determine the level of the constant component relative to the steady-state mode of the stand-alone oscillator, which is due to the action of an external synchronising signal only. The second terms in these expressions are associated with the effect of the reflected microwave emission; they determine the creation of the dependences of the instantaneous values of the variations in the oscillation amplitude and phase on the delay time of the reflected emission.
The first dependence in the AD theory is usually called the amplitude characteristic [9], the second one is called the phase characteristic of SAD. In contrast to the characteristics of ordinary (non-synchronised) AD, the SAD characteristics, as can be seen from (10), (11), are harmonic functions of the delay time  of the reflected emission. Fig. 2 presents dependency diagrams of the coefficients stabilized) AD, which contributes to the expansion of their dynamic range. It is demonstrated that in the synchronized AD by introducing the initial detuning between the frequencies of the external oscillator and the autodyne frequency within the synchronisation band, there is the possibility of the significant increase in the transfer coefficient of the autodyne signal compared to conventional AD.