On the base of developed mathematical model of microwave oscillator interaction with the strong re ected emission, with attraction of numerical methods, the research results are presented for the features of the autodyne signal characteristics formation. Researches are ful lled for the microwave oscillator model with a single-circuit oscillating system, taking into account its non-isochronity and non-isodromity. An in uence of inherent parameters and the re ected emission delay phenomenon onto the autodune response formation is revealed. Graphs of normalized signal charcteristics are obtained showing the autodyne response shape, and its spectral analysis is performed. Harmonic coe cients and amplitudes of spectrum harmonic components are calculated together with averaged value levels of the autodyne response as a function of re ection coe cient modulus at various distance of the radar object.
Research results of autodyne microwave oscillator (MO) at strong re ected emission represent the practical interest in various areas of science and engineering. They are claimed at determination of output signal formation features, at estimation of the dynamic range, for analysis of parameter measurement errors and for a choice of the optimal operation mode of the short-range radar sensors, which use the autodyne principle of the transceiver architecture.
Speci cs of these sensor functioning at problem solution, for instance,
determination of electric-physical and dynamic parameters of the radar objects
consists in the fact that the distance between a sensor and an object can be
extremely small (down to zero) under several conditions. At that, the re ected
emission level may turn out to be commensurable with the level of probing
emission. In contrast to the enough studied case of the weak re ected emission, the
case of the MO impact of strong emission was investigated insu ciently. It is
examined in the known literature on the base of experimental data and modeling
results on the analog computers only [
In this paper, on the base of developed mathematical model with involving of numerical approaches, we present the analysis results of the single-circuit autodyne, which partially ll a mentioned gap. 2
Mathematical model of the autodyne at strong re ected emission The functional block diagram of the radar sensor with the autodyne architecture principle of the transceiver is presented in g. 1. Electromagnetic oscillations produced by MO are emitted through the receiving-transmitting antenna towards the radar object. Microwave emission re ected from the object returns through the same antenna into MO and causes the autodyne e ect.
As we know [
The equivalent diagram of the autodyne MO can be represented in the form
of parallel connection of averaged (over the oscillation period) AE conductivity
YAE = YAE(A; !), depended on the amplitude A and the current frequency !
of oscillations, the oscillating system (OS) YOS(!) and the load YL = YL(!).
Oscillation equation for this circuit has a form [
YAE(A; !) + YOS(!) + YL(!) :
A di culty of analytical solution nding of this equation consist in the presence of nonlinear dependences of all its terms upon oscillation parameters. At that, we should note that the YL(!) conductivity in (1) also depends on the delay time of the emission, at that
YL(!)
YL(!; ) = GL(!; ) + jBL(!; ) : (1)
Calculation and analysis of signal characteristics Main signal autodyne characteristics are functions of relative variations of the amplitude a and the frequency of oscillations versus variations of delay time is a modulus of the re ection coe cient reduced to MO terminals, which characterizes the emission attenuation at its propagation to the radar object and back; (!; ) is the total phase incursion; G0 is the load conductivity at re ected emission absence.
To simplify analysis of (1) without loss of generality, we take some assumptions. We shall limit the present research by the case of autodyne response extraction of oscillation amplitude variation with the help of the external detector.
In addition, we shall examine the processes in MO in the form of variations of oscillation parameters in the vicinity of its steady-state, when at = 0 we have: A = A0, ! = !0 For this, we represent the oscillation amplitude and frequency in the form: A = A0 + A, ! = !0 + ! where A and ! are appropriate variations of the MO steady-state at 6= 0.
Then, acting in accordance the accepted approach [
a + " + a + QL + 1 + 1 +
cos (!; )
2 + 2 cos (!; )
sin (!; )
2 + 2 cos (!; )
= 0 ;
= 0 ;
where ,", are dimensionless parameters determining the limit cycle strength,
non-isodromity and non-isochronity of MO, relatively [
Under real functioning conditions of the autodyne sensor, the high level of
re ected emission is observed on the extremely short distance to the radar object.
Under such autodyne operation conditions, it is really acceptable to assume:
(!; ) = ! [
(!; ) = ( ; n) = 2 (1 + )(N + n) ;
where n = !0 is normalized time; N = 2l is the integer number of
halfwavelengths, which falls between a sensor and the radar object.
(3)
(4)
(5)
where
n of re ected emission [
Bc( ; n)
1 ( n)
Bs( ; n)
Bc( ; n) + = 0
; Bc( ; n)
QL(1
Bs( ; n) ) (6) (7) Bc( ; n) = Bs( ; n) = 1 + 1 +
cos 2 (1 + )(N + n) 2 + 2 cos 2 (1 + )(N + n) sin 2 (1 + )(N + n) 2 + 2 cos 2 (1 + )(N + n) ; ; = , = Q"L are non-isochronity and non-isodromity coe cients of MO, QL is the loaded Q - factor.
The solution of equations (6), (7) we nd using mathematical packet MathCAD. For this, at rst, we nd the solution for autodyne frequency variations of the transcendent equation (7) by the secant method o the iteration algorithm realized in the root function. After substitution of obtained values of into equation (6), we obtain values of a variable by the same approach. a) b) c) d)
Then, performing a search of local extreems in functions a = a( n) and = ( n) with the help of embedded Maximize (f; x1; :::xm) function, we obtain maximal values of autodyne variation deviations amax and max. After that, we perform normalization of a = a( n) and = ( n) function with respect to extreme values amax and max obtained. At the end, we obtain the required signal characteristics in the normalized form: an = aa(manx) and n = (manx) .
The developed calculation algorithm according to (6) and (7) was veri ed
for obtained results convergence at the small signal, when < 1, with results of
signal characteristics calculations, which were obtained starting from the
smallsignal analysis ful lled in [
At rst, we perform the analysis for the case of the extremely short distance to the radar object assuming N = 1. Then, we reveal the time delay in uence on features of autodyne response formation assuming N = 100. At that, for each case, we introduce variations in inherent MO parameters ful lling calculations, at rst, for isochronous and isodromous case ( = = 0), and then for nonisochronous and non-isodromous autodyne MO, when 6= 0 and 6= 0.
Plots of harmonic coe cients KAF C and KACC , the level of harmonic
components of the autodyne response spectra on the frequency n(F n) and of
amplitude an(F n) variations, as well as the average value an(0) versus the re ection
coe cient modulus of the isochronous MO are presented in Figs. 3(a) and (b).
Here we show plots of feedback (FB) parameter CF B de ning as a product of
the delay time of re ected emission by the autodyne frequency deviation [
For positive ( = 1:5; = 0:1) and negative ( = 1:5; = 0:1) coe cients of non-isochronity and non-isodromity of MO, Figs. 4(a) (d) show AFC n( n) and AAC an( n) which are calculated at the previous value of distance (N = 1), but at di erent values of the re ection coe cient : = 0:01 (curves 1) and = 0:5 (curves 2).
Spectra n(Fn) and an(Fn) are presented in Figs. 4(e) and (f). Functions KAF C ( ), KACC ( ) and CF B( ) are shown in Fig. 5 (a) and (b). Here, we present curves of the harmonic components level n(Fn), an(Fn) (at n = 1; :::5) and n(0), an(0) (n = 0) versus coe cient. The shape of all these plots, as we see from Fig. 5, are broken at = 0:5.
a) b) c) d)
The in uence of the radar object on the shape and the spectrum of the autodyne signal for the case of isochronous MO is presented on plots in Fig. 6. We observe AFC n( n) (a), AAC an( n) (b) and their spectra (c), (d) calculated at = 0:1 and various values of the half-wavelengths number N from the sensor to the radar object: N = 1 (curves 1) and N = 100 (curves 2). Figures 7(a) and (b) show the plots of harmonic coe cients KAF C and KACC , the FB parameter CF B and the level of spectra harmonic components n(Fn) (a) and an(Fn) (b) (when n = 1; :::5), as well as average values of the autodyne response (at n = 0) versus of the re ection coe cient modulus .
For the case of non-isochronous and non-isodronous MO AFC n( n), AAC an( n), calculated at various number of half-waves: N = 1 (curves 1) and N = 100 (curves 2), are presented in plots in Fig. 8 (a) and (b). Spectra calculation results for the case N = 100, are presented in Figs. 8(c) and (d). Functions KAF C , KACC , CF B, as well as levels n(Fn), an(Fn) (n = 1; :::5), and n(0), an(0) at(n = 0) versus coe cient are presented in Figs. 9(a) and (b). As we see from Fig. 9, in this case plots are broken at =0.08. 4
Conclusions The resume of analysis of numerical modeling of the autodyne response formation processes in MO obtained results consist in the following.
Plots of normalized functions of reactive bn( n) and resistive gn( n)
components of the MO load conduction versus the normalized time n are presented in
Figs. 2(a) and (b) of the paper [
Spectra of reactive bn(Fn) and active gn(Fn) load conduction (see Figs. 3(c),
(d) of the paper [
Characteristic comparison results presented here allow conclusion that
autodyne response distortions in the case of strong re ected emission are caused
predominantly by the action of the load nonlinearity rather than the signal
restriction by the AE electronic conductance, as was assumed in [
MO non-isochronity, as can be seen from comparison of AFC n( n) and AAC an( n) in Figs. 2(a), (b) and Figs. 4(a)-(d), causes the phase o set of n( n) characteristics by the angle = tan 1( ), which for chosen parameters ( = 1:5) for the positive value of , is 1, while for the negative values 1. MO non-isodromity causes the AAC phase o set by the angle = tan 1( ). Since QL1 >> 1, the angle is comparatively small in value and usually does not exceed 0:5. In here considered case, its value is 0:1 at = 0:1 (see Fig. 4(b)) and 0:1 at = 0:1 (see Fig. 4(d)). These phase o sets, as wee see from the curves in Figs. 4(a) (d), cause additional clutter of the autodyne response, which are expressed in appearance of characteristic wave tilts to one or another side depending on the ratio of magnitudes and signs of coe cients of non-isochronity and non-isodromity of MO.
Besides, MO non-isochronity in the case of strong re ected emission causes on AFC the increase of frequency deviation and appearance of DC component, the sign and magnitude of which depend on the sign and the magnitude of coe cient. The rst phenomenon leads to increase the autodyne CF B parameter, while the second one lead to the o set of the central frequency of oscillation. Therefore, during increase of the re ection emission level of the isochronous MO, the appearance of signal jumps (see Figs. 5(a), (b)) is observed at lesser re ection coe cient ( = 0:5), than for the isochronous MO ( = 0:7) (see Figs. 3(a), (b)). We should also note that the amplitude spectrum picture for sign change of non-isochronity coe cient does not practically change (see Figs. 4(e), (f)). This means that for sign change of or movement direction change of the radar object, variations of phase relations occur for harmonic components in the autodyne response spectrum.
From analysis of above cases, it follows that for strong re ected emission,
when the re ection coe ient in commensurable with one, jumps in the process
of signal formation begin not for CF B = 1, as in the case of weak signals [
The increase of radar object distance causes the appropriate growth of the CF B parameter and appearance of signal jumps at lower level of re ected emission. Therefore, at studying of the distance variation in uence (the number of half-wavelengths N) on features of the autodyne response formation in the mode of strong re ected emission, the range of analyzed values of the re ection coe cient are proportionally narrowed and passes towards the area of weak signals.
Such a situation found its interpretation in values of chosen for calculations on plots in Figs. 6-9. From these plots, we also see that the presence of the DC component in autodyne frequency variations of the isochronous is absent as in the case of extremely small values of N , whereas for the non-isochronous MO this dependence happens at radar object distance increase. There are no other qualitative di erences in features of autodyne signal formation at distance increase.
We should note that obtained results of theoretical studies are well agreed
with experimental data published in [