ISSN 10637842, Technical Physics, 2015, Vol. 60, No. 3, pp. 420–426. © Pleiades Publishing, Ltd., 2015. Original Russian Text © V.Yu. Konev, A.I. Klimov, O.B. Koval’chuk, V.P. Gubanov, V.Yu. Kozhevnikov, A.V. Kozyrev, 2015, published in Zhurnal Tekhnicheskoi Fiziki, 2015, Vol. 85, No. 3, pp. 103–109.
RADIOPHYSICS
Phase Stabilization of Nanosecond Microwave Oscillations in GunnDiodeBased Oscillators V. Yu. Koneva, A. I. Klimova, O. B. Koval’chuka, V. P. Gubanova, V. Yu. Kozhevnikovb, and A. V. Kozyrevb * a
Institute of HighCurrent Electronics, Siberian Branch, Russian Academy of Sciences, Akademicheskii pr. 4, Tomsk, 634055 Russia b National Research Tomsk State University, pr. Lenina 36, Tomsk, 634050 Russia *email:
[email protected] Received May 22, 2014
Abstract—The “intrusion” of the phase of a Gunndiode nanosecond microwave oscillator by applying a modulating voltage pulse is numerically simulated. The dependences of the microwave oscillation phase on the spread of the pulse rise time and modulating pulse amplitude are revealed. The standard deviation of the phase lag time in a 3cmrange oscillator relative to a fixed level at the leading edge of the modulating phase is measured. Phase synchronization between two electrodynamically uncoupled oscillators that are simulta neously excited by a single modulator is studied experimentally. DOI: 10.1134/S1063784215030147
INTRODUCTION Phase synchronization between microwave oscilla tors has been and is still of great applied interest [1, 2]. Gunndiode oscillators with high phase stability can be effectively used in shortrange radar systems offer ing a high range resolution [3, 4]. Gaining insight into the phase stabilization of microwave pulses from such oscillators would help in improving the recognition capability of radar systems through coherent accumu lation of received signals and also in designing active phased antenna arrays [5]. Of special interest is the way of fixing the phase of 3cmrange milliwatt nanosec ond Gunndiode oscillators (GDOs) [6–9]. This is the method of phase synchronization between two electrodynamically uncoupled oscillators in which synchronization is achieved by applying steep modu lating voltage pulses to oscillator diodes [6]. The authors of [6] think that the initial phase of microwave oscillations is set by “shock” excitation of the cavity with a current spike and remains unchanged if the rise time of voltage pulses applied to the Gunn diodes (GDs) is comparable to the oscillation period, that is, is on the order of 100 ps. Later, however, it was shown [10] that in higher power (about 30 W) nanosecond 3cmrange GDOs, the microwave oscillation phase is also stable at a fixed point on the leading edge of the voltage pulse if its rise time τe ≈ 4.5 ns, which is much longer than the oscil lation period. Because of this, the phase stabilization in 3cmrange 30W nanosecond GDOs is studied in greater detail experimentally and by numerical simu lation.
GUNN DIODE MODEL Electronic processes in the GD were numerically simulated with a 1D local field model [11–13]. The practical use of this model is limited largely because of the assumption that the velocity of electrons is an instantaneous function of the local electric field. It is known [11, 12] that if the electron energy relaxation time (in GaAs this time is τ ~ 10–12 s) exceeds the duration of the processes under study, such an assump tion is invalid. In this work, numerical calculations were carried out for a microwave oscillation frequency of about 10 GHz, which corresponds to an oscillation period of 100 ps. Therefore, our problem completely satisfies the applicability condition of the local field model. It was supposed that a GD was made on an ntype GaAs substrate with two ohmic contacts on opposite sides. An equation for electric field E(x, t) in the semi conductor can be derived from the continuity equa tion for electrons in which the electron concentration is expressed through the Poisson equation [13] 2
∂E ∂E ∂E = D 2 – V ( E ) ∂x ∂t ∂x
(1) ∂n I e + ⎛ D 0 – n 0 V ( E ) + ⎞ . eS⎠ ε 0 ε ⎝ ∂x Here, V(E) is the dependence of the electron drift velocity on the local electric field strength, D is the electron diffusion coefficient, n0(x) is the donor con centration distribution along the sample, I is the cur rent in the external circuit, e is the electron charge, ε0ε is the GaAs permittivity, S is the crosssectional area of
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the semiconductor, and x the coordinate along the structure. The electric field in the semiconductor (diode’s base) was assumed to satisfy a given initial condition [13] and the Neumann boundary conditions [14]
∂E ∂x
x=0
∂E = ∂x
I2
IGD
M
R1
L1
C2
I3
I4
L3
R2
C1
GD UGD(t)
x=l
∫ E ( x, t ) dx. 0
In the GD base, the dopant profile was assumed to be uniform with donor concentration n0. A highfield domain was initiated by making a “notch,” i.e., by locally decreasing the donor concentration by 10% relative to n0 near the cathode. The notch l2 wide was made at distance l1 from the cathode with l1, l2 Ⰶ l. In model calculations, the parameters of the semi conductor were assigned the following values: the diameters of the structure was set equal to d = 300 μm, base length l = 12.5 μm, donor concentration n0 = 1015 cm–3, impurity concentration in the notch n1 = 0.9 × 1015 cm–3, notch width l2 = 0.6 μm, and notch– cathode distance l1 = 0.6 μm. As in [13], electron dif fusion coefficient D = 200 cm2/s was assumed to be constant and fieldindependent. Mean drift velocity V(E) of electrons and local field strength E were assumed to obey the model relationship [13] 4
4
V ( E ) = [ μ n E + V m ( E/E m ) ]/ [ 1 + ( E/E m ) ].
(3)
Here, Vm = 107 cm/s is the drift velocity at which the characteristic of the diode saturates in a high field, E Ⰷ Em (Em = 4 kV/cm). The electron mobility in the lowfield approximation was set equal to μn = 8000 cm2/(V s). Simulation was carried out in the MATLAB R2012b environment. Equation (1) was solved using standard implicit absolutely stable difference schemes [15], which are applicable to solving quasilinear hyperbolic equations. Spatial step Δx was taken to be roughly 100 times smaller than the size of the highfield domain. Time step Δt was taken to be much shorter than the characteristic transient time of the electric field in the diode, i.e., was equal to several hundred femtoseconds [13]. Vol. 60
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Fig. 1. Equivalent circuit of the GDO with regard to its electrodynamic system.
SIMULATION OF THE OSCILLATOR PERFORMANCE
l
TECHNICAL PHYSICS
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(2)
= 0
and conductivity nonuniformities at contacts were neglected in calculations. In (2), ϕT = kT/e is the tem perature potential and l is the length of the diode’s base. Field E(x, t) and voltage drop U(t) across the diode were assumed to be related by the wellknown relationship U GD ( t ) =
L2
U0(t)
1 ∂n 0 E ( x, t = 0 ) = ϕ T , n 0 ∂x
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First, we simulated the simplest GDO the equiva lent circuit of which includes seriesconnected ele ments: a GD, voltage pulse source U0(t) (modulator) with a given pulse shape, and currentlimiting resistor R = 1 Ω. The GD represented either a singlelayer semiconductor structure or a bilayer structure made up of two seriesconnected identical singlelayer structures. The current in this circuit is found from Kirchhoff’s equation l
1 I ( t ) = U 0 ( t ) – E ( x, t ) dx , R
∫
(4)
0
which closes the set of equations in the theoretical model. To take into account the influence of the micro wave field of the cavity on processes in the semicon ductor structure, the GD circuit should be comple mented by an oscillating loop. Therefore, at the sec ond stage, we numerically analyzed a GD with an oscillating loop (see Fig. 1) with regard to basic design features of the oscillator’s electrodynamic system [16]. Resistor R1 is a currentlimiting resistor, which can control the voltage across the oscillating loop. The L1– C1 network simulates a microwave filter providing the microwave decoupling between the oscillating loop and modulator. Capacitor C2 takes into account the capacitance of the GD body and the headerto ground capacitance. Simultaneously, it provides the rf decoupling between resistor R2, simulating power losses in the electrodynamic system of the GD, and the power escaping into the output waveguide. Capac itor C2 and inductor L2 constitute an oscillating loop, which replaces a combined oscillatory system made up of the GD’s coaxial circuit and the controlled waveguide cavity connected to the GD’s resonant chamber. Element L3 stands for the GD inductance.
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U0
Ust
U0
ΔU0
ΔU0
Δτe Δτe
Δtst Δtst
Ust
Δtst Δtst tst
tst
τe
τe Fig. 2. Shape of the modulating voltage pulse used in the simulation.
0.016 ns 3
0.022 ns 0.022 ns
(a)
(b)
2
0.016 ns
1 I, A
I4, A
2 1
0
0
−1 0.2
0.4
0.6 t, ns
0.8
1.0
−2 0.8
1.0
1.2
1.4 t, ns
1.6
1.8
Fig. 3. Time variation of the (a) GD current in the singlelayer semiconductor structure with a resistive load and (b) current through load R2 in the circuit with an oscillating loop.
When simulating processes in the GD with an oscillating loop, we used Eq. (1) to find the electric field in the semiconductor structure. Boundary and initial conditions were taken in the form of (2), and the diode voltage was determined from the relationship 1
U GD ( t ) =
∫ E ( x, t ) dx 0
Also, boundary conditions for the external circuit that follow from Kirchhoff’s laws were applied, I1 = I2 + I3 ,
I 2 = I 4 + I GD , t
dI 1 U 0 ( t ) = I 1 R 1 + L 1 3 + I 3 dt, dt C 1
∫ 0
t
t
∫
∫
dI dI 1 t dt + I R , 1 I 3 dt + L 1 3 = L 2 2 + 4 4 2 C1 dt dt C 2 0
0
t
dI GD 1 L 3 + U GD ( t ) = I 4 dt + I 4 R 2 . C2 dt
∫ 0
A modulating voltage pulse from the voltage source was set in a trapezoidal form (Fig. 2) with variable rise time τe and amplitude U0. The mean value of U0 was equal to 20 V in the circuit with a resistive load for both the singlelayer and bilayer structures, as well as in the circuit with an oscillating loop for the singlelayer struc ture. In the circuit with an oscillating loop for the bilayer structure, the mean value of U0 was equal to 100 V. In the course of simulation, the parameters of the equivalent circuit and the modulating voltage pulse amplitude were optimized. The circuit parameters that were optimal for the process analysis were the follow ing. For the singlelayer semiconductor structure, R1 = 1 Ω, R2 = 0.5 Ω, L1 = 0.5 nH, C1 = 0.5 pF, L2 = 1.2 nH, C2 = 1.2 pF, and L3 = 0.5 nH. For the bilayer structure, R1 = 1 Ω, R2 = 0.5 Ω, L1 = 0.8 nH, C1 = 0.5 pF, L2 = 0.5 nH, C2 = 1.5 pF, and L3 = 0.9 nH. These values, combined with the amplitudes of the modulating pulse mentioned above, provided oscilla TECHNICAL PHYSICS
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tions at a frequency of 10 GHz. In Fig. 2, Ust is the modulating voltage at which the GD acquires a nega tive differential resistance at time instant τst and a highfield domain arises. We calculated the time dependences of the diode voltage, UGD(t); the current through the diode, IGD(t); and the current through load resistance R2, I4(t). Figure 3a plots the calculated time dependences of the current in the GD circuit with the resistive load, and Fig. 3b shows the same dependences for current I4 through resistor R2 in the circuit with the oscillating loop. In both cases, the modulating pulse amplitude was constant, U0 = 20 V. Calculations showed that for pulse rise time fluctuation Δτe = ±0.05 ns (at a mean rise time of 1 ns), the oscillation phase fluctuation in the case of the singlelayer structure is Δtph = 0.016 ns for the circuit with the resistive load and Δtph = 0.022 ns for the circuit with the oscillating loop. In the case of the bilayer structure, the phase fluctuation was Δtph = ±0.027 ns for the circuit with the resistive load and Δtph = ±0.011 ns in the circuit with the oscillating loop. These results indicate that the phase fluctuation depends on a ratio between the initial, Ust, and modu lating, U0, voltages (Fig. 2). The smaller the UsttoU0 ratio, the smaller the phase fluctuation. Also, the numerical simulation showed that irrespective of the GD connection scheme, the oscillator is started in a hard mode: the amplitude reaches a stationary value within the first oscillation halfcycle. The amplitude of the precursor pulse arising in the oscillating loop because of its “shock” excitation turned out to be two orders of magnitude lower than the current oscillation amplitude in the GD. From the calculated time varia tion of the current through the GD [17], it follows that these fluctuations influence little the phase of funda mental oscillations, which depends on time instant tst when the semiconductor structure acquires a negative differential resistance. This means that inner noise in the circuit (if any) has a negligible effect on the sta tionary oscillation phase. Using the singlelayer semiconductor structure of the GD and both its equivalent circuits, phase fluctu ation Δtph of microwave oscillations was studied in greater detail for a fixed amplitude of the modulating pulse (U0 = 20 V) and pulse rise time fluctuation Δτe = ±0.1 ns. Pulse rise time τe was varied from 0.4 to 2.0 ns. In addition, for the same variation of the rise time without fluctuations (Δτe = 0), pulse amplitude U0 was varied within 20 ± 2 V in the case of the circuit with the resistive load and within 20 ± 1 V in the case of the cir cuit with the oscillating loop. The smaller amplitude variation in the latter case was necessary to minimize an error in phase determination. This error, associated with the dependence of the stationary oscillation fre quency on the voltage amplitude, was greater in the case of the circuit with the oscillating loop. The calcu lated phase shift was compared with the calculated shift of microwave oscillation settling time tst. For time instant tst, we took the instant the first highfield TECHNICAL PHYSICS
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domain arose. For the latter, we took the instant the field at the cathode reached its critical value, 4 kV/cm. Calculations showed that phase fluctuation Δtph lin early depends on pulse rise time variation Δτe for both equivalent circuits of the oscillator. A theoretical estimate of microwave oscillation phase fluctuation Δtph resulting from amplitude fluctu ation ΔU0 and rise time fluctuation Δτe can be obtained under the assumption that oscillations at the leading edge of the modulating pulse start once the GD volt age reaches threshold (stationary) value Ust. From the shape of the pulse in Fig. 2, shift Δ t *st of the microwave oscillation start time can be expressed as (5) Δt *st = τ e ( U st /U 0 ) ( Δτ e /τ e – ΔU 0 /U 0 ). Unfortunately, comparing the results obtained by formula (2) with numerical simulation data, we found that this formula can be useful only for qualitative analysis and primarily in the case of the circuit with a resistive load. This is because the shape of the voltage pulse applied to the GD differs from the trapezoidal shape of the modulating voltage pulse. In the circuit with an oscillating loop, the specific features of oscil lation excitation are an additional influencing factor. Yet, from (4) conclusions of practical importance fol low: when the amplitude of the modulating pulse is stable (ΔU0 = 0), fluctuation Δ t *st of the time of onset of oscillations will depend not on pulse rise mean time τe but on its fluctuation Δτe. If the pulse rise time is sta ble (Δte = 0), fluctuation Δ t *st of the time of onset of oscillations will linearly depend on ΔU0 and grow in proportion to mean rise time τe. Numerical simulation data for phase fluctuation Δtph that were obtained for mean rise time τe varying from 0.4 to 2.0 ns, Δτe = 0.1 ns, and ΔU0 = 0 indicate that Δtph correlates with fluctuation Δtst of the time of onset of oscillations in both equivalent circuits. In the circuit with the resistive load, quantity Δtph ≈ 32.3 ps is almost independent of the voltage pulse rise time and completely coincides with the estimate by formula (4), Δ t *st ≈ 32 ps. Figure 4 confirms the above conclusions. When the modulating pulse amplitude varies with the pulse rise time being constant (Δτe = 0), phase fluctu ation Δtph increases linearly with the rise time varying from 0.4 to 2.0 ns and depends on the sign of ampli tude deviation ΔU0 from its mean value, ΔU0 = 20 V. This conclusion follows from the geometrical con structions shown in Fig. 2b. Fluctuations Δtst and Δtph are larger when U0 decreases rather than when U0 rises. Phase deviation Δtph correlates well with deviation Δtst of the instant of onset of oscillations in the circuit with the resistive load (Fig. 4a). In the circuit with the oscil lating loop, Δtph markedly exceeds Δtst (Fig. 4b). This may be because the settling of oscillations depends on the modulating pulse amplitude. From the numerical simulation data, one can conclude that the stability of
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80 (a)
(b)
20
0
20 20
40 Δtst, ps
60
0 80
V 21 20−
40
Δtph, ps
40
Δtph, ps
40
50
60
V 22 20− V 20 − 8 1
Δtph, ps
Δtph, ps
60
40
V 20 19−
30 20 10
15
20
25 30 Δtst, ps
35
30 40
Fig. 4. (a) Phase fluctuation Δtph of the GD current in the circuit with the resistive load and (b) phase fluctuation Δtph of the cur rent through load R2 in the circuit with the oscillating loop vs. fluctuation Δtst of the onset time of microwave oscillations.
R(Itr)
Cable
R1 20 dB
R1
20 dB
R1
Tr
PS
R1
To oscilloscope R2
C
R2
MU
Fig. 5. Measuring circuit: PS, pulse shaper; C, storage capacitor of the modulator; Tr, DE 275 switch transistor with nonlinear resistance R(Itr); and MU, microwave unit of the GDO.
the microwave oscillation phase at different rise times of the modulating pulse is considerably affected by the instability of its amplitude. If the pulse amplitude is stable, oscillations with a wellcontrolled phase can be initiated even if the pulse rise time is much longer than the oscillation period. This finding is of special impor tance for designing arrays of impulsive phasesynchro nized Gunn oscillators.
Attenuators WRA Triggering N
Attenuator Dummy loads
Oscilloscope
M Modulation
O
Directional couper
MU
Fig. 6. Circuit for measuring the standard deviation of the microwave oscillation delay relative to a fixed point on the voltage pulse leading edge: MU, microwave unit of the oscillator and WRA, waveguidetoradio adapter.
EXPERIMENTS WITH GUNNDIODE OSCILLATORS The measuring circuit used in the experiments is depicted in Fig. 5. Its basic element is a modulator based on a voltage pulse shaper with a storage capaci tor (C = 12.6 pF), which partially discharges using a DE275 switch transistor. The circuit diagram of the modulator was described in detail elsewhere [16]. It was complemented by a B547 power source, whereby the modulating voltage amplitude across a 50Ω load could be controlled within 0–300 V. The time varia tion of voltage UGD(t) at the input to the microwave unit of the GDO [16] (hereinafter this voltage is taken to be roughly equal to the GD voltage) is measured by a circuit consisting of RK50311 and RG58 cables and two attenuators (RADIALL and R41620000) connected to them. The operating bandwidth and the attenuation factor of either attenuator were 4 GHz and 20 dB. The voltage pulse rise time measured on a 50Ω load by this circuit using an Agilent DSO9254A with an operating bandwidth of 2.5 GHz and a sampling rate of 20 G/s was found to be τe = 1.9 ns at a level of 0.1–0.9 of the amplitude (the microwave unit was dis connected). Experiments were conducted with one or two seriesconnected 3A762 GDs. The respective standard deviations, σt1 and σt2, of the microwave oscillation phase delay relative to a fixed point at the leading edge of the GD voltage pulse. Measurements were taken according to the circuit depicted in Fig. 6. TECHNICAL PHYSICS
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Le Croy
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UGD, V 125
Delay 100 1 75
1
2 50 (b)
Le Croy
2
0
1
In this circuit, the modulating voltage was applied to the microwave unit directly from the modulator. The output of the microwave oscillator was connected to a waveguide directional coupler; a D51 waveguide attenuator; a waveguidetoradio adapter; and a piece of an RG58 cable, through which an rf signal attenu ated to an acceptable level by two RADIALL 18GHz attenuators was applied to a LeCroy WaveMaster 830Zi digital oscilloscope with a bandwidth of 30 GHz and a sampling rate of 40 Gs/s. The oscilloscope was started by the voltage pulse applied to the GD. Time intervals were determined using the delay function block of the oscilloscope. The voltage pulse amplitudes on one or two series connected GDs were equal to 98 and 165 V, respec tively, and the pulse rise time in both cases was 4.1 ns at a level of 0.1–0.9 of the amplitude (for details of the measuring technique, see [18]). It was found that the measured value of σt1 and σt2 are considerably affected by the instability of the oscil loscope triggering time (instability is due to microwave noise in the GDO). This noise arises when a micro wave signal having penetrated through a lowpass filter serving for microwave decoupling between the modu lator and resonance chamber falls into the triggering circuit of the oscilloscope [16]. Narrowing the trigger ing circuit operating bandwidth to 1 GHz made it pos sible to adequately solve this problem. In addition, the measuring data are influenced by the sensitivity limit of the oscilloscope and its trigger voltage. The minimal values of σt1 and σt2 were obtained by applying trigger voltage Utr = 500 mV at the instant immediately pre ceding the onset of microwave oscillations. The sensi tivity of the triggering circuit was 5–10 mV/div, and the operating bandwidth of the microwave signal recording circuit was 13 GHz. The time interval of microwave signal phase variation selected by the oscil Vol. 60
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Fig. 8. GD voltage pulse waveforms (1) in the absence of an additional inductor and (2) in the presence of (2) 8.2nH and (3) 82nH chip inductors.
20 ps
Fig. 7. (1) Systematic sample of 2500 waveforms (oscillo grams) of the GDO’s microwave signal and (2) the same sample of the GD voltage waveforms in the case of (a) one and (b) two seriesconnected GDs.
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loscope had to cover no more than one minimumto maximum (or, vice versa, maximumtominimum) transition in the signal sinusoid [18]. Oscilloscopic errors in measuring time intervals were taken into account. Figure 7 plots the statistical samples of 2500 accu mulated oscillograms in the case of one (Fig. 7a) and two seriesconnected (Fig. 7b) GDs. For single and doublediode GDOs, the standard deviations were σt1 ≈ 2.1 ps and σt2 ≈ 0.8 ps, respectively. These values correspond to the maximum measuring capacity of the LeCroy WaveMaster 830Zi oscilloscope under the conditions of our experiment. To see how σt1 depends on the rate of rise of voltage (or, in other words, on the voltage pulse rise time), dUGD/dt, a chip inductor was inserted in between the modulator and microwave unit. The oscilloscope was started within the initial portion of the voltage pulse leading edge below the plateau, where dUGD/dt 0. Figure 8 demonstrates the waveforms of the voltage pulses applied to the GD. If the modulating pulse amplitude slightly fluctuates and the rise time of the pulse increases (dUGD/dt decreases), the spread of the microwave oscillation phase grows. The minimal value of σt1 was measured to be 2.4 ps in the first case and 14.5 ns in the second one. In the third case, the spread was so large that the waveforms (oscillograms) almost completely filled the display of the oscilloscope. These results are in good qualitative agreement with the above results of mathematical simulation. In [17], experiments on phase synchronization of two 30W oscillators were carried out. The oscillators were simultaneously excited by a single modulator through striplines with a geometrical length of 120 cm, which were isolated by glassfiber plastic with permit tivity ε = 6. In this way, crosstalk between the oscilla tors through the modulator was excluded. Measure ments were taken according to the technique described in [10, 17]. The rise time of the pulse applied to the GD was equal to 6.4 ns. Experiments showed that the microwave oscillation phases of the oscillators
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are independent of the phase of the weak signal shock exciting the oscillating loop. Even if a slight phase mis match between the oscillators was observed at the early stage of oscillation buildup, the phases equalize within several periods. The microwave signal waveforms remained unchanged after the disconnection of one GD. Thus, the mutual synchronization of the oscilla tors is excluded. The phase synchronism of microwave oscillations persists up to the last period for an FWHM of the pulse of 16 ns. CONCLUSIONS Numerical simulation and experiments convinc ingly demonstrated that the GD may effectively “intrude” the oscillator’s phase. We called this phe nomenon “the phase stabilization effect.” The initial phase of pulsed microwave oscillators several tens of watts in power that are built, for exam ple, around 3A762 oscillator diodes may be stable even if the leading edge of the modulating pulse is consider ably longer than the microwave oscillation period. As follows from numerical simulation data, the regular formation of a highfield domain (when the GD acquires a negative differential resistance at the cath ode) and the motion of this domain toward the anode under the action of an applied voltage generate micro wave oscillations the initial phase of which is always the same. Even after the first oscillation period, the oscillation amplitude considerably exceeds the ampli tude of GD’s inner noise due to different known mechanisms [17]. This means that noise oscillations have an insignificant effect both on the time instant of oscillation excitation and on the initial phase of oscil lations. Therefore, whether the phase will be stable depends on only the stability of the parameters of a modulating pulse applied to the GD. Standard devia tion σt of the microwave oscillation phase delay rela tive to a fixed point on the leading edge of the modu lating pulse applied to the GD (which results from the pulse amplitude instability) decreases with increasing dUGD/dt because of a decrease in the spread of the microwave oscillation onset time. The minimal value of σt measured in our experiments was 0.8 ps. which corresponds to σϕ = 2.9°. The typical value of σt was not greater than 2.5 ps. This value meets well the crite rion of phase instability of radiation sources used in designing active phased antenna arrays [19].
ACKNOWLEDGMENTS This work was supported by the Russian Founda tion for Basic Research, grant nos. 120831171 mol_a and 110800041a. REFERENCES 1. Microwave Radio Equipment Based on Synchronized Generators, Ed. by N. N. Fomin (Radio i Svyaz’, Mos cow, 1991) [in Russian]. 2. L. V. Kasatkin, Izv. Vyssh. Uchebn. Zaved., Radioelek tron. 49 (4), 38 (2006). 3. N. N. Badulin, A. P. Batsula, V. P. Gubanov, et al., Prib. Tekh. Eksp., No. 6, 111 (1998). 4. V. A. Vdovin, V. V. Kulagin, E. V. Mitrofanov, et al., Zh. Radioelektron., No. 12, 1 (2012). 5. Active Phased Antenna Arrays, Ed. by D. I. Voskresen skii and A. I. Kanashchenkov (Radiotekhnika, Mos cow, 2004), pp. 11–31 [in Russian]. 6. V. Yu. Vvedenskii, A. V. Andriyanov, and E. A. Ermilov, Prib. Tekh. Eksp., No. 1, 114 (1975). 7. V. Yu. Vvedenskii and V. S. Syuvatkin, Radiotekh. Ele ktron. (Moscow), No. 3, 664 (1981). 8. V. Yu. Vvedenskii, A. B. Zuev, D. D. Karimbaev, et al., Prib. Tekh. Eksp., No. 3, 123 (1985). 9. V. Yu. Vvedenskii, V. S. Syuvatkin, and A. A. Khrustalev, Radiotekh. Elektron. (Moscow), No. 10, 2063 (1985). 10. Yu. V. Konev, V. P. Gubanov, A. I. Klimov, et al., Prib. Tekh. Eksp., No. 6, 37 (2011). 11. M. E. Levinshtein, Yu. K. Pozhela, and M. S. Shur, Gunn Effect (Sov. Radio, Moscow, 1975) [in Russian]. 12. S. I. Domrachev and A. A. Kuznetsov, Tech. Phys. 46, 422 (2001). 13. G. I. Veselov, Microelectronic Microwave Devices, Ed. by G. I. Veselov (Vysshaya Shkola, Moscow, 1988), pp. 124–160 [in Russian]. 14. A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Pergamon, Oxford, 1964). 15. J. G. Ruch and G. S. Kino, Phys. Rev. E 174, 921 (1969). 16. V. P. Gubanov, A. I. Klimov, O. B. Koval’chuk, et al., Prib. Tekh. Eksp., No. 5, 95 (2010). 17. V. Yu. Konev, A. I. Klimov, O. B. Koval’chuk, et al., Tech. Phys. Lett. 39, 957 (2013). 18. V. Yu. Konev, Report for RFBR project No. 1208 31171mol_a (2012). 19. M. J. Howes and D. V. Morgan, Microwave Devices (Wiley, New York, 1976).
Translated by V. Isaakyan
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