RU2136010C1 - Method determining parameters of slow-wave structures - Google Patents

Abstract

FIELD: measurement technology. SUBSTANCE: method involves excitation of slow- wave structure placed in cylindrical cavity resonator which ends are left open in specified frequency range, measurement and recording of all resonance frequencies f_om1 and Q-factors Q_om1 on these frequencies in specified range. Then length of slow- wave system is shortened by value of Δl_sp≤ 0,1l_sp and all resonance frequencies f_om2 in this frequency range are measured again. Moderation factors n_m on resonance frequencies and electrodynamic parameters of slow-down structure are determined by computation by given formulas and dispersion dependencies a=a(f) are plotted, where a is any electrodynamic parameter while measured resonance frequencies f_om are taken as argument f. EFFECT: provision for measurement of resonance frequencies by method of electrodynamic parameters needed for design of slow-wave structures. 5 dwg, 4 tbl

Description

 The present invention relates to measuring technique, more specifically to the field of measurements in microwave electronics. It can be used in measurements of electrodynamic parameters (EHP) of various transmission lines (PL) of electromagnetic waves (EMW), in particular, slowing systems (ZS).

где m - номер заданной дискретной частоты, m = 1,2,3...;
λ_om - длина волны в свободном пространстве, λ_om= c/f_m, с = 3 • 10⁸ м/с, и построение ДХ n=n(f) в заданной полосе по рассчитанным n_m.A known method of measuring the dispersion characteristic (DC) of ES in the mode of standing waves [1, p. 265; 2, p. 410], including the excitation of a shorted circuit at the end by microwave oscillations through a high-frequency measuring path (HFIT) in a given frequency band, determination using a probe inserted into a closed circuit and moving along the closed circuit, adjacent points of the minimum indicator readings on given discrete frequencies f _m located within a given range, and fixing these points X _1m and X _2m , determining the deceleration coefficient n _m at each frequency f _m according to the formula

where m is the number of a given discrete frequency, m = 1,2,3 ...;
λ _om is the wavelength in free space, λ _om = c / f _m , s = 3 • 10 ⁸ m / s, and the construction of the DC n = n (f) in a given band from the calculated n _m .

 A device for measuring HF contains [2, p. 410, Fig. 11.29] a microwave generator, a wave meter connected to the first output of the microwave generator, a variable attenuator connected in series and shorted at the end of the ZS, connected to the second output of the microwave generator, connected in series to the probe introduced into the ZS, the detector section and the indicator device.

The measurement of the deceleration coefficient n _m at a given discrete frequency f _m is as follows. Gather a measurement scheme. Excite ZS at a given frequency f _m within a given frequency range, setting the frequency f _m by the wavemeter. Moving the probe along the GL, find and fix the point X _1m , at which the indications of the indicator device are minimal. Again, move the probe along the ES and find to the right or left of the point X _{1m the} neighboring point X _2m with the minimum indications of the indicator device and fix it. Calculate n _m at a frequency f _m according to the formula (1). Minimum points are measured at other frequencies, n _{m is} calculated at these frequencies. Based on the calculated values of n _m, they construct the DC n = n (f).

 The disadvantage of the analogue is the complexity of the measurements and significant errors associated with the need to coordinate the input of the ES with RFIT before receiving the traveling wave mode in the entire specified frequency range.

где m - номер резонанса, m = 1,2,3...;
λ_om - длина волны в свободном пространстве, соответствующая m-й резонансной частоте, λ_om= c/f_om, с = 3 • 10⁸ м/с;
lsp -длина ЗС,
и построение ДХ ЗС n=n(f) по рассчитанным значениям n_m.There is a known resonant method for measuring the DC of an HF [3, p. 23], including the excitation of a HF placed in a cylindrical cavity resonator (OCR) and shorted to its end walls, by microwave oscillations from the microwave generator in a given frequency range, tuning the frequencies of the microwave generator until it is found the first resonant frequency f ₀₁ and registration of this frequency, tuning the frequency of the microwave generator until the second f ₀₂ , third f ₀₃ , etc. resonance frequencies are found and registered, determining the deceleration coefficient n _m at each of the resonant frequencies f _0m by the formula

where m is the resonance number, m = 1,2,3 ...;
λ _om is the wavelength in free space corresponding to the m-th resonant frequency, λ _om = c / f _om , s = 3 • 10 ⁸ m / s;
lsp is the length of the AP,
and constructing the DC of the LC n = n (f) from the calculated values of n _m .

 A device for measuring DC by a resonant method contains [1, p. 268, Fig. 168] a microwave generator, a wave meter connected to the first output of the microwave generator, serially connected to the first probe, the OCR with a closed circuit shorted to the end walls of the OCR, a second probe, a detector section and an indicator device connected to the second output of the microwave generator, while the connection of the first and the second probes with ES in the CRO carry out weak using communication loops.

The measurement of the deceleration coefficient n _m at the resonant frequencies is as follows. Gather a measurement scheme. The OCR with ES is excited by the microwave generator through the first probe in the form of a communication loop. The microwave generator is tuned in frequency to obtain the first resonance in the ES at a frequency of f ₀₁ . The presence of resonance is determined by the maximum readings of the indicator device. Frequency f _{01 is} recorded using a wave meter. The frequency generator is tuned in frequency to obtain a second f ₀₂ , a third f ₀₃ , etc. resonances in the CS in a given frequency range. Record these frequencies on the wavemeter. The deceleration coefficient n _{m is} determined at each resonant frequency by the formula (2) and the DC is built n = n (f) from the calculated values of n _m .

 The disadvantages of the second analogue include the work only with shorted to the ends OTSR ZS and the inability to determine other ZDZ ZS, necessary during its design, other ZDZ ZS, necessary during its design and operation. However, the second analogue is close in technical essence to the proposed technical solution and is selected by the authors as a prototype.

 The problem to which the invention is directed is the measurement of the EAF of the ES by the resonance method in idle mode (xx) of both ends of the ES.

 The technical result of the proposed solution is the possibility of measuring the resonance method of a number of EHPs necessary for the design of the ES.

β_zm = n_m•β_om (5)
α_om = β_zm/2Q_om (6)
Z_om = R_om/2α_om (7)

L_om = β_zm•Z_om/ω_om (9)
C_om = β_zm/Z_omω_om (10)
где m - номер резонанса, m = 1,2,3...;
lsp₁ - длина замедляющей системы при первом измерении резонансных частот f_0m1 и добротностей Q_0m1;
lsp₂ - длина замедляющей системы при втором измерении резонансных частот f_0m2, после укорочения первоначальной длины lsp₁ на Δl_sp, м,
Δl_кр - кажущееся удлинение замедляющей системы в режиме холостого хода, м;
λ_om - длина волны в свободном пространстве, соответствующая m-й измеренной резонансной частоте, м, λ_om = c/f_om, с = 3 • 10⁸, м/с;
β_zm - фазовая постоянная распространения электромагнитной волны в замедляющей системе, рад/м;
β_om - фазовая постоянная распространения электромагнитной волны в свободном пространстве, рад/м, β_om = 2π/λ_om;
α_om - постоянная затухания электромагнитной волны в замедляющей системе на резонансной частоте, Нп/м;
Z_0m - волновое сопротивление замедляющей системы на резонансной частоте, Ом;
R_0m - заданное погонное сопротивление проводников замедляющей системы на резонансной частоте, Ом/м;
R_cbm - сопротивление связи на оси замедляющей системы на резонансной частоте, Ом;
P_m - поперечное волновое число замедляющей системы на резонансной частоте рад/м,

L_0m - погонная индуктивность замедляющей системы на резонансной частоте, Гн/м;
C_0m - погонная емкость замедляющей системы на резонансной частоте, Ф/м,
и строят дисперсионные зависимости a=a(f), где a - любой из электродинамических параметров, принимая за аргумент f измеренные резонансные частоты f_0m.This technical result is achieved by the fact that in the method for determining the parameters of slowing systems, including the excitation of a slowing system located in a volumetric cylindrical resonator, microwave oscillations in a given frequency range, measuring all resonant frequencies f _0m in this range, determining the deceleration coefficients n _m at resonance frequencies by calculation and construction of dispersion characteristic n = n (f) from the calculated values n _m, new is that the ends of the retardation system, placed in a volume CYLINDRICAL th resonator left open, the measured and record all the resonant frequencies f _0m1 Q and Q _0m1 at these frequencies in a predetermined range, the shortened length of the delay network by an amount Δ lsp ≤ 0,1 lsp measured again and all the resonant frequency f _0m2 in this same range frequencies, determine the electrodynamic parameters of the retarding system according to the formulas:

β _zm = n _m • β _om (5)
α _om = β _zm / 2Q _om (6)
Z _om = R _om / 2α _om (7)

L _om = β _zm • Z _om / ω _om (9)
C _om = β _zm / Z _om ω _om (10)
where m is the resonance number, m = 1,2,3 ...;
lsp ₁ is the length of the slowdown system in the first measurement of the resonant frequencies f _0m1 and Q factors Q _0m1 ;
lsp ₂ is the length of the retarding system in the second measurement of resonance frequencies f _0m2 , after shortening the initial length lsp ₁ by Δl _sp , m,
Δl _cr - the apparent lengthening of the decelerating system in idle mode, m;
λ _om is the wavelength in free space corresponding to the m-th measured resonant frequency, m, λ _om = c / f _om , s = 3 • 10 ⁸ , m / s;
β _zm is the phase constant of propagation of the electromagnetic wave in the slowing system, rad / m;
β _om — phase constant of propagation of an electromagnetic wave in free space, rad / m, β _om = 2π / λ _om ;
α _om is the attenuation constant of the electromagnetic wave in the decelerating system at the resonant frequency, Np / m;
Z _0m is the wave impedance of the slowing system at the resonant frequency, Ohm;
R _0m - the specified linear resistance of the conductors of the slowing system at the resonant frequency, Ohm / m;
R _cbm is the coupling resistance on the axis of the slowing system at the resonant frequency, Ohm;
P _m - the transverse wave number of the slowing system at the resonant frequency rad / m,

L _0m is the linear inductance of the decelerating system at the resonant frequency, GN / m;
C _0m is the linear capacity of the slowing system at the resonant frequency, f / m,
and build the dispersion dependences a = a (f), where a is any of the electrodynamic parameters, taking the measured resonance frequencies f _0m as argument f.

 The set of essential features of the proposed technical solution allows us to measure the listed ED of the ES (4) - (10) and their frequency dependencies along with the DC n = n (f) using the resonance method in xx mode.

In FIG. 1 shows a structural diagram of a device for measuring the EAF of an ES by the proposed method, FIG. 3, 2, 4 - measured (solid lines) and calculated (dashed) frequency dependences of the deceleration coefficient n, wave resistance Z ₀ and attenuation constant α _o, respectively, in FIG. 5 - frequency dependence of the apparent elongation Δl _cr (f).
A device for measuring the ESF of an ES in accordance with the proposed method comprises (Fig. 1) a microwave generator 1, a wave meter 2 connected to the first output of the microwave generator 1, series-connected VCHITZ, the first probe 4, OCR 5 with ZS 6 in xx mode at both ends, the second probe 7, the detector section 8 and the indicator device 9 connected to the second output of the microwave generator 1. The first and second zones 4 and 7 are connected to the OCR 5 and ZS 6 using communication loops 10.

 As a microwave generator 1, wavemeter 2, RFIT 3, detector section 8 and indicator device 9, an industrial meter of complex transmission coefficients P4-37 can be used in the measurement mode of the passage parameters of four-terminal devices [4]; as probes 4 and 7, wire probes with short-circuited communication loops 10 that excite the magnetic components of the fields inside the CCR; as OCR 5, a cylindrical metal resonator of a given length and internal diameter.

The determination of the EAF ES by the proposed method is as follows. The measurement circuit of FIG. 1. On the oscillator of the oscillating frequency of the P4-37 meter, the specified frequency oscillation range ΔF is set, in which the ESF of the ES is measured, for example, ΔF = (100 = 800) MHz and the device switches to the automatic oscillation mode. On the screen of the measuring unit of the P4-37 meter, the amplitude-frequency characteristic (AFC) of the GC 6 transmission coefficient is displayed. The frequencies of all resonances f _0m1 in the oscillation band ΔF are measured and recorded. The resonance frequencies are determined by the maximum frequency response. The Q factors Q _{0m1 are} measured and recorded at all resonant frequencies f _{0m1 by} any known method, for example, by the resonance method [5, p. 58]. Turn off the measurement circuit. Reduce the length of the GL 6 lsp ₁ by the value Δl _sp = (0.5 -0.1) lsp ₁ , get lsp ₂ . Again, the measurement circuit of FIG. 1 and include it in the automatic frequency sweep mode in the same range ΔF _o . The frequencies of all resonances f _0m2 in the ΔF band are measured and recorded by the frequency response of ES 6. Turn off the measurement circuit. The linear resistance R _{om1 of the} conductors of the ЗС 6 is calculated at the resonant frequencies f _{0m1 by} any known method, for example, by the methods of [6, p. 132]. The _{ESF of the ES is} calculated using formulas (3) - (10) for each resonant frequency f _0m1 . The DC is built a = a (f), taking the measured resonant frequencies f _0m1 as the argument f. In order to confirm the feasibility of the proposed method and to achieve a technical result of measurements, a laboratory model for measuring the parameters of a spiral decelerating system (SES) according to the proposed method was manufactured at the enterprise. As a microwave generator 1, wavemeter 2, VCHITZ, detector section 8 and indicator device 9, the P4-37 meter was used in the mode of measuring the passage parameters of the four-terminal devices. CRO laboratory setup is a metal cylinder of length L _i = 220 mm with an inner diameter D _p = 120 mm. Inside the cylinder, an SZS of a copper conductor is coaxially attached using three dielectric plexiglass rods with a diameter of d _st = 8 mm and a length of l _st = 220 mm. The ends of the CES are open, i.e. are in xx mode with respect to the chassis.

The geometrical dimensions of the studied SES are given in table. 1. (see tab. 1-4 at the end of the description)
In the table. 1 marked: D _cf - the average diameter of the SES; d _CR - the diameter of the conductors SZS; lsp ₁ - the length of the SES during the first measurement of resonant frequencies and Q factors; lsp ₂ - the length of the SES in the second measurement of resonant frequencies; N is the number of turns in the CES; h - step (period) of the CES; n _g - geometric slowdown of the SES.

 The results of measurements and calculations of deceleration according to formulas (3) and (11) are given in table. 2 and 3, the calculation of the remaining parameters of the CES is given in Table 4.

In the table. 2-3 are indicated: f ₀₁ - the measured resonant frequencies during the first measurement of the SES with a length of lsp ₁ ; f ₀₂ - the measured resonant frequencies in the second measurement of the CES length lsp ₂ ; λ _o - wavelength in free space corresponding to the measured frequency, λ _o = c / f _o , c = 3 • 10 ⁸ m / s; β _o = 2π / λ _o ; n _1,2 - deceleration coefficients calculated by the formula (11); n = CES deceleration coefficient calculated by formula (3); Δl _cr - the apparent elongation of the CES at resonant frequencies; α _a is the decay constant of EMW in the CES; β _z is the phase constant of EMW propagation in the CES; Z ₀ - wave impedance of the SES; R _St - communication resistance on the axis of the SES; R ₀ , L ₀ , C ₀ - linear resistance, inductance and capacitance of the SES, respectively.

 Note that all EHPs have a dispersion dependence.

In FIG. 3, 2, 4 show the measured (solid lines) and theoretically calculated (dashed) dispersion dependences of the deceleration coefficient n, wave resistance Z ₀ and damping constant α _o . The experimental and theoretically calculated EHFs satisfactorily coincide, the differences in the EHEs are explained by errors in the manufacture and installation of SES in the resonator body, as well as the inaccuracy of the SES theory in the low-frequency region [11, p. 140]. In FIG. 5 shows the frequency dependence of the apparent elongation Δl _{cr of the} CES in xx mode.

 We show that the proposed method is technically implemented, is a technical solution and allows you to measure the ESF of the ES.

где m - номер резонанса, m = 1,2,3...;
λ_om - длина волны в свободном пространстве, соответствующая m-й измеренной резонансной частоте;
lsp - геометрическая длина ЗС.According to [1, p. 267; 2, p. 411; 3, p. 23], the deceleration coefficient n _m with a resonant measurement method is determined by the formula

where m is the resonance number, m = 1,2,3 ...;
λ _om is the wavelength in free space corresponding to the m-th measured resonant frequency;
lsp is the geometric length of the CS.

Formula (11) gives the correct results for the short circuit mode (short circuit) of the AP at both ends. It was experimentally established [8] that in the XX end-of-end mode, there is some “elongation” of the electric length lsp ₉ ЗС compared with the geometric length lsp by a certain value Δl _cr from each end of the ЗС. This “extension” is associated with the run-out of the electromagnetic field beyond the end boundaries of the CS: the field does not break off immediately at the geometric boundary of the CS, but falls to zero at this distance at a certain distance Δl _cr from the boundary of the end of the CS. This phenomenon of field run-out over the end boundaries is also called either the edge effect or the influence of edge capacities. In the XX mode of both ends of the CS, the extension of the CS is 2Δl _cr . For each measurement of the resonance frequencies f _{0m, the} quantity 2Δl _{cr is} unknown and is not taken into account when calculating n _m according to formula (11), which leads to significant calculation errors (see lines n ₁ and n _{2 of} Tables 2-4). To eliminate these errors, it is necessary to take into account the influence of Δl _cr . Accounting for these Δl _cr allows you to make the proposed method.

В формуле (12) учтен краевой эффект в виде слагаемого 2Δl_кр. Однако эта формула не позволяет однозначно определить n_m, так как здесь одно уравнение и два неизвестных: n_m и Δl_крm. Для однозначного определения двух неизвестных нужно второе независимое уравнение. Для получения этого второго уравнения укоротим длину lsp₁ до величины lsp₂ на некоторую необходимую длину Δl_sp, такую, которая изменила бы резонансную частоту f_0m1 на f_0m2, но на которой замедление n_m и кажущееся "удлинение" Δl_крm остались практически такими же, как и при первом измерении f_0m1. Для частоты f_0m2 и соответствующей ей длины волны λ_om2 уравнение (11) можно записать так:

Уравнения (12) и (13) и дают систему из двух уравнений с двумя неизвестными n_m и Δl_крm

Решение этой системы позволяет определить однозначно n_m и Δl_крm

Формулы (15) и (16) суть формулы (3) и (4).Let a resonance frequency f _{0m1 be} measured at a known length lsp ₁ in the XX ZS mode. Given the field run-out by 2Δl _cr , formula (11) should be written as follows

Formula (12) takes into account the edge effect in the form of the term 2Δl _cr . However, this formula does not allow unambiguous determination of n _m , since here there is one equation and two unknowns: n _m and Δl _km . To uniquely determine two unknowns, a second independent equation is needed. To obtain this second equation, we shorten the length lsp ₁ to lsp ₂ by some necessary length Δl _sp , such that it would change the resonant frequency f _0m1 to f _0m2 , but at which the deceleration n _m and the apparent "elongation" Δl _krm remained almost the same , as in the first measurement f _0m1 . For the frequency f _0m2 and the corresponding wavelength λ _om2, equation (11) can be written as follows:

Equations (12) and (13) give a system of two equations with two unknowns n _m and Δl _crm

The solution of this system allows us to uniquely determine n _m and Δl _krm

Formulas (15) and (16) are formulas (3) and (4).

где с = 3 • 10⁸ м/с;
U_ф - фазовая скорость распространения ЭМВ в ЗС;
λ_o - длина волны в свободном пространстве, λ_o= c/f_o;
f₀ - частота;
λ_z - длина волны в ЗС, соответствующая частоте f₀, λ_z= λ_o/n;
β_o= 2π/λ_o;
β_z - фазовая постоянная распространения ЭМВ в ЗС, β_z= 2π/λ_z.
Из формулы (17) следует формула (5) для любой частоты
β_z= β_o•n (18)
В реальной ЗС, как линии передачи с малыми потерями, постоянную затухания α_o можно выразить через погонные параметры R₀, L₀, C₀, G₀ [5, стр. 204; 9, стр. 292]:

где Z₀ - волновое сопротивление ЗС,

R₀ - погонное сопротивление проводников ЗС, рассчитывается, например, по методике [6, стр. 132]. Из формулы (19) следует формула (7) для произвольной заданной частоты
Z_o= R_o/2α_o. (20)
В теории длинных линий доказывается [9, стр. 298], что добротность Q₀ линии на любой частоте f₀ связана с погонными параметрами l₀ и R₀ соотношением

Представим L₀ в формуле (21) следующим образом:

получим

так как

по определению, а 2α_o= R_o/Z_o из формулы (19). Следовательно, из (20), (21) и (22) следует формула (6) для любой частоты
α_o= β_z/2Q_o (23)
Перемножим

на

получим для любой частоты ω_o= 2πf_o,

Отсюда для любой частоты ω_o следует формула (9):
L_o= β_z•Z_o/ω_o (24)
Разделим

на

получим

Отсюда для любой частоты ω_o следует формула (10):
C_o= β_z/Z_o•ω_o (25)
Наконец, в [10] показано, что волновое сопротивление Z₀ ЗС и сопротивление связи R_св на оси той же ЗС связаны соотношением

Формула (26) суть формулы (8).By definition, the deceleration coefficient n ZS is the ratio [6, p. 6]:

where c = 3 • 10 ⁸ m / s;
U _f - phase velocity of propagation of EMW in the AP;
λ _o - wavelength in free space, λ _o = c / f _o ;
f ₀ is the frequency;
λ _z is the wavelength in ZS corresponding to the frequency f ₀ , λ _z = λ _o / n;
β _o = 2π / λ _o ;
β _z is the phase constant of EMW propagation in the CS, β _z = 2π / λ _z .
From formula (17) follows formula (5) for any frequency
β _z = β _o • n (18)
In a real LC, as transmission lines with low losses, the damping constant α _o can be expressed in terms of linear parameters R ₀ , L ₀ , C ₀ , G ₀ [5, p. 204; 9, p. 292]:

where Z ₀ - wave impedance ZS,

R ₀ - linear resistance of the conductors of the AP, calculated, for example, according to the methodology [6, p. 132]. From formula (19) follows formula (7) for an arbitrary given frequency
Z _o = R _o / 2α _o . (20)
In the theory of long lines it is proved [9, p. 298] that the quality factor Q _{0 of a} line at any frequency f ₀ is related to the linear parameters l ₀ and R _{0 by the} relation

We represent L ₀ in formula (21) as follows:

we get

as

by definition, and 2α _o = R _o / Z _o from formula (19). Therefore, from (20), (21) and (22) follows formula (6) for any frequency
α _o = β _z / 2Q _o (23)
Multiply

on the

we obtain for any frequency ω _o = 2πf _o ,

From here for any frequency ω _o follows the formula (9):
L _o = β _z • Z _o / ω _o (24)
Divide

on the

we get

From here for any frequency ω _o follows the formula (10):
C _o = β _z / Z _o • ω _o (25)
Finally, in [10] it was shown that the wave impedance Z ₀ ZS and the bond resistance R _cv on the axis of the same ZS are related by the relation

Formula (26) is the essence of formula (8).

 The analysis and experimental results of the table. 2-4 and FIG. 2 to 5 show that the proposed method meets the criteria of "novelty", "inventive step", is a technical solution, is technically implemented and can be used to measure electrodynamic parameters and characteristics of slowing down systems of microwave electronics.

Sources of information
1. Taranenko Z.I., Trohimenko Y.K. Slow down systems. Kiev, 1965.

 2. Lebedev I.V. Microwave engineering and instruments 4.1. M., High School, 1970.

 3. "Electromagnetic slowdown systems" (method for measuring characteristics). Ed. Doctor of Technical Sciences Deryugina L.N. M., MAI, Oborongiz, 1060.

 4. P4-37. Measuring instrument of complex transmission coefficients, TO and IZ, TsU1, 400.245TO.

 5. Aseev B.P. "Vibrational chains." - M., Communication and Radio, 1955.

 6. Silin R.A., Sazonov V.P. Slow down systems. M., Sov. Radio, 1966.

 7. Zernov N.V., Karpov V.G. Theory of radio circuits. L., Energy, 1972.

 8. Sledkov V.A., Ryazanov V.D. "Measurement of the dispersion of phase velocities in transmission lines", RE, 1989, v. 34, No. 11, pp. 2429-2433.

 9. Losev A.K. Linear electronic circuits. M., Higher school., 1971.

 10. Pchelnikov Yu. N. "On the relationship between wave resistance and coupling resistance", RE, 1983, v. 28, N 10, pp. 1981-1985.

 11. Sovetov N.M. Microwave technology. M., High School, 1976.

Claims (1)

1. A method for determining the parameters of retardation systems, including the excitation of a retardation system located in a cylindrical cavity resonator, microwave oscillations in a given frequency range, the measurement of all resonant frequencies f _om in this range, the determination of deceleration coefficients n _m at resonant frequencies by calculation and the construction of the dispersion characteristics n = n (f) according to the calculated values of n _m , characterized in that the ends of the retardation system located in the cylindrical cavity resonator are left open, measured and register all resonant frequencies f _om1 and Q factors Q _om1 at these frequencies in a given range, shorten the length of the slowing system by Δl _sp ≤ 0.1l _sp and again measure all resonant frequencies f _om2 in the same frequency range, determine the electrodynamic parameters of the slow system formulas

β _zm = n _m • β _om ,
α _om = β _zm / 2Q _om ,
Z _om = R _om / 2α _om ,

L _om = β _zm • Z _om / ω _om ,
C _om = β _zm / Z _om ω _om ,
where m is the resonance number, m = 1,2,3 ...;
l _sp1 is the length of the slowdown system during the first measurement of the resonant frequencies f _om1 and Q factors Q _om1 , m;
l _sp2 is the length of the slowing system in the second measurement of the resonance frequencies f _om2 after shortening the initial length l _sp1 by Δl _sp , m;
Δl _krt - the apparent lengthening of the decelerating system in idle mode, m;
λ _om is the wavelength in free space corresponding to the m-th measured resonant frequency, m, λ _om = c / f _om ;
β _zm is the phase constant of propagation of the electromagnetic wave in the slowing system, rad / m;
β _om — phase constant of propagation of an electromagnetic wave in free space, rad / m, β _om = 2π / λ _om ;
α _om is the attenuation constant of the electromagnetic wave in the decelerating system at the resonant frequency, Np / m;
Z _om - wave resistance of the slowing system at the resonant frequency, Ohm;
R _om - the specified linear resistance of the conductors of the decelerating system at the resonant frequency, Ohm / m;
Rсв _m is the coupling resistance on the axis of the slowing system at the resonant frequency, Ohm;
P _m - the transverse wave number of the slowing system at the resonant frequency, rad / m,

L _om is the linear inductance of the decelerating system at the resonant frequency, GN / m;
C _om is the linear capacity of the slowing system at the resonant frequency, f / m,
and build the dispersion dependences a = a (f), where a is any of the electrodynamic parameters, taking the measured resonance frequencies f _om as argument f.

Images