CN102298658B - Method for simulating beam wave interaction of traveling wave tubes - Google Patents

Abstract

The invention belongs to a traveling wave tube simulation technology and discloses a method for simulating the dominant wave interaction of traveling wave tubes. Specific to the problems existing in the process of simulating the dominant wave interaction of various traveling wave tubes with the conventional semi-analytical and semi-numerical parameter theoretical model, high-frequency field distributions in high-frequency structures of traveling wave tubes are obtained with a pure numerical method in the method, and simultaneous differential equations are established by combining a high-frequency field equation, a particle phase equation and a motion equation, so that a dominant wave interaction parameter theoretical model suitable for various traveling wave tubes is established. The method can be used for simulating the dominant wave interaction processes in various traveling wave tubes, has high efficiency of the parameter theoretical model, and can be applied to numerical simulation of the interactions of electron beams and high-frequency fields in various traveling wave tubes with periodic high-frequency structures.

Description

Simulation method for wave injection interaction of traveling wave tube

Technical Field

The invention belongs to the traveling wave tube simulation technology, and particularly relates to a method for simulating the wave injection interaction of a traveling wave tube.

Background

The traveling wave tube is one of the most widely used vacuum electronic devices, and is widely applied to the fields of satellite communication, radar, electronic countermeasure and the like. At present, the adoption of Computer Aided Design (CAD) technology is one of the main means for saving cost, improving Design and improving the overall performance of the traveling wave tube. In the traveling wave tube CAD technology, the method has important significance for analyzing the interaction between the electron beam in the traveling wave tube and the high-frequency electromagnetic field (namely the wave injection interaction). The beam interaction in the traveling wave tube is a self-consistent process: the high-frequency electromagnetic field in the high-frequency structure of the traveling wave tube and the space charge field generated by the particle clustering push the particles to move, the particle movement forms high-frequency current, and the high-frequency current excites the high-frequency electromagnetic field in the high-frequency structure in turn.

At present, two methods are mainly used for simulating the wave injection interaction process in the traveling wave tube, namely a pure numerical particle simulation method and a parameter theoretical model utilizing semi-analytic semi-numerical values. The pure numerical particle simulation method has universality and can be used for simulating the wave injection interaction process of various traveling wave tubes, but the calculation is long in time consumption and high in memory consumption, and hours or even days are often needed for calculating the wave injection interaction process once. The parameter theoretical model only needs a few minutes or even tens of seconds to calculate the one-time wave injection interaction process. However, for different types of traveling-wave tubes, such as a helix traveling-wave tube, a coupled cavity traveling-wave tube, a folded waveguide traveling-wave tube, etc., the difference of the high-frequency structures thereof causes inconsistent electromagnetic field distribution, so that different theoretical models of the wave-injection interaction parameters need to be established. For some traveling wave tubes with complex high-frequency structures such as a coupled cavity structure and a folded waveguide structure, an effective wave injection interaction parameter theoretical model cannot be established.

Disclosure of Invention

The invention aims to solve the problems of the existing semi-analytic half-numerical parameter theoretical model in simulating various traveling wave tube wave injection interaction processes, and provides a traveling wave tube wave injection interaction simulation method.

The technical scheme of the invention is as follows: a method for simulating the wave-injecting interaction of a traveling wave tube comprises the following steps:

A. obtaining a high-frequency field equation according to a generalized high-frequency field expression in the high-frequency structure of the traveling wave tube;

B. acquiring passive high-frequency field distribution in the high-frequency structure of the traveling wave tube by using a pure numerical method;

C. calculating a space charge field;

D. using a Lorentz force equation to push the particles to move, and obtaining a phase equation and a motion equation of the particles;

E. and (4) establishing a differential equation set for describing the wave-injection interaction process of the traveling wave tube by using the high-frequency field equation, the phase equation of the particles and the motion equation obtained in the step A, B, C, D, setting an interaction initial condition, and solving the established differential equation set until the wave-injection interaction is finished, so that the simulation of the wave-injection interaction process can be completed.

The invention has the beneficial effects that: the method of the invention obtains passive high-frequency field distribution in the high-frequency structure of the traveling wave tube by utilizing a pure numerical method, and constructs a differential equation set by combining a high-frequency field equation, a phase equation of particles and a motion equation, namely, establishes a wave injection interaction parameter theoretical model suitable for various traveling wave tubes

Drawings

FIG. 1 is a schematic flow diagram of the process of the present invention.

FIG. 2 is a schematic diagram of the distribution of power of the fundamental wave 5GHz and the harmonic wave 10GHz of the spiral traveling wave tube solved by the Christine theoretical model.

FIG. 3 is a schematic diagram showing the comparison of the saturation output power of the coupled cavity traveling wave tube solved by the method of the present invention and the test result.

Detailed Description

The invention is further described with reference to the following figures and specific examples.

The flow schematic diagram of the simulation method of the traveling wave tube wave-injection interaction is shown in fig. 1, and the method comprises the following steps:

A. and obtaining a high-frequency field equation according to the generalized high-frequency field expression of the high-frequency structure of the traveling wave tube.

In the traveling wave tube high frequency structure having axial periodicity, the generalized high frequency electromagnetic field can be expressed as formula (1.1)

$\left\{\begin{array}{c}{E}_{\mathrm{rf}}(x_{\&perp;},z,t)=a\left(z\right)/\sqrt{\stackrel{\&OverBar;}{{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="HDA0000084077250000012.png,HDA0000084077250000021.png"\; state="\{\{state\}\}">p</figure-callout>}}_0}}e(x_{\&perp;},z)\exp (-\mathrm{i\&omega;t})\\ H_{\mathrm{rf}}(x_{\&perp;},z,t)=a\left(z\right)/\sqrt{\stackrel{\&OverBar;}{{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="HDA0000084077250000012.png,HDA0000084077250000021.png"\; state="\{\{state\}\}">p</figure-callout>}}_0}}h(x_{\&perp;},z)\exp (-\mathrm{i\&omega;t})\end{array}\right.$ Formula (1.1)

The first formula in equation (1.1) is the high frequency electric field E_rfSecond expression is a high-frequency magnetic field H_rfThe expression of (1); where vector x represents the spatial position vector, subscript "-" represents the lateral component, and variables z and t represent axial position and time, respectively.

a (z) is the high frequency field amplitude, which is a complex function of the axial position variable z; using the relation p (z) ═ a (z) according to the high-frequency field amplitude a (z)²Power can be obtained.

Is the intrinsic power, e (x)_⊥Z) and h (x)_⊥And z) are distribution functions of a passive high-frequency electric field and a passive high-frequency magnetic field respectively, and satisfy a passive Maxwell equation set. i is an imaginary factor, ω -2 π f is an angular frequency, and f denotes an eigenfrequency.

Applying the high-frequency electric field and high-frequency magnetic field expression in the formula (1.1) to an active Maxwell equation set, utilizing the Poynting theorem, simultaneously averaging in a high-frequency structure period l and a time period T of 2 pi/omega, and finally obtaining a high-frequency field equation which is satisfied by the high-frequency field amplitude a (z):

$[\frac{d}{\mathrm{dz}}+\mathrm{\&sigma;}\left(z\right)]a\left(z\right)=-\frac{1}{2\sqrt{\stackrel{\&OverBar;}{{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="HDA0000084077250000012.png,HDA0000084077250000021.png"\; state="\{\{state\}\}">p</figure-callout>}}_0}}}\underset{z}{\overset{z+l}{\&Integral;}}\frac{\mathrm{dz}}{l}\underset{t}{\overset{t+T}{\&Integral;}}\frac{\mathrm{dt}}{T}\underset{S}{\&Integral;\&Integral;}\mathrm{dsj}(x_{\&perp;},z)\&CenterDot;e^{*}(x_{\&perp;},z)\exp \left(\mathrm{i\&omega;t}\right)$ formula (1.2)

Wherein,

is a differential operation with respect to the axial position z. σ (z) is an attenuation amount of the high-frequency field per unit length, i.e., an attenuation coefficient; l is the axial period length of the high-frequency structure, T2 pi/omega is the time period corresponding to the signal frequency,

representing the integral along the cross section of the high frequency structure. j (x)_⊥And z) is the current density distribution, derived from the state of motion of the particles. Upper label "^*"means taking conjugation, the physical meaning of other symbols is the same as in equation (1.1).

The formula (1.2) has universality, is suitable for travelling wave tubes with various high-frequency structures, and is only suitable for the high-frequency structures with different types and passive high-frequency electric field distribution e (x)_⊥Z) have different distribution morphologies.

B. And acquiring passive high-frequency field distribution in the high-frequency structure of the traveling wave tube by using a pure numerical method.

Knowing from step A that the high-frequency field equation is required to be solved, the passive high-frequency electric field distribution e (x) in the periodic structure of the traveling wave tube needs to be obtained_⊥Z). However, the high-frequency structures adopted by different types of traveling-wave tubes are different, resulting in a passive high-frequency electric field distribution e (x) inside the high-frequency structure_⊥Z) there is no uniform analytical expression. Therefore, instead of analyzing the field distribution e (x), consider using a pure numerical method to obtain the numerical field distribution in the high-frequency structure of any traveling-wave tube_⊥Z) making the theoretical model versatile.

The pure numerical method can be realized by three-dimensional electromagnetic field numerical simulation software or experimental test of a passive high-frequency field in a high-frequency structure.

The traveling wave tube has a complex structure and a long interaction region, passive high-frequency field distribution in the whole interaction region is obtained through three-dimensional electromagnetic field numerical simulation software or experimental tests of a passive high-frequency field, and the consumption of resources such as memory and time is high.

Here, as a preferred scheme, the characteristic of the periodicity of the traveling wave tube high-frequency structure may be utilized, the numerical high-frequency field distribution in one axial period of the traveling wave tube high-frequency structure is obtained through three-dimensional electromagnetic field numerical simulation software or experimental tests of a passive high-frequency field, and then the numerical high-frequency field distribution at any position in the traveling wave tube high-frequency structure is obtained by using the frocquie theorem.

According to Froquini theorem, in periodic structures

e(x_⊥，z)＝e(x_⊥，ρ)exp[(n-1)iφ]Formula (1.3)

In equation (1.3), φ represents a monocycle phase shift corresponding to an angular frequency ω; the integer n represents the number of high-frequency structural cycles of the current axial position z, rho represents the position offset of the starting position of the space cycle of z and z, and the physical quantities satisfy the following conditions: and z is (n-1) l + rho, wherein l is the axial period length of the high-frequency structure.

C. Calculating a space charge field;

in the present embodiment, a space charge wave model is used to calculate the space charge field.

In the one-dimensional case, the axial space charge field can be obtained by summing the m space charge waves to obtain the real part, i.e.

$E_{\mathrm{sc},z}=-\mathrm{Re}\underset{m}{\mathrm{\&Sigma;}}\frac{8I_{b}i}{{\mathrm{\&omega;}}_m(r_{\mathrm{bo}}^2-r_{\mathrm{bi}}^2)}{R}_m<e^{\mathrm{im\&omega;t}}>e^{-\mathrm{im\&omega;t}}$ Formula (1.4)

In the formula (1.4), E_sc，zIs an axial space charge field; m is the number of space charge waves, omega_mIs the angular frequency of m space charge waves and satisfies omega_m＝mω；r_biAnd r_boThe inner radius of the electron beam and the outer radius of the electron beam are respectively; i is_bThe current is the electron beam current; i is an imaginary factor; r_mA plasma frequency reduction factor of the m-th space charge wave

$R_m=1-\frac{2}{(r_{\mathrm{bo}}^2-r_{\mathrm{bi}}^2)}\left\{\right[r_{\mathrm{bo}}{K}_1\left({\mathrm{\&gamma;}}_m^{\left(e\right)}{r}_{\mathrm{bo}}\right)+r{I}_1\left({\mathrm{\&gamma;}}_m^{\left(e\right)}r\right){|}_{r_{\mathrm{bi}}}^{r_{\mathrm{bo}}}{K}_0\left({\mathrm{\&gamma;}}_m^{\left(e\right)}{r}_s\right)/I_0\left({\mathrm{\&gamma;}}_m^{\left(e\right)}{r}_s\right)].$ Formula (1.5)

$\left[r{I}_1\left({\mathrm{\&gamma;}}_m^{\left(e\right)}r\right){|}_{r_{\mathrm{bi}}}^{r_{\mathrm{bo}}}\right]-r_{\mathrm{bi}}{I}_1\left({\mathrm{\&gamma;}}_m^{\left(e\right)}{r}_{\mathrm{bi}}\right)\left[r{K}_1\left({\mathrm{\&gamma;}}_m^{\left(e\right)}r\right){|}_{r_{\mathrm{bi}}}^{r_{\mathrm{bo}}}\right]\}$

In the formula (1.5), r is the radial position, I₀(. and I)₁(. is) the first-order and zeroth-order metamorphosis Bessel function, K₀(. and K)₁(. cndot.) are the zeroth and first order second class metamorphosis Bessel functions, respectively.

Is a space charge wave propagation factor, satisfies

${\mathrm{\&gamma;}}_m^{\left(e\right)}=m{[{\mathrm{<figure-callout\; id="0"\; label="k"\; filenames="HDA0000084077250000012.png,HDA0000084077250000021.png"\; state="\{\{state\}\}">k</figure-callout>}}_0^2-{(\mathrm{\&omega;}/c)}^2]}^{1/2}$ Formula (1.6)

Wherein k is₀＝ω/v₀，v₀The initial axial velocity of the macro-particles, c the speed of light,<e^imωt>the average excitation of the m-th space charge wave by all macro-particles at the axial position z satisfies

$<e^{\mathrm{im\&omega;t}}>=\frac{1}{N_{\mathrm{\&lambda;}}}\underset{v=1}{\overset{N_{\mathrm{\&lambda;}}}{\mathrm{\&Sigma;}}}{e}^{\mathrm{im\&omega;}{t}_{\mathrm{zv}}}$ Formula (1.7)

Wherein N is_λIs the total number of macro particles, t_zvThe time when the v-th macro particle reaches the axial position z.

The calculation of the space charge field in two or three dimensions is similar and will not be described in detail.

D. And (4) pushing the particles to move by using a Lorentz force equation to obtain a phase equation and a motion equation of the particles.

Under the action of the high-frequency electromagnetic field and the space charge field, the particles are pushed to move according to the Lorentz force equation, and the equation describing the particle motion comprises a phase equation and a motion equation.

In the one-dimensional case, the phase equation for the v-th particle is:

$\frac{d{\mathrm{\&psi;}}_v}{\mathrm{dz}}=\frac{\mathrm{\&omega;}}{v_{z,v}}$ formula (1.8)

Wherein,

is a differential operation with respect to the axial position z, #_vIs the phase of the v-th particle, ω is the angular frequency, v_z，vIs the axial velocity of the v-th particle.

The equation of motion of the particles under one-dimensional condition obtained from the lorentz force equation is as follows:

$m_0{c}^2\frac{d{\mathrm{\&gamma;}}_v}{\mathrm{dz}}=-\left|e\right|\mathrm{Re}[\frac{a\left(z\right)}{\sqrt{\stackrel{\&OverBar;}{{\mathrm{<figure-callout\; id="0"\; label="p"\; filenames="HDA0000084077250000012.png,HDA0000084077250000021.png"\; state="\{\{state\}\}">p</figure-callout>}}_0}}}{<\hat{z}\&CenterDot;e(x_{\&perp;},z)>}_{\mathrm{beam}}{e}^{-i{\mathrm{\&psi;}}_v}-\frac{8I_{b}i}{(r_{\mathrm{bo}}^2-r_{\mathrm{bi}}^2)}\underset{m}{\mathrm{\&Sigma;}}\frac{R_m}{{\mathrm{\&omega;}}_m}{e}^{-\mathrm{im}{\mathrm{\&psi;}}_v}<e^{\mathrm{im}{\mathrm{\&psi;}}_v}>]$ equation (1.9)

Wherein e and m₀Charge and static mass of electrons, respectively, gamma_vIs a relativistic factor of macro particles.

The first term on the right of equation (1.9) is the high frequency electric field force to which the macroparticle is subjected,

is the average value of the axial electric field on the electron beam cross section

${<\hat{z}\&CenterDot;e(x_{\&perp;},z)>}_{\mathrm{beam}}=\frac{1}{\mathrm{\&pi;}(r_{\mathrm{bo}}^2-r_{\mathrm{bi}}^2)}\underset{S_{\mathrm{beam}}}{\&Integral;\&Integral;}\mathrm{ds}\hat{z}\&CenterDot;e(x_{\&perp;},z)$ Formula (1.10)

Wherein S is_beamShowing the cross-section of the electron beam,is an axial unit vector.

The second term on the right of equation (1.9) is the space charge force experienced by the macro-particles.

Corresponding to equation (1.7), is the average excitation of the m-th space charge wave by all the macro-particles, i.e.:

$<e^{\mathrm{im\&omega;}{t}_{\mathrm{zv}}}>=<e^{\mathrm{im}{\mathrm{\&psi;}}_v}>=\frac{1}{N_{\mathrm{\&lambda;}}}\underset{v=1}{\overset{N_{\mathrm{\&lambda;}}}{\mathrm{\&Sigma;}}}{e}^{\mathrm{im\&omega;}{t}_{\mathrm{zv}}}=\frac{1}{N_{\mathrm{\&lambda;}}}\underset{v=1}{\overset{N_{\mathrm{\&lambda;}}}{\mathrm{\&Sigma;}}}{e}^{\mathrm{im}{\mathrm{\&psi;}}_v}$

the other symbols appearing in equation (1.9) are identical to those appearing in the previous equations and will not be elaborated upon here.

The equations of motion of the particles in the two-dimensional or three-dimensional case are similar and will not be described in detail.

E. And (3) establishing a differential equation set for describing the wave-injection interaction process of the traveling wave tube by using the high-frequency field equation, the phase equation and the motion equation of the particles obtained in the step A, B, C, D, namely establishing the differential equation set by using simultaneous formulas (1.2), (1.8) and (1.9), setting an interaction initial condition, and solving the established differential equation set until the wave-injection interaction is finished, so that the simulation of the wave-injection interaction process can be completed.

Here, the established system of differential equations may be solved step by step using the longguratan method. To solve the established system of differential equations, initial conditions must be set. The initial conditions for each differential equation are set as follows:

for the high frequency field equation, the initial conditions of the high frequency field amplitude are:

$\mathrm{Rea}(z=0)=\sqrt{p_{\mathrm{in}}}\cos \left({\mathrm{\&theta;}}_{\mathrm{in}}\right)$ formula (1.11)

$\mathrm{Ima}(z=0)=\sqrt{p_{\mathrm{in}}}\sin \left({\mathrm{\&theta;}}_{\mathrm{in}}\right)$

In the formula (1.11), p_inAnd theta_inRespectively the power and phase of the input signal.

In the one-dimensional case, N is divided within the period length of one high-frequency structure_zEach grid is provided with N_λParticles, total N_z×N_λEach particle corresponds to one phase equation and one motion equation. N on each grid_λParticles having an initial phase uniformly distributed within 0 to 2 pi, all particles having the same initial relativistic factorNamely, the initial phase condition and the initial relativistic factor condition as follows.

The initial phase conditions of the particle phase equation are:

${\mathrm{\&psi;}}_v=\frac{2\mathrm{\&pi;}}{N_{\mathrm{\&lambda;}}}v,v=(\mathrm{0,1,2},...N_{\mathrm{\&lambda;}}-1)$ formula (1.12)

The initial relativistic factor condition of the particle equation of motion is:

${\mathrm{\&gamma;}}_0=\frac{-\left|e\right|}{m{c}^2}{U}_b$ formula (1.13)

Wherein, U_bThe electron beam voltage.

FIG. 2 is a schematic diagram of the power distribution along the axis of the fundamental wave 5GHz and the second harmonic 10GHz, which are solved when the method and Christine theoretical model of the invention are used for simulating the wave injection interaction of the helix traveling wave tube.

FIG. 3 is a schematic diagram of solving the saturated output power of the coupled cavity traveling-wave tube and comparing the test results.

In fig. 2, curve 1 is the output power of the 5GHz signal calculated using the method of the present invention, curve 2 is the power of the harmonic excitation of 10GHz calculated using the method of the present invention, curve 3 is the output power of the 5GHz signal calculated using the Christine theoretical model, and curve 4 is the power of the harmonic excitation of 10GHz calculated using the Christine theoretical model.

In fig. 3, curve 5 is the saturation output power calculated using the method of the present invention, and curve 6 is the saturation output power of the actual coupling cavity test.

It can be seen from fig. 2 that curves 1 and 3, and curves 2 and 4 are respectively overlapped, and when the wave injection interaction of the helix traveling wave tube is calculated, the calculation result of the method is consistent with that of the Christine theoretical model. Fig. 3 shows that curves 5 and 6 are very close to each other, and the difference is caused by factors such as process errors, which indicates that the method can also be applied to the injection wave interaction simulation of the coupled cavity traveling wave tube with a complex structure, and the accuracy is high.

The method of the invention obtains the high-frequency field distribution in the high-frequency structure of the traveling wave tube by utilizing a pure numerical method, and constructs a differential equation set by combining a high-frequency field equation, a phase equation of particles and a motion equation, namely, establishes a wave injection interaction parameter theoretical model suitable for various traveling wave tubes.

The simulation method of the invention also belongs to a semi-analytic semi-numerical parameter theoretical model, has high calculation speed, and can calculate the simulation of the wave injection interaction process only in a few minutes on a common PC, so that a traveling wave tube designer can quickly and effectively design and optimize various traveling wave tubes, the working performance of the traveling wave tubes is improved, and the research and development period and the cost are greatly reduced.

It will be appreciated by those of ordinary skill in the art that the embodiments described herein are intended to assist the reader in understanding the principles of the invention and are to be construed as being without limitation to such specifically recited embodiments and examples. Those skilled in the art can make various other specific changes and combinations based on the teachings of the present invention without departing from the spirit of the invention, and these changes and combinations are within the scope of the invention.

Claims (4)

1. A simulation method for the wave injection interaction of a traveling wave tube is characterized by comprising the following steps:

A. obtaining a high-frequency field equation according to a generalized high-frequency field expression in the high-frequency structure of the traveling wave tube, wherein the high-frequency field equation specifically comprises the following steps:

$[\frac{d}{\mathrm{dz}}+\mathrm{\&sigma;}\left(z\right)]a\left(z\right)=-\frac{1}{2\sqrt{\stackrel{\&OverBar;}{p_0}}}\underset{z}{\overset{z+l}{\&Integral;}}\frac{\mathrm{dz}}{l}\underset{t}{\overset{t+T}{\&Integral;}}\frac{\mathrm{dt}}{T}\&Integral;\underset{S}{\&Integral;}\mathrm{dsj}(x_{\&perp;},z)\&CenterDot;e^{*}(x_{\&perp;},z)\exp \left(\mathrm{i\&omega;t}\right),$

wherein,

is a differential operation with respect to the axial position z, and σ (z) is an attenuation amount, i.e., an attenuation coefficient, of the high-frequency field per unit length; l is the axial period length of the high frequency structure, T =2 pi/ω is the time period corresponding to the signal frequency,

representing the integral along the cross-section of the high-frequency structure, j (x)_⊥Z) is the current density distribution, obtained from the state of motion of the particles, superscript "^*"represents taking conjugation, subscript" - "represents a lateral component, variables z and t represent axial position and time, respectively, and a (z) is a high-frequency field amplitude and is a complex function of an axial position variable z;

is the intrinsic power, e (x)_⊥Z) is the distribution function of the passive high-frequency electric field, i is an imaginary factor, ω =2 π f is the angular frequency, f denotes the eigenfrequency;

B. acquiring passive high-frequency field distribution in the high-frequency structure of the traveling wave tube by using a pure numerical method;

C. calculating a space charge field;

D. using a Lorentz force equation to push the particles to move, and obtaining a phase equation and a motion equation of the particles;

E. and (4) establishing a differential equation set for describing the wave-injection interaction process of the traveling wave tube by using the high-frequency field equation, the phase equation of the particles and the motion equation obtained in the step A, B, C, D, setting an interaction initial condition, and solving the established differential equation set until the wave-injection interaction is finished, so that the simulation of the wave-injection interaction process can be completed.

2. The method for simulating traveling wave tube beam interaction according to claim 1, wherein the pure numerical method in step B is specifically: numerical value high-frequency field distribution in one axial period of the traveling wave tube high-frequency structure is obtained through three-dimensional electromagnetic field numerical simulation software or experimental tests of a passive high-frequency field, and then numerical value high-frequency field distribution at any position in the traveling wave tube high-frequency structure is obtained by utilizing the Frouq's theorem.

3. The method according to claim 2, wherein the space charge field calculated in step C is calculated by using a space charge wave model.

4. The method for simulating traveling wave tube beam interaction according to claim 2 or 3, wherein the set of differential equations created by solving in step E is implemented by Runge Kutta.

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