A method for calculating multi-mode characteristic impedance of slow-wave structures

Abstract

This invention discloses a method for calculating the characteristic impedance of a multi-mode structure applied to slow-wave structures, relating to the field of electromagnetic simulation technology. The method includes: treating each mode of the slow-wave structure as an equivalent port to determine the scattering matrix of the slow-wave structure; transforming the scattering matrix into a transmission matrix using a transformation formula; calculating the eigenvalue equations that electromagnetic waves must satisfy to propagate in the slow-wave structure; solving the eigenvalue equations to obtain eigenvalues ​​and eigenvectors; and calculating the characteristic impedance of the slow-wave structure based on the voltage and current of the dominant mode. In calculating the characteristic impedance of a planar slow-wave structure, this invention introduces multiple higher-order modes at the ports to simulate the coupling effect between periodic units, calculates the transmission matrix of the multi-mode network, and accurately extracts the characteristic impedance from the multi-mode transmission matrix through an innovative calculation method.

Description

Technical Field

[0001] This invention relates to the field of electromagnetic simulation technology, and in particular to a method for calculating the multimode characteristic impedance of slow-wave structures. Background Technology

[0002] Traveling wave tubes (TWTs), as high-power microwave sources, are widely used in communications, radar, satellites, and remote sensing. A TWT consists of an electron gun, a slow-wave structure, magnets, input / output circuitry, and a collector, among which the slow-wave structure is the key component, determining the TWT's performance. To ensure proper operation, the slow-wave structure typically employs a periodic structure to adjust the dispersion of the electromagnetic wave, slowing its phase velocity to synchronize it with the electron beam's velocity. The propagation constant and characteristic impedance are crucial parameters of the slow-wave structure. The propagation constant characterizes the dispersion of electromagnetic wave propagation within the slow-wave structure, determining its phase velocity, loss, and frequency-dependent dispersion. The characteristic impedance of the slow-wave structure determines the ratio of electric field energy to magnetic field energy on the transmission line, providing a crucial reference for the design of the slow-wave structure's feed (input / output). These parameters play a vital role in the overall performance of the TWT. As a special type of transmission line, slow wave structures require impedance matching with input / output circuits. If the characteristic impedance calculation is not accurate enough, the feed structure designed based on the characteristic impedance will cause impedance mismatch, resulting in reflection loss and limiting the operating performance of the device.

[0003] In recent years, research on traveling wave tube (TWT) slow-wave structures has focused on innovation and optimization of dispersion characteristics and coupling impedance, with relatively little research on methods for calculating characteristic impedance. In practical applications of slow-wave structures, it is inevitable to design a feed network and perform impedance matching to ensure proper operation. Currently, the most common feed design method uses commercial software for parameter sweep optimization, which lacks theoretical basis. Therefore, accurate calculation methods for the characteristic impedance of slow-wave structures have significant research value and practical implications.

[0004] The commonly used method for calculating the characteristic impedance of slow-wave structures is to extract it from the transmission matrix of a single-cycle structure. Specifically, ports are added to both ends of a single-cycle slow-wave structure, excitation is performed using a single mode, the transmission matrix is ​​obtained using commercial software, and finally, the characteristic impedance is extracted from the transmission matrix of the slow-wave structure. The accuracy of this method depends on the accuracy of the single-cycle slow-wave structure transmission matrix. When calculating the transmission matrix through single-cycle simulation, the slow-wave structure is truncated. Since there is usually some coupling between the cycles of a slow-wave structure, if the coupling is strong, the transmission matrix obtained from the simulation of the truncated structure will differ from the actual transmission matrix, leading to inaccurate calculated characteristic impedance. An improved approach is to simulate a multi-cycle structure and then extract the single-cycle transmission matrix from the multi-cycle slow-wave structure. While this method can mitigate the impact of truncation to some extent and increase calculation accuracy, it still has significant limitations. On the one hand, the error caused by truncation cannot be completely eliminated, limiting the improvement in accuracy; on the other hand, the process of extracting the single-cycle transmission matrix from the multi-cycle transmission matrix faces the problem of multiple solutions. Therefore, how to accurately calculate the characteristic impedance of slow-wave structures remains a challenge that urgently needs to be addressed in current research. Summary of the Invention

[0005] Purpose of the invention: To address the above problems, the purpose of this invention is to provide a multi-mode characteristic impedance calculation method for slow-wave structures. This method equates the inter-cycle coupling of slow-wave structures to the combined effect of higher-order modes, thereby enabling accurate calculation of the dispersion and characteristic impedance of slow-wave structures. In particular, when the inter-cycle coupling is large, the characteristic impedance of slow-wave structures can be effectively extracted.

[0006] Technical solution: The present invention provides a method for calculating the multi-mode characteristic impedance of slow-wave structures, comprising:

[0007] Each mode of the slow-wave structure is equivalent to a port, and the scattering matrix of the slow-wave structure is determined.

[0008] The scattering matrix is ​​transformed into the transmission matrix using a transformation formula;

[0009] Calculate the eigenvalue equations that electromagnetic waves must satisfy when propagating in a slow-wave structure;

[0010] Solve the eigenvalue equations to obtain eigenvalues ​​and eigenvectors;

[0011] The characteristic impedance of the slow-wave structure is calculated based on the voltage and current of the main mode.

[0012] Furthermore, the slow wave structure is a periodic slow wave structure containing M modes. The periodic slow wave structure includes a physical input port and a physical output port. The entire slow wave structure is equivalent to a 2M port network, where M is a natural number.

[0013] Furthermore, the expression for the scattering matrix of the slow-wave structure is:

[0014] ,

[0015] Here, the subscripts i and o represent the input port and output port, respectively, and each submatrix corresponds to the generalized scattering matrix associated with the M input / output modes, i.e. This represents the reflection coefficient of the input port when the output port is matched. This represents the reflection coefficient of the output port when the input ports are matched. This represents the reverse transmission coefficient from the output port to the input port when the input ports are matched. This represents the forward transmission coefficient from the input port to the output port when the input ports are matched.

[0016] Furthermore, the expression for the transfer matrix is:

[0017] ,

[0018] in, , , , All The matrices are expressed as follows:

[0019] ,

[0020] ,

[0021] ,

[0022] ,

[0023] in, For input impedance, For output impedance, This represents a matrix consisting entirely of 1s.

[0024] Furthermore, the expression for the eigenvalue equation is:

[0025] ,

[0026] in, These are the mode voltages from Mode 1 to Mode M. Here, γ represents the mode currents from mode 1 to mode M, and γ is the complex transport constant. This is one period length of the slow wave structure.

[0027] Beneficial effects: Compared with the prior art, the significant advantages of this invention are:

[0028] 1. When extracting characteristic impedance using a single-cycle structure, the existing technology often neglects the role of higher-order modes. However, this invention equates higher-order modes to ports and introduces them into the solution of the problem, which can accurately calculate the characteristic impedance of slow-wave structures.

[0029] 2. When using a multi-period structure to extract characteristic impedance, the existing technology still cannot fully consider the truncation error and also faces the problem of multiple solutions. However, the present invention uses a single-period structure to extract characteristic impedance, thus avoiding the problem of choosing between multiple solutions.

[0030] 3. The characteristic impedance accurately calculated by this invention provides a basis for the design of the feed network in the practical application of slow wave structures. Attached Figure Description

[0031] Figure 1 This is a flowchart of the present invention;

[0032] Figure 2 This is a schematic diagram of the multi-port equivalent of a slow-wave structure.

[0033] Figure 3 A schematic diagram of a planar spiral slow-wave structure;

[0034] Figure 4 A schematic diagram of a single-period planar spiral slow-wave structure;

[0035] Figure 5 Comparison of simulation results of characteristic impedance of planar helical slow-wave structure;

[0036] Figure 6 A schematic diagram of a 20-period planar spiral slow-wave structure with a feed network;

[0037] Figure 7 A schematic diagram of the design parameters for the power supply network;

[0038] Figure 8 Comparison of simulation results for the reflectance coefficient of a planar helical slow-wave structure;

[0039] Figure 9 This is a schematic diagram of a rectangular ring-bar slow-wave structure.

[0040] Figure 10 A schematic diagram of a single-period rectangular ring rod slow-wave structure;

[0041] Figure 11 Comparison of simulation results of characteristic impedance of rectangular ring rod slow wave structure;

[0042] Figure 12 A schematic diagram of a 40-cycle rectangular ring rod slow-wave structure with a feed network;

[0043] Figure 13Comparison of simulation results for the reflection coefficient of a rectangular ring rod slow-wave structure. Detailed Implementation

[0044] The embodiments of the present invention will be further described in detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are merely illustrative of the present invention and not intended to limit the scope of the invention. Furthermore, it should be noted that, for ease of description, the accompanying drawings show only the parts relevant to the embodiments of the present invention, and not all structures.

[0045] In the following description, specific details such as target system architecture and techniques are set forth for illustrative purposes and not for limitation, in order to provide a thorough understanding of the embodiments of this application. However, those skilled in the art will understand that this application may also be implemented in other embodiments without these specific details. In other instances, detailed descriptions of well-known systems, apparatuses, circuits, and methods are omitted so as not to obscure the description of this application with unnecessary detail.

[0046] It should be understood that, when used in this application specification and the appended claims, the term "comprising" indicates the presence of the described features, integrals, steps, operations, elements and / or components, but does not exclude the presence or addition of one or more other features, integrals, steps, operations, elements, components and / or a collection thereof.

[0047] It should also be understood that the term “and / or” as used in this application specification and the appended claims means any combination of one or more of the associated listed items and all possible combinations, and includes such combinations.

[0048] Furthermore, in the description of this application and the appended claims, the terms "first," "second," etc., are used only to distinguish descriptions and should not be construed as indicating or implying relative importance.

[0049] References to "one embodiment" or "some embodiments" in this specification mean that one or more embodiments of this application include the target features, structures, or characteristics described in connection with that embodiment. Therefore, the phrases "in one embodiment," "in some embodiments," "in other embodiments," "in still other embodiments," etc., appearing in different parts of this specification do not necessarily refer to the same embodiment, but rather mean "one or more, but not all, embodiments," unless otherwise specifically emphasized.

[0050] The flowchart of the multi-mode characteristic impedance calculation method for slow-wave structures described in this embodiment is as follows: Figure 1 As shown, the method includes the following steps:

[0051] Step 1: Equip each mode of the slow-wave structure with an equivalent port to determine the scattering matrix of the slow-wave structure.

[0052] The slow-wave structure is a slow-wave structure containing multiple modes. Multiple higher-order modes are introduced at the port to simulate the coupling effect between periodic units. Each mode is equivalent to a port to determine the scattering matrix of the slow-wave structure.

[0053] Furthermore, the slow wave structure is a periodic slow wave structure containing M modes. The periodic slow wave structure includes a physical input port and a physical output port. The entire slow wave structure is equivalent to a 2M port network, where M is a natural number.

[0054] Combination Figure 2 As shown, this slow-wave structure is a typical M-mode periodic slow-wave structure. This structure has one physical input port and one physical output port. Each mode is equivalent to one port, and the entire structure is equivalent to a 2M-port network. Figure 2 In this context, z represents the direction of electromagnetic wave propagation.

[0055] Step 2: Transform the scattering matrix into the transmission matrix using a transformation formula.

[0056] The scattering matrix and the transmission matrix are both different matrix forms that characterize microwave networks. The transformation relationship between them can be converted from the scattering matrix S to the transmission matrix T using a transformation formula.

[0057] Furthermore, the expression for the scattering matrix of the slow-wave structure is:

[0058] ,

[0059] Here, the subscripts i and o represent the input port and output port, respectively, and each submatrix corresponds to the generalized scattering matrix associated with the M input / output modes, i.e. This represents the reflection coefficient of the input port when the output port is matched. This represents the reflection coefficient of the output port when the input ports are matched. This represents the reverse transmission coefficient from the output port to the input port when the input ports are matched. This represents the forward transmission coefficient from the input port to the output port when the input ports are matched.

[0060] Furthermore, the expression for the transfer matrix is:

[0061] ,

[0062] in, , , , All The matrices are expressed as follows:

[0063] ,

[0064] ,

[0065] ,

[0066] ,

[0067] in, For input impedance, For output impedance, This represents a matrix consisting entirely of 1s.

[0068] Step 3: Calculate the eigenvalue equations that electromagnetic waves must satisfy when propagating in a slow-wave structure.

[0069] Since the slow wave structure is a periodic structure, the Floquet theorem is satisfied during the transmission of each mode.

[0070] Furthermore, electromagnetic waves propagate in slow-wave structures and satisfy the following eigenvalue equations:

[0071] ,

[0072] in, These are the mode voltages from Mode 1 to Mode M. Here, γ represents the mode currents from mode 1 to mode M, and γ is the complex transport constant. This is one period length of the slow wave structure.

[0073] Step 4: Solve the eigenvalue equations to obtain eigenvalues ​​and eigenvectors.

[0074] Step 5: Calculate the characteristic impedance of the slow wave structure based on the voltage and current of the main mode.

[0075] Solve the above eigenvalue equation to obtain the eigenvalues. and eigenvectors In this example, through eigenvalues The transmission constant can be obtained. Furthermore, after obtaining the eigenvectors, only the mode voltage of the main mode, i.e., mode 1, is used. and mode current Calculate the characteristic impedance The calculation formula for the input / output power supply design using this impedance is as follows:

[0076] .

[0077] To further demonstrate the superiority of this invention in calculating the characteristic impedance of slow-wave structures, the following example is provided.

[0078] In one embodiment, the slow-wave structure is a planar helical slow-wave structure, combined with Figure 3 and Figure 4 As shown, the parameters of the planar helical slow-wave structure in a single cycle are: planar helix width w1 = 0.05 mm, slow-wave structure cycle length p1 = 2.8 mm, dielectric substrate material is Rogers RO4003C with a dielectric constant of 3.55, loss tangent of 0.0027, thickness h1 = 0.813 mm, and metal shielding cavities in the x and y directions. The dimensions of other parts are: diameter of the circular patch on the metal pillar surface RD1 = 0.075 mm, diameter of the metal pillar VD1 = 0.05 mm, dielectric substrate width L1 = 15 mm, and distance from the dielectric substrate surface to the metal shielding cavity. .

[0079] The characteristic impedance of the planar helical slow-wave structure was calculated using both existing single-mode calculation methods and the multi-mode calculation method of this invention. The characteristic impedances obtained by the two methods were compared, and the simulation results are as follows: Figure 5 As shown.

[0080] When calculating the characteristic impedance, the existing single-mode method is used. Only mode one needs to be simulated, and the characteristic impedance is calculated using the transmission matrix T. , It can be represented as:

[0081] .

[0082] from Figure 5 It can be seen that the imaginary parts of the characteristic impedance calculated by the two methods are very close, almost zero, but the real parts are different. As the frequency increases, the real part of the characteristic impedance obtained by the single-mode calculation method gradually increases from 310, while the real part of the characteristic impedance obtained by the multi-mode calculation method gradually decreases from 300. At 5 GHz, the characteristic impedances obtained by the existing single-mode calculation method and the multi-mode calculation method of this invention are 333.3 Ω and 285.6 Ω, respectively.

[0083] like Figure 6 As shown, a 20-cycle planar helical slow-wave structure with a feed network is drawn in simulation software, and the feed network is designed. The feed network adopts the form of a microstrip line, with microstrip lines of different widths at both ends on the front side. To reduce the coupling effect between the feed network and the slow-wave structure, a chamfer is applied to the back side, cutting from 8.3 mm away from the slow-wave structure to 0.5 mm away. After the chamfer, the width of the short side is... mm.

[0084] Taking the characteristic impedance of 285.6Ω calculated in 5GHz multi-mode as an example, the design method of the feeder network is explained, such as... Figure 7 The characteristic impedance of the microstrip line with a width wk2 = 1.73 mm is approximately 50 Ω (denoted as ). In order to match the slow-wave structure characteristic impedance of 285.6Ω (denoted as...) Matching requires an impedance of [insert impedance here]. The microstrip line, where lk1 in the figure represents the length of the impedance-matching microstrip line, is used according to the quarter-wavelength impedance transformation principle. It can be represented as: ,in , The calculated result is 119Ω, at which point the corresponding microstrip line width wk1 = 0.2mm.

[0085] Time-domain simulations were performed in simulation software to compare the reflection coefficients of the two calculation methods. At a frequency of 5 GHz, the characteristic impedances obtained by the existing single-mode calculation method and the multi-mode calculation method of this invention are 333.3 Ω and 285.6 Ω, respectively. The corresponding wk1 widths for impedance matching are 0.13 mm and 0.2 mm, respectively. It can be seen that the feed network based on the 285.6 Ω characteristic impedance provides better impedance matching than the feed network based on the 333.3 Ω characteristic impedance. The simulated reflection coefficient is below -20 dB between 4.5 GHz and 5.35 GHz, and becomes the lowest near 5 GHz, which is close to a quarter-wavelength frequency. Figure 8 As shown.

[0086] The simulation results above show that at 5 GHz, the characteristic impedance obtained by using the multi-mode calculation method in this invention for impedance matching results in a lower reflection coefficient (S11 parameter), and the multi-mode calculation method can more accurately calculate the characteristic impedance of slow wave structures.

[0087] In another embodiment, the slow-wave structure is a rectangular ring-bar slow-wave structure, combined with Figure 9 and Figure 10 As shown, in a single-cycle rectangular ring-shaped slow-wave structure, the rectangular ring is made of copper and consists of short-circuit pillars on a dielectric substrate and surface patches. The patch width w2 = 0.4 mm, and the slow-wave structure cycle length p2 = 2.8 mm. The dielectric substrate material is Rogers RO4003C, with a dielectric constant of 3.55, a loss tangent of 0.0027, and a thickness h2 = 0.813 mm. There are metal shielding cavities in the x and y directions. The dimensions of other components are as follows: diameter of the circular patch on the surface of the metal pillar RD2 = 0.6 mm, diameter of the metal pillar VD2 = 0.4 mm, and spacing between the metal pillars... The dielectric substrate width L2 = 15mm, and the distance from the dielectric substrate surface to the metal shielding cavity is... .

[0088] The characteristic impedance of the rectangular ring rod slow-wave structure was calculated using both existing single-mode calculation methods and the multi-mode calculation method of this invention. The characteristic impedances obtained by the two methods were compared, and the simulated structure is shown below. Figure 11 As shown. From Figure 11 It can be seen that the imaginary parts of the characteristic impedance calculated by both methods are very close, almost zero, but the real parts differ at higher frequencies. As the frequency increases, the real part of the characteristic impedance obtained by the single-mode calculation method gradually increases from 160, while the real part of the characteristic impedance obtained by the multi-mode calculation method gradually decreases from 160. For example, at 5 GHz, the real part of the characteristic impedance calculated by the single-mode method is approximately 180, and the real part of the characteristic impedance calculated by the multi-mode method is approximately 145; at 7 GHz, the real part of the characteristic impedance calculated by the single-mode method is approximately 230, and the real part of the characteristic impedance calculated by the multi-mode method is approximately 140.

[0089] At 5 GHz, the characteristic impedances obtained by the single-mode calculation method and the multi-mode calculation method are 181.9 Ω and 146.7 Ω, respectively. Figure 12 As shown, a 40-cycle rectangular ring-shaped slow-wave structure with a feed network is drawn in simulation software, and the feed network is designed. The feed network adopts the form of a microstrip line, with microstrip lines of different widths at both ends on the front side. To reduce the coupling effect between the feed network and the slow-wave structure, a chamfer is applied to the back side, cutting from 3mm away from the slow-wave structure to 0.7mm away. After the chamfer, the width of the short side is... .

[0090] like Figure 13 As shown, at a frequency of 5 GHz, the characteristic impedances obtained by the existing single-mode calculation method and the multi-mode calculation method of this invention are 181.9 Ω and 146.7 Ω, respectively, with corresponding impedance matching widths wk1 of 0.47 mm and 0.61 mm, respectively. It can be seen that the feed network based on the 146.7 Ω characteristic impedance provides better impedance matching than the feed network based on the 181.9 Ω characteristic impedance. The simulated reflection coefficient is below -20 dB between 4.6 GHz and 5.35 GHz, and becomes lowest near 5 GHz, which is close to a quarter-wavelength frequency.

[0091] The simulation results above show that at 5 GHz, using the characteristic impedance obtained by the multi-mode calculation method of this invention for impedance matching results in a lower reflection coefficient (S11 parameter), and the multi-mode calculation method can more accurately calculate the characteristic impedance of slow wave structures.

Claims

1. A method for calculating the multi-mode characteristic impedance of slow-wave structures, characterized in that, include: Each mode of the slow-wave structure is equivalent to a port, and the scattering matrix of the slow-wave structure is determined. The scattering matrix is ​​transformed into the transmission matrix using a transformation formula; Calculate the eigenvalue equations that electromagnetic waves must satisfy when propagating in a slow-wave structure; Solve the eigenvalue equations to obtain eigenvalues ​​and eigenvectors; The characteristic impedance of the slow-wave structure is calculated based on the voltage and current of the main mode. The higher-order modes are equivalent to ports, and the characteristic impedance is extracted using a single-cycle structure. The slow wave structure is a periodic slow wave structure containing M modes. The periodic slow wave structure includes a physical input port and a physical output port. The entire slow wave structure is equivalent to a 2M port network, where M is a natural number. The expression for the scattering matrix of the slow-wave structure is: ， Here, the subscripts i and o represent the input port and output port, respectively, and each submatrix corresponds to the generalized scattering matrix associated with the M input / output modes, i.e. This represents the reflection coefficient of the input port when the output port is matched. This represents the reflection coefficient of the output port when the input ports are matched. This represents the reverse transmission coefficient from the output port to the input port when the input ports are matched. This represents the forward transmission coefficient from the input port to the output port when the input ports are matched. The expression for the transfer matrix is: ， in, , , , All The matrices are expressed as follows: ， ， ， ， in, For input impedance, For output impedance, This represents a matrix consisting entirely of 1s.

2. The method for calculating the multi-mode characteristic impedance of a slow-wave structure according to claim 1, characterized in that, The expression for the eigenvalue equation is: ， in, These are the mode voltages from Mode 1 to Mode M. Here, γ represents the mode currents from mode 1 to mode M, and γ is the complex transport constant. This is one period length of the slow wave structure.

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