High frequency circuit of extended interaction klystron and design method thereof
By optimizing the position and radius of the electron beam channel, stable operation at high power was achieved, solving the problem of stable operation of single mode in existing technologies.
Patent Information
- Application Number
- CN202210097020.X
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2022-01-26
- Publication Date
- 2026-02-13
- Estimated Expiration
- 2042-01-26
AI Technical Summary
In the existing technology, the technical problem that high-power terahertz klystrons cannot effectively solve in the frequency band is how to achieve stable single-mode operation.
This technology employs a dual-symmetric electron beam channel and multiple resonant cavities design, and optimizes the position and radius of the electron beam channels to achieve stable single-mode operation.
Efficient technical means were implemented, and stable high-power operation was achieved by optimizing the position and radius of the electron beam channel.
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Figure CN114639581B_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The present application relates to the field of microwave vacuum electron devices, in particular to an extended interaction circuit of high-power terahertz velocity modulation tube. BACKGROUND
[0002] Terahertz technology has important application value in the field of future communication and radar. At present, the main obstacle restricting its development is that the power level of the radiation source is far from meeting the requirements of application scenarios. Although compared with solid-state devices, vacuum electron devices have obvious advantages in power level, however, when the working frequency is raised to the terahertz frequency band, due to the influence of the scale effect, the power level of the device is still seriously reduced.
[0003] The extended interaction technology is a kind of distributed interaction technology with multi-gap resonant cavity as the interaction circuit. It breaks through the limitation of the single-gap resonant cavity gap transit angle on the characteristic impedance (R / Q) improvement in the traditional velocity modulation tube, and has the advantages of R / Q proportional to the number of gaps, high unit length gain, large power capacity and high efficiency. In theory, any section of slow wave structure with both ends short-circuited to form a periodic resonant structure can be used as an extended interaction circuit. However, in the terahertz frequency band, considering the problem of machining implementation, generally the ladder circuit is adopted. Figures 1A-1C An extended interaction velocity modulation tube is shown, which includes an input resonant cavity 1, a plurality of intermediate resonant cavities 2 and an output resonant cavity 3. The cross section of each resonant cavity is a dumbbell-shaped structure with equal area, and the longitudinal section has a plurality of interaction gaps, which is called a multi-gap cavity. These interaction gaps are electrically and magnetically coupled through the connected coupling cavities, and in the whole they exhibit a certain working mode. However, the number of modes is quite large, and they exist in the longitudinal and transverse directions at the same time. Therefore, preventing mode competition and maintaining single-mode working stability are the core problems in the design of the interaction system based on multi-gap cavities.
[0004] Traditionally, the extended interaction circuit works in the fundamental mode, that is, it has the maximum field strength in the center, as shown in Figure 2 Due to the overall device architecture and focusing system, the electron beam is generally placed in the center of the structure. However, when working in the fundamental mode, the transverse size of the terahertz circuit is very small. Taking the 220GHz frequency band as an example, under the condition of the fundamental mode, the transverse size of the circuit AH1 is about 0.7-0.8mm. The corresponding electron beam channel and the size of the electron beam are also smaller, thus limiting the allowed working current. This is the fundamental factor leading to the serious reduction of the power level of the device.
[0005] In recent years, researchers have begun to consider using high-order modes to solve this problem. At the same frequency, the circuit operating in high-order modes can have a larger lateral size. This not only increases the electron beam size and operating current, thereby increasing the power level, but also has a stronger heat treatment capability. However, from the existing research examples, it can be found that even if high-order modes are used, the interaction circuit still uses a symmetric mode with a maximum central field strength. According to the mode classification, these modes are all odd modes, that is, TM 11 , TM 31 ,... TM 2n+1,1 modes. However, taking TM 31 mode as an example, using a single electron beam, the power level cannot be greatly improved; using three electron beams, other high-order modes are easily excited, and thus the mode competition problem is serious. In order to obtain single-mode stable operation, the interaction circuit often needs to use external means such as dielectric loading, which reduces the reliability of the device operation.
[0006] Therefore, it is necessary to provide a high-power terahertz klystron that can stably operate. SUMMARY
[0007] To achieve the above object, one aspect of the present application provides an extended interaction klystron high-frequency circuit, comprising an electron beam channel and a plurality of resonant cavities,
[0008] The electron beam channel comprises a first electron beam channel and a second electron beam channel,
[0009] The plurality of resonant cavities comprises an input resonant cavity, a plurality of intermediate resonant cavities and an output resonant cavity, each resonant cavity comprising a first coupling cavity and a second coupling cavity symmetrically arranged with respect to the first and second electron beam channels, and a plurality of gap cavities respectively connecting the first and second electron beam channels and the first and second coupling cavities,
[0010] The positions and radii of the electron beam channels are such that the resonant cavities operate in TM 21 modes.
[0011] Preferably, each gap cavity in the high-frequency circuit has the same structure.
[0012] Preferably, the number of gap cavities in each resonant cavity is the same or different.
[0013] Preferably, the resonant cavities have a dumbbell-shaped cross section, and the electron beam channels extend perpendicular to the resonant cavity cross section.
[0014] Preferably, the lateral width of the gap cavities is such that the TM 21 mode of the resonant cavities has an ideal mode separation from other modes.
[0015] Preferably, the channel spacing and channel radius of the first and second electron beam channels are such that the resonant cavity TM 21 The characteristic impedance of the mode is higher than that of other modes.
[0016] Preferably, the channel spacing and channel radius of the first and second electron beam channels are such that the first and second electron beam channels are located in the peak area of transverse electric field of the resonant cavity TM 21 mode.
[0017] Preferably, the high frequency circuit further comprises an input waveguide in communication with the input resonant cavity and an output waveguide in communication with the output resonant cavity.
[0018] Preferably, the output waveguide is coupled to the penultimate gap cavity of the output resonant cavity.
[0019] According to another aspect of the present application, there is provided a method of designing a high frequency circuit of an extended interaction klystron as described above, the method comprising the steps of
[0020] 1) plotting curves of different mode frequencies versus gap cavity transverse width, and determining the gap cavity transverse width according to the mode separation;
[0021] 2) adjusting the channel spacing and channel radius of the two electron beam channels, and plotting curves of resonant cavity characteristic impedance of different modes versus electron beam channel spacing;
[0022] 3) determining the electron beam channel radius and channel spacing based on the plotted curves, so that the resonant cavity TM 21 The characteristic impedance of the mode is higher than that of other modes.
[0023] The present application proposes an extended interaction klystron high frequency circuit operating in an anti-symmetrical mode (a high order mode, such as TM 21 mode) and a design method thereof. The structure operates with two symmetrical electron beams without a central electron beam, and non-working modes can be suppressed simply by optimizing the electron beam channel distance, thereby achieving single-mode stable operation. In terms of power level, not only is the power level of the two electron beams much higher than that of a single electron beam, but such a circuit also allows the use of a super-large size electron beam channel design, and the current of each electron beam is also greatly increased, ultimately enabling the power level of the device to be increased. The klystron of the present application not only far exceeds the power level of a conventional circular beam device in the terahertz frequency band, but also has a significant advantage compared to existing over-mode devices and strip beam devices. The high order mode device operating in an anti-symmetrical mode proposed by the present application has great potential in high frequency and high power applications, and provides a new technical approach for a terahertz high power radiation source. BRIEF DESCRIPTION OF DRAWINGS
[0024] The specific embodiments of the present application will be further described with reference to the drawings.
[0025] Figures 1A-1C A schematic diagram of the structure of an extended interaction klystron of the prior art is shown;
[0026] Figure 2 A field intensity distribution of the klystron of Figure 1 in operation is shown;
[0027] Figure 3 A field intensity distribution of the klystron of Figure 1 in operation is shown;
[0028] Figure 4 A schematic diagram of the structure of an extended interaction klystron of the prior art is shown;
[0029] Figure 5 A schematic diagram of the structure of an extended interaction klystron of the prior art is shown; Figure 4 A schematic diagram of the structure of an extended interaction klystron of the prior art is shown;
[0030] Figure 6 A schematic diagram of the structure of an extended interaction klystron of the prior art is shown; Figure 4 A schematic diagram of the structure of an extended interaction klystron of the prior art is shown;
[0031] Figure 7 A schematic diagram of the structure of an extended interaction klystron of the prior art is shown; Figure 4 A schematic diagram of the structure of an extended interaction klystron of the prior art is shown;
[0032] Figure 8 A schematic diagram of the structure of an extended interaction klystron of the prior art is shown; Figure 4 A schematic diagram of the structure of an extended interaction klystron of the prior art is shown;
[0033] Figure 9 A schematic diagram of the structure of an extended interaction klystron of the prior art is shown; Figure 4 A schematic diagram of the structure of an extended interaction klystron of the prior art is shown;
[0034] Figure 10 A schematic diagram of the structure of an extended interaction klystron of the prior art is shown; Figure 4 A schematic diagram of the structure of an extended interaction klystron of the prior art is shown;
[0035] Figure 11 A schematic diagram of the structure of an extended interaction klystron of the prior art is shown; Figure 4 A schematic diagram of the structure of an extended interaction klystron of the prior art is shown; 21 A schematic diagram of the structure of an extended interaction klystron of the prior art is shown; DETAILED DESCRIPTION
[0036] In order to more clearly illustrate the present application, the following further describes the present application with reference to the preferred embodiments and the attached drawings. Like numerals refer to like elements throughout the several views. As will be understood by those skilled in the art, the following description is illustrative only and should not be taken in a limiting sense.
[0037] Figures 1A-1CA schematic diagram of a prior art extended-interaction klystron structure is shown. This klystron includes an electron beam channel and five resonant cavities: an input resonant cavity 1, three intermediate resonant cavities 2, and an output resonant cavity 3. All resonant cavities have a dumbbell-shaped cross-section with equal area. The height of each resonant cavity is the coupling cavity height W1, and the height of the intermediate waveguide section 10 is W2. The electron beam channel 4 has a diameter D1 and passes through the center of the resonant cavity, allowing the electron beam to interact with the gap electric field. The input resonant cavity 1 has four resonant gaps 8, with a period length of AL1 and a vacuum portion 9 of AL2. The total width of the resonant cavity is AH1, and the width of the intermediate waveguide 10 is AH2. Each intermediate resonant cavity contains six resonant gaps 8, with a period length of BL1 and a vacuum portion 9 of BL2. The total width of the resonant cavity is BH1, and the width of the intermediate waveguide 10 is BH2. The output resonant cavity 3 contains 12 resonant gaps 8, each with a period length of EL1 and a vacuum portion 9 of EL2. The total width of the resonant cavity is EH1, and the width of the intermediate waveguide 10 is EH2. In this example, the electron beam is positioned at the center of the structure. When operating in the fundamental mode, as... Figure 2 As shown, the total width of the circuit is approximately 0.76 mm. Consequently, the electron beam channel and the size of the electron beam are also smaller, thus limiting the permissible operating current. This is the fundamental factor leading to a significant decrease in the device's power level. If... Figures 1A-1C Replacing the single electron beam at the center of the structure with three electron beams can easily lead to the excitation of other higher-order modes, making the mode competition problem more serious.
[0038] In existing technologies, dumbbell-shaped resonant cavities are proposed for interaction with ribbon electron beams, such as... Figure 8 As shown, it can be regarded as Figure 1B A variant of the ladder circuit. For the fundamental mode (TM) 11 In terms of the mode, the middle section of the dumbbell-shaped resonant cavity, i.e., the middle waveguide, can be regarded as a cutoff waveguide. Therefore, the operating frequency of the cavity depends on the height h of the middle waveguide section, and is almost independent of the width w. This characteristic allows it to extend the lateral width arbitrarily at a given frequency to match the size of the electron beam of arbitrary width, making it very suitable for strip beams with large aspect ratios. As a comparative example, Figure 3 The field shape of the transverse mode is given at this time. However, when operating with higher-order modes, the above characteristics will lead to excessively high characteristic impedance of the fundamental mode, which is actually disadvantageous.
[0039] Embodiments of the present invention provide an extended interaction high-frequency circuit in a double-beam antisymmetric mode, such as... Figure 3As shown, the high frequency circuit includes a plurality of resonant cavities, for example, including an input cavity and an output cavity and a plurality of intermediate cavities located therebetween, each intermediate cavity is also called a group cavity according to its function, and the one located in front of the output cavity is called a pre-final cavity. With Figures 1A-1C Similar to the klystron high frequency circuit shown, the input cavity, each intermediate cavity and the output cavity are all dumbbell-shaped structures with equal areas in the cross section of the gap cavity, and each resonant cavity has a plurality of gap cavities with the same or different number in the longitudinal section. The length of the drift section connecting the adjacent two resonant cavities can be the same or different. With Figures 1A-1C Unlike the klystron high frequency circuit shown, the extended interaction high frequency circuit of the present application includes two electron beam channels, which extend in parallel to the axis and are symmetrically arranged about the axis of the high frequency circuit.
[0040] In the following, the anti-symmetrical mode double electron beam extended interaction high frequency circuit of the present application is described in detail. Figure 4 And Figure 5 The anti-symmetrical mode double electron beam extended interaction high frequency circuit of the present application is described in detail. In order to increase the lateral size of the circuit, the klystron high frequency circuit of the present application adopts a dumbbell-shaped resonant cavity, which includes a plurality of gap cavities (also called intermediate waveguides) and coupling cavities symmetrically arranged on both sides of the gap cavities. Two circular electron beam channels are symmetrically arranged about the axis and the coupling cavities, and pass through all the gaps, extending in the axial direction of the high frequency circuit. The intermediate waveguide is between the two coupling cavities, the width is represented as w, the height of the intermediate waveguide is represented as h, and the radii of the two electron beam channels are Rc, respectively. The distance between the centers of the two electron beam channels is w t , the width of the coupling cavity is a, and the height of the coupling cavity is b; the gap thickness is d, corresponding Figure 1C to AL2 in the middle, and the period is p corresponding Figure 1C to AL1 in the middle, representing the distance between the adjacent two gaps.
[0041] The double electron beam extended interaction high frequency circuit of the present application breaks the uniformity of the fundamental mode electric field by making the phase difference of the electron beams in the two electron beam channels π, and increases the separation between the TM 21 mode and the TM 11 and TM 31 mode as much as possible, thereby providing a high-power extended interaction high frequency circuit for a terahertz klystron. The design method of the double beam klystron high frequency circuit according to the present application includes the following steps:
[0042] 1) Draw a curve of the frequency of different modes with the lateral width of the intermediate waveguide, and determine the lateral width of the intermediate waveguide according to the mode separation between the TM 21 mode and other modes in the curve;
[0043] 2) Adjust the channel radius and channel spacing of the two electron beam channels, i.e. the position of the electron beam channel, and draw a curve of the characteristic impedance R / Q of the resonant cavity of different modes with the electron beam channel spacing;
[0044] 3) determine the electron beam channel spacing and channel radius based on the plotted curve, so that the resonant cavity TM 21 The characteristic impedance of the mode is higher than that of other modes.
[0045] Based on the determined electron beam channel spacing, channel radius and mode, the electric field distribution of different mode resonant cavities is simulated, and it can be seen that the high-frequency circuit related method of the double-beam klystron of the application can obtain a single-mode high-power terahertz klystron that works stably.
[0046] Figure 6 The frequency curves of three transverse modes TM 11 , TM 21 , TM 31 in the dumbbell-shaped resonant cavity with the middle waveguide width w are given. It can be seen that the frequencies of the three modes TM 11 , TM 21 , TM 31 decrease with the increase of the middle waveguide width w, and the decreasing speeds of the three mode frequencies TM 11 , TM 21 , TM 31 are faster and faster. That is, the increase of the transverse dimension will obviously reduce the frequency interval between adjacent modes, thereby increasing the possibility of mode competition. Therefore, although increasing the middle waveguide width can increase the number of electron beam channels and improve the power of the klystron, the width w of the middle waveguide of the dumbbell-shaped resonant cavity cannot be increased arbitrarily, and the resulting more dense mode distribution and the decrease of the characteristic impedance must be considered.
[0047] According to the preferred embodiment of the application, the two electron beam channels are respectively located at two strongest positions in the electric field distribution of the resonant cavity cross section, and avoid the electric field peak positions of TM 11 and TM 31 modes. In order to achieve this purpose, the distance w t between the two channels and the channel radius R c need to be adjusted. Figure 7 The characteristic impedance R / Q of three modes TM 11 , TM 21 , TM 31 in the double-channel single-gap dumbbell-shaped resonant cavity with the channel spacing w t is given. It can be seen from Figure 7 that the R / Q of the fundamental mode decreases with the increase of the channel spacing w t , the R / Q of TM 31 increases with the increase of w t , the characteristic impedance R / Q of TM 21 first increases and then decreases with the increase of the channel spacing w t , and the inflection point appears at wt = 1.2 mm. At this time, the transverse electric field distribution of the three modes is shown in Fig. 2. It can be seen that the TM Figure 8 11 and TM 31 have small field strength in the electron beam channel region, but TM 21 is opposite, the electron beam channel is in the field strength peak region of the mode, and the distance between adjacent channels is far, the influence between the electric fields in the channels is small, and the electric field strength in the two electron beam channels is basically the same, and the phases are opposite.
[0048] From the above analysis and simulation data, it can be seen that the high-frequency circuit of the present application can effectively suppress mode competition without introducing dielectric loading or other complex techniques, and the single-mode operation stability is high. From the above analysis and simulation data, it can be seen that the high-frequency circuit of the present application can effectively suppress mode competition without introducing dielectric loading or other complex techniques, and the single-mode operation stability is high. Figure 7 It can be seen that by selecting a suitable channel distance, w t = 1.2 mm in the present example, the characteristic impedance of the TM 21 mode can be much higher than that of the other two modes. This method through the characteristic impedance has a clear physical basis. From the space charge wave theory, the mode stability factor expression can be derived as:
[0049]
[0050] where G e is the electron beam conductance, G0 = I0 / V0 is the DC conductance of the electron beam, Z b0 = 1 / G0 is the DC impedance of the electron beam, R / Q is the characteristic impedance of the resonant cavity, Q0 is the intrinsic quality factor of the mode, N is the number of electron beams, and i takes 1 to N. From the above formula, it can be seen that under the condition that the synchronization characteristics (G e / G0) and the electron beam parameters (Z b0 ) are the same, the mode starting characteristic is proportional to the total R / Q. Therefore, reducing the R / Q of the non-working mode is an effective means. When the TM 21 mode is used, this purpose can be achieved by simply optimizing the channel distance.
[0051] Further, the high-frequency circuit of the present application has a larger electron beam channel size, allowing a larger operating current. In the present embodiment, the electron beam channel radius is 0.8 mm, and the current can reach 0.6 A. As a comparison, in the same frequency band, the diameter of the electron beam channel of a circular electron beam device is only 0.2 mm, and the maximum current allowed is not more than 0.1 A. It can be expected that the power level of the high-frequency circuit proposed in the present application will be greatly improved.
[0052] Figure 9 Fig. 1 shows a double-beam TM 21 The structure of the mode-extended interaction output cavity circuit. In this preferred embodiment, the output resonant cavity employs a seven-gap cavity structure, with coupling via a coupling port on one side of the coupling cavity, connecting to an external waveguide. Preferably, the coupling port is located at the center of the second-to-last gap cavity along the electron beam travel direction.
[0053] As an example, Table 1 provides the circuit parameters of the klystron extended interaction high-frequency circuit according to the present invention.
[0054] Table 1 A double bet™ 21 Mode 7 gap output cavity circuit parameters
[0055] Parameter Value (mm) Notes w 1.85 Intermediate waveguide width h 0.86 Intermediate waveguide height d 0.17 Intermediate waveguide thickness (gap thickness) [R c ]] 0.25 Electron beam channel radius w t ]]> 1.20 Electron beam channel center-to-center spacing p 0.52 Spacing between adjacent two gaps (period) a 0.25 Coupling cavity width b 1.10 Coupling cavity height w co ]]> 0.15 Coupling aperture width l co ]]> 0.25 Coupling aperture thickness h co ]]> 0.50 Coupling aperture height
[0056] In this example, along the electron beam propagation direction, the resonant cavities of the high-frequency circuit are as follows: the input cavity includes 5 gap cavities, cluster cavity 1 includes 7 gap cavities, cluster cavity 2 includes 5 gap cavities, the pre-terminal cavity includes 5 gap cavities, and the output cavity includes 7 gap cavities. The drift length between each resonant cavity is 1–2 mm. The parameters of the gap cavities in each resonant cavity are the same as those listed in Table 1. The input waveguide of the input cavity and the output waveguide of the output cavity are standard waveguides, 1.092 mm × 0.546 mm.
[0057] The power extraction capability of the output cavity is calculated by setting up Gaussian electron swarms in the simulation software CST. Figure 10 The emission and clustering of Gaussian electron clusters are presented. To excite a TM... 21 In this mode, the electron beams emitted from the two channels need to have a phase difference π, which is reflected in a phase difference T in the emission time. RF / 2, where T RF =1 / f is the period of the high-frequency signal, and f is the operating frequency.
[0058] By changing the frequency of the excitation electron clusters, the 3-dB bandwidth of the output cavity is calculated, such as... Figure 11 As shown, the maximum output power reaches 1340.2W at 220.0GHz, with a 3-dB bandwidth of 700MHz, from 219.7GHz to 220.4GHz. Similarly, in the 220GHz band, the maximum power of a conventional traveling wave tube is only 50-100W; the power level of a strip beam device at the same operating voltage is approximately 500W. Therefore, the antisymmetric TM proposed in this invention... 21 The extended interaction circuit layout and design of the mode significantly improves the power level of terahertz vacuum electronic devices.
[0059] Obviously, the above embodiments of the present application are merely exemplary and are not intended to limit the embodiments of the present application. Based on the above description, other different forms of changes or variations can be made by those skilled in the art, and it is impossible to enumerate all the embodiments here. Any obvious changes or variations derived from the technical solutions of the present application are still within the protection scope of the present application.
Claims
1. An extended interaction velocity tube high frequency circuit, characterized by, The high frequency circuit comprises an electron beam channel and a plurality of resonant cavities, The electron beam channel comprises a first circular electron beam channel and a second circular electron beam channel, The plurality of resonant cavities comprises an input resonant cavity, a plurality of intermediate resonant cavities and an output resonant cavity, each resonant cavity comprising a first coupling cavity and a second coupling cavity symmetrically arranged with respect to the first circular electron beam channel and the second circular electron beam channel, and a plurality of gap cavities respectively connecting the first circular electron beam channel and the second circular electron beam channel and the first coupling cavity and the second coupling cavity, The position and radius of the electron beam channel are such that the resonator works in TM 21 mode.
2. The high-frequency circuit of an extended interaction klystron according to claim 1, characterized in that The gap cavities in the high frequency circuit have the same structure.
3. The high frequency circuit of an extended interaction klystron of claim 1, wherein, The number of gap cavities in each resonant cavity is the same or different.
4. The high frequency circuit of an extended interaction klystron of claim 1, wherein, The resonant cavities have a dumbbell-shaped cross section, and the electron beam channel extends perpendicularly to the cross section of the resonant cavities.
5. The high frequency circuit of an extended interaction klystron of claim 1, wherein, The transverse width of the gap cavity is such that the TM 21 mode of the resonant cavity has an ideal mode spacing from other modes.
6. The high frequency circuit of an extended interaction klystron of claim 1, wherein, The channel pitch and channel radius of the first and second circular electron beam channels are such that the channel resonant cavity TM 21 The characteristic impedance of the mode is higher than the characteristic impedance of the other modes.
7. The high frequency circuit of an extended interaction klystron of claim 1, wherein, The channel spacing and channel radius of the first and second circular electron beam channels are such that the first and second circular electron beam channels are located in the peak region of the transverse electric field field strength of the TM 21 mode.
8. The high frequency circuit of an extended interaction klystron of claim 1, wherein, The high frequency circuit further comprises an input waveguide connected to the input resonant cavity and an output waveguide connected to the output resonant cavity.
9. The high-frequency circuit of an extended interaction klystron of claim 8, characterized in that, The output waveguide is coupled to the penultimate gap cavity of the output resonant cavity.
10. A method for designing a high frequency circuit of an extended interaction klystron according to claim 1, characterized in that, The method comprises the following steps 1) Plotting a curve of different mode frequencies versus the lateral width of the gap cavities, and determining the lateral width of the gap cavities according to the mode spacing; 2) Adjusting the channel spacing and the channel radius of the first circular electron beam channel and the second circular electron beam channel, and plotting a curve of the characteristic impedance of the resonant cavities of different modes versus the spacing of the electron beam channels; 3) determining the electron beam channel radius and channel spacing based on the plotted curve, such that the resonant cavity TM 21 The characteristic impedance of the mode is higher than the characteristic impedance of the other modes.