J Infrared Milli Terahz Waves (2010) 31:237–248 DOI 10.1007/s10762-009-9582-7
Validation of HFCS-I on Calculation of High-Frequency Parameters of Helical Slow-Wave Structures XiaoFang Zhu & ZhongHai Yang & Bin Li & JianQing Li & Li Xu
Received: 4 January 2009 / Accepted: 24 September 2009 / Published online: 8 October 2009 # Springer Science + Business Media, LLC 2009
Abstract To validate HFCS-I, a newly developed design tool for high frequency circuits of microwave tubes, the high-frequency parameters (including dispersion, interaction impedance and attenuation constant) of a typical helical slow-wave structure (SWS) for millimetre wave travelling-wave tube are calculated by HFCS-I and MAFIA. Both the direct calculation method and the Non-Resonant Perturbation (NRP) technique are adopted to get the interaction impedance. The obtained high-frequency parameters from HFCS-I and MAFIA are compared in detail and the consistency has proved the reliability and validity of HFCS-I. Keywords Non-resonant perturbation technique . Direct calculation method . High-frequency parameter . Helical slow-wave structure . HFCS-I . MAFIA
1 Introduction HFCS (acronym of High Frequency Circuit Simulator), one module of MTSS [1], is developed as an advanced three-dimensional (3D) electromagnetic (EM) simulation tool for microwave sources. It has two versions: HFCS-I (HFCS based on the Finite Integration Technique) [2] and HFCS-E (HFCS based on the Finite Element Method) [3]. In this paper, HFCS-I is concerned. Just like MAFIA [4, 5], HFCS-I is developed on the Finite Integration Technique (FIT) [6–8], which transforms the Maxwell’s equations in integral form into the equivalent Maxwell Grid Equations (MGEs) using a first-order approximation, whereas the allocation of field components to grid G (shown in Fig. 1) uses the Yee lattice with the electric field components allocated at the mid-points of the sides of the rectangular cells and the
X. Zhu (*) : Z. Yang : B. Li : J. Li : L. Xu Vacuum Electronics National Laboratory, School of Physical Electronics, University of Electronic Science and Technology of China, NO.4, Section 2, North Jianshe Road, Chengdu, People’s Republic of China 610054 e-mail:
[email protected]
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e Fig. 1 Allocation of Electric and Magnetic Field Components in Grid G and G.
~ magnetic field components at the centre of each face, defining a dual grid G. In orthogonal * * coordinate system r ¼ rðu; v; wÞ, grid G is described as G ¼ ui ; vj ; wk : u1 ui uI ; i ¼ 1; . . . I; ð1:1Þ : v1 vj vj ; j ¼ 1; . . . J ; : w1 wk wk ; k ¼ 1; . . . K;g All nodes of G are numbered linearly by n ¼ 1 þ ði 1ÞMu þ ðj 1ÞMv þ ðk 1ÞMw with Mu ¼ 1; Mv ¼ I; Mw ¼I J . ~ Using the dual grid G; G , the Maxwell equations in integral form are transformed into the MGEs and one has [6–8] I ZZ * * @B * d A , CDs e ¼ DA b ð1:2Þ E d *s ¼ @A A @t I @A
*
ZZ *
H ds ¼ A
ZZ ZZ
*
*
@V
*
* * @D * ~~ ~ þ J d A , C Ds h ¼ DA d þ j @t
*
*
ð1:3Þ
B d A ¼ 0 , SDA b ¼ 0
ð1:4Þ
* * @D * ~~ þ J d A ¼ 0 , S DA d þ j ¼ 0 @t
ð1:5Þ
@V
*
D ¼ " E , d ¼ D" e; B ¼ m H , b ¼ Dm h;
*
*
J ¼ k E þ r v* , j ¼ Dk e þ Dr v *
*
*
*
*
ð1:6Þ
Where e, b, d, h and j are discrete analogs of E; B; D; H and J . These vectors ðdimension 3N ; N ð¼ I J K Þ number of mesh points) contain field components on each mesh cell. Diagonal matrices Ds, DA contain mesh step sizes and cell surface areas and
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D" ; Dm ; Dk contain the material properties. C and S denote discrete curl and div operator respectively and all the matrices marked with a tilde are attached to dual grid ~ Gcorrespondingly. C and S are defined as 0 1 0 Pw Pv C ¼ @ Pw 0 Pu A; S ¼ ðPu jPv jPw Þ ð1:7Þ Pv Pu 0 Mx
1 here Pu, Pv , Pw satisfy Px
:
1
~ ~ and C ¼ C H ; S ¼ PuH PvH PwH
1 1 x u /v /w
where the superscript denotes transpose and conjugate operation. When time harmonic fields concerned, one has b ¼ jwb; d ¼ jwd, and for a loss-free medium with no driving current, combining the MGEs, one obtains an eigenvalue equation, whose eigenvalues are the resonant frequencies squared [6–8], 0 ~ 0 ~ T ð1:8Þ D CD D CD e~ ¼ w2 e~ ~ 1=2 1=2 ~ 1=2 1=2 0 1=2 1=2 ~ 1=2 ~ and e ¼ D" DA Ds e. Where D ¼ Dm Ds DA ; D ¼ D" Ds DA To simulate an actual high frequency circuit, boundary conditions have to be considered to confirm the exclusive field solution of the structure. Presently, HFCS-I is ready to deal with electric-wall, magnetic-wall and periodic boundary conditions. The electric-wall and the magnetic-wall boundary condition is encapsulated in material matrices D" and Dm respectively by setting elements corresponding to fields on corresponding boundaries to zero, that is, setting the tangential electric and normal magnetic field components on electricwall boundary, and the normal electric and tangential magnetic components on magnetic-wall boundary to be zero [9]. To perform periodic boundary condition, the Floquet theorem is adopted and only field components on plane k=2,3,...K are calculated. The field components on plane k=1 is easily obtained by those on plane k=K by multiplying exp ( jβL). At this situation, Pw with dimension N ¼ I J ðK 1Þ is redefined as [10] 0 1 1 1 . B .. C 1 1 B C B C jbL ð1:9Þ Pw ¼ B e ::: C @ A jbL 1 e ejbL 1 When appropriate boundary conditions for a certain high-frequency circuit are applied and the corresponding eigenvalue equation is solved by the specially developed eigensolver [11], the resonant frequencies of interest and the discrete EM fields will be obtained. By post-processing, high-frequency parameters including dispersion, interaction impedance and attenuation constant and other quantities of high-frequency circuits are obtained further. To validate the performance of HFCS-I, the dispersion, interaction impedance and attenuation constant of a typical helical slow-wave structure (SWS) for millimetre wave travelling-wave tube is calculated by HFCS-I and MAFIA, in which the direct calculation method [12, 13] and the Non-Resonant Perturbation (NRP) technique [13, 14] are both
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adopted to get the interaction impedance. The results from HFCS-I and MAFIA are compared in detail and the consistency has proved the reliability and validity of HFCS-I.
2 Basic theory of simulation Computer simulation of helix travelling-wave tube requires accurate information about phase velocity, interaction impedance and attenuation constant of helical slow-wave structures. And fortunately, the advent of advanced 3D EM simulation tool made it possible to get accurate high-frequency parameters without performing costly and time-consuming experimental measurements [13]. 2.1 Dispersion Using the resonant frequency-domain model of 3D EM simulation code, normal mode frequency f corresponding to a specified phase advance θ can be obtained for periodic slowwave structures and multiple simulations can therefore generate the dispersion curve which is generally expressed by the relation between phase velocity vp normalized by light speed c and mode frequency f [12–14], vp w wL 2pf L ¼ ¼ ¼ bc qc qc c
ð1:10Þ
where L is the dimensional period of certain periodic slow-wave circuit and β is the fundamental axial propagation constant, satisfying β=θ/L. 2.2 Interaction impedance Interaction impedance is an important parameter of slow-wave circuits which is closely related to the gain and efficiency of traveling-wave tubes. Generally, using 3D EM simulation tool such as HFCS-I and MAFIA, the direct calculation method and the perturbation technique can be used to determine interaction impedance of a helical slow-wave circuit. 2.2.1 Direct calculation method Using 3D EM simulation code, the direct calculation of interaction impedance for the nth spatial harmonic is based on its definition, which can be expressed as [12, 13, 16] Kc;n ¼
jEzn ð0Þj2 2bn 2 P
ð1:11Þ
Where bn ð¼ b þ 2pn=LÞis the axial phase constant of the nth spatial harmonic. Ezn(0) is the on-axis longitudinal electric field magnitude of the nth spatial harmonic which can be obtained by doing a Fourier analysis on the total on-axis axial electric field Ez(0) [12, 13, 16], that is 1 Ezn ð0Þ ¼ L
ZL Ez ð0Þ expðjbn zÞdz 0
ð1:12Þ
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The time-averaged RF power flow P is defined as Z * * 1 P ¼ Re E H * dA 2
ð1:13Þ
Where dA is area integral evaluated along any cut transverse to z axis. Based on the power flow theorem of periodic circuits, P can be calculated by multiplying group velocity vg and time-averaged stored energy W, that is [13, 14] P ¼ vg
W L
ð1:14Þ
where vg can be calculated by taking the slope of dispersion curve and W is easy to get by a volume integral of discrete EM fields by ZZZ ZZZ * * 1 1 * ** 1 * * * 1 dV ¼ ð1:15Þ "E E þ mH H " E E * dV W ¼ 2 2 2 2 V
V
For there is no assumptions involved in the direct calculation method, the interaction impedance from it is of great accuracy. 2.2.2 Non-resonant perturbation method The perturbation technique, including Resonant Perturbation (RP) and Non Resonant Perturbation (NRP) technique is in vogue for the experimental characterization of slowwave structures. For the difficulty of a helical slow-wave structure to be perfectly shorted at its ends, the NRP technique is more suitable than the RP technique and is widely applied to determine interaction impedance of helical slow-wave circuits [13, 14]. Inserting a cylindrical dielectric rod on the central axis of a helical slow-wave structure and obtaining difference in axial phase constant Δβ between the perturbed and unperturbed circuits, the on-axis interaction impedance can be then calculated by [13, 14] Kc ¼
1 1 2Δb Gp;nu;r Gs wb2 p"0 ð"r 1Þrp2
ð1:16Þ
Where εr and rp are relative permittivity and radius of the cylindrical dielectric rod respectively. Gp,nu.r is the correction factor introduced to account for the finite perturbation of the axial electric field inside the dielectric-rod perturber, the presence of a radial electric field and the non-uniformity of fields over the rod cross section, which can be expressed as [13, 14] b2 ð1:17Þ Gp;nu;r ¼ p1 p2 þ 0 p3 gg g and g 0 are theffi unperturbedqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and perturbed radial propagation constant given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0 g ¼ b w2 m0 "0 , and g ¼ b2 w2 m0 "0 "r with μ0,ε0 the free space permeability and permittivity respectively. p1,p2 and p3 satisfy p1 ¼
1 g 0 grp K1 grp I0 g rp þ g 0 "r K0 grp I1 g 0 rp
ð1:18Þ
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p2 ¼ I0 grp I0 g 0 rp
"
!# 2I1 g 0 rp 2I1 grp 1 g 2 2I1 g 0 rp þ ð1:19Þ "r 1 k g 0 rp I0 g 0 rp g 0 rp I0 g 0 rp grp I0 grp
! 2I1 g 0 rp 2I1 grp g 0 rp I0 g 0 rp grp I0 grp
I0 grp I0 g 0 rp p3 ¼ "r 1
ð1:20Þ
Where I0 and I1 are the modified Bessel function of first kind and K0, K1 of second kind. When considering space-harmonic effects, Gs should be introduced. In presence of harmonics 1; 2; . . . ; n, one has [14] Gs ¼ 1 þ
n X
2 S0;q þ
q¼1
n X
2 S0;q
ð1:21Þ
q¼1
where S0,±q is the relative±qth space harmonic axial electric-field amplitude at the structure axis, which can be conducted through tape-helix analysis for a helix surrounded by a dielectric tube of an effective relative permittivity "' r and enclosed in a metal envelope [14]:
S0;q ¼
qb cos y g 2q Kq g q a sin y gq2 a q
g 20 K0 ðg 0 aÞ sin y
Fq sin bq d=2 b F0 sinðbd=2Þ bq
ð1:22Þ
"
Fq
# " !#1 0 Iq g q a Kq g q c I 0 q g q a Kq g q c 0 ¼ 1 : 1 ð" r 1Þg q aIq g q a K q g q a : 1 0 Kq g q a Iq g q c K q g q a Iq g q c
ð1:23Þ
I0 ðg 0 aÞK0 ðg 0 cÞ I 0 0 ðg 0 aÞK0 ðg 0 cÞ 1 0 0 : 1 ð" r 1Þg 0 aI0 ðg 0 aÞK 0 ðg 0 aÞ: 1 0 F0 ¼ 1 K0 ðg 0 aÞI0 ðg 0 cÞ K 0 ðg 0 aÞI0 ðg 0 cÞ ð1:24Þ
g q ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2q k 2 ; g 0 ¼ b2 k 2 ; bq ¼ b q cot y=a; k 2 ¼ w2 m0 "0
ð1:25Þ
Here y, δ and a are the helix pitch angle, tape width and mean radius respectively, and c radius of the metal envelope. I±q and K±q are the modified Bessel functions of the first and second kinds of order ±q respectively, ant the primes denote their derivatives with respect to their arguments. When the dielectric-rod perturber is quite small, the axial electric field can be assumed to be uniform in the transverse direction and the transverse electric fields can be neglected.
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Without considering the space-harmonic effect further, the interaction impedance can be simplified as [13, 14] Kc ¼
2Δb wb p"0 ð"r 1Þrp2 2
ð1:26Þ
which is generally referred as the first order expression of the NRP technique. When using the NRP technique to determine interaction impedance of helical slow-wave structures, the dimension and the dielectric permittivity of the perturber are of most responsibility of the simulation accuracy. Theoretically, when the dimension is small and the relative permittivity is near that of vacuum, that is to say, the perturbation is small enough, the interaction impedance by the NRP technique converge to that by the direct calculation method [15]. In practice, limited by fabrication, equipment and material, the perturber is of finite size and a certain dielectric permittivity is selected. 2.3 Attenuation characteristics Attenuation constant α is a loss parameter used to picture the resistive wall heating, which is an important consequence of device operation and ohmic dissipation [16]. a¼
p1 p1 pL ðNp=mÞ ¼ 8:686 ðdb=mÞ ¼ 8:686 ðdb=mÞ 2P 2P 2PL
ð1:27Þ
Where P is the time-averaged RF power flow, p1 is the rate of energy dissipation and pL the total energy dissipation at metal surfaces over structure period L. For walls that are good conductors, the resistive loss may be calculated from the solutions for perfectly conducting walls by using a perturbative procedure [16]. The fields are computed as though all the metal walls are perfectly conducting. The magnetic field H at the wall varies discontinuously to zero at the surface of the metal. This jump in H is just the surface current density in the perfect conductor. For a good conductor, approximately the same current would flow in a skin depth of the surface. The finite resistivity rð¼ 1=s Þ of the wall is then used to compute the rate of energy dissipation of the surface * 1 1 1 * 1 * pffiffiffiffiffiffiffiffiffiffiffi ð1:28Þ jJs j2 Rs ¼ jnˆ H t j2 Rs ¼ j H t j2 Rs ¼ j H t j2 pf mr W =m2 2 2 2 2 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi Where Rs ¼ pf mr ¼ pf m=s is the surface resistance due to the finite conductivity σ of a good conductor. The rate of energy dissipation, together with the power flow in the circuit, determines the attenuation constant α. p1 ¼
3 Simulation and discussion To validate performance of HFCS-I, a typical helical slow-wave structure for millimetre wave travelling-wave tube is analyzed, whose parameters are listed in Table 1. Using the resonant frequency domain model and periodic boundary condition, the long helical slow-wave structure is assumed as a periodic constant-pitch helix and the geometrical model is reduced to a single pitch treated as a resonator. The geometrical model and the mesh model with a single pitch in HFCS-I are illustrated in Fig. 2, in which Cartesian coordinate system is used and staircase approximation is adopted to approximate the arc boundaries.
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Table 1 Parameters of the helical SWS (unit: SI). Helix inner radius
0.35e-3
Helix outer radius
0.46e-3
Helix width Helix pitch angle
0.25e-3 0.2129
Envelope inner radius
1.10e-3
Rectangular Support Width
0.5e-3
Relative permittivity of Support
6.5
In this simulation, the helix and the envelope are treated as perfect conductor and the electric-wall boundary condition is applied. The periodic boundary condition is used to simulate the periodicity of the helical SWS and a certain phase advance is specified. Once the final eigenvalue equation is formed and solved, the resonant frequency and discrete EM fields are obtained and then the high-frequency parameters of interest are at hand. First of the simulation, convergence process driven by mesh refinement is studied, during which the phase advance is fixed as βl=70(Deg) and lowest mode frequencies corresponding to different mesh density are calculated and compared with those from MAFIA with almost the same mesh density. The results are listed in Table 2, in which fMAFIA indicates results of MAFIA, fHFCS-I notes results of HFCS-I and EH-M denotes discrepancy between results of these two codes. From data in Table 2, the convergence process is plotted in Fig. 3, where the abscissa denotes the different sets of mesh density in Table 2. It is shown that mode frequency from HFCS-I and MAFIA both tends to converge and the discrepancy between decreases gradually with increased mesh density. It is also seen that mode frequency around mesh density I J K ¼ 101 101 26 is almost convergent. Based on this fact, as well as computation time and storage, mesh density I J K ¼ 101 101 26 is selected to mesh the helical SWS and then to calculate the dispersion, interaction impedance and attenuation constant in succession. To get the dispersion performance, lowest mode frequencies corresponding to different phase advance are calculated on model with mesh density I J K ¼ 101 101 26, from which the dispersion curve of HFCS-I is plotted. For comparison, mode frequencies to the same phase advance of model with mesh density I J K ¼ 101 101 26 and a
Fig. 2 (a) Geometrical model and (b) discretized mesh model of helical SWS with a single pitch.
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Table 2 Compare of mode frequency from HFCS-I and MAFIA with different mesh density. 22 22 6 43 43 11 64 64 16 82 82 21 101 101 26 130 130 33
I×J×K
fMAFIA(GHz) 16.18595
16.95455
17.24318
17.41663
17.48510
fHFCS-I(GHz) 15.7202
16.79488
17.045189
17.31749
17.34656
17.42881
−0.94175
−1.148228
−0.569220
−0.792338
−0.271967
−2.87745
EH-M(%)
17.47634
total six million meshes are both obtained by MAFIA. The discrepancy of mode frequency and the dispersion curves are plotted in Fig. 4. It is obvious that the frequency discrepancy between HFCS-I and MAFIA with same mesh density I J K ¼ 101 101 26 is around -0.8% steadily and the dispersion curves are almost parallel to each other at this situation. It is also shown that the mode frequency from HFCS-I with mesh density I J K ¼ 101 101 26 is an average 2% lower compared to that from MAFIA simulation with six million meshes and such a difference can be expected to decrease if denser mesh density is adopted in HFCS-I. To get the interaction impedance of the helical SWS by HFCS-I and MAFIA, the direct calculation method is performed for mesh model with mesh density I J K ¼ 101 101 26. Using MAFIA, this is also done for mesh model with six million meshes. Besides the direct calculation method, the NRP technique is also adopted to get the interaction impedance of the helical SWS. For the dimension and the permittivity of the perturber are of most responsibility of the accuracy of the NRP technique, they should be carefully decided before the simulation. In this paper, a relatively small radius and an appropriate relative permittivity of the perturber are used, which is set to be 0.1 to average helix radius and 13 respectively. For the perturber is relatively small, local mesh refinement is performed to describe the perturber and an un-uniform mesh density I J K ¼ 111 111 26 is used in HFCS-I. To guarantee that the mesh models before and after perturbation only differ in the material property in the perturbed region, mode frequency corresponding to the refined mesh model of the perturbed circuit f2 is first calculated and then the filled material in the perturbed region is changed into vacuum and mode frequency
Discrepancy of frequency (%)
Phase = 70(Deg)
Frequency f ( GHz )
17.5 17 MAFIA HFCS-I
16.5 16 15.5
1
2
3 4 mesh
(a)
5
6
Phase = 70(Deg)
0 -0.5 -1 -1.5
HFCS-I~MAFIA -2 -2.5 -3
1
2
3 4 mesh
5
(b)
Fig. 3 (a) Convergence process and (b) discrepancy of mode frequency from HFCS-I and MAFIA.
6
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Normalized Phase Velocity vp/c
Frequency Discrepancy( % )
HFCS-I 101*101*26 -0.8 -1 MAFIA 101*101*26 MAFIA 6e6
-1.2 -1.4 -1.6 -1.8 -2 -2.2 60
80
100 120 140 160 Phase advance( Deg )
HFCS-I 101*101*26 MAFIA 101*101*26 MAFIA 6e6
0.165 0.16 0.155 0.15 0.145
180
20
25 30 35 Frequency f ( GHz )
(a)
40
(b)
Fig. 4 (a) Frequency discrepancy vs phase advance and (b) dispersion curves of HFCS-I and MAFIA.
f1 to the same phase advance is obtained. Frequency change Δf ð¼ f2 f1 Þ is then transformed into change in axial phase constant by Δf vg Δb ¼ f vp b
ð1:29Þ
The NRP technique can then be applied to calculate the interaction impedance. For the effect of Gs is generally small, equation (1.16) with neglected Gs is used for the perturbation and equation (1.26) for the first-order expression. Using MAFIA, the perturbation procedure is only duplicated on mesh model with six million meshes to ensure that the perturber is well represented and the result is convergent. The obtained interaction impedance using HFCS-I and MAFIA by the direct calculation method and the NRP technique are plotted in Fig. 5, which have shown consistency between results of HFCS-I and MAFIA by a certain calculation method. It is even found that using the direct calculation method, the interaction impedance from HFCS-I with mesh density I J K ¼ 101 101 26 is around -3% lower than that from MAFIA with Fig. 5 Interaction impedance of HFCS-I and MAFIA with different calculation method.
50
Interaction Impedance Kc ( Ohm )
45 HFCS-I,direct calculation(101*101*26) MAFIA,direct calculation(101*101*26) MAFIA,direct calculation(6e6) MAFIA, first order expression(6e6) MAFIA,perturbation(6e6) HFCS-I, first order expression(111*111*26) HFCS-I,perturbation(111*111*26)
40 35 30 25 20 15 10 5 0 15
20
25
30
Frequency f (GHz)
35
40
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same mesh density and an average difference of −7% is found comparing to that from MAFIA with six million meshes. Using the NRP technique, whatever the perturbation or the first order approximation is concerned, the interaction impedance from HFCS-I with refined mesh density I J K ¼ 111 111 26 is in quite agreement with that from MAFIA with six million meshes and a largest discrepancy is found to be −6%. Using HFCS-I and MAFIA, the attenuation constant due to the finite conductivity of the helix and the envelope are obtained by post-processing the discrete EM fields of mesh model with mesh densityI J K ¼ 101 101 26. At the same time, the attenuation characteristics of mesh models with six million meshes are obtained by MAFIA simulation. The results and the discrepancy between are plotted in Fig. 6. Obviously, the attenuation characteristics from HFCS-I and MAFIA are consistent with each other. For mesh models with mesh densityI J K ¼ 101 101 26, the discrepancy between is less than 6.5% over the whole frequency range. For mesh model with I J K ¼ 101 101 26 of HFCS-I and mesh model with six million meshes of MAFIA, the attenuation constant from HFCS-I is an average of 4% lower than that of MAFIA simulation. The discrepancy of the high-frequency parameters of the helical slow-wave structure between HFCS-I and MAFIA can be mostly explained by the difference of the way to approximate arc boundaries. In HFCS-I, staircase approximation is adopted, while in MAFIA, a better way combing rectangular mesh and triangular mesh is used.
4 Conclusion
Attenuation Constant ( dB/m )
40 35 30 25 HFCS-I 101*101*26 MAFIA 101*101*26 MAFIA 6e6
20 15 15
20
25 30 35 Frequency f (GHz)
(a)
40
Discrepancy of Attenuation Constant ( % )
To validate HFCS-I, an actual helical slow-wave structure for millimeter-wave travelingwave tube is analyzed using both HFCS-I and MAFIA. Convergence process driven by mesh refinement is firstly studied and the high-frequency parameters for mesh model with I J K ¼ 101 101 26 are obtained by HFCS-I. And then, the results are compared with those from MAFIA for mesh model with same mesh density I J K ¼ 101 101 26 and six million meshes respectively. The discrepancies are analysed and the consistency of the results have proved the availability and reliability of HFCS-I. HFCS-I 101*101*26
8 6 4
MAFIA 101*101*26 MAFIA 6e6
2 0 -2 -4 -6 15
20
25 30 35 Frequency f ( GHz )
(b)
Fig. 6 (a) Attenuation characteristics and (b) attenuation discrepancy between HFCS-I and MAFIA.
40
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Furthermore, the accordance of the interaction impedance by the direct calculation method and the NRP technique also confirms the NRP technique in decision of interaction impedance of helical SWSs. Acknowledgement This work was supported by Vacuum Electronics National Laboratory, National Natural Science Foundation of China (Grant No. 60601004) and National Key Technology R&D Program.
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