accurate and fast finite-element modeling of attenuation in slow-wave structures for traveling-wave...

1534 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 59, NO. 5, MAY 2012

Accurate and Fast Finite-Element Modelingof Attenuation in Slow-Wave Structures

for Traveling-Wave TubesLi Xu, Zhong-Hai Yang, Member, IEEE, Jian-Qing Li, and Bin Li, Member, IEEE

Abstract—This paper presents a novel 3-D finite-element mod-eling technique for the arbitrary lossy slow-wave structure (SWS)of a traveling-wave tube (TWT). By using this technique, we canaccurately and quickly calculate not only dielectric losses butalso conductivity losses of the SWS. In this modeling technique,a new frequency-specified eigenmode analysis (FSEA) for SWSsis proposed and utilized. Unlike the traditional phase-advance-specified eigenmode analysis for SWSs, which has to solve anonlinear generalized eigenvalue problem (GEP), the new FSEAapproach only needs to solve a linear GEP and is capable of obtain-ing the attenuation constant more accurately and directly withoutany postprocessing when simulating the lossy SWSs. Moreover,to further significantly improve the efficiency of modeling lossySWSs, three advanced techniques are introduced in the standardimplicit restarted Arnoldi method (IRAM) and an improved in-exact IRAM is proposed. By simulating many practical SWSs,the accuracy and highly efficient performance of this modelingtechnique have been validated. It is shown that this modelingtechnique would be very useful to design a low-loss SWS forhigh-efficiency TWTs.

Index Terms—Attenuation, cold parameters, finite element, loss,slow-wave structure (SWS), traveling-wave tube (TWT).

I. INTRODUCTION

WHEN DESIGNING a high-efficiency traveling-wavetube (TWT), accurate estimation of the attenuation in a

slow-wave structure (SWS) by computational electromagnetics(CEM) methods, such as the finite-element method (FEM),is one of the crucial steps. Compared to the lossless case,the computational cost of the finite-element (FE) modelingof lossy SWSs is significantly higher for the following threereasons. First, like the lossless case, the resulting FE matricesare not only large scale but also indefinite. Second, and differentfrom the Hermitian property of FE matrices in the losslesscase, in the presence of a periodic boundary condition (PBC),

Manuscript received October 25, 2011; revised December 27, 2011,January 12, 2012, and January 22, 2012; accepted January 23, 2012. Dateof publication February 22, 2012; date of current version April 25, 2012.This work was supported in part by the National Natural Science Foundationof China under Grant 60801029, Grant 10876005, Grant 60931001, Grant61071030, and Grant 10905009 and in part by the China Scholarship Council.The review of this paper was arranged by Editor R. Carter.

The authors are with the National Key Laboratory of Science and Tech-nology on Vacuum Electronics, University of Electronic Science and Tech-nology of China, Chengdu 610054, China (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TED.2012.2186140

either a complex dielectric constant or an imposed surfaceimpedance boundary condition (SIBC) would lead to the com-plex nonsymmetric property of FE matrices. Third, instead ofa linear generalized eigenvalue problem (GEP) in the loss-less case, a nonlinear GEP has to be solved in all previousworks about accurately modeling metallic–lossy SWSs withvarious CEM methods or 3-D CEM simulators [1]–[3]. Thisis because the surface impedance in SIBC, which is imposedto accurately calculate the conductivity losses of SWSs, isinherently frequency dependent, but the primary purpose ofthe conventional procedure applied in these works is to solvefor the frequency. In the conventional procedure, first, a phaseadvance is specified, and before the attenuation is able to beevaluated in postprocessing, the eigenfrequency of the SWSmust be obtained. Such a procedure is referred to as the phase-advance-specified eigenmode analysis (PSEA) for SWSs. Toavoid solving a nonlinear GEP in the PSEA for modelingmetallic–lossy SWSs, an alternative perturbation theory [4] isutilized, which can estimate the conductivity losses after aneigenmode analysis for the SWS in the assumed lossless case.However, the accuracy of the perturbation theory will quicklybecome unreasonable when the frequency increases and thegeometry of SWS is complex.

In this paper, we present a new 3-D FE modeling techniquefor an arbitrary lossy SWS to accurately and quickly calculatenot only dielectric losses but also conductivity losses, in ad-dition to dispersion and coupling impedance. To completelyaddress solving a nonlinear GEP in modeling lossy SWSs, inSection II, we propose a novel approach called the frequency-specified eigenmode analysis (FSEA) for SWSs, in which thefrequency is specified first and then the attenuation and phaseconstants are obtained as the real and imaginary parts of theeigenvalue, respectively. The FSEA for modeling lossy SWSswill lead to a standard linear GEP. Moreover, there are anotherthree advantages using the new FSEA rather than the traditionalPSEA in modeling lossy SWSs. First, by using the FSEA,the attenuation constant is the direct output from the solutionof the linear GEP without any assumptions and accumulatednumerical errors that occur in postprocessing of the PSEA.Second, the FSEA is more efficient than the PSEA whenmodeling materials, in which the dielectric loss tangent varieswith the frequency, because in the PSEA, the loss tangent ofsuch material has to be determined from a real frequency, whichis obtained in the additional assumed dielectric-lossless case,whereas in the FSEA the loss tangent can be exactly determined

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XU et al.: ACCURATE AND FAST FE MODELING OF ATTENUATION IN SWSs for TWTs 1535

without any additional dielectric-lossless modeling. Third, theFSEA is also more efficient for the subsequent simulation ofbeam-wave interaction (BWI) in SWSs than the PSEA becausein the simulation of BWI, the cold parameters, which arerequired at a particular frequency, can be directly obtained bymodeling an SWS only once at this frequency in the FSEA,whereas in the PSEA, we have to perform modeling of the SWSmany times at different phase advances and then interpolateon the modeling results to approximately evaluate these coldparameters at the required frequency.

Although the resulting GEP from the FSEA for lossy SWSsbecomes linear, the solution of this linear GEP is still veryinefficient when using the implicitly restarted Arnoldi method(IRAM) [5] because of the first and second reasons aforemen-tioned. To solve the required GEP faster and more efficiently, inSection III, we introduce three new techniques in the standardIRAM and propose an improved inexact IRAM. First, a relax-ation strategy is introduced to relax the solution tolerance ofthe linear systems within the IRAM so that the solution of thesesystems, which is the most time-consuming step in IRAM, canbe inexact. Consequently, the solution time of these systems issignificantly reduced. Second, we propose a tuning techniqueto seek a very good approximate solution as the initial guessfor the iterative solution of these linear systems. This tuningtechnique requires little computational cost and will remarkablyspeed up the convergence of solving these systems. Lastly,based on second-order hierarchical basis functions, an effectivep-type multigrid preconditioner is constructed to solve theselinear systems efficiently, and hence, the convergence rate ofsolving these systems is nearly independent of the dimensionsof FE matrices.

In Section IV, a few numerical results and discussion showthat combined with the FSEA approach and the improved inex-act IRAM, the proposed novel FE modeling technique for lossySWSs makes the calculation of attenuation in SWSs accurateand quite efficient. Finally, we conclude in Section V.

II. FSEA FOR MODELING LOSSY SWSS WITH FEM

We begin by defining a boundary value problem (BVP) forthe FE modeling of lossy SWSs. It may be written as

⎧⎪⎪⎨⎪⎪⎩

∇× μ−1r ∇× E − k2

0ε∗rE = 0 in Ω

OR (n̂ × (Es × n̂))= n̂ × (Em × n̂)e−(α+jβ)L/M on ΓPBC

n̂ × μ−1r ∇× E = (jk0η0/Zs)n̂ × (E × n̂) on ΓSIBC

(1)

where n̂ is the outgoing normal on the boundary surfaces. Thesecond equation in (1) indicates the imposed rotated PBC [3],which employs both azimuthal and longitudinal periodicitiesof the SWS to further reduce the computational domain fromone period in the standard PBC to 1/N of one period (N isthe order of the azimuthal symmetry). In this equation, Es

and Em are the electric fields on slave–face and master–facewith PBC, respectively; α is the attenuation constant; β isthe phase constant; and L is the periodic length. Moreover, inthe standard PBC, the rotation operator OR = 1 and M = 1,

whereas in the rotated PBC, OR represents rotating the electricfields on slave–face clockwise by 2π/N around the axis andM = N .

Complex relative permittivity ε∗r is defined by εr (1 −j tan δ), and the imaginary part of ε∗r represents the dielectriclosses of dielectric materials constituting the SWS, such asthe support rods of a helical SWS. In general, the dielectricloss tangent tan δ varies with frequency. The last equationin (1) is the imposed SIBC. Since the conducting materialsconstituting the SWS are always good conductors, imposingthe SIBC on BVP (1) is sufficiently accurate to simulate theconductivity losses arising from the finite conductivity valuesof these materials. In the SIBC, surface impedance Zs isgiven by

Zs = (1 + j)√

πfμ/σ (2)

where f denotes the operating frequency and σ denotes the bulkconductivity. Obviously, Zs is frequency dependent.

Following the Galerkin procedure, we can obtain the weakformulation for BVP (1). After this, the computational domainΩ is discretized into a number of tetrahedral elements Ωh. Eh

and wh denote the discrete trial functions and test functions,respectively. In this paper, wh are adopted on the second-order hierarchical Whitney basis [6] and Eh is expanded interms of the basis. The chosen basis function space Xh be-longs to the curl-conforming functions space, viz., H(curl;Ωh), where electric and magnetic fields reside. We now havethe finite-dimensional weak statement corresponding to BVP(1) as follows: If an operating frequency f of the SWS isspecified, then seek Eh ∈ Xh ⊂ H(curl; Ωh) and α, β ∈ R

such that

(∇× wh, μ−1r ∇× Eh)Ωh − k2

0(wh, ε∗rE

h)Ωh + (jk0η0/Zs)

×⟨n̂ × (wh × n̂), n̂ × (Eh × n̂)

⟩Γh

SIBC= 0 ∀wh ∈ Xh

(3)

subject to OR(n̂×(Ehs ×n̂))=(n̂×(Eh

m×n̂))e−(α+jβ)L/M onΓh

PBC where the volume and surface integrals of two complex-valued vector functions u and v are defined, respectively, asfollows:

(u,v)Ωh =∫Ωh

u∗ · vdv and 〈u,v〉ΓhSIBC

=∫

ΓhSIBC

u∗ · vds.

(4)

Let Ni represent a basis function described in [6], Eh isthen expanded by Eh =

∑xiNi, where xi is the unknown

coefficient. We then partition xi and Ni into three parts anduse superscript t to distinguish the three parts. When t = I , xi

and Ni are inside the computational domain; when t = M , theyare on the master surfaces; when t = S, they are on the slavesurfaces. We substitute xS

i by its corresponding xMi using the

rotated PBC [3], and then the discrete system (3) can be castas a matrix equation for the unknown coefficients xI and xM


in (5), shown at the bottom of the page, where Φ = (α +jβ)L/M and the block matrices are given by

Ktt(f) =Stt − f2 · T tt

Sttij =

(∇× Nt

i, μ−1r ∇× Nt

j

)Ωh

T ttij = 4π2μ0ε0 ·

(Nt

i, ε∗rN

tj

)Ωh

CIIij = 2j

√πμ0σ/(1 + j)

·⟨n̂ ×

(NI

i × n̂), n̂ ×

(NI

j × n̂)⟩

ΓhSIBC

. (6)

It is clearly shown in (5) that under the existence of term√f CII, when a phase advance is specified in the traditional

PSEA approach, frequency f has to be obtained from a non-linear GEP, which will be solved much more inefficiently thana linear GEP. However, if we take a contrary way of thinking,i.e., that frequency f is specified whereas attenuation constant αand phase constant β are required to be solved, then a standardlinear GEP will be deduced from (5). First, assume that thereis not a tetrahedral element touching both master and slavesurfaces. As a result, matrices KMS and KSM are zero. Aftermultiplying the second row of (5) by e−Φ, a standard linear GEPwith e−Φ as its eigenvalue can be written as[

KII(f) +√

fCII KIM(f)KSI(f) 0

]︸︷︷︸

A

[xI

xM

]

= −e−Φ

[0 KIS(f)

KMI(f) KMM(f) + KSS(f)

]︸︷︷︸

B

[xI

xM

]. (7)

Once the eigenvalue λ = e−Φ and eigenvector x are obtainedthrough solving the linear GEP (7), the cold parameters of thelossy SWS can be calculated by the following procedure. SinceΦ = (α + jβ)L/M , we have

β = (−Im(ln λ) + 2nπ) · (M/L) n = ±0, 1, 2, . . . (8)

α = −Re(lnλ) · (M/L) (9)

where Re(ln λ) and Im(ln λ) denote the real and imaginaryparts of ln λ, respectively. Equation (8) accurately gives thephase constant of an eigenmode of the SWS at a specifiedfrequency f . As f ranges across the working frequency band ofSWS, the corresponding β and the normalized phase velocityat each working frequency point are obtained. Consequently,the dispersion curve of the SWS is determined by the FSEAmethod. It is clearly shown by (8) that β has the periodicity of2nπM/L. This coincides very well with the periodic propertyof SWS. The attenuation constant α of a lossy SWS is accu-

rately given by (9), which is the direct output from solving theGEP (7), whereas in the traditional PSEA, α has to be evaluatedthrough complicated postprocessing in which accumulated nu-merical errors occur. Since the eigenvectors x are identical inthe FSEA and the PSEA, the coupling impedance of the SWScan be calculated in the same way in the two methods [3].By the way, if we just need to roughly estimate the workingfrequency band of SWS, the PSEA may be used to model theSWS in the lossless case at 0◦ and 180◦ phase-shift points.In (7), matrices A and B are complex nonsymmetric. Moreseriously, they are both singular and indefinite. Therefore, effi-ciently solving the GEP (7) is the key to successful applicationof the FSEA for modeling lossy SWSs. In the next section,an improved inexact IRAM will be proposed, which is verysuitable for solving the GEP (7) and significantly reduces thecomputational cost.

III. IMPROVED INEXACT IRAM IN THE

FE MODELING OF LOSSY SWSs

The IRAM is widely used to compute a few eigenpairs of thelarge-scale linear GEP, such as (7) [5]. In (7), both A and B aresingular, but the solution of (7) using the IRAM requires eitherA or B to be invertible. Moreover, the desired eigenmodesof SWSs are always in the interior of the spectrum of theGEP (7); however, it is well known that the Arnoldi iterationfavors the outlying eigenvalues [5]. For the above two reasons,a shift–invert transformation is applied to (7) to transform itinto a standard eigenvalue problem, i.e., (A − ϕB)−1Bx =1/(λ − ϕ)x, where shift ϕ = exp(−jΦ∗) and Φ∗ is a roughestimate of phase advance. It is a good practice that Φ∗ isroughly estimated by a user only in the initial point of frequencysweep, whereas at other frequency points, Φ∗ is chosen to bethe phase advance solved in the previous frequency point. LetP = (A − ϕB). As a result, P becomes nonsingular and theeigenvalues of (7) close to ϕ are mapped to the eigenvalues ofthe largest magnitude of the transformed equation. In the IRAMalgorithm, a linear system Py = Buj has to be solved withineach Arnoldi step, where uj is the Arnoldi vector generated inthe previous Arnoldi step. Since P is large scale, linear systemshave to be solved via an iterative method. Furthermore, thehighly ill-conditioned property of P makes the iterative solutionof these linear systems very inefficient. Consequently, solving aseries of linear systems is the most time-consuming step in theIRAM. If we can improve its efficiency, the overall efficiencyof modeling lossy SWSs using the FSEA will be significantlyimproved. Then, the question is: how can the iterative solutionsof the linear systems be accelerated? We discuss three answersto this question: 1) properly choose a larger solution tolerance;2) obtain a good approximate solution as the initial guess;and 3) construct an effective preconditioner. Correspondingto above three answers respectively, three techniques will be

[KII(f) +

√fCII KIM(f) + KIS(f)e−Φ

KMI(f) + KSI(f)eΦ KMM(f) + KSS(f) + KSM(f)eΦ + KMS(f)e−Φ

]· [xI xM ]T = 0 (5)


proposed and applied in the standard IRAM. Combination ofthese new techniques results in an improved inexact IRAM witha remarkable performance for solving the GEP (7).

A. Relaxation Strategy for the IRAM

When a tolerance ε of the outer Arnoldi iteration in theIRAM is assigned, what is the best tolerance l for efficientlysolving the inner linear systems in the IRAM? In the standardIRAM, l is suggested to be 10−2ε [5]. However, in the recentresearch [7], it has been observed that the linear systems mustbe solved with high accuracy in the initial Arnoldi steps,whereas accuracy can be relaxed as the IRAM proceeds withoutobviously affecting the convergence of approximate eigenpairs.Since the linear systems are solved with allowable errors, suchIRAM is referred to as the inexact IRAM. Assume the linearsystems are inexactly solved for m Arnoldi steps with anerror lj (1 ≤ j ≤ m) introduced at each step, and let D =(A − ϕB)−1B, thus, we have the following inexact Arnoldifactorization:

DUm + Lm = UmHm + hm+1,mum+1eTm (10)

where Lm = [l1, . . . , lm], Um ∈ Cn,m is the matrix that con-

sists of the Arnoldi vectors in m steps, Hm is an upper Hes-senberg matrix with dimension m, and eT

m = (0, 0, . . . , 0, 1) ∈R

m. We now consider the error that has been introduced inthe Ritz residual. Let Hm have the partial Schur decompositionHmQk = QkTk, where Tk ∈ C

k,k is an upper triangular matrixin which k wanted eigenvalues of Hm appear as diagonalelements and the columns of Qk ∈ C

m,k form an orthonormalbasis for an k-order invariant subspace of Hm corresponding tothese k wanted eigenvalues. Then, we have

DUmQk − UmQkTk︸︷︷︸R∗

m

= hm+1,mum+1eTmQk︸︷︷︸

Rm

−LmQk (11)

where R∗m denotes the true Ritz residual that is not available

during the computations and Rm denotes the computed Ritzresidual that, in turn, is available during the iterations. Unlikethe exact IRAM, these two quantities are no longer equal inthe inexact IRAM. To obtain accurate results with the inexactIRAM, the difference between R∗

m and Rm should be smallenough. Hence, we consider

‖R∗m − Rm‖ = ‖LmQk‖ ≤ ‖Lk‖ + ‖Lk+1:m‖‖Qk+1:m

k ‖(12)

where ‖ · ‖ denotes the matrix two-norm and Qk+1:mk is the

last m − k entries of the wanted Ritz vectors Qk. It is shownby (12) that, for a given k-step inexact Arnoldi factorizationwith small-enough errors, the upcoming m − k inexact Arnoldisteps do not have to be very small as long as the magnitudeof Qk+1:m

k is small enough. Fortunately, it has been provedin [7] that the magnitude of Qk+1:m

k is proportional to thecomputed Ritz residual at step k, viz., Rk. This means that,as the IRAM proceeds, Rk gets smaller and then the errors ofthe inexact Arnoldi factorization are allowed to be larger. Based

on this theory, the following practical estimate of the allowabletolerance for the linear systems in IRAM is used:

∥∥∥l(i)j

∥∥∥ ≤ 10−2ε

2k(i = 1, 1 ≤ j ≤ m) (13)

∥∥∥l(i)k+j+1

∥∥∥ ≤ 10−2ε

2(m − k)

min∣∣∣λW

(H

(i)m

)− λU

(H

(i)m

)∣∣∣∥∥∥R(i)k

∥∥∥× (i > 1, 0 ≤ j ≤ m − k − 1) (14)

where λW (H(i)m ) represents the wanted eigenvalues whereas

λU (H(i)m ) are the unwanted eigenvalues of H

(i)m . In our adopted

inexact IRAM, we only need to solve the linear systems exactlyusing the tolerance (13) in the first m Arnoldi steps, whereasfrom the first restart onward, the linear systems are inexactlysolved using the tolerance (14), which will get very large as theIRAM proceeds. Hence, the total iteration cost of solving thelinear systems in IRAM will be significantly reduced.

B. Tuning Technique for the IRAM

Although a series of different linear systems Py = Buj

has to be solved within the IRAM, there are two importantproperties that exist among these linear systems. First, matricesP and B are the same for all linear systems and only uj on theright-hand side varies. Second, uj is generated not randomly,but from the previous Arnoldi vector in the IRAM. In viewof these two important properties, a new tuning technique isproposed to find a good approximate solution y1 as the initialguess for the iterative solution of the linear system. Assume thatwe are in the ith IRAM cycle and then compute Du

(i)k+j+1 by

solving Py = Bu(i)k+j+1 during the current step. Consider the

following set:

Y (i,l)p =

[DU (i−l)

m ,DU(i−l+1)k+1:m , . . . , DU

(i−1)k+1:m,DU

(i)k+1:k+j

](15)

where U(i−1)k+1:m represents the (k + 1)-th through the mth

columns of U(i−1)m and p = (m − k)(l − 1) + m + j is the

number of vectors in Y(i,l)p . Y

(i,l)p is called the set of solution

vectors, which contains the solution vectors of previous p linearsystems within current and previous l IRAM cycles. We thenhave another set, i.e.,

PY (i,l)p =

[BU (i−l)

m , BU(i−l+1)k+1:m , . . . , BU

(i−1)k+1:m, BU

(i)k+1:k+j

].

(16)

This set is referred to as the set of right-hand sides becauseits columns are the right-hand sides in the previous p linearsystems. We introduce span{Y (i,l)

p } and span{PY(i,l)p } to stand

for the subspaces spanned by the column vectors of Y(i,l)p and

PY(i,l)p , respectively. It is a very important relation between

Bu(i)k+j+1 and span{PY

(i,l)p } that if p > m, as p gets large,

Bu(i)k+j+1 approximately lies in span{PY

(i,l)p }, as proved in

[8]. This important relation implies that, for a large p, Bu(i)k+j+1


is roughly a linear combination of the right-hand sides of theprevious p solved systems. Since matrix P is the same in alllinear systems, this important relation motivates us to seekan approximate solution of the current linear system from thesubspace spanned by the solution vectors of previous p linearsystems, viz., span{Y (i,l)

p }. This process can be easily achievedby solving the following least squares minimization problem:

minf

∥∥∥Bu(i)k+j+1 − PY (i,l)

p f∥∥∥ (17)

which requires little additional computer resource due to thesmall column number of PY

(i,l)p . After solving (17), we have

a very good approximate solution y1 = Y(i,l)p f as the initial

guess for the iterative solution of the current system. Conse-quently, the convergence time of solving this linear system willbe further reduced. In the practical application of the tuningtechnique, a proper value of p should be chosen consideringboth convergence rate and memory (p = 6 is our suggestedvalue by numerous numerical experiments).

C. p-Type Multigrid Preconditioner for the IRAM

Since matrix P has complex nonsymmetric and highly ill-conditioned properties, an effective preconditioner is very valu-able in the iterative solution of the linear systems within IRAM.However, as the dimension of P increases, the efficiency ofwidely used preconditioners such as incomplete LU factor-ization (ILUF) quickly deteriorates [9]. The essential reasonis that, in the iterative solution of the linear systems, thehigh-frequency components of the iterative errors damp veryrapidly whereas the low-frequency components are difficult todamp. The multigrid method utilizes the coarse-grid correctionscheme to efficiently eliminate these low-frequency compo-nents of the iterative errors because they manifest themselvesas the high-frequency components of the iterative errors onthe coarser grid. Hence, the convergence rate is enabled to beindependent of the dimension of P . In [6] and [9], using FEM,a p-type multigrid preconditioner is successfully applied in thedriven-frequency modeling of electromagnetic (EM) scatteringand radiation problems. Here, we introduce it as an effectivepreconditioner to efficiently solve the linear systems in theIRAM. To the best of our knowledge, this is the first time thismultigrid preconditioner has been used together with IRAMto solve the GEP arising from the FE modeling of SWSs.This preconditioner is different from the traditional multigridmethod because it exploits the hierarchical property of thehierarchical basis to form a two-level grid, instead of the geo-metrical multigrid in the traditional multigrid method. When weadopt the second-order hierarchical basis in [6], matrix P canbe partitioned in terms of the basis order as

P =[

P11 P12

P21 P22

](18)

where P11 is completely constituted by the first-order basis;hence, it is viewed as the coarser grid matrix, and P is viewedas the finer grid matrix. We apply the Schur decomposition forthe 2 × 2 matrix P and make an approximation for the inverse

Fig. 1. (a) One-cavity geometry of coupled-cavity SWS. (b) Distributions ofelectric vector field of cavity mode simulated by FSEA-lossy(C) at 61.232 GHz.

TABLE ICOMPARISON BETWEEN COLD PARAMETERS OF CAVITY MODE FROM

THE PSEA AND THE FSEA IN THE LOSSLESS AND LOSSY CASES

of P , and then this p-type multigrid preconditioner can befound as

M−1 =[

I −(L11U11)−1P12

0 I

] [(L11U11)−1 0

0 (L22U22)−1

]

×[

I 0−P21(L11U11)−1 I

](19)

where L11U11 and L22U22 stand for the ILUF of P11 and P22,respectively, and the drop threshold is 10−5 for P11 and 10−2

for P22. Since P is the same in all linear systems, we do theILUF of P11 and P22 only once before performing the IRAM.

IV. NUMERICAL RESULTS AND DISCUSSION

A. Accuracy Study of the FSEA Versus the PSEA

We first compare the accuracy values of modeling lossySWSs using the proposed new FSEA method and the traditionalPSEA method through an example of coupled-cavity SWStaken from [10] (see Fig. 1). To make the comparison moreconvenient, we first specify a phase advance 270◦ and employthe PSEA method to compute a frequency that is used asthe specified frequency for the FSEA, and then a computedphase advance will be obtained by the FSEA method. Theerror between the specified and computed phase advances isexactly the difference of dispersions calculated by the PSEAand the FSEA. Table I shows the cold parameters of the cavitymode calculated using this procedure in the lossless and lossycases. All the results shown in Table I are obtained in the same


converged meshes. Moreover, in this table, (C) represents thecase of only considering the conductivity losses due to the finiteconductivity of cavity σ = 5.8 × 107 S/m, whereas (D) denotesthe case of only considering the dielectric losses, where weassume a dielectric material with εr = 1.0 and material havingloss tangent tan δ = 3e − 3 fills the coupled-cavity SWS. Itis clearly shown in Table I that the dispersions and couplingimpedance values Kc calculated by the PSEA and the FSEA arealmost the same due to adopting identical meshes and bases. Inevery case, the differences are less than 1.11e-4% for the disper-sion and 0.07% for Kc. The frequency and coupling impedancemeasured by the experiment at a phase advance 270◦ are61.36 GHz and 7.08 Ω, respectively [10]. The errors betweenthe experimental data and the FSEA are 1.05% for Kc and only0.16% for the dispersion in the lossy(C) case, which is closer toreality.

With respect to the calculation of attenuation, in the FSEA,the conductivity and dielectric losses can be directly obtainedwithout any postprocessing, whereas in the PSEA, only con-ductivity losses can be estimated by the perturbation theory inthe lossless case and the conductivity and dielectric losses areevaluated by complicated postprocessing in the lossy case. Inour postprocessing of the PSEA, first, the total power loss ofthe SWS is calculated from the Q factor, and then the totalstored EM energy and the time-averaged RF power flow arecalculated by numerical integrations, and finally, the attenua-tion is obtained. Moreover, to avoid solving a nonlinear GEPwhen using the PSEA in the lossy(C) case, in this paper, wepropose an alternative PSEA method according to the followingtwo-phase procedure. First, we use the PSEA to calculate areal frequency of the SWS in the lossless case. Second, thisreal frequency is substituted in the imposed SIBC (2), and acomplex nonsymmetric linear GEP with the frequency as itseigenvalue is be deduced from (5). Consequently, we can solvethis linear GEP to get a complex frequency, which is used tofurther evaluate the attenuation constant. Although such PSEAin this paper alleviates the inefficiency of solving a nonlinearGEP in the usual PSEA, it is still not efficient enough comparedto the FSEA for the reason that we have to model the sameSWS in both lossless and lossy cases, which will be shownin a later subsection. In addition, we also use the convergedresults simulated by high frequency structure simulator (HFSS)(version 13.0) as the benchmark for our results. It should benoted that the eigenmode solver type adopted by the HFSSis the PSEA and that HFSS evaluates the attenuation throughsame postprocessing as our PSEA.

The conductivity and dielectric losses evaluated by the HFSSare 26.56 and 89 dB/m, respectively. It is shown in Table I thatthe conductivity losses estimated by the perturbation theory inthe PSEA-lossless case are lower than those from HFSS by13.02%; hence, the perturbation theory is not accurate enoughin this case. The FSEA agrees very well with HFSS on theconductivity losses with a difference 0.88%. If the real fre-quency is accurately obtained enough in the first phase losslessmodeling, the PSEA will have almost the same conductivitylosses with the FSEA. In Table I, the difference is 0.13%. Inthe comparison of the dielectric losses, the results obtained bythree eigensolvers agree very well.

Fig. 2. Attenuation versus frequency for a helix SWS [11] obtained by theFSEA and validated against the experiment, the HFSS, and other analyticalmethods.

B. Accuracy Study of the FSEA Versus Other Methods

Next, we simulate a practical helix SWS, for which thedimensional details and measured conductivity losses are givenby Gilmour et al. [11]. The SWS uses a copper-plated helixand is enclosed in a copper barrel without vanes. We comparethe conductivity losses calculated by the FSEA against thosefrom the experiment, HFSS, and other analytical methods.To increase the accuracy of modeling the curved surfacesof the SWS, particularly those of helix, we use curvilinearelements and the new tau meshing engine in HFSS 13.0.Although rectilinear elements are used in our codes, we utilizean advanced curved surface approximation technique basedon Simmetrix’s meshing kernel [3] to improve the modelingaccuracy. Compared to the analytical methods of Jain andBasu [12] and Gilmour et al. [11], Fig. 2 shows that ourFSEA gives better agreement with the experiment. The averagedifference between the FSEA and the HFSS is 0.94%. It isinteresting that our FSEA also agrees very well with the recentanalytical method proposed by Rao et al. [13] with a differenceof less than 3%, except for the last frequency point. The FSEAagrees with the experiment by a mean difference 13.23%.The differences in the computed and experimental results areattributed to the reduction in the conductivity of the helix andbarrel due to their surface roughness by Gilmour et al. [11].If it is assumed that the effective conductivity of the helix andbarrel is 4.3 × 107 S/m instead of 5.8 × 107 S/m, the FSEAwould agree very well with the experiment, as the dashed lineshows in Fig. 2.

C. Computational Performance Study

We now investigate the computational performance of theFSEA and various advanced techniques adopted in the im-proved inexact IRAM for modeling lossy SWSs. Table II givesthe computational statistics for simulating the same lossy helixSWS with the subsection B at 12 GHz using different technicalcombinations. Meanwhile, we also give the statistical data forHFSS with the same basis and rectilinear elements to validate


TABLE IIPERFORMANCE AND ACCURACY COMPARISONS OF VARIOUS

TECHNIQUES ADOPTED IN EIGENSOLVER FOR MODELING

A LOSSY HELIX SWS [11] AT 12 GHZ

the efficiency of our methods. The relaxation strategy, thetuning technique, and the p-type multigrid preconditioner forthe improved inexact IRAM are labeled as Relax, Tuning, andPMP, respectively, in Table II. All simulations in the table werecarried out on a Windows 7 64-bit Intel Core i7 1.6-GHz and8-GB RAM laptop. All simulations with the FSEA and PSEAare computed with the same final converged mesh (15 749elements and matrix dimension 102,386). The inner linearsystems in the IRAM are solved via the generalized conjugateresidual method. We also make the simulations herein in avery rigorous condition, which would be needed sometimes forpractical application. In the simulations herein, the estimatedphase advance for the FSEA is very rough and the toleranceshave very high accuracy values (10−12 for the outer Arnoldiiteration and 10−14 for the inner linear systems solution inthe IRAM). It is shown in Table II that the FSEA approachequipped with the PMP preconditioner is nearly 1.8 times fasterthan that equipped with the widely used ILUF preconditioner(the drop threshold of the ILUF is 10−2) and only uses nearly55% of the memory of the latter. When applying the same PMPpreconditioner, our proposed PSEA approach with the two-phase procedure requires 31 s in the lossless modeling phase,which is performed using the fast eigensolver for lossless SWSsproposed in [14], and uses 119 s in the lossy modeling phase.Compared to such a PSEA, the FSEA completely avoids thecomputational cost in the lossless modeling phase of the PSEA,which will become very huge when the frequency increases andthe geometry of the SWS is complex. Moreover, the applicationof the relaxation strategy and the tuning technique can furtherimprove the computational performance of the FSEA. Com-pared to only employing the PMP preconditioner, the FSEA,with either the relaxation strategy or the tuning technique,provides a gain of more than 2 times in the CPU times, asshown in Table II. When the FSEA is combined with thethree advanced techniques, a very remarkable computationalperformance is achieved. The combination of the FSEA andthe improved inexact IRAM is more than 3 times faster thanthe FSEA plus PMP, more than 4 times faster than the PSEAplus PMP, and almost 5.4 times faster than the FSEA plusILUF. It should be noted that the accuracy values of the coldparameters are not affected when the solution times of modelinglossy SWSs are significantly reduced by the improved inexactIRAM. It is clearly shown in Table II that the errors betweenvarious technical combinations with the FSEA are less than

Fig. 3. (a) Attenuation constant, including conductivity and dielectric losses,versus frequency for a sector-vane-loaded helix SWS obtained by the FSEA andvalidated against the HFSS. Geometry of the SWS and the one-third structureof the SWS. (b) Dispersion and coupling impedance versus frequency for theSWS obtained by the FSEA and validated against the HFSS. Distributions ofelectric magnitude field simulated by the FSEA at 17.54 GHz.

7e-5% for the normalized velocity and 0.0057% for the atten-uation constant. To make a fair comparison with our FSEA,we only consider the computational statistics for HFSS in thefinal converged mesh (14 524 elements and matrix dimension94,648). It is shown that the FSEA equipped with the improvedinexact IRAM consumes far less time and memory than HFSS.

D. Lossy Sector-Vane-Loaded Helical SWS Application

The final example is a sector-vane-loaded helical SWS thatuses a tungsten helix and is enclosed in a copper barrel. More-over, the material of the dielectric support rods in the SWS isberyllia with loss tangent 6e-4. We simulate this lossy SWS viathe FSEA method combined with the improved inexact IRAMand also validate the results against the HFSS. Fig. 3(a) givesthe geometry of the SWS and its one-third structure, which iscalculated when applying the rotated PBC in our simulator. Theattenuation constants, including both conductivity and dielec-tric losses, computed by the two simulators agree very well, asshown in Fig. 3(a). The averaged difference is 1.36%. Fig. 3(b)also shows good agreement of the FSEA and the HFSS for the


dispersion and coupling impedance. The difference between theFSEA and the HFSS is less than only 0.42% for the dispersionand 1.72% for the Kc.

V. CONCLUSION

This paper has presented a novel 3-D FE modeling techniquefor an arbitrary lossy SWS, which incorporates the FSEAapproach and the improved inexact IRAM. Through simulatingseveral practical SWSs, this new modeling technique hasbeen validated against the experiment, HFSS, and analyticalmethods. It has been shown that the new FSEA method issuperior to the traditional PSEA method in the modeling oflossy SWSs because the FSEA not only avoids solving anonlinear GEP but also can obtain the attenuation accuratelyand directly without any postprocessing. Moreover, it hasbeen also shown that the improved inexact IRAM provides asignificant improvement in the overall efficiency of modelinglossy SWSs. By using this new modeling technique, wecan accurately and very quickly calculate the attenuationconstant, including both the dielectric and conductivity losses,in addition to the dispersion and coupling impedance of thearbitrary SWS. This is expected to be valuable for the designof low-loss SWS for high-efficiency TWTs.

ACKNOWLEDGMENT

The authors would like to thank Mr. P. Raja Ramana Raofor many helpful discussions on the accurate calculation of theattenuation in lossy slow-wave structures.

REFERENCES

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[14] L. Xu, Z. Ye, Z. Yang, J. Li, and B. Li, “A novel approach of removingspurious dc modes in finite-element solution for modeling microwavetubes,” IEEE Trans. Electron Devices, vol. 57, no. 11, pp. 3181–3185,Nov. 2010.

Li Xu received the B.S. degree in applied physicsin 2005 from the University of Electronic Scienceand Technology of China, Chengdu, China, wherehe is currently working toward the Ph.D. degree inphysical electronics.

Zhong-Hai Yang (M’93) received the Ph.D. degreein physical electronics and optical electronics in1984 from the University of Electronic Science andTechnology of China, Chengdu, China, where he iscurrently a Professor.

His research is in vacuum and plasma electronics.

Jian-Qing Li received the Ph.D. degree in physicalelectronics in 2003 from the University of ElectronicScience and Technology of China, Chengdu, China,where he is currently a Professor.

His research interests include modeling and simu-lation of vacuum electronic devices.

Bin Li (M’06) received the Ph.D. degree in physicalelectronics in 2003 from the University of ElectronicScience and Technology of China, Chengdu, China,where he is currently a Professor.

His research interests include modeling and simu-lation of vacuum electronic devices.

accurate and fast finite-element modeling of attenuation in slow-wave structures for traveling-wave...

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