Partially Loaded Cavity Analysis by Using the 2-D FDTD Method

Partially Loaded Cavity Analysis by Using the 2-D FDTD Method

?

YAO Bin( )1,2,ZHENG Qin-Hong( )1,2??,PENG Jin-Hui( )3,ZHONG Ru-Neng( )2,

XIANG Tai( )2,XU Wan-Song( )2

1

College of Physical Science and Technology,Yunnan University,Kunming 650092

2

School of Physics and Electronic Information,Yunnan Normal University,Kunming 650092

3

Faculty of Materials and Metallurgical Engineering,Kunming University of Science and Technology,Kunming 650093

(Received 24June 2011)

A compact two-dimensional (2-D)?nite-di?erence time-domain (FDTD)method is proposed to calculate the resonant frequencies and quality factors of a partially loaded cavity that is uniform in the z -direction and has an arbitrary cross section in the x –y plane.With the description of z dependence by k z ,the three-dimensional (3-D)problem can be transformed into a 2-D problem.Therefore,less memory and CPU time are required as compared to the conventional 3-D FDTD method.Three representative examples,a half-loaded rectangular cavity,an inhomogeneous cylindrical cavity and a cubic cavity loaded with dielectric post,are presented to validate the utility and e?ciency of the proposed method.

PACS:84.40.?x,03.50.De,41.20.?q DOI:10.1088/0256-307X/28/11/118401

Since the ?nite-di?erence time-domain (FDTD)method was ?rst introduced by Yee,[1]it has been widely used to solve electromagnetic problems.[2,3]Among them,endeavors have been made to achieve faster resonator computation and some numerical techniques have been combined with the FDTD method for faster computation.The numerical tech-niques include digital ?ltering,the modern spectrum estimation technique,Prony analysis,Padéapproxi-mation and the Baker algorithm.[4?8]Bene?ting from these techniques,the analysis time of resonators has been greatly reduced.However,there are two de?cien-cies in the above-mentioned literature.First,these ap-proaches are still based on a three-dimensional (3-D)mesh,which needs numerous computational resources.Second,these approaches lack a clear physical mean-ing and suitable mathematic demonstration.Partic-ularly,the computational accuracy heavily depends on the samples of signals.In some cases,the ac-curate results are di?cult to obtain.Recently,the compact two-dimensional (2-D)FDTD method has

Consequently,this 2-D method is very suitable for the e?cient computation of resonant frequencies and the quality factors of a partially loaded cavity,which is uniform in the z -direction and has an arbitrary cross section in the x –y plane.

x

y

ε2

ε1

Fig.1.Cross section of a partially loaded cavity which is uniform in the z -direction.

Although an electromagnetic resonant cavity can be of any shape whatsoever,an important class of cav-ities is produced by placing end surfaces on a length of waveguide.Figure 1shows the cross section of a partially loaded cavity,which is uniform in the z -direction.The electromagnetic ?eld components in this cavity can be expressed as

If the plane boundary surfaces are at z=0and z=l, then the boundary conditions can be satis?ed at each surface only when

k z=pπ

l

,(p=0,1,2,···)(4)

where l is the length of cavity in the z-direction.Then, the?eld components can be written as

E(x,y,z,t)=?e(x,y,t)e i pπl z(5)

H(x,y,z,t)=?h(x,y,t)e i pπl z.(6) Finally,instead of??z by i pπl,Maxwell’s curl equations can be expressed as

?H z ?y ?i

pπ

l

H y=ε

?E x

?t

+σE x,

i pπ

l

H x?

?H z

?x

=ε

?E y

?t

+σE y,

?H y ?x ?

?H x

?y

=ε

?E z

?t

+σE z,

?E z ?y ?i

pπ

l

E y=?μ

?H x

?t

?σm H x,

i pπ

l

E x?

?E z

?x

=?μ

?H y

?t

?σm H y,

?E y ?x ?

?E x

?y

=?μ

?H z

?t

?σm H z,(7)

whereσis the electric conductivity,εis the permittiv-ity,σm is the magnetic conductivity andμis the per-meability.For complex factor i in Eq.(7),we can also remove it in two ways.[13,14]However,Eq.(7)will be invalid under some conditions if factor i is removed.[15] Therefore,Eq.(7)is directly used.From Eq.(7),the time-domain di?erence equations of E x and E z can be derived as

E n+1

x

(i+

1

2

,j)

=

1?

σ(i+1/2,j)Δt

2ε(i+1/2,j)

·

1+

σ(i+1/2,j)Δt

2ε(i+1/2,j)

?1·E n x(i+

1

2

,j)+

Δt

ε(i+1/2,j)

·

1+

σ(i+1/2,j)Δt

2ε(i+1/2,j)

?1

·

H n+1/2

z

i+

1

2

,j+

1

2

?H n+1/2

z

i+

1

2

,j?

1

2

·

1

(Δy)

?i

pπ

l

H n+1/2

y

i+

1

2

,j

,(8)

E n+1

z

(i,j)=

1?

σ(i,j)Δt

2ε(i,j)

1+

σ(i,j)Δt

2ε(i,j)

?1

·E n z(i,j)+

Δt

ε(i,j)

1+

σ(i,j)Δt

2ε(i,j)

?1

·

H n+1/2

y

i+

1

2

,j

?H n+1/2

y

i?

1

2

,j

·

1

(Δx)

?

H n+1/2

x

i,j+

1

2

?H n+1/2

x

i,j?

1

2

·

1

(Δy)(9) where p is an input parameter that corresponds to di?erent mode series,such as TEδ0,TEδ1,TEδ2,etc. The corresponding time-domain di?erence equations of the remaining equations(7)are similar to Eq.(8) and(9),which,for brevity,are not given here.

https://www.wendangku.net/doc/0e12278683.html,parison of the results computed by HFSS(A),the proposed2-D FDTD with the elapsed CPU time2624s(B),the conventional3-D FDTD with the elapsed CPU time5241s(C),and the US-FDTD(D)(Ref.[16]).

A B C D

Resonant Quality

p Resonant Quality Resonant Resonant frequencies(MHz)factors frequencies(MHz)factors frequencies(MHz)frequencies(MHz)

18.618128.8118.624128.518.60718.610

To

As the

lar

method

methods,a uniform meshΔl=0.1m and time-step Δt=Δl/2c are used,where c is the speed of light in vacuum.A comparison of the computed resonant fre-quencies and the quality factors is presented in Table 1.From the comparison,one can?nd a good agree-ment.Moreover,the CPU time cost is5241s for the conventional3-D FDTD method,whereas it is only 2624s for the proposed2-D FDTD method.Obvi-ously,in this example,about half of the CPU time is saved and only one?fteenth memory cost of the3-D FDTD method is required.

y

z

x

εr/1εr/64

Fig.2.Cavity half?lled with air and the other half?lled with the dielectric material.

Fig.3.The cross section of the cylindrical cavity loaded with a dielectric cylinder and two metallic https://www.wendangku.net/doc/0e12278683.html,parison of resonant frequencies(in GHz)com-puted by HFSS(A),the proposed2-D FDTD with the elapsed CPU time126min(B),and the conventional3-D FDTD with the elapsed CPU time5432min(C).The cavity radius R= 30mm and height H=100mm.Parameters of the post: d=10mm,r1=5mm,εr=5,r2=10mm and?=60?.

HFSS results

Proposed2-D FDTD,

the elapsed CPU time

is126min

Conventional3-D

FDTD,the elapsed

CPU time is5432min

0.983210.98810.9881

1.48201 1.4913 1.4912

1.89992 1.9030 1.9030

2.75153 2.7447 2.7447

2.96241 2.9551 2.9551

2.98312 2.9735 2.9645

3.16951 3.1472 3.1472

3.56254 3.5498 3.5498

3.92262 3.9066 3.9065

4.07412 4.0560 4.0555

The second example is a dielectric cylinder with two metallic strips placed in the cavity,[17]which is shown in Fig.3.The resonant frequencies of this cavity are also calculated by HFSS,the proposed2-D FDTD method and the conventional3-D FDTD method,respectively.For comparison of the CPU time cost,the same parameters(meshΔl=0.5mm, time-stepΔt=Δl/2c,and time iterations217)are used by the2-D FDTD and3-D FDTD methods. The results are compared in Table2.The compari-son shows that the resonant frequencies computed by HFSS,the proposed2-D FDTD method and the con-ventional3-D FDTD method agree with each other well.However,the CPU time cost by the proposed 2-D FDTD method is only about2.3%of the conven-tional3-D FDTD method.

https://www.wendangku.net/doc/0e12278683.html,parison of the normalized wave number computed by HFSS and the proposed2-D FDTD method.

b/a Method Normalized wave number ka

Mode1Mode2Mode3Mode4Mode5Mode6Mode7 3/

HFSS 3.227 3.467 3.757 4.508 4.783 4.818 5.005

The last example is a fabricated cubic cavity loaded with dielectric posts.Figure4shows the cross section in the x–y plane of the cavity.The lowest seven normalized wave numbers versus b/a are calcu-lated by HFSS and the proposed compact2-D FDTD method.The calculated results are presented in Table 3.From the comparison,one can?nd that the results agree with each other very well.

Fig.4.The cross section of a cubic cavity loaded with

a dielectric post.The cavity is uniform along the z-axis

and the length of the cavity in the z-direction is1.5a.The relative permittivityεr=9.8.

In summary,an e?cient2-D FDTD method is proposed.With the description of z dependence by wave number k z,this method can transform the3-D problem into a2-D model.To validate the pro-posed method,three representative examples have been computed.The numerical results show that the proposed method has advantages of CPU time and memory saving over the conventional3-D FDTD method for the analysis of partially loaded cavity,which is uniform in z-direction and has an arbitrary cross-section in the x–y plane. References

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