复旦大学：《电动力学》学生课堂报告_谐振腔 Discussion about resonant cavity and graphical simulation with COMSOL

Discussion of resonant cavity and the simulation by software COMSOL Wang Juan Lin yihua Dec 25th 2010

Wang Juan Lin Yihua Dec 25th,2010 Discussion of resonant cavity and the simulation by software COMSOL

Content 1.Theoretical deduction of tm wave function 2. Practical simulation of te TM wave in resonant cavity by COMSOL 3.Some interesting questions in the process of simulation

Content • 1.Theoretical deduction of TM wave function • 2.Practical simulation of TE & TM wave in resonant cavity by COMSOL • 3.Some interesting questions in the process of simulation

I. Theoretical deduction E E。 cOS x sin a k 2y-0 ah Eo cos/mn xsin n兀 E sin xcOS y/eg-i nt k SIn xcOS ve b k ik n兀 yeg+ Eo sin -ik SIn x sin x sin Bo nZL SIn x cOS yle g+e sin x cos/z ye 6 ck .m丌 E cos x leg+Eo cos X sin y e ck

I. Theoretical deduction 0 0 0 2 2 cos sin cos sin g g g g ik z ik z x c c m m n m m n k k E i E x y e i E x y e a k a b a k a b               − = −                 ’ 0 0 0 2 2 sin cos sin cos g g g g ik z ik z y c c n m n n m n k k E i E x y e i E x y e b k a b b k a b               − = −                 ’ 0 0 0 sin sin sin sin g g ik z ik z z m n m n E E x y e E x y e a b a b             − = +                 ’ 0 0 0 0 2 sin cos sin cos g g ik z ik z x c n m n m n k B i E x y e E x y e b ck a b a b                − = − +                     ’ 0 0 0 0 2 cos sin cos sin g g ik z ik z y c m m n m n k B i E x y e E x y e a ck a b a b                − = +                     ’

I. Theoretical deduction According to: Eox=0=0 Z=0: M7 b∥/Eek i cOS x sin Eoe ! z=d ik. d cOS x sin 6 y Eo/e"d 0 k2 ∴k2a=nn→ksPn ,(P=0,,2,3…)

0 0 z 0 x z a E = = According to: = 2 cos sin 0 ( 0 0 ) g g g ik z ik z c m m n k i x y E e E e a k a b        −     − =     ’  = E E 0 0 ’ 2 cos sin 0 0 ( ) g g g ik d ik d c m m n k i x y E e e a k a b        −     − =     ,( 0 1,2,3 ) g g p k d p k p d   =  = =  ， …… Z=0: Z=d: I. Theoretical deduction

I. Theoretical deduction 28 mT n丌 E n兀 b元 -lwt 0 COS xsin y sin le b E 2-h nD兀 c6 Sin XcOS y Sin b n7 1元 E=2E Sin nt x sin y COS 2|e 1D兀 1D兀 B.=-2 e sin X cos V/coS/pr -It 2|e ck- b B=-2 k 1D兀 1D兀 E。cos nt x sin V COS z e ck b B.=0

2 0 2 cos sin sin g iwt x c k m m n p E E x y z e k a a b d           − = −             2 0 2 sin cos sin g iwt y c k n m n p E E x y z e k b a b d           − = −             0 2 sin sin cos iwt z m n p E E x y z e a b d          − =             0 2 0 2 sin cos cos iwt x c k n m n p B i E x y z e ck b a b d           − = −             0 2 0 2 cos sin cos iwt y c k m m n n B i E x y z e ck a a b d           − = −             0 B z = I. Theoretical deduction

I. Theoretical deduction h Resonance frequency 0=C丌 =2/ In a cubical resonant cavity: o=Cx q vm+n+p n=2a/vm+n+p Discussion of degree of degeneracy Define: n 2 +n2+p2=X 1 3 the degree of degeneracy is 1 2.m2 A,P2=B(2A+B=X, A+B), the degree of degeneracy is 3 B,p2=C(A≠B≠C,A+B+C=X),the degree of degeneracy is b;

I. Theoretical deduction Resonance frequency: In a cubical resonant cavity: Define: Discussion of degree of degeneracy: 1. , the degree of degeneracy is 1; 2. , the degree of degeneracy is 3; 3. , the degree of degeneracy is 6; 2 2 2 2 2 2 m n p c a b d   = + + 2 2 2 2 2 2 2 / m n p a b d  = + + c 2 2 2 m n p a   = + + 2 2 2  = + + 2 / a m n p 2 2 2 m n p X + + = 2 2 2 3 X m n p = = = 2 2 2 m n A p B = = = , (2 , A B X A B + =  ) 2 2 2 m A n B p C = = = , , ( A B C A B C X   + + = , )

I. Practical simulation C3×10 TE(0, 1, 1 Mode: U= ×10Hz 元√2×0.3V2

• TE (0,1,1) Mode: II. Practical simulation 8 3 10 1 9 10 2 0.3 2 c  Hz   = = =  

I. Practical simulation Section : z=0 Section x=0 Section: y=0 a=0,b=1 1,c=1 Ck兀 B。S SIn 2 k C

0 2 0 E 2 B sin sin iwt x c ck y z e k a a a        − =         Section: z=0 a=0, b=1 Section: y=0 a=0, c=1 Section: x=0 b=1, c=1 II. Practical simulation

I. Practical simulation B.=-2 B。sin COS C Section: y=0 0,C=1

Section: y=0 a=0, c=1 g 2 0 2 B sin cos iwt y c k B i y z e k a a a        − = −         II. Practical simulation

I. Practical simulation 3×10 TE(2, 1, 1 ) Mode 1.5×10H 0.3 ction: y a=2,b=1

• TE (2,1,1) Mode: Section: y=0 a=2, b=1 8 3 10 9 1.5 10 2 0.3 3 c  Hz   = = =   II. Practical simulation