Chapter 6 Microwave Resonators 1 (1) Series Resonant Circuit Zrjljcin  

6.1 Series and Parallel Resonant Circuits

Chapter 6 Microwave Resonators 1 (1) Series Resonant Circuit ZRjLjCin  

Part I 11* 2 2Pin PVIIZPjWWZiiliin in loss2   m e  in  1. Series and Parallel Resonant Circuits 22 I 2

1 2 2. Loss and Q Factor of a Resonant Circuit Power dissipation : PIR oss 2 3. Various Waveguide Resonators 1 2 Energy stored in LW : m  IL 4. Coupling to a Lossy Resonator 4 I 2 Energy stored in CW :  e 4 2C Part II  RtR esonance occurs a t WWZR , , and  Time-Domain Analysis of Open Cavities mein

0 1.LC Part III  Quality factor :

 Spectral-Domain Analysis of Open Cavities Average energy stored2WLm 0 1 Q  0 ElEnergy loss per second PRRCloss 0

1 2

Series Resonant Circuit Parallel Resonant Circuit

NRNear Resonance  0 , is sma ll.  1 2 V ZRjLin 1 2 1  LC Power dissipation : Ploss  2 R  22 RjL 0 1 2 2 Energy stored in CW : e  CV  4 QR  L 2 Rj 2  LRj  2  , Q 0 1 V Energy stored in LW : m  2 0 R 1 4  L 11 ZjC  j in  Complex frequency :  1 RjL 00  Complex power delivered to the resonator 2Q  2  If R=0 or lossless case , 11V 1112  *  Pin VI*  V  j C 22Zin 2RjL ZjLjLin 22  0 PjWWloss2  m  e  j 0 L jL21 00  jL 2  2QQ  Input impedance at resonance, jL2 0 R 1 ZRWWin, m e , 0   A resonator with loss can be treated as a lossless resonator whose resonant LC  frequency ω0 is replaced by a complex frequency ω0(1+j/2Q) 3 4 Parallel Resonant Circuit Parallel Resonant Circuit

The loss factor can be accounted by Average energy stored Q   using 0012.j Q ElEnergy loss per second 2Wm R QRC00 2Wm R PLloss 0 00RC 1 PLloss 0  Fractional bandwidth 2  0 Q R 1 Z   11 1 in if RZ, in  2 Zjin  C , near resonance  0  j2  C RjL   1 jL  jR 1 22CjC  12  0  1 LC  jC  jL 1 1 11  0 2 L  jC0   jC2 12j Q LC 1, RjL R 00 0    R  2  j  jC21 1  RR 1 0   10 Quadratic equation R 2Q   00Q  12jjjRC  j 2  C 12 jQ 0C 1  jC22 0  jC 11 1 1 0 QR 4 12 2 QQQ2 5 00 6

Loaded and Unloaded Q Summary

 Unloaded Q (Load resistance R L  ) (1) Model for a Series Resonant Circuit  L 1 Q 0 Series Resonance: QRCLR1 00 RRC0

Zin R jjj22LR  j  0 L PlllParallel Resonance: QRCRL00 • A lossless series resonant circuit is  0 L short-circuited at ω and has Q    , fifor series resonant tiit circuits; 0  RL  ExternalQQ, e   R  L , for parallel resonant circuits. (2) Model for a Parallel Resonant Circuit  0 L R QRC 0  Loaded Q, or QL , of a resonant circuit with a load of RL 0 L  L 111 11 0 YjCjC 22   Series Resonance: QL  in 0 RR LLe Q Q Q R R RR// 111 1 11 • A lossless parallel resonant circuit is Parallel Resonance: Q L , L short-circuited at ω and has Q  0 LQQQRRRRLe// L L 0

7 8 6.2 Transmission Line Resonators 6.2 Transmission Line Resonators

 (1) Short-circuit λ/2 line section  All realistic resonators have a finite Q=f0 /BW     Nonzero bandwidth Resonator is lossy (a) tanh , (b) is small. 0  1 tan   vp 00 Transmission line resonator has an   0  tanh j tan  Lossy transmission line Z Z tanh j  Z in 001 j tanh tan      j    ZZj0    00  (1) Short circuit λ/2 line series resonance 1 j  0  If loss is small, 1, tan   . 0

 Near resonance,  0   ,  is small. RjL2   What is the equivalent circuit? Equivalent circuit :

 Z0 1 RZ0, L, C 2 2 00 L  L  Q 0 R 22 

9 10

Transmission Line Resonators Transmission Line Resonators

(2) Short-circuit λ/4 line parallel resonance (3) Open-circuit λ/2 line parallel resonance

 If loss is sma ll,  1t1, tan .  If loss is sma ll,  1t1, tan .

 Near resonance, 0 , is small.  Near resonance, 0 , is small.

1tanhcot j  1 j tanh tan  ZZtanh  j  Z ZZin 00coth  j  Z in 00tanh j cot tanh j tan       1cottan1, cot tan     1,tantan1,  tan tan  222000  000  Z 1 0 Equivalent circuit : Z0 1 Equivalent circuit : Zin  Z    1 in  1  jjC2    j2 jC 2 0 R R 0  Z 1  Z 1 CRL,,, 0 , CRL, 0 , 4 Z  2C 2 ZC 2 00 0 00 0    QRC0 ,  at resonance. QRC,   at resonance. 42  2 0 2

11 12 Example 6.2 6.3 Rectangular Waveguide Cavity AhalfA half-wave microstrip resonator 50-Ω line,  /2 resonator, df 1.59 mm, 2.2, tan 103 , 5 GHz.  r TE , TM modes : Calculate its Q value. mn mn  jmnz j mn z Exyzt  ,, exyAe , Ae  Sol: 22 Zd0 =50 Ω, =1.59 mm, and  rreff =2.2 W =4.9 mm,  =1.87. 2 mn  mn k   ab  22cf reff   21.9 mm 2 143.2 rad/m For a resonant cavity, Ezdt 0, at 0, c RZW0 0.075 Np/m mnd   ,  1, 2, 3, ... d ... 0.0611 Np/m 222 mn  Q 22   c  d   526. kmn  abd

c kmn fmn  2 rr

13 14

Example 6.3 Q-Factor of a TE10l Cavity Design of a Rectangular Waveguide Cavity

WW, , P, and P 4  Total fields:  Q factor  emc d ab40 mm, 20.0 mm, r  2.25, tan  4 10 . If f 5 GHz, find d and Q val ues f or  1.  xz  2 abd 2 EE sin sin WEdvEe   0 y 0 ad 416 Sol: E xz 2 9 0  2 abd 2 1    2510  1 Hjx  sin cos WHdvEme0   W  2222 k r 10 2.25  1.5708 cm ZadTE 416Zka TE  c 310 jE  xz 0 22 22  Hjz   cos sin 1 2 RE  ab bd a d ka a d PHdss 0   dd 23. 05 mm, f1for 1. ct222 222da 2 walls 8 da 2  k   3 a  Total losses = Pc + Pd 2We kad b 1 Qc 0   P 2233233 2 120 c 22Rs a b 2 bd a d  ad 1 Rs 1.84 10 Ω(copper@5GHz),  251.3 Ω   11 2   abd 2 r Q   PEdEPEdd  v E0 QQcd 224 1 v QQdctan 2500, 8370,  1. 2W   1 Q e tan  1 1 d 0 P   11 1 1 d Q    1925,  1. QQcd 8370 2500 15 16 6.6 Excitation of Resonators Coupling to Microwave Resonators

Gap-Coupling to a Probe-coupling to Micro resonator cavity A rectangular WG cavity

(a) A microstrip transmission line resonator gap coupled to a microstrip feed line. (1) The penetration depth h is tunable (b) A rectangular cavity resonator fed by a coaxial probe. For impedance adjustment. (c) A circular cavity resonator aperture coupled to a rectangular waveguide. (()2) The ppggrobe can be sliding along z. (()d) A dielectric resonator coupppled to a microstrip feed line. (e) A Fabry-Perot resonator fed by a waveguide horn antenna.

17 18

A Gap-Coupled Microstrip Resonator The Coupling Coefficient g bZCc  0 1     Z0 cot  Zin C zjin  j ZZ00 b  tan  j c bc tan   2 dzsec2  d  1 bc 1   Temporarily treat the lossy resonator as in jjjj    dbdtan  222 vv losses and apply the concept of complex cppbbbccc1 frequency to evaluate its loss term R 1 111 j dzin   j 2Q j 1 zz           1 in in 11d 1222  Z cot   11bQbbccc2 Zb0  tan  1 bZC ; zin jC j c cin0 ZZ btan  00 c 2 Z   ZQ2 b The input resistance Rg0 and coupling cofficient 0  c . 2 R  2Qbc  Resonance occurs at zbin 0, ctan 0, tan  b c , bQgc   2 ,  1, undercoupled  If b c  1,  1 is close to the first resonant frequency of the unloaded resonator.  The coupling of the capacitor C will lower its resonant frequency . bQgc  2, 1, critically coup led

bQgc   2 ,  1, overcoupled 19 20 Smith chart for the gap-coupled microstrip resonator Example 6.6 DiDesign of fG Gap-CldMitiRtCoupled Microstrip Resonator 50-Ω microstrip feedline, 50-Ω microstrip  /2 resonator with   21.75 mm,

 reff 1.9,  0.001 dB/mm. Find the coupling capacitor and the resonant

frequency f1.

Sol: 11 vp c 310 f0   5 GHz g 2  reff 2 21.75 1.9  Q  628(or 200  ) 22  b  0.05 c 22200Q   

bc 0.05 C 9 0.032 pF Z0 251050 

f1  4.918 GHz(obtained by a root-searching process) 21 22

Smith chart illustrating coupling to 6.6 Excitation of Resonators series RLC circuit Critical coupling : A resonator is matched to a feedline to have max power transfer at resonant ffqrequency .

ZRjLin 2  Max ppjggower transfer Conjugate matching PP 0, See Chapter 2. RXin in * ZZRRXin g, in  g , in  X g

 At resonance,  00,,, or 0, ZRZin

Unloaded QLR 0

External QLZe  00

 Coupling coefficient gQQZRe 0 g < 1, resonator is undercoupled to the feedline. g = 1, resonator is critically coupled to the feedline. g > 1, resonator is overcoupled to the feedline. 23 24 Part II Time-Domain Analysis of Open Cavities

Lecture Notes and Computational Exercises＊ for Graduate Course "Microwave Physics and Applications" Department of Physics, National Tsing Hua University, Hsinchu, Taiwan, R.O.C. g{x XÇw Éy V{t ÑA I and Graduate Research and Training in Advanced Microwave and MM Wave Thermionics University of California, Los Angeles, California, U.S.A.

*Developed under the Agreement on Academic Exchange and. Cooperation between Center for High Frequency Electronics (The University of California] Los Angeles) and Higgqyh Frequency Electrody namics Laboratory (National Tsing Hua University)

The PowerPoint file is based on Prof . K . R . Chu ’ s lecture notes – Time-Domain Analysis of Open Cavities.

25 26

Time-Domain Analysis of Open Cavities I. Introduction Caption: Illustrative model of an open

1. Introduction cavity. Section z  z2 is a cutoff waveguide (for electron beam entrance). The section

2. Formulation between z2 and z3 comprises the main body of the cavity. The section between z3 and z4 3. Numerical Algorithm is sliggyhtly tapered to provide partial reflection back to the cavity and partial 4. A Fortran Exercise transmission into the output waveguide (z 

z4). 5. Diiiscussion • Open cavities are employed in gyrotrons for generation and 6. Appendix extraction of high power millimeter/terahertz radiation. • Dispersion relation for a lossy waveguide • The resonant characteristics depends on the structure. This is a • Complex-root finding by Muller’s method (Fortran) low-Q resonant circuit due to the open-end structure. • IifdiffiliiIntegration of differential equation using Runge-Kutta (Fortran ) • Spectral domain analysis of open cavity • The materials are self-contained and self-explanatory. • Solution to exercise • A complementary spectral -domain model can be found in Appendix. 27 28 II. Formulation Time-Domain Analysis Consider a typical open cavity formed of multiple sections of The time dependence of a field component (say Bz ) is

uniform and linearly tapered it it itr structures. Find the field profile Bez ~  ee and the Q-faccotor. 2 3 assumptions: Field energy ~Be ~2it , ( <0) • The wavegu i de rad ius c ha nges sl owl y an d th er e i s n o m ode zi conversion. d Power loss ~ (field energy) = 2  (field energy) • A resonant mode is initially present in the cavity. All fields vary dt i with time as exp(-iωt). • The end sections are uniform to ensure the correctness of calculation. QlitftQuality factor of fth the cavit y:  field energy  rii (() where i < 0.) Q r r power loss 2 i 29  30

Characteristics of TEmn mode Field Profile ik z ik z For a circular waveguide with slowly varying radius r (z) , the TE Aezz Be ,if c w fz( )rcmn , where mn mode wave equation is expressed in a cylindrical coordinate  zz cmn Cezz De , if  rw system as,  rcmn im  i t The complex function of f(z) takes the general form, BfzJkzrez  ()mmn ()   iz() Applying the boundary condition on the side wall, fff ()z  f ()ze  x The dependences of f(z) and Φ(z) on z indicate the nature of the Bk 0()0  ()z  mn , z rr w() z mn wave. rrzw() where xnmn is the -th root of Jx m () ( ) 0 • For a pure traveling wave ( A =0or= 0 or B =0)|= 0), |f(z)| is independent of z, Substituting to the wave equation, we obtain but Φ(z) is a linear function of z. • For a ppg(ure standing wave (A = ± B), | f(z)| is a sinusoidal function  2 x2 kz2( )mn above cutoff of z. 2 z 22 d 2  crzw () kz() fz () 0, where  • For decaying waves at both ends, Φ(z) is independent of z. 2 z 2 2 dz  2 xmn  z ()z  below cutoff rz22() c Use the boundary conditions to determine |f(z)| and Φ(z).  w 31 32 Boundary Conditions Numerical Procedure Out-going wave boundary conditions: Initially there is a field ppygyrofile satisfying all the boundary conditions and then deca ygying  With a proper guess of the value of ωr and Q (ω= ωr +iωi). with time.  f is given at z=z1 and f ′ is set accordingly. AbAt bot h en ds, ADA=D=0 at z=z and CBC=B=0 at z . 1 5  Integrate from z1 to z5 using Runge Kutta method

 ik()() z f z, if  (() z )  Check the boundary condition at z and using Muller ’s method f ()z  zrcmn11 1 5 1  to guess the next root of ω and Q. zrcmn()(),z11fz if () z 1 r  Iterative integration, each time with an improved guess for ω, ik()(),if z f z () z  zrcmn55 5 will eventually converge to a correct solution for ω, and f(z) fz()5   zrcmn()()if()()zfz55, if (() z 5 ) will sati sf y all th e b ound ary conditi ons.

33 34

Complex Boundary Condition (cbc sub-rountine) Comments

Boundary condition at z=z1 is given. However the boundary  It is clear that the solution for should be independent of the condition at zzz=z needs to be checked . 5 positions of z1 and z5 as long as they are in the uniform end sections.  ik()(),if z f z () z fz() zrcmn55 5 5   Validity of the evanescent wave boundary condition requires zrn()(),ifzf55z  cm () z 5 that the end waveguide radius (Rl or R2) be smaller than the  fz()555 ikzzrcmn ()()()()if fz, if (() z5 ) cavityradius (R). or D()    fz()555zrcmn ()(), z fz if (z 5 )  It should also be noted that the assumption of slowly varyyging Standard root-finding algorithms such as Muller’s method can be cross-section is violated at z=z2 (Fig.1). This is justifiable only readily used. if the left end waveguide (z  z2) is cutoff to the cavity mode. In this case, total reflecti on from the lfleftend takes pllace just as a There are a series of discrete solutions for ω corresponding to more exact model would predict. different axial modes ((gassuming that the transverse mode number m and n are given).

35 36 III. Numerical Algorithm III. Numerical Algorithm Initial Boundary conditions at z How to integrate a differential equation? 1  2 x2 kz2( )mn above cutoff 2 z 22 d 2  crzw () The boundary conditions at z = z1 can be written, kz() fz () 0, where  2 z 2 2 dz  2 xmn  z (()z ) below cutoff The Runge-Kutta method: rz22() c  fr (z1 ) arbitrary real constant  w  The second order equation shown above can be decomposed into  fzi (1 ) arbitrary real constant the form of coupled real differential equations of the first order. kzfzkzfz()() () (),if () z  d zr11 i zi 1 r 1 r cmn 1 ff  fzr()1   ffriif dz rr zr()zfz11 r () zi ()(),if zfz 11 i  r cmn () z 1  ffifri  d  kfz() kfz (),if  () z ffii  zr r11 zi i r cmn 1 22 2 dz fzi ()1   kkikzzR(Re() ) Im (() z )   zif r (zf11)()if())()zrf izz, if r cmn ( 1 )   d fkfkf Re()Im()22  kkzzr  kzi dz r zr zi   i     d zzr  zi  fkfkf Im()Re()22   izrzi dz 37 38

III. Numerical Algorithm IV. A Fortran Exercise Final Boundary Conditions at z5 The program (named CAVITY.f) consists of a main program and the following subprograms.  A guessed value for ω can now be integrated from z to z . 1G1. Genera l purpose sub programs. l 5 MULLER: finding the complex roots of an arbitrary complex function (see Appendix B).  The resulting functions fr(z5), fi(z5), fr'(z5)and) and fi'(z5)give) give f(z5) RKINT: per form ing in tegra tion o f s imu ltaneous differen tia l equa tions of th e fi rst and f'(z5). order by the Runge-Kutta method (see Appendix C). SSCALE and SPLOT (or BSCALE and BPLOT): plotting data conveniently in  The procedure is to be repeated with an improved guess of characters (see Appendix D) . until the required accuracy is achieved. 2. Subprograms written for CAVITY (It is recommended to go over the contents closely). CBC: evaluating the function D( ) in Eq . (19) by integrating Eqs. (24)-(27) with initial values given by Eqs. (28)-(31). DIFEQ: evaluating the derivatives in Eqs. (24)-(27). RADIUS: evaluating the cavity wall radius as a function of z. RHO: evaluating the wall resistivity as a function of z. CLOSS: evaluating the wall loss factor derived in Appendix E (The loss factor has been incorpppp)orated into the formalism in Appendix F.)

39 40 Procedures for Running Program Cavity.f Cavity Dimensions and Calculated Results  To begin, the cavity dimensions , mode of interest, and Cavity dimensions used for numerical example numerical instructions, etc. are specified in the main program.  A guessed value of is then input into MULLER which calls CBC to evaluate D(ω). Subprogram CBC calls RKINT to

perform the integration from zl to z5. Subsequently, RKINT calls DIFEQ to evaluate the derivatives at every z-step of the integration.  Finally, MULLER returns the solution for to the main program whic h pr int s all th e i nf ormati on of i nt erest and call s SSCALE and SPOLT (or BSCALE and BPLOT) to plot and .  CblktilldfiftiCommon blocks are extensively employed for information sharing (e.g. the cavity dimensions specified in the main

program and the field profile calculated in subprogram CBC) TE111 mode field profile | f| and phase angle Φ as functions of z for between the main program and subprograms. the cavity shown above. 41 42

Check the Validity of the Results Wall Resistivity and Loss Factor (Convergence Test) (i) When calling MULLER to solve for the root of a function (e.g. CF in  Ohmic wall losses have been included in CAVITY through Appendix B) , alway s monitor the number of times (e. g. ICONT in subprograms CLOSS and RHO . Appendix B) that the function has been evaluated. For a well behaved  Formulation of wall loss can be found in Appendix E. function, the number should be small (less than 10 per root). When the number becomes too large or in the case MULLER is unable to find a  As a first exercise, we can ignore this effect (hence CLOSS root, it is a warning signal of some numerical difficulty due to, for and RHO) by setting the wall resistivity to zero in the main example, erratic behavior of the function (discontinuities and sharp program. spikes, etc.) or the presence of many closely spaced roots. The result  The TE mode dispersion relation for a vacuum filled may be in question or the physics may be unexpected. The warning mn signal can not be ignored . waveguide, (ii) After MULLER returns a root, always insure that the resulting function value (e.g. VALUE1 and VALUE2 in Appendix B) is vanishingly small  22 2222 c m  relative to the largest terms of the function. For example, if the function kczcmn1(1)  i  1   0 r 222 10 2 w xmmn mn is composed of terms of the order of 10 , a function value of 10 may be considered vani shi ngly l sma ll (beware of th e numb er of sig inifi cant digit di its the computer is capable of handling). 43 44

 Check the Validity of the Results III Check the Validity of the Results II Figures below show a typical convergence test. The resonant frequency and quality factor Q are plotted as functions of the total number of steps in the z-integration (named IZSTEP in (iii) A valid root is not necessarily the desired root. For example, program CAVITY). Note that the positions of the junction points (z2, z3,andz4 in Fig. 1) and hence the cavity dimensions, as resolved on the uniformly spaced axial grid points for the z- we provide a guessed value for the l=2 root and call integration, are subject to an uncertainty in the magnitude of the step size. This is the MULLER to search around it for the correct =2 root. primary reason for the fluctuations of and Q with respect to IZSTEP in the approach to MULLER will return a different root (e .g . l=1or3)ifthe1 or 3) if the convergence. The slow convergence shown in Fig. 4 is predominantly due to the uncertainty of resolving junction point z2 (where there is a discontinuity in wall radius) on the discrete guessed value happens to be a better guess for that root. A grid points. Generally speaking, the minimum IZSTEP required for good convergence reliable wayyy to verify the l number of the root returned by depends on the circuit ggyeometry and the ratio of the total circuit length to the guide MULLER is to count the number of peaks in the versus z. wavelength. Too large an IZSTEP can also bring in accumulation of round-off errors.

(iv) Even wi th all th ese ch eck s, th ere i s still no guarantee th at th e results are free from numerical errors. We must also check whether the step size in the z-integration is sufficiently fine to insure convergence of results.

45 46

V. Discussion Does the Numerical Results Make Sense?  Computer programs bdbased on thetime didomain fliformalism (h(such as program CAVITY) are extremely effective in that they directly evaluate the resonant Even computed correctly, numerical results can not be trusted frequency, Q, and field profile. They are essential tools for gyrotron designs. unless they make sense physically. We may start by asking some Many runs can be rapidly made to achieve the desired resonant frequency and universal questions: Is the energy conserved (see the slightly Q, to optimize the field profile and maximize mode separation, etc. However, positive slope of in Fig. 3a)? Do the results reduce to well known because of its inability to scan the freqqy,uency, the time domain formalism does not present a complete physical picture of the open cavity. The low Q nature limits [see Exercises (1) below]? Do they exhibit reasonable of the open cavity brings about some issues that can only be clarified with a parametric dependence [see Exercises (2)-(4)]?Dothey conform to spectral domain analysis [see Exercises (8)-(11) and Appendix F]. known scaling laws [see Exercise (5)]? Obtaining answers to these questions is a sure way to become familiar with the problem. We  Resonances of the type taking place in an open cavity are common in arenowreadytogodeeperintotheproblem [see Exercises (6)-(11)] microwave circuits which often contain slightly mismatched junctions between various circuit elements. Single path reflection from one mismatched and, for the best reward of all, let our imagination take us to the junction results in a standing wave pattern (measured by VSWR). Multiple unexplored territories of research. reflecti ons btbetween two mithdismatched jtijunctions result in resonances, much like those of the open cavity. Thus, a circuit with multiple mismatched junctions Always check carefully! behaves like coupled open cavities. The resulting circuit resonances are seen on an oscilloscope as multiple spikes superimposed on a swept frequency signal. 47 48 Exercise (1) Exercise (2) 2. Use program CAVITY to show how the quality factor Q of a 1. For the open cavity of Fig. 1 with dimensions given in Table I, given mode varies with the output taper angle of the open the resonant frequency of the TE mode is 9.839 GHz (see Fig. 111 cavityy( (keep ing other parameters fixed) . Interp ret the results 4). For an enclosed cylindrical cavity with the same radius (0 .9 qualitatively. cm) and length (11.7 cm) as those of the main body of the open cavity, the resonant fre quenc y of the TE mode is 9.851 GHz. 111 SlLSol: Larger θ resulilts in more refl fliection f rom th e open end , and Explain the difference qualitatively. hence lower diffraction loss and higher Q. Sol: Because of the fringe field, the open cavity has an effective length longer than L, hence the resonant frequency (of the  >0 modes) is lower than that of an enclosed cavity of length L. It is worth noting that for the  =0 (TM) modes of an enclosed cavity for which the axial field pp,pgrofile is uniform, an opening at either end will impose an axial mode structure and therefore increase the resonant frequency.

49 50

Exercise (()3) Exercise (4) 4. Use program CAVITY t o sh ow h ow th e qualit y f act or of a 3. For the cavity dimensions provided in Table I, the quality given mode varies with the cavity length L (keeping other factors of the first three modes ( =1, 2, 3) are, respectively,  parameters fixed) . Give three reasons to explain the rapid 439, 116, and 56 (see output data in Appendix A). Give two increase of Q with L. reasons to explain the rapid drop of Q with the axial mode number . Sol: A shorter cavity stores less field energy which further reduces the Q value. Sol: Higggqher l number modes have higher resonant frequencies which result in (i) less reflection from the open end and (ii) higher group velocity of the wave. Both effects lead to greater diffraction loss through the open end and hence the decrease in Q values

51 52 Exercise (5) Exercise (6) and (7) 5. With reference to Fig. 1, assume that a traveling wave propagating to the left 6. Assume that the outppgut waveguide section of the o pypen cavity is totally reflected at z=z2 and a traveling wave propagating to the right is (see Fig. 1, z  z4) is terminated in a slightly mismatched load. partially reflected at z=z3 with reflection coefficient . Show by the multiple reflection approach (see R.E. Collin, II Foundations for Microwave Explain qualitatively how the load will affect the resonant Engineering", 1st edition, pp. 340-343 and Eq. (49) in Ref.10 of Appendix F) frequency and Q of the cavity. that the diffraction Q is approximately given by 1/2 7. Use program CAVITY to verify  4 L Q  ()2 your answer to Exercise (6) by 1   adding a smooth bump on the where λ is the free space wavelength of the resonant mode. Compare this wall of th e out put waveguid e t o relation with the scaling of Q with respect to θ, l,andL as considered in simulate the effects of the Exercises (2)-(4). Note that depends on the taper angle and resonant frequ ency and that is a function of and L. If wall losses are included , show mismatched load. that the combined diffractive/ Ohmic Q is given by the above equation with

 replaced by exp(－2kzL), where is the attenuation constant which can be evaldluated from Eq. (10) of AppendixE.

53 54

Exercise (8) Exercise (9) 8. The lack of a sharp boundary at the output end of the open 9. In Exercise (8) , if the fixed -position spectrum is measured at cavity is expected to result in a frequency sensitive field profile different points along the length of the cavity, will the spectral for a resonant mode, as shown in Fig. 6 of Appendix F. How shappge of a given mode be different from p oint to p oint? Does would the frequency dependent field profile affect the spectral this imply that probes located at different axial positions in the shape of a resonant mode, measured at a fixed position in the cavity will measure different resonant frequencies and Q's for cavity, i n response t o a swept f requency source i ncid ent f rom the same mode? the output waveguide into the cavity?

55 56 Exercise (10) Exercise (11) 11. Program CAVITY calculates Q by its time domain definition (denoted by  superscript “t”) Q (t)  r 10. Write a computer program based on the spectral domain formalism in Sec. II of 2  Appendix F to: i • verify your answers to Exercises (8) and (9) While the spectral domain formalism yields Q by its spectral domain definition ()  • calculate the reflection coefficient assumed in Exercise (5) as a function of (denoted by superscript “ω”) Q   the wave freqqyuency. • calculate the reflection coefficient at the smooth bump assumed in Exercise where Δ ω is th e FWHM b and width . C ompare numeri cal runs mad e with (7) as a function of the wave frequency. program CAVITY with those made with the program developed in Exercise (10) to show that, for the same mode of a low Q open cavity, the two definitions of Q do not yield the same result and Sol: RFS or RFS2 QQ()t  ( ) Exppqylain this result qualitatively.   itj t j ()t 2Qj ft(,)xx  fj () e j 11 i fftedtf(,xx ) (,)it () x 0  j ()t 22j jjjiQ2  2 11ii ff(,xx ) () f* () x jj()tt () 22jj j  iQ jj22  j iQ jj 57 ... Q()t Q ( ) 58  

Appendix: Muller’s Method Appendix: Runge-Kutta’s Method (RKINT)

Note: Muller’s method is discussed in Conte and de boor, Elementary Numerical rd Analyy,(sis”, (3 edition,,) Sec. 3.7).

59 60 Part III Appendix: Dispersion Relation for a Lossy Waveguide Spectral-Domain Analysis of Open Cavities Der ivati on of th e l oss f act or i n E q. (19) of A ppendi x F . Reason: Cold tests of open cavities almost always employ the method of frequency sweeping and Q is measured by its spectral domain definition ((ypphence denoted by superscript ω) where Δω is the Jackson, Chap. 8 full width between the half maxima of the resonant line. QlitftQuality factor of fth the cavity it: 22 Time-domain definition: 2222 c m kczcmn1(1)  i  1  0 r 222 ()t fie ld energy r w xmmn  mn Q r  power loss 2 i Frequency-domain definition:  Q()  0 What is the difference  between these two definitions?

61 62

II. Numerical Approaches for the Spectral Model Numerical Model

3 assumptions:  With a proper guess of the value of Γ (Γ = Γr +i Γi). • The waveguide radius changes slowly and there is no mode  fz() eikzzzz()11 e ikzz () 11 conversion.  1  where rcmn  ikzzzz()11 ikzz () 11 • A resonant mode is initially present in the cavity. All fields vary  fz()1  ikez ( e ) with time as exp( -iωt). • The end sections are uniform to ensure the correctness of  f is given at z=z1 and f ′ is set accordingly. calculation.  Integrate from z to z using Runge-Kutta method Time-domain model: 1 5

ri i  (where i < 0 .)  Check the boundary condition at z5 and using Muller’s method to guess the next root of Γ.

Frequency-domain model:  Iterative integration, each time with an improved guess for ω, ? will eventually converge to a correct solution for ω, and f(z) will ClComplex refl fltiffiitection coefficient satisfy all the bou ndary conditions. Γ 63 64 Frequency Response Numerical Results Numerical results obtained under the temporal and spectral models for the cavity dimensions in Table I with different output

taper angles θ . TE11l were calltdlculated. fc is thecutfftoff frequency of the main body of the cavity.

65 66

Example I Example II Freqqyuency Tunable Terahertz G yrotron 394 GHz Freqqyuency Tunable Terahertz Gy rotron 203 GHz Using Open Cavity Structure With Mode Selectivity NlNovel mechihanism: RfltdReflected gyrotron bkbackwar d-wave oscillat or An on-going research with JFkiUiJapan Fukui Univ.

N. C. Chen, T. H. Chang*, C. P. Yuan, T. Idehara and I. Ogg,awa, “Theoretical investigation of a high efficiency and broadband sub-terahertz gyrotron", Appl. Phys. Lett. 96, 161501 (2010).

T. H. Chang*, T. Idehara, I. Ogawa, L. Agusu, C. C. Chiu, and S. Kobayashi, “Frequency tunable gyrotron using backward-wave components”, J. Appl. Phys. 105, 063304 (2009). 67 68 g{x XÇw Éy bÑxÇ Vtä |àç