STUDY OF SCATTERING PARAMETERS OF WAVEGUIDE DEVICES USING COMSOL MULTYPHYSICS

Transcription

1 STUDY OF SCATTERING PARAMETERS OF WAVEGUIDE DEVICES USING COMSOL MULTYPHYSICS A thesis submitted in patial fulfillment of the equiements fo the degee of Bachelo of Science in Electical, Electonics & Communication Engineeing Submitted by Tanjila Ahmed Student No Samiul Islam Student No Shajid Islam Student No Supevised by D. Md. Shah Alam Associate Pofesso Depatment of Electical and Electonics Engineeing Bangladesh Univesity of Engineeing & Technology Dhaka-1000, Bangladesh Depatment of Electical, Electonics and Communication Engineeing Militay Institute of Science and Technology Mipu Cantonment, Dhaka 1216 Decembe

2 Declaation It is heeby declaed that this thesis o any pat of it has not been submitted elsewhee fo the awad of any degee o diploma (Tanjila Ahmed) (Samiul Islam) (Shajid Islam) Supeviso D. Md. Shah Alam Associate Pofesso, Depatment of Electical and Electonic Engineeing, Bangladesh Univesity of Engineeing and Technology (BUET), Dhaka 1000, Bangladesh. 2

3 Acknowledgement We thank the Almighty Allah fo the successful completion of ou thesis. We would like to expess ou heatiest gatitude, pofound indebtedness and deep espect to ou supeviso D. Shah Alam, fo his supevision, continuous encouagement and valuable suggestions as well as constant guidance though the couse of wok. Ou special thanks also go to ou head of the EECE depatment, Col. Moin Uddin and ou espected couse coodinato si fo thei valuable assistance and help. Ou Special thanks also go to all ou fiends and classmates fo thei help and advices. 3

4 Table of contents Acknowledgement Declaation Table of content Abstact III IV V VI Chapte 1 Intoduction S-paametes Classification Objective of the thesis Oganization of the thesis 05 Chapte 2 Mathematical desciption of S-paamete Netwok based desciption Requiement of calculation of scatteing Paametes PDE fomulation fo plane waves PDE fomulation fo the TE waves PDE fomulation fo TM waves 14 Chapte 3 Use of COMSOL Multiphysics Waveguide H-bend Waveguide adapte 22 4

5 Chapte 4 Results and Discussion Waveguide H-bend Waveguide adapte 38 Chapte 5 Conclusion 49 Refeences 51 5

6 Abstact S-paametes efe to the scatteing matix. The concept was fist populaized aound the time that Kaneyuke Kuokawa of Bell Labs wote his 1965 IEEE aticle Powe Waves and the Scatteing Matix. It helped that duing the 1960s, Hewlett Packad intoduced the fist micowave netwok analyzes. The scatteing matix is a mathematical constuct that quantifies how RF enegy popagates though a multi-pot netwok. The S-matix is what allows us to accuately descibe the popeties of incedibly complicated netwoks as simple "black boxes. One difficulty associated with tansmission paametes is that in a pactical high-fequency cicuit it is extemely difficult to obtain the pefect opencicuit and shot-cicuit conditions equied to make good measuements. The S- paamete method cicumvents the open/shot difficulties by always connecting both pots to tansmission lines with stable, well-defined impedances. To the extent that such a test setup bette epesents the actual woking conditions of you cicuit, the S-paamete method geneates a moe accuate model. These ae the easons why this thesis topic is chosen. In this thesis pape, an idea about what is S-paametes, it s classification and usefulness is descibed elaboately. Then hee the mathematical and netwok based analysis is done which helps to know about S-paametes moe accuately. Hee S- paametes ae expessed with mathematical equations. Examples ae being done by the help of the softwae named FEMLAB. FEMLAB is a poweful, inteactive envionment fo modeling and solving scientific and engineeing poblems based on patial diffeential equations. It has helped to know the detailed status of waves in diffeent types of waveguides. These detailed statuses ae being descibed at the end of this thesis. 6

7 Chapte 1 Intoduction 1.1 S-paamete Scatteing paametes o S-paametes ae popeties used in electical engineeing, electonics engineeing, and communication systems engineeing descibing the electical behavio of linea electical netwoks when undegoing vaious steady state stimuli by small signals. S-paametes ae a mathematical subset of a moe geneal non-linea fomulation called X-paametes. An electical netwok to be descibed by S-paametes may have any numbe of pots. Pots ae the points at which electical cuents eithe ente o exit the netwok. Sometimes these ae efeed to as pais of 'teminals'. So fo example, a 2-pot netwok is equivalent to a 4-teminal netwok, though this teminology is unusual with S- paametes, because most S-paamete measuements ae made at fequencies whee coaxial connectos ae moe appopiate. S-paametes descibe the esponse of an N-pot netwok to voltage signals at each pot. The fist numbe in the subscipt efes to the esponding pot, while the second numbe efes to the incident pot. Thus S 21 means the esponse at pot 2 due to a signal at pot 1. The most common "N-pot" in micowaves is one-pots and two-pots. Thee-pot netwok S-paametes ae easy to model with softwae such as Agilent ADS, but the thee-pot S-paamete measuements ae extemely difficult to pefom with accuacy. The S-paamete matix descibing an N- pot netwok will be squae of dimension 'N' and will theefoe contain 7 2 N elements. At the test fequency each element o S-paamete is epesented by a unit less complex numbe, thus epesenting magnitude and angle, o amplitude and phase. The complex numbe may eithe be expessed in ectangula fom o, moe commonly, in pola fom. The S-paamete magnitude may be expessed in linea fom o logaithmic fom. When expessed in logaithmic fom, magnitude has the "dimensionless unit" of decibels. The S-paamete angle is most fequently expessed in degees but occasionally in adians.

8 Any S-paamete may be displayed gaphically on a pola diagam by a dot fo one fequency o a locus fo a ange of fequencies. If it applies to one pot only (being of the fom S nn ), it may be displayed on an impedance o admittance Smith Chat nomalized to the system impedance. The Smith Chat allows simple convesion between the paamete, equivalent to the voltage eflection coefficient and the associated (nomalized) impedance (o admittance) seen at that pot. The following infomation must be defined when specifying any S-paamete: The chaacteistic impedance (often 50 Ω). The allocation of pot numbes. Conditions which may affect the netwok, such as fequency, tempeatue, contol voltage, and bias cuent, whee applicable. When we ae talking about netwoks that can be descibed with S-paametes, we ae usually talking about single-fequency netwoks. Receives and mixes ae not efeed to as having S-paametes, although we can cetainly measue the eflection coefficients at each pot and efe to these paametes as S-paametes. The touble comes when we wish to descibe the fequency-convesion popeties, this is not possible using S-paametes. S-paametes ae membes of a family of simila paametes used in electonics engineeing, othe examples being: Y-paametes, Z-paametes, H-paametes, T- paametes o ABCD-paametes. They diffe fom these, in the sense that S-paametes do not use open o shot cicuit conditions to chaacteize a linea electical netwok; instead matched loads ae used. These teminations ae much easie to use at high signal fequencies than open-cicuit and shot-cicuit teminations. Moeove, the quantities ae measued in tems of powe. Many electical popeties of netwoks o components may be expessed using S-paametes, such as gain, etun loss, voltage standing wave atio (VSWR), eflection coefficient and amplifie stability. Snn The tem 'scatteing' is moe common to optical engineeing than RF engineeing, efeing to the effect obseved when a plane electomagnetic wave is incident on an obstuction o passes acoss dissimila dielectic media. In the context of S-paametes, scatteing efes to the way in which the taveling cuents and voltages in a tansmission line ae affected when they meet a discontinuity caused by the insetion of a netwok into 8

9 the tansmission line. This is equivalent to the wave meeting impedance diffeing fom the line's chaacteistic impedance. Although applicable at any fequency, S-paametes ae mostly used fo netwoks opeating at adio fequency (RF) and micowave fequencies whee signal powe and enegy consideations ae moe easily quantified than cuents and voltages. S-paametes change with the measuement fequency so this must be included fo any S-paamete measuements stated, in addition to the chaacteistic impedance o system impedance. S-paametes ae eadily epesented in matix fom and obey the ules of matix algeba. Histoically, an electical netwok would have compised a 'black box' containing vaious inteconnected basic electical cicuit components o lumped elements such as esistos, capacitos, inductos and tansistos. Fo the S-paamete definition, it is undestood that a netwok may contain any components povided that the entie netwok behaves linealy with incident small signals. It may also include many typical communication system components o 'blocks' such as amplifies, attenuatos, filtes, couples and equalizes povided they ae also opeating unde linea and defined conditions. 1.2 Classification Thee ae diffeent types of S-paametes which ae: Small signal S-paamete They ae what we ae talking about 99% of the time. By small signal, we mean that the signals have only linea effects on the netwok, small enough so that gain compession does not take place. Fo passive netwoks, small-signal is all we have to woy about, because they act linealy at any powe level. Lage signal S-paamete They ae moe complicated. In this case, the S-matix will vay with input signal stength. Mixed-mode S-paamete They efe to a special case of analyzing balanced cicuits. 9

10 Pulsed S-paamete They ae measued on powe devices so that an accuate epesentation is captued befoe the device heats up. 1.3 Objective of thesis Netwok scatteing paametes ae poweful tools fo the analysis and design of high fequency and micowave netwoks. Scatteing Paametes, o s-paametes, ae the eflection and tansmission coefficients between the incident and eflection waves. They descibe completely the behavio of a device unde linea conditions at micowave fequency ange. Each paamete is typically chaacteized by magnitude, decibel and phase. The s-paametes ae voltage atios of the waves. The advantage of s-paametes does not only lie in the complete desciption of the device pefomance at micowave fequencies but also the ability to convet to othe paametes such as hybid (H) o admittance (Y) paametes. Additionally, stability facto (K) and many gain paametes can be computed also. In this thesis we have chosen a softwae named FEMLAB to find out S-paamete of micowave devices. FEMLAB is a poweful, inteactive envionment fo modeling and solving scientific and engineeing poblems based on patial diffeential equations. It can easily extend conventional models that addess one banch of physics to state-of-the at multiphysics models that simultaneously involve multiple banches of science and engineeing. Multiphysics teat simulations that involve multiple physical models o multiple simultaneous physical phenomena. Fo example, combining chemical kinetics and fluid mechanics o combining finite elements with molecula dynamics. Multiphysics typically involve solving coupled systems of patial diffeential equations. Accessing this powe, howeve, does not equie an in-depth knowledge of mathematics o numeical analysis. Indeed, one can build many useful models simply by defining the elevant physical quantities athe than defining the equations diectly. FEMLAB then intenally compiles a set of PDEs epesenting the poblem. FEMLAB also allows ceating equation-based models. When solving the PDEs that descibe a model, FEMLAB applies the finite element method (FEM). FEMLAB uns that method in conjunction with adaptive meshing and eo contol as well as with a vaiety of numeical solves. 10

11 Woking in an easy-to-use gaphical inteface o fom a command line, uses choose fom seveal ways to descibe thei poblems in 1D, 2D, and 3D. A paticula stength of this softwae is its PDE modeling capability, enabling equations fom vaious fields such as stuctual mechanics, electomagnetics, fluid flow, and chemisty to be linked and solved all in the same model and all at the same time. These and many othe featues make FEMLAB an unpecedented modeling envionment fo eseach, poduct development and education. 1.4 Oganization of thesis This thesis consists of fou chaptes. A bief desciption of each chapte is given below. In chapte 1, hee is the definition and shot desciption of S-paametes. A classification of S-paametes egading its signal condition is also mentioned hee. By the end of this chapte the objective of this thesis and its oganization patten is being descibed. In chapte 2, hee is a netwok based analysis of S-paametes. Its mathematical epesentation in tems of powe flow is also deived hee. Need of calculation of S- paametes, its PDE fomulation fo TE and TM waves ae descibed at the end. In chapte 3, pocess of using COMSOL Multiphysics 3.2 is noted pefoming two examples. One is waveguide H-bend and the othe one is waveguide adapte. In chapte 4, the possible esults of the two examples descibed in the pevious chapte ae given. Along with these esults individual discussions ae given also. In chapte 5, hee is the conclusion of this thesis. A bief idea about what is done so fa in the thesis is given in this conclusion pat. 11

12 Chapte 2 Mathematical Desciption of S -Paametes 2.1 Netwok based desciption S-paametes ae a set of paametes descibing the scatteing and eflection of taveling waves when a netwok is inseted into a tansmission line. S -paametes ae nomally used to chaacteize high fequency netwoks, whee simple models valid at lowe fequencies cannot be applied. The scatteing matix is a mathematical constuct that quantifies how RF enegy popagates though a multi-pot netwok. Fo an RF signal incident on one pot, some faction of the signal bounces back out of that pot, some of it scattes and exits othe pots and is pehaps even amplified, and some of it disappeas as heat o even electomagnetic adiation. S -paametes ae nomally measued as a function of fequency, so when looking at the fomulae fo S -paametes it is impotant to note that fequency is implied, and that the complex gain (i.e. gain and phase) is also assumed. Fo this eason, S -paametes ae often called complex scatteing paametes. Fig. 2.1 A two pot netwok. A two-pot device shown in Fig. 2.1 can be descibed by a numbe of paamete sets. We e familia with the H-, Y-, and Z-paamete sets. All of these netwok paametes 12

13 elate total voltages and total cuents at each of the two pots. All these paametes ae applicable in lowe fequency. We cannot apply these paametes fo the analysis of highe fequency because moving to highe and highe fequencies, some poblems aise: Equipment is not eadily available to measue total voltage and total cuent at the pots of the netwok. Shot and open cicuits ae difficult to achieve ove a boad band of fequencies. Active devices, such as tansistos and tunnel diodes, vey often will not be shot o open cicuit stable. Some method of chaacteization is necessay to ovecome these poblems. The logical vaiables to use at these fequencies ae taveling waves athe than total voltages and cuents and thus the concept of S -paametes come. As we said ealie that in high fequency we conside waves (both taveling and eflected) instead of voltage and cuent. Whee eflected waves ae the function of incident waves. Let us look at E 2 in Fig. 2.2 whee we see that it is made up of that potion of E i2 eflected fom the output pot of the netwok as well as that potion of E i1 that is tansmitted though the netwok. Each of the othe waves ae similaly made up of a combination of two waves. Fig. 2.2 High fequency two pot netwok. It should be possible to elate these fou taveling waves by some paamete set. While the deivation of this paamete set will be made fo two-pot netwoks, it is applicable fo n-pots as well. Let s stat with the H -paamete set fo a two pot netwok. We can wite 13

14 V1 h11i1 h12v2 (2.1) I2 h21i1 h22v2 (2.2) Fo the Fig.2.2, we can wite fo the total cuent and total voltage V E E (2.3) 1 i1 1 V E E (2.4) I 2 i2 2 E E i1 1 1 (2.5) Z0 I E E i2 2 2 (2.6) Z0 By substituting the expessions fo total voltage and total cuent on a tansmission line into this paamete set, we can eaange these equations such that the incident taveling voltage waves ae the independent vaiables; and the eflected taveling voltage waves ae the dependent vaiables. E f ( h) E f ( h) E (2.7) 1 11 i1 12 i2 E f ( h) E f ( h) E (2.8) 2 21 i1 22 i2 The functions f11, f21 and f12, f 22 epesent a new set of netwok paametes elating taveling voltage waves athe than total voltages and total cuents. It is appopiate that we call this new paamete set S -paametes, since they elate those waves scatteed o eflected fom the netwok to those waves incident upon the netwok. These scatteing paametes will commonly be efeed to as S -paametes. So now the equation becomes E S E S E (2.9) 1 11 i1 12 i2 E S E S E (2.10) 2 21 i1 22 i2 Now let us divide the above equation by Z 0, the chaacteistic impedance of the tansmission line, the elationship becomes b1 S11a1 S12a2 (2.11) b2 S21a1 S22a2 (2.12) Hee is the matix algebaic epesentation of 2-pot S-paametes 14

15 b b 1 2 S S S S a a (2.13) whee the incident waves ae designated by the lette a, whee n is the pot numbe of n the netwok. Fo each pot, the incident (applied) and eflected wave ae measued. The eflected wave is designed byb n, whee n is the pot numbe. S S S S b 1 11 a2 0 a1 b 2 22 a1 0 a2 b 2 21 a2 0 a1 b 1 12 a1 0 a2 a a b E wave incident on pot 1 i1 1 (2.14) Z E Z 0 0 wave incident on pot 2 i2 2 (2.15) E Z Z 0 0 wave eflected fom pot (2.16) b E Z Z 0 0 wave eflected fom pot (2.17) Z Z 0 0 = Input eflection coefficient with the output pot teminated by a matched load ( ZL Zo sets a2 0 ). = Output eflection coefficient with the input pot teminated by a matched load( Zs Zo sets a1 0 ). = Fowad tansmission gain with the input pot teminated in a matched load( a 2 0 ). = Revese tansmission gain with the input pot teminated in a matched load ( a 1 0 ). Now, we need to calculate s-paametes by the means of electic field coesponding to the voltage. To convet an electic field patten on a pot to a scala complex numbe coesponding to the voltage in tansmission line theoy we need to pefom an eigenmode expansion of the electomagnetic fields on the pots. Let us assume that we have pefomed eigenmode analyses on the pots 1, 2, 3 and so on and that we know the

16 electic field pattens E1, E2, E 3 of the fundamental modes on these pots. The fields should have zeo phases. Futhemoe, let us assume them to be nomalized with espect to the integal of the powe flow acoss each pot coss section espectively. Let us note that this nomalization is fequency dependent unless we ae dealing with TEM modes. The pot excitation is assumed to be of unity powe and applied using the fundamental eigenmode. The computed electic field eflected field. The S-paametes is given by S 11 pot1 (( E c pot1 E ). E 1 1 ( E. E * 1 * 1 ) da ) da 1 E c on the pot consists of the excitation plus the 1 (2.18) S 21 pot 2 (( E ( E pot 2 2 c. E. E * 2 * 2 ) da ) da 2 2 (2.19) S 31 pot3 ( E ( E pot3 c 3. E. E * 3 * 3 ) da 3 ) da 3 (2.20) S-paametes in tems of powe flow: It is possible to calculate the S-paametes fom the powe flow though the pots. Such a definition is only the absolute value of the S-paametes which does not have any phase infomation. The advantage of using the powe flow to calculate the S-paametes is that the modes need not to be known. When the modes ae degeneated, as fo example is the case fo cicula pots, it is geneally impossible to know how the wave is oiented. It is impossible to find the coect supeposition of the degeneated modes to match the calculated wave. Fist we see the elation between the vaiables a1, a2, b1, b 2 and vaious powe waves. 16

17 2 a1 powe incident on the input of the netwok powe available fom a souce of imedance Z 2 a2 powe incident on the output of the netwok b powe eflected fom the load 2 1 powe efl ected fom the input pot of the netwok powe available fom a Z souce minus the powe deliveed to the input of the netwok 2 b2 powe eflected o emanating fom the output of the powe incident on the load = powe that would be deliveed to a Z load netwok Hence S paametes ae simply elated to powe gain and mismatch loss, quantities which often ae of moe inteest than the coesponding voltage function. 2 powe eflected fom the netwok input S11 powe incident on the netwok input 2 powe eflected fom the netwok output S22 powe incident on the netwok output 2 powe deliveed to a Z 0 load S21 powe available fom Z souce tansduce powe gain with Z load and souce S evese tansduce powe gain with Z load and souce 0 S-paamete magnitudes ae pesented in one of two ways, linea magnitude o decibels (db). Because S-paametes ae a voltage atio, the fomula fo decibels in this case is S ij ( db) 20 log S ( magnitude) (2.21) ij 2.2 Requiement of calculation of scatteing paametes Scatteing paametes mainly descibes an electical netwok. Moe specifically, many electical popeties of netwoks may be expessed using scatteing paametes; such as, gain, etun loss, voltage standing wave atio (VSWR), eflection co-efficient and amplifie stability. It is a way to descibe the opeation of a linea, time invaiant two-pot cicuit. Scatteing paametes ae vey popula as a way to chaacteize connectos and 17

18 cables fo high-speed applications above Gbps. Moeove, some poblems occu when not using scatteing paametes: A boadband open o shot cicuit is difficult to achieve at high fequencies. Active devices vey often will oscillate when teminated in a eactive load. At high fequencies, total voltage and total cuent cannot be measued diectly. Scatteing paametes, howeve, ae measued with esistive teminations and ae detemined by measuing incident and eflected waves. 2.3 PDE fomulation fo plane waves In ode to calculate the S-paametes of a micowave netwok, we need to find the patial diffeential equation solving which we can calculate ou equiements. Fo PDE fomulation in case of plane waves, we conside a situation whee we have no vaiation in the z diection, and the electomagnetic field popagates in the modeling x-y plane. The application mode can handle: Tansvese electic (TE) waves. Tansvese magnetic (TM) waves. Hybid-mode waves. A hybid-mode wave is simply a linea combination of a TE and a TM wave. Woking with hybid-mode waves allows fo specifying elliptical o cicula waves as incoming waves. We will only conside TE and TM waves hee in the following sub sections PDE fomulation fo the TE waves As the field popagates in the modeling x-y plane, a TE wave has only one electic field component in the z diection, and the magnetic field lies in the modeling plane. Thus the tansient and time-hamonic fields can be witten j t E ( x, y, t) E ( x, y, t) E ( x, y) e e (2.22) z z z H( x, y, t) H ( x, y, t) e H ( x, y, t) e ( H ( x, y) e H ( x, y) e ) e jt x x y y x x y y (2.23) 18

19 To be able to wite the fields in this fom, it is also equied that, σ, and ae nondiagonal only in the x-y plane. denotes a 2-by-2 tenso, and zz and zz ae the elative pemittivity and conductivity in the z diection. Given the above constaints, the equation is whee ( E) k E 0 (2.24) c c j (2.25) 0 can be simplified to a scala equation fo whee E.( ~ 2 E ) k E 0 (2.26) z T z czz 0 z ~ (2.27) det( ) 2 Using the elation n, whee n is the efactive index, the equation can altenatively be witten. E z n 2 zz k 2 0 E z 0 (2.28) when the equation is witten using the efactive index, the assumption is that 1 and σ = 0. The wave numbe in vacuum k 0 is defined by k (2.29) c 0 whee c 0 is the speed of light in vacuum. The equation in the time-domain A A ( D ) ( ( A B )) 0 (2.30) t t t can be simplified to a scala equation fo A z, A A D t t t z z ( z ) ( ( z )) 0 A B (2.31) 19

20 Hee, the constitutive elations used ae B 0H B and D 0E D. Othe constitutive elations can also be handled fo tansient poblems. Using the elation n 2, whee n is the efactive index, the equation can altenatively be witten 2 Az 00 ( n.( Az B ) 0 (2.32) t t when using the efactive index, the assumption is that constitutive elations fo linea mateials can be used. = 1 and σ = 0 and only the PDE fomulation fo TM waves TM waves have a magnetic field with only a z component and an electic field in the x-y plane. Thus the fields can be witten as j t H( x, y, t) H ( x, y, t) H ( x, y) e e z z z (2.33) E( x, y, t) E ( x, y, t) e E ( x, y, t) e ( E ( x, y) e E ( x, y) e ) e jt x x y y x x y y (2.34) To wite the fields in this fom, it is also equied that and ae nondiagonal only in the x-y plane. and σ denote 2-by-2 tensos, and diection. Given the above constaints the time-hamonic equation 1 2 c 0 20 zz is the elative pemeability in the z ( H) k H 0 (2.34) can be simplified to a scala equation fo whee z H ~ 2.( H ) k H 0 (2.35) c T z czz 0 z ~ c c (2.36) det( ) c 2 Using the elation n, whee n is the efactive index, the equation can altenatively be witten.( ~ 2 2 n H ) k H 0 (2.37) z 0 In the time-domain the equation fo TM wave equation is simplified to z A A ( D ) ( ( A Bz )) 0 (2.38) t t t

21 Hee, the constitutive elations B 0H B and D 0E D ae used. Othe constitutive elations can also be handled by the application mode. Using the elation n 2, whee n is the efactive index, the equation can altenatively be witten 2 A 00 ( n ) ( A ) 0 (2.39) t t In this case only the constitutive elations fo linea mateials can be used. 21

22 Chapte 3 Use of COMSOL Multiphysics A vey useful method to calculate the scatteing paamete is Finite Element Method. Again, COMSOL Multiphysics 3.2 is an efficient softwae to calculate the scatteing paametes using Finite Element Method. We have caied out thee diffeent tutoials on COMSOL Multiphysics 3.2 fo diffeent types of waveguides and saw the esults. Diffeent steps of the tutoials ae shown below. 3.1 Waveguide H-bend Model Libay path: Electomagnetics_Module/ RF_and_Micowave_Engineeing/hbend_3d 3D Modeling Using the Gaphical Use Inteface: Model navigato: 1. In the Model Navigato, 3D was selected in the Space dimension list. 2. In the list of application modes, Electomagnetics Module>Electomagnetic Waves>Hamonic popagation was selected. 3. OK was clicked. Options and settings: Fig. 3.1 Constants dialogue box. 22

23 Table 3.1 Table of constants Expession Desciption Fc 4.3e9 Cut-off fequency fq1 1.2fc 20% above cut-off Geomety modeling: 1. Fom the Daw menu, the Wok Plane Settings dialog box was opened. OK was clicked to obtain the default wokplane in the x-y plane. 2. Fom the Options menu, Axes/Gid Settings was chosen. 3. In the Axes/Gid Settings dialog box, the following axis and gid settings wee specified: Table 3.2 Axes/ Gid settings Axis Gid x min x spacing 0.05 x max Exta x y min -.04 y spacing 0.05 y max 0.2 Exta y The waveguide width 2* coesponds to a cut-off fequency of 4.3 GHz. 4. Thee lines wee dawn. Afte selecting the line tool, (0, ), (-0.1, ), (-0.1, ), and (0, ) was clicked. 5. The 2nd Degee Bezie Cuve button was clicked. Afte that (0.0576, ) and (0.0576, 0.075) wee clicked. 6. The Line button was clicked. So was (0.0576, 0.175), (0.0924, 0.175), and (0.0924, 0.075). 7. The 2nd Degee Bezie Cuve button was clicked. Then the point (0.0924, ) was clicked. 8. The ight mouse button was clicked to close the bounday cuve and ceate a solid object. 23

24 Fig. 3.2 Two dimensional solid object befoe extuding. 9. Extude was selected fom the Daw menu. The object was extuded using a distance of The Zoom Extents toolba button was clicked. Fig. 3.3 Thee dimensional view of the waveguide afte extuding. 24

25 Physics Modeling: Scala Vaiables: 1. Fom the Physics menu, Scala Vaiables was chosen. 2. In the Application Scala Vaiables dialog box, the fequency nu_emw was set to fq1. Bounday Conditions: 1. Fom the Physics menu, Bounday Settings was chosen Boundaies 2 8 and 10 wee selected. 2. In the Bounday condition list, pefect electic conducto was set as the bounday condition. These boundaies epesent the inside of the walls of the waveguide which is plated with a metal, such as silve, and consideed to be a pefect conducto. 3. On boundaies numbe 1 and 9, the Pot bounday condition was specified. On the Pot tab the values accoding to the table 3.3 wee set. Sub domain Settings: Default values fo, and can be used because the waveguide is filled with ai. Mesh Geneation: 1. In the Mesh Paametes dialog box was typed in the Maximum element size field. Fig. 3.4 Mesh paametes dialogue box. 25

26 2. Remesh was clicked to geneate the mesh. Fig. 3.5 Thee dimensional view afte doing emesh. Computing the Solution: The Solve button was clicked to solve the poblem. Post Pocessing and Visualization: The default plot is a slice plot of the z component of the electic field. To bette see the popagating wave the position of the slices wee changed. 1. On the Slice tab in the Plot Paametes dialog box the Slice positioning x levels was set to 0, the y levels to 0, and the z level to 1. 26

27 Fig. 3.6 Z component of the slice view of electic field. 2. A convenient way to visualize the good tansmission is to plot the total enegy density. Reflections give ise to a wave patten in the enegy distibution. Total enegy density, time aveage was selected as the slice data. 3. To compae the powe flow of the incident wave to the powe flow of the outgoing wave, bounday integation was pefomed. The entance pot was excited using the pot bounday condition which automatically nomalizes the excitation to unity powe (1 W). To obtain the powe outflow fom the exit pot, powe outflow, time aveage was integated ove bounday 9. The esult was W. Table 3.3 Pot bounday specifications Bounday 1 9 Pot numbe 1 2 Wave excitation at this pot selected cleaed Mode specification ectangula ectangula Mode type Tansvese electic(te) Tansvese electic(te) Mode numbe

28 Fig. 3.7 Total enegy density, time aveage of slice view. 3.2 Waveguide adapte Model Libay path: Electomagnetics_Module/RF_and_Micowave_Engineeing/waveguide_adapte Modeling Using the Gaphical Use Inteface: The waveguide adapte model is set up in two steps. Fist, the bounday mode analysis is pefomed on the elliptical pot of the adapte; and then the wave popagation poblem is solved using the mode shape as a bounday condition. The file waveguide_adapte_geomety.mph was opened in the Model Libay. The geomety in the file is a waveguide tansition with one elliptical end and one ectangula end. The following steps wee used to calculate the S-paametes fo this geomety when the wave entes fom the ectangula end and exits at the elliptical end. 1. The Model Navigato was opened fom the Multiphysics menu. 28

29 2. In the Model Navigato the Electomagnetics Module> Bounday Mode Analysis> TE Waves application mode was selected and added it to the model. 3. Then the Electomagnetics Module> Electomagnetic Waves> Hamonic popagation application mode was selected and added it to the model. 4. OK was clicked to close the Model Navigato. Fist, the Bounday Mode Analysis application mode was used to compute the eigenmode of the elliptical end. This eigenmode is then pat of the bounday condition fo the electomagnetic wave popagation simulation. Physics settings: Fist, the bounday mode analysis poblem was set up. Bounday Mode Analysis was selected fom the Multiphysics menu. Scala Vaiables: In the Scala Vaiables dialog box, the fequency nu_emweb was set to 7e9. This is a fequency above the cut-off fequency, which ensues that you find a popagating wave mode when doing the mode analysis. Bounday Settings: In the Bounday Settings dialog box, the Active in this domain check box was cleaed fo all boundaies except bounday 6. This bounday is the elliptical end of the waveguide, whee the mode analysis will be done. Edge Settings: In the Edge Settings dialog box the Pefect electic conducto bounday condition was selected at edges 5, 6, 10, 14, 29, 39, 52 and 55. These ae the exteio edges of the ellipse. Mesh geneation: 1. In the Mesh Paametes dialog box, was enteed in the Maximum element size edit field. 29

30 2. The mesh was initialized. Computing the solution: 1. The Solve Paametes dialog box was opened. 2. On the Geneal tab, the Eigenvalue solve was selected and enteed 1 in the Seach fo effective mode indices aound edit field. 3. OK was clicked to close the dialog box. 4. The Solve Manage dialog box was opened and selected only the Bounday Mode Analysis application mode on the Solve Fo tab. 5. The Solve button was clicked. Postpocessing and visualization: The eigenmodes wee visualized as a bounday plot. 1. In the Plot Paametes dialog box, the Slice check box was cleaed and selected the Bounday check box. 2. The Effective mode index was selected On the Bounday tab, the Tangential electic field, x component was selected. Thee ae two field vaiables one fo each of the two application modes. The tex_emweb vaiable was selected. 4. OK was clicked to plot the field and the Go to XY view button was clicked to bette see the field. 5. The same type of plot was made of the y component of the electic field, tey_emweb. 6. To see the nomal component of the magnetic field, Nomal magnetic field (wev) was selected in the Pedefined quantities list. 7. On the Geneal tab, 90 was enteed in the Solution at angle (phase) edit field. This compensates fo the phase shift of 90 degees between the electic and the magnetic field. 8. OK was clicked to see the plot. 30

31 Now the 3D poblem was continued to set up to do the S-paamete analysis. Then Go to Default 3D View button was clicked in ode to see the full geomety. Fig. 3.8: The y component of the electic field fo the fist eigenmode of the elliptical pot. Bounday Settings: 1. Electomagnetic Waves was selected in the Multiphysics menu. 2. The Bounday Settings dialog box was opened. 3. At the ectangula end, bounday 13, the Pot bounday condition was selected. On the Pot tab the values wee set accoding to the table below: Table 3.4 Pot bounday specifications fo pot 1 Desciption Value Pot no. 1 Wave excitation at this pot Selected Mode specification Rectangula Mode type Tansvese Electic (TE) Mode No

32 4. At the elliptical end, bounday 6, the Pot bounday condition was selected. On the Pot tab, the values wee set accoding to the table 3.5. Table 3.5 Pot bounday specifications fo pot 2 Desciption Value Pot no. 2 Wave excitation at this pot Cleaed Mode specification Numeic Mode type Automatic Use numeic data fom Bounday mode analysis, TE waves (wev) 5. The default Pefect electic conducto bounday condition was selected at all othe boundaies. Computing the solution: 1. The Solve Manage dialog box was opened. On the Initial Value tab, Stoe Solution was clicked. In the dialog box that appeaed, the solution was clicked with the effective mode index and OK was clicked. 2. The Initial Value was set to stoed solution. 3. On the Solve Fo tab only the Electomagnetic Waves application mode was selected. 4. OK was clicked to close the Solve Manage. 5. In the Solve Paametes dialog box, the Paametic linea solve was selected. 6. Name of paamete was set to feq and List of paamete values to linspace(6.6e9,1e10,50). 7. In the Linea system solve list, Diect (SPOOLES) was selected. 8. The Symmetic was selected in the Matix symmety list in ode to save memoy. 9. On the Advanced tab, the Use Hemitian tanspose check box was cleaed. 10. OK was clicked to close the Solve Paametes dialog box. 11. The Application Scala Vaiables dialog box was opened fom the Physics menu and the Synchonize equivalent vaiables check box was cleaed. This allows us to set diffeent values fo the two fequency vaiables nu_emwev and nu_emw. 32

33 nu_emweb is the fequency vaiable fo the Bounday Mode Analysis application mode. It is impotant that it etains its value of 7e9 in ode fo the data fom the mode analysis solution to be evaluated coectly. The fequency vaiable nu_emw was set to feq. This is the fequency vaiable of the 3D Electomagnetic Waves application mode. 12. The eigenvalue vaiable lambda fom the mode analysis solution is not available in postpocessing afte solving the second poblem. Theefoe we needed to define a constant lambda with the value This lambda value coesponds to the effective mode index Lambda was defined in the Constants dialog box. To find the value of lambda, we could also use the Data Display dialog box afte doing the mode analysis but befoe solving the second poblem. 13. The Solve button was clicked. Postpocessing and visualization: 1. To visualize the x component of the electic field in the waveguide, the Plot Paametes dialog box was opened. 2. On the Geneal tab, the Solution at angle (phase) text field eads 0 was made sue. 3. The Bounday check box was cleaed, and selected the Slice check box. 4. The Slice tab was clicked and Electic field, x component (weh) was selected in the Pedefined quantities list. 5. Unde Slice Positioning, the numbe of x levels, y levels, and z levels wee set to 1, 1, and 0 espectively. 6. OK was clicked to see the plot. The pot bounday conditions geneate two S-paamete vaiables, S11_emw and S21_emw as well as two vaiables S11dB_emw and S21dB_emw, which ae the S- paametes on a db scale. The names of the vaiables ae deived fom the pot numbes enteed in the Bounday Settings dialog box. A domain plot was made in a point to plot the S-paametes as functions of fequency. 1. On the Point tab in Domain Plot Paametes dialog box point 1 was selected and S-paamete db (S11) (emw) was selected. Apply was clicked to plot S11 on a db scale. 33

34 2. When plotting S21 fist the two fist data points wee deselected on the Geneal tab to bette see the vaiation. The vaiable S-paamete db (S21) (emw) was selected and OK was selected to make the plot. 34

35 Chapte 4 Results and Discussion Thee wee many paametes which can be found using the COMSOL MULTIPHYSICS fo the thee dimensional waveguide H- bend, waveguide adapte and thee-pot feite ciculato. Some of them ae shown below: 4.1 Waveguide H-bend Fig. 4.1 The stuctue of waveguide H-bend This example illustates how to ceate a model that computes the electomagnetic fields and tansmission chaacteistics of a 90-degee bend fo a given adius. This type of waveguide bends changes the diection of the H field components and leaves the diection of the E field unchanged. The waveguide is theefoe called an H-bend. Hee, the side of coodinate -.1 indicates the input pot while the side of coodinate.15 indicates the output pot. In the input, a signal of 1 watt is applied in all the cases of this expeiment. This model examines a TE wave, one that has no electic field component in the diection of popagation. Moe specifically, fo this model we select the fequency 35

36 and waveguide dimension so that TE10 is the single popagating mode. In that mode the electic field has only one nonzeo component a sinusoidal with two nodes, one at each of the walls of the waveguide. One impotant design aspect is how to shape a waveguide to go aound a cone without incuing unnecessay losses in signal powe. The output figue of the analysis and explanation of the figues ae given below: In Fig. 4.2, we can see that the electic field component passes between top and bottom of the wave guide. It is maximum at cente and zeo at conducting walls so the enegy must flow though cente of the waveguide. The magnetic field foms closed path suounding the electic displacement cuents so the magnetic field is maximum at the bounday as shown in the Fig The Fig. 4.4 shows the time aveage of total enegy density of the waveguide. Total enegy should be maximum at the cente while zeo at the walls as we used hee pefectly electic conducto as bounday condition to maintain the loss minimum. The figue also shows that with this type of design and cutoff fequency the objective was achieved. In the Fig. 4.5, we can see the electic field in z diection. The applied wave is TE wave which has electic field component in the diection of popagation which is z diection. It is seen in the figue that the vaiation happens as a cosine function. In the Fig. 4.6 we can see the x component of the magnetic field. As the diection of popagation is in z diection and magnetic field is pependicula with electic field so we can see the sinusoidal vaiation of magnetic field with pependicula of electic field. This is the figue of x component so the magnetic field in y diection is not clealy visible. In the Fig. 4.7 we can see the y component of the magnetic field. As the diection of popagation is in z diection and magnetic field is pependicula with electic field so we 36

37 can see the sinusoidal vaiation of magnetic field with pependicula of electic field. This is the figue of y component so the magnetic field in x diection is not clealy visible. In the Fig. 4.8, we can see fo x component the powe exists in x diection as we bend the waveguide in y diection thee ae no powe othe than x diection and the same esult also seen fom the y diection in Fig In the Fig. 4.10, electic field displacement in z diection is shown. As the electic field moves by bouncing the walls of the waveguide so we see in the figue that thee once occu positive pick and then negative peak as cosine wave which is expected. The Fig 4.11 shows the suface cuent density in z component. We know that this is the diection of popagation and the input is TE10 wave. So thee is no magnetic field component in z diection and as a esult we didn t see any cuent in the uppe and lowe suface (though lowe suface is not visible in the figue). Thee ae cuent component in two side of the stuctue as it follows the magnetic field. The Fig 4.12 shows the suface cuent density nomalized which includes both x and y component of the suface cuent. So we can see the suface cuent ciculates in the total suface and as thee is no suface cuent in z component so the nomalized value of the suface cuent in the cente is low to some extent. In Fig. 4.13, the bounday plot of z component of tangential electic field of the waveguide is shown. The waveguide walls ae typically plated with a vey good conducto, such as silve. In this example the walls ae consideed to be made of a pefect conducto, implying that n E = 0 on the boundaies. This bounday condition is efeed to as a pefect electic conducto (PEC) bounday condition. 37

38 Fig. 4.2 Time aveage of electical enegy density taking a slice of the waveguide. Fig. 4.3 Time aveage of magnetic enegy density taking a slice of the waveguide. 38

39 Fig. 4.4 Time aveage of total enegy density taking a slice of the waveguide. Fig. 4.5 Z component of the electic field taking a slice of the waveguide. 39

40 Fig. 4.6 X component of magnetic field taking a slice of the waveguide. Fig. 4.7 Y component of the magnetic field taking a slice of the waveguide. 40

41 Fig. 4.8 Time aveage of the x component of powe flow taking a slice of the waveguide. Fig. 4.9 Time aveage of the y component of powe flow taking a slice of the waveguide. 41

42 Fig Subdomain plot of z component of electic displacement. Fig Bounday plot of z component of suface cuent density. 42

43 Fig Bounday plot of nomal component of suface cuent density. Fig Bounday plot of z component of tangential electic field. 43

44 4.2 Waveguide adapte This is a model of an adapte fo micowave popagation in the tansition between a ectangula and an elliptical waveguide. Such waveguide adaptes ae designed to keep enegy losses due to eflections at a minimum fo the opeating fequencies. To investigate the chaacteistics of the adapte, the simulation includes a wave taveling fom a ectangula waveguide though the adapte and into an elliptical waveguide. The S-paametes ae calculated as functions of the fequency. The involved fequencies ae all in the single-mode ange of the waveguide, that is, the fequency ange whee only one mode is popagating in the waveguide. The output figue of the analysis and explanation of the figues ae given below: In the Fig. 4.14, we can see the bounday of the elliptical pat which is passive but the eigenmode is still used in the bounday condition. We excite the input pot with TE10 wave. So in the diection of popagation thee is only electic field. As pefect electic conducto in the bounday is used, so we found the maximum electic powe in the cente which is shown in the coss-section of the elliptical suface. In the Fig. 4.15, we can see the magnetic enegy density. We know magnetic field always flows pependiculaly with the electic field. So as electic enegy density is in the cente of the adapte, the magnetic enegy density is maximum at the sides which ae expected. In the Fig. 4.16, the total enegy density is shown. As the electic field is taveling in the diection of popagation and thee is no electic field in the boundaies so it is expected that the total enegy will flow though the cente. This is what is veified by the figue. As the enegy density flows though the cente, so the powe also flows in the figue which is shown in the Fig

45 In the Fig. 4.18, as the magnetic field flows pependiculaly with the electic field and hee electic field flows with the diection of popagation, so the magnetic field is pependicula with that. As the electic field vaies sinusoidally, so the magnetic field also vaies sinusoidally as in the figue shown the magnetic field occus in two sides with maximum and minimum value. In the Fig 4.19, we can see the nomal component of magnetic flux density of the waveguide which is maximum in the input pot (ectangula side) and at a lowe value at the othe side due to some loss. It is to be noted that we conside hee only dielectic loss and no conducto loss ( =0). In the Fig 4.20, we can see the nomal component of electic field of the waveguide which is maximum in the input pot (ectangula side) and at a lowe value at the othe side due to changes of mode in elliptical suface. It is to be noted that we conside hee only dielectic loss and no conducto loss ( =0). In the Fig. 4.21, we see the nomal component of time aveage of powe flow taking a slice of the waveguide. The magnitude of powe is the maximum in the ectangula pat of the waveguide. The magnitude of powe flow is less in the elliptical side due to change of mode cause hee it woks in a hybid mode. In the Fig. 4.22, we see the sub domain plot of time aveage of electic enegy density. This is the maximum in the ectangula side in the cente as TE10 mode is thee and showing lowe value in elliptical pat as thee popagating hybid mode of the wave. The coesponding magnetic field density was shown in the Fig In Fig. 4.24, we see the sub domain plot of z component of magnetic flux density. As the electic field bounces in the x diection so thee is a magnetic field tavelling sinusoidally in the z diection as shown in the figue which is gaphically same with z component of magnetic field shown in the Fig

46 Fig Bounday plot of time aveage of electic enegy density. Fig Bounday plot of time aveage of magnetic enegy density. 46

47 Fig Bounday plot of time aveage of total enegy density. Fig Bounday plot of time aveage of nomal powe flow. 47

48 Fig Bounday plot of nomal magnetic field. Fig Nomal component of magnetic flux density taking a slice of the waveguide. 48

49 Fig Nomal component of electic field taking a slice of the waveguide. Fig Nomal component of time aveage of powe flow taking a slice of the waveguide. 49

50 Fig Subdomain plot of time aveage of electic enegy density. Fig Subdomain plot of time aveage of magnetic enegy density. 50

51 Fig Subdomain plot of z component of magnetic flux density. Fig Subdomain plot of nomal component of electic field. 51

52 Fig Subdomain plot of z component of magnetic field. In the Fig. 4.25, we see the sub domain plot of electic field, nom. We know electic field popagates by bouncing in x diection so the esult in the figue is expected. The side walls also show that the component is zeo thee. Result of the s-paamete analysis: We have found the gaph fo two s-paametes with espect to fequency ( S11 and S 21 ) which ae shown below. Fo S 11, we can see that the S-paamete is the lowest at cetain fequency and thee ae some othe lowe values of S 11 at cetain fequencies. Similaly, fo S 21, thee ae cetain fequencies whee the S-paamete is high. We can use these fequencies as windows fo tansmission as they involve lowe loss. 52

53 Fig 4.27 S 11 vesus Fequency. 53

54 Fig 4.28 S 21 vesus Fequency. 54

55 Chapte 5 Conclusion A scatteing matix (S-paamete matix) is one way to descibe the opeation of a linea, time-invaiant two-pot cicuit. In this thesis pape it was aimed to develop a basic and elaboate idea about S-paametes. The S-paamete matix is apidly becoming vey popula as a way to chaacteize connectos and cables fo high-speed applications above 1 Gb/s. Many electical popeties of netwoks o components may be expessed using S- paametes, such as gain, etun loss, voltage standing wave atio (VSWR), eflection coefficient and amplifie stability. As fo high fequency analysis and micowave netwoks thee is no altenative of S-paametes so it is must to study about it. At the beginning of this pape S paametes with thee shot histoy and a significant classification is being mentioned. Then it was tied to find out the eason why we have to study about S-paametes and what is the basic diffeences with othe netwok paametes. The notation which is used to epesent a S-paamete of N-pot netwok and it s significant is mentioned in this vey intoductoy pat of this thesis. Ou main objective of this thesis is descibed vey paticulaly. FEMLAB is used to have a pactical knowledge about what is actually happening in these waveguides and how the waves ae being diven in them.. FEMLAB is a poweful, inteactive envionment fo modeling and solving scientific and engineeing poblems based on patial diffeential equations. It can easily extend conventional models that addess one banch of physics to state-of-the at multiphysics models that simultaneously involve multiple banches of science and engineeing. Fo bette undestanding two examples ae shown hee. One is of waveguide H-bend and the othe one is of waveguide adapte. The step by step pocess of doing the examples is mentioned hee. Finally the esults of these examples and pobable desciption of these esults ae given in bief. 55