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Presented at the COMSOL Conference 2008 Hannover<br />
ResonancesinTaperedDouble-Port<br />
TEMWaveguides<br />
Prof.Dr.-Ing.JensPeterKärst<br />
HochschulefürAngewandteWissenschaftundKunst<br />
FakultätN<br />
COMSOLMultiphysicsKonferenz6.November2008
Contents Introduction<br />
Analyticalcalculation<br />
Simulation<br />
Measurements<br />
Conclusion<br />
JensPeterKärst ResonancesinTEMWaveguides
Introduction<br />
Analyticalcalculation<br />
Simulation<br />
Measurements<br />
Conclusion<br />
Introduction<br />
Whatarewetalkingabout?<br />
Whatdidwecalculate?<br />
Whycouldthisbeinterestingforyou?<br />
JensPeterKärst ResonancesinTEMWaveguides
Smalltapereddouble-portTEMwaveguides<br />
◮ µTEMcell,PTB,German�E-fieldstandard<br />
◮ µC 3 cell,proposedCircularCoaxialCalibrationcell<br />
JensPeterKärst ResonancesinTEMWaveguides
Mechanicalconstruction,CrawfordTEMcell<br />
◮ Rectangularcrosssection<br />
◮ Flatinnerconductor(Septum)<br />
JensPeterKärst ResonancesinTEMWaveguides
Mechanicalconstruction,µC 3 cell<br />
◮ Circularcrosssection<br />
◮ Roundinnerconductor(Septum)<br />
JensPeterKärst ResonancesinTEMWaveguides
Introduction<br />
Analyticalcalculation<br />
Simulation<br />
Measurements<br />
Conclusion<br />
Analyticalcalculation<br />
Generalizedtelegraphist’sequations<br />
Resonancecalculation<br />
JensPeterKärst ResonancesinTEMWaveguides
Fieldmodes,transversal�E<br />
TEMmode<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
−0.1<br />
−0.2<br />
−0.3<br />
−0.4<br />
TEM−Mode<br />
−0.5 0 0.5<br />
JensPeterKärst ResonancesinTEMWaveguides
Fieldmodes,transversal�E<br />
TM01mode<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
−0.1<br />
−0.2<br />
−0.3<br />
−0.4<br />
E−Mode, k (E)<br />
/(2a) = 11.0242<br />
c<br />
−0.5 0 0.5<br />
JensPeterKärst ResonancesinTEMWaveguides
Fieldmodes,transversal�H<br />
TM01mode<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
−0.1<br />
−0.2<br />
−0.3<br />
−0.4<br />
E−Mode, k (E)<br />
/(2a) = 11.0242<br />
c<br />
−0.5 0 0.5<br />
JensPeterKärst ResonancesinTEMWaveguides
TransmissionlinemodelofataperedTEMwaveguide<br />
Setup<br />
V<br />
ZC<br />
TEMmode<br />
Z (TEM)<br />
W<br />
V·u<br />
V (TEM)<br />
outerconductor<br />
innerconductor(septum)<br />
coupling coupling<br />
I (TEM)<br />
γ (TEM) ,Z (TEM)<br />
W<br />
ZC<br />
0z0 z z<br />
(TE)<br />
c z (TM)<br />
c zK zsym zL<br />
JensPeterKärst ResonancesinTEMWaveguides<br />
Z (TEM)<br />
W
TransmissionlinemodelofataperedTEMwaveguide<br />
TEMmode<br />
Z (TEM)<br />
W<br />
V·u<br />
TEmode<br />
Z (TE)<br />
W (z(TE)<br />
V (TEM)<br />
CTEM,TE<br />
undercutoff overcutoff<br />
I (TEM)<br />
γ (TEM) ,Z (TEM)<br />
W<br />
V (TE)<br />
CTE,TEM<br />
I (TE)<br />
γ (TE) ,Z (TE)<br />
W<br />
Z (TEM)<br />
W<br />
c ) Z (TE)<br />
JensPeterKärst ResonancesinTEMWaveguides<br />
W
TransmissionlinemodelofataperedTEMwaveguide<br />
TEMmode<br />
Z (TEM)<br />
W<br />
V·u<br />
TMmode<br />
V (TEM)<br />
Z (TM)<br />
W (z(TM)<br />
Cp,TM<br />
I (TEM)<br />
γ (TEM) ,Z (TEM)<br />
W<br />
V (TM)<br />
CTM,q<br />
γ (TM) ,Z (TM)<br />
W<br />
I (TM)<br />
Z (TEM)<br />
W<br />
c ) Z (TM)<br />
undercutoff overcutoff undercutoff<br />
JensPeterKärst ResonancesinTEMWaveguides<br />
W
TransmissionlinemodelofataperedTEMwaveguide<br />
TMmode<br />
Setup<br />
Z (TM)<br />
W (z(TM)<br />
V<br />
ZC<br />
V (TM)<br />
undercutoff overcutoff undercutoff<br />
outerconductor<br />
I (TM)<br />
c ) Z (TM)<br />
γ (TM) ,Z (TM)<br />
W<br />
innerconductor(septum)<br />
W<br />
coupling coupling<br />
ZC<br />
0z0 z z<br />
(TE)<br />
c z (TM)<br />
c zK zsym zL<br />
JensPeterKärst ResonancesinTEMWaveguides
Generalisedtelegraphist’sequationsformodep<br />
Usingpropagationconstantγ (p) andwaveimpedanceZ (p)<br />
W :<br />
dV (p)<br />
dz =−γ(p) (z)Z (p)<br />
W (z)I (p) ∞�<br />
(z) + Cpq(z)V (q) (z)<br />
dI (p)<br />
dz =−γ(p) (z)<br />
Z (p)<br />
W (z)V (p) (z)−<br />
q=1<br />
∞�<br />
Cqp(z)I (q) (z)<br />
q=1<br />
Neglectingmodecouplingusingwavenumberk=ω √ µε:<br />
d 2 V (TE)<br />
dz<br />
d 2 I (TM)<br />
k<br />
= 2<br />
k<br />
= 2<br />
dz<br />
�<br />
k (TE)2<br />
c (z)−k 2�<br />
V (TE)<br />
k (z)<br />
�<br />
k (TM)2<br />
c (z)−k 2�<br />
I (TM)<br />
k (z)<br />
JensPeterKärst ResonancesinTEMWaveguides
I (TM) (z)ofTM01atresonanceinaµC 3 -cell<br />
I (TM) zsym<br />
0.05<br />
0<br />
−0.05<br />
0 0.02 0.04 0.06 0.08 0.1<br />
Firstresonanceatfres =4.456GHz<br />
JensPeterKärst ResonancesinTEMWaveguides<br />
z<br />
m
I (TM) (z)ofTM01atresonanceinaµC 3 -cell<br />
I (TM) zsym<br />
0.05<br />
0<br />
−0.05<br />
0 0.02 0.04 0.06 0.08 0.1<br />
Secondresonanceatfres =5.001GHz<br />
JensPeterKärst ResonancesinTEMWaveguides<br />
z<br />
m
Introduction<br />
Analyticalcalculation<br />
Simulation<br />
Measurements<br />
Conclusion<br />
Simulation<br />
MATLAB/NAGsimulation<br />
FEMLABsimulation<br />
JensPeterKärst ResonancesinTEMWaveguides
Simulationusinggeneralisedtelegraphist’sequations<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
0<br />
0 1 2 3 4 5 6<br />
MATLABwithNAGtoolbox,S21,range0
Longitudinalcomponentoftheelectricfield�E<br />
Belowresonanceatf =1GHz<br />
JensPeterKärst ResonancesinTEMWaveguides
Longitudinalcomponentoftheelectricfield�E<br />
AtfirstTM01resonancefTM011 =4.4GHz<br />
JensPeterKärst ResonancesinTEMWaveguides
Longitudinalcomponentoftheelectricfield�E<br />
AtsecondTM01resonancefTM012 =4.9GHz<br />
JensPeterKärst ResonancesinTEMWaveguides
SimulationusingFEMmodel,FEMLABwithRFtoolbox<br />
1<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
0 1 2 3 4 5 6<br />
x10 9<br />
S11,S21,range0
Introduction<br />
Analyticalcalculation<br />
Simulation<br />
Measurements<br />
Conclusion<br />
Measurements<br />
Comparisonwithanalyticalcalculations<br />
Comparisonwithsimulationresults<br />
JensPeterKärst ResonancesinTEMWaveguides
SimulationusingFEMmodel,FEMLABwithRFtoolbox<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
0<br />
0 1 2 3 4 5 6<br />
S11,S21,range0
Introduction<br />
Analyticalcalculation<br />
Simulation<br />
Measurements<br />
Conclusion<br />
Conclusion<br />
JensPeterKärst ResonancesinTEMWaveguides
Conclusion<br />
◮ Ananalyticalmethodforthecalculationofresonant<br />
frequencieshasbeenpresented<br />
◮ Acomparisonofresultsobtainedusingtwodifferent<br />
simulationmethods<br />
1. Generalisedtelegraphist’sequations(MATLABwithNAG<br />
toolbox)<br />
2. FEMmodel(FEMLABwithRFtoolbox)<br />
andmeasurentsshowscloseagreement<br />
◮ Simulationscanbebenchmarkedusingthisanalyticalmethod<br />
JensPeterKärst ResonancesinTEMWaveguides
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