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# Class Note for ECE 6341 with Professor Jackson at UH

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Date Created: 02/06/15
ECE 6341 Spring 2009 Prof David R Jackson ECE Dept Notes 37 SteepestDescent Path Physics kx 2 k0 sin Q kyo 2 k0 0034 Note No branch points in Q plane cosQ is analytic Both sheets of the kx plane get mapped into a single sheet of the Q plane Examine kyo to see where the Q plane is proper and improper ky0 2 k0 0034 14 2 k0 cosQ r coshQ l jsinQ r sinhQ i 1m kyO 2 k0 sin Q r sinh Q l SDP Physics cont 1m kyo 2 k0 sin fr sinh 4 Cf talk talk P proper I improper Mapping of quadrants in kx plane SDP Physics cont kx kosin k0sin rcosh i jcos rsinhgj 2 X Nonphysical growing LW I I pole SDP Physics cont SDP 00592 6coshg l 1 A leakywave pole is considered to be physical if it is captured when deforming to the SDP SDP Physics cont LWP captured 9 gt 9 SDP The angle 6b represents the boundary for which the leakywave field exists Note SDP Physics cont Behavior of LW field AZ 2 Fkxe fky yekaxdkx IF ejkpcos 9k0 coscfdgquot C AZW 2 1 RCSFQ ko cos p ejkpcosgp6 w In rectangular coordinates LW 39k 39kxx AZ Ae We where kxp ja SDP Physics cont Examinethe exponential term 5 005617 6 cosg rp 6J ip cosg rp 6 cosh 117 jsin rp 6sinh ip Hence kop sin rp 6sinh gay lwle koplsinh quotlp 311109 9317 since Ci lt 0 SDP Physics cont Ram lllllll aymg lwl2eltkopgtlsmh pside cm Power Flow gRegRerx 130 Reik0 sinlt rp jgjp2k0 cos rp 1401 2 k0 isingjp coshgjp icos rp coshgjp Power Flow cont k0 ising p coshgylp 2cos m coshgjp Also g ism 6L 2c036L Note that 3 tan 9L x tangy y y Hence ESDP Extreme SDP The ESDP is important for evaluating the fields on the interface which determines the farfield pattern We can show that the ESDP divides the LW region into slowwave and fastwave regions 12 ESDP ESDP cont To see this cos Q 6 coshcfl 1 SDP sing coshg i 1 ESDP Recall that kxp 2 k0 8111 p k0 sinr m Hence Re kxp 2 k0 sin m cosh 1p ESDP cont Hence 2 sm Cm cosh I 0 Fastwave region k lt 1 sin m COSh 11 lt 1 O Slowwave region k gt1 Sin Cm COSh Ci gt1 0 Compare with ESDP sin a cosh Cl 1 14 ESDP cont The ESDP thus establishes that for fields on the interface a leakywave pole is physical captured if it is a fast wave 4139 SWP LWP captured 15 ESDP 67r2 LWP not captured SDP in kx Plane We now examine the shape of the SDP in the kx plane kx 2 k0 sin 2 k0 Sin r so that kw k0 sin a coshg kxi k0 00er sinh g SDP 0036 9coshgquotl 1 The above equations allow us to numerically plot the shape of the SDP in the kx plane 16 SDP in kx Plane cont 7239 9 y 2 see the appendix for a proof Fields on Interface k xi XI The leakywave pole is captured if it is in the fastwave region ESDP 18 Fields on Interface cont AZ A AZCS km AfW AfW AfW k0 SW LW The contribution from the ESDP ESDP is called the spacewave field or the residualwave RW field It is similar to the lateral wave in the halfspace problem 19 Asymptotic Evaluation of ResidualWave Field cont AZRW jejk0x 00 F k0 jsesxds je kox EF k0 jsesxds AfW jejk x 00 F k0 jsesxds jejk x fF k0 jsesxds Asymptotic Evaluation of ResidualWave Field cont Then AZRW je1k0x FHS6sde 0 Q X for X gt 00 Assume HSAS as S gt0 Watson s ARW N e jkox Ara 1 lemma 2 J xa1 Asymptotic Evaluation of ResidualWave Field cont It turns out that Hence Note For dipole we have AfW A 23 Discussion of Asymptotic Methods We have now seen two ways to asymptotically evaluate the fields on an interface as x gt oo 1 steepestdescent Q plane There are no branch points in the steepestdescent plane The fields on the interface correspond to a higherorder saddlepoint evaluation since the functionf4 is zero at 4 0 6 7z2 2 wavenumber kx plane The SDP becomes an integration along a vertical path that descends from the branch point at kx k0 The integrand is not analytic at the endpoint of integration branch point since there is a squareroot behavior at the branch point Watson s lemma is used to asymptotically evaluate the integral 24 Summary of Waves continuous spectrum Z jk0x LW jk Wx RW N SW jk Wx A2 ALW 6 A2 ARW x32 A ASW e 25 Interpretation of RW Field The RW field is a sum of lateralwave fields y ex Appendix Proof of Angle Property Proof of angle ro ert k k k p p y ta11 xr 0 xr The last identity er 2 k0 sin y cosh 4 1 er follows from 2 tan gr km 2 k0 cos 4 smh 1 km Hence 7 r or 77Z39 r 28 fr gt 9 the asymptote On SDP As Ii gt00 Proof cont To see which choice is correct ESDP 6 E 2 In the kx plane this corresponds to a vertical line for which 7 0 29

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