ECE Spring Prof. David R. Jackson ECE Dept.

Transcription

1 ECE 634 Spring 26 Prof. David R. Jacson ECE Dept. Notes Notes 42 43

2 Sommerfeld Problem In this set of notes we use SDI theory to solve the classical "Sommerfeld problem" of a vertical dipole over an semi-infinite earth. Goal: Find E on the surface of the earth in the air region. ε Il h r x ε εε rc 2

3 Planar vertical electric current from Notes 39: V s t J where s s J xy,, J xy, δ h the impressed source current For a vertical electric dipole of amplitude I l, we have J x, y Il δ x δ y δ h Hence δ J x, y Il δ x y s Therefore, we have V s t Il 3

4 TEN: h - Z V s V s t Il Z The vertical electric dipole excites waves only. 4

5 Vertical Field Find E x,y, inside the air region >. H j E H y H x E j x y E j H j H x y y x j H x y H y x 5

6 Example cont. ˆ ˆ ˆ ˆ H H u x H v x x u v u cosφ H v sinφ H x y Hu Hv t t y, t x y ˆv û ˆ ˆ ˆ ˆ H H u y H v y y u v H u sinφ H v cosφ y x H u H v t t φ x 6

7 Example cont. Hence cancels y x x y E x H u H v y H u H v t t t t or 2 2 E H v x y t or E H t v 7

8 Hence E I,, x y t We use the Michalsi normalied current function: t I Iv Vs Iv Il The v subscript indicates a V series source. We need to calculate the Michalsi normalied current function at since we want the field on the surface of the earth. 8

9 Calculation of the Michalsi normalied current function Z Z - h V h - V I v Z I v Z Z Z 2 2 t 2 2 t 9

10 I v h - Z Z V Z This figure shows how to calculate the Michalsi normalied current function: it will be calculated later in these notes.

11 Return to the calculation of the field: t E x, y, t Iv Il Hence we have t j xx yy E x, y, 2 I Il e d d 2π t v x y or Il j xx yy 2 E x, y, I 2 2 e d d 2π v t x y

12 Il j xx yy 2 E x, y, I 2 2 e d d 2π v t x y Iv is only a function of t Note: Change to polar coordinates: x y ρcosφ ρsinφ x y t t cosφ sinφ d d d dφ x y t t 2

13 Il j xx yy 2 E x, y, I 2 2 e d d 2π v t x y E x y I e d d 2π Il 3 j t cosφρcosφ t sinφρsinφ,, 2 2 v t φ t 2π Switch to polar coordinates d d d dφ x y t t Il E x y I e d d 2π 3 j cos,, 2 2 t ρ φ φ v t φ t 2π 3

14 j t ρ cos φ φ E xy I e dφ Use 2π Il 3,, 2 2 v t 2π α φ φ 2π 2π φ 2π j ρ cos φ φ j ρ cosα j ρ cosα φ α α t t t e d e d e d φ We see from this result that the vertical field of the vertical electric dipole should not vary with angle φ. Integral identity: 2π ρ j t cosα I e dα 2πJ t ρ 4

15 Hence we have Il 3 E ρ, J ρ I d 2π 2 t v t t This is the Sommerfeld form of the field. 5

16 We now return to the calculation of the Michalsi normalied current function. I v h Z Γ Z - Vv - V Z v V Ae Be j j j j e Γ B e Γ Z Z < < h B Iv e e Z j j Γ A B Z Z 6

17 I v h Z Γ Z - Vv - V Z v j V Ce > h C Iv e Z j Here we visualie the transmission line as infinite beyond the voltage source. 7

18 I v h Z Γ Z - Vv - V Z Boundary conditions: v v V h V h v v I h I h 8

19 Hence we have v v V h V h Ce B e e Γ j h j h j h Z C v v I h I h B e e Γ e jh jh jh Z 9

20 Substitute the first of these into the second one: Ce B e Γ e j h j h j h Z C B e e Γ e jh jh jh Z This gives us Z j h j h j h j h B e Γ e e Γ e Z B 2

21 Z B Γ Γ Z j h j h j h j h B e e e e j h j h j h j h B e Γ e B e Γ e j h B 2e cancels Hence we have B e j h 2 2

22 For the current we then have B Iv e e Z with Hence j j Γ B e j h Iv e e e 2Z 2 j h j j Γ For : I v Γ e j 2Z h 22

23 We thus have Il 3 E ρ, J ρ I d 2π with Hence 2 t v t t Iv h e 2Z, Γ j Il 3, j h E ρ J ρ Γ e d π 2 t t t 2 2Z or Il 3, j h E ρ J ρ Γ e d 2π 2 2 t t t h 23

24 Il 3, j h E ρ J ρ Γ e d 2π 2 2 t t t Il 3, j h E ρ J ρ Γ e d t t t 4π 24

25 Final Result Il h ε r x ε εε rc Il 3, j h E ρ J ρ Γ e d t t t 4π Γ Γ t Z Z Z Z Z Z ` 2 2 t 2 2 t 25

26 ti Note: A box height of about.5 is a good choice. C tr Zennec-wave pole tp ε ε rc rc derivation on next slide ε rc complex relative permittivity of the earth accounting for the conductivity. 26

27 Zennec-wave pole The TRE is: ε rc ε ε ε tp rc rc rc Z Z Z Z Note: Both vertical wavenumbers and are proper for the Zennec wave proof omitted. ε rc tp tp ε ε tp rc rc tp ε ε rc rc ε rc tp 2 ε rc 2 2 tp 2 2 rc tp rc 2 ε rc ε rc ε ε ε rc ε rc εrc εrc 27

28 Alternative form The path is extended to the entire real axis. ρ, ρ Odd E J d t t t Transform the H term: We use 2 J x H x H x 2 2 H x H x 2 Odd t H tρ dt Odd t H tρ dt 2 ρ Odd H d Odd Odd t t t 2 ρ H d t t t 2 ρ H d t t t Use t t 28

29 Hence we have E Il H e d 2 j h 3 ρ, tρ t t 4π 2 Γ This is a convenient form for deforming the path. 29

30 ti tp C tr tp tp ε ε rc rc The Zennec-wave is nonphysical because it is a fast wave, but it is proper. It is not captured when deforming to the ESDP vertical path from. 3

31 ti The Riemann surface has four sheets. tp ESDP Original path C tp tr Top sheet for both and The ESDP from the branch point at is usually not important for a lossy earth. Bottom sheet for Bottom sheet for Bottom sheet for both and 3

32 ti tp tr tp This path deformation maes it appear as if the Zennec-wave pole is important. 32

33 Throughout much of the 2 th century, a controversy raged about the reality of the Zennec wave. Arnold Sommerfeld predicted a surface-wave lie field coming from the residue of the Zennec-wave pole 99. People too measurements and could not find such a wave. Hermann Weyl solved the problem in a different way and did not get the Zennec wave 99. Some people Norton, Niessen blamed it on a sign error that Sommerfeld had made, though Sommerfeld never admitted to a sign error. Eventually it was realied that there was no sign error Collin, 24. The limitation in Sommerfeld s original asymptotic analysis which shows a Zennec-wave term is that the pole must be well separated from the branch point the asymptotic expansion that he used neglects the effects of the pole on the branch point the saddle point in the steepest-descent plane. When the asymptotic evaluation of the branch-cut integral around includes the effects of the pole, it turns out that there is no Zennec-wave term in the total solution branch-cut integrals pole-residue term. The easiest way to explain the fact that the Zennec wave is not important far away is that the pole is not captured in deforming to the ESDP paths. 33

34 R. E. Collin, Hertian Dipole Radiating Over a Lossy Earth or Sea: Some Early and Late 2 th -Century Controversies, AP-S Magaine, pp , April

35 ti Alternative path efficient for large distances ρ tp C tr tp This choice of path is convenient because it stays on the top sheet, and yet it has fast convergence as the distance ρ increases, due to the Hanel function. 35