Field and Network Theories in ECE

1. Field and Network Theories in ECE Levent SEVGİ 1Electronics and Communication Engineering Dept, DOĞUŞ University, Zeamet Sokak, No 21, Acıbadem / KADIKOY - Istanbul

2. Well-posed Problems • A problem is said to be well-posed if the following conditions are met: • Existence; implies that the conditions do not over determine the problem and the solution exists, • Uniqueness; implies that the conditions do not under determine the solution and it assures that the solution, no matter how obtained, is the correct and the only one. • Stability; implies that arbitrary small perturbations in data, e.g., sources, do not, for any physical quantity, let to infinity. • Completeness, is to represent any excitation in terms of building blocks called Modes (i.e., eigensolutions.

3. Complexity Architecture • Wave interaction with complex environmens poses major challenges to the analytic modeler. Complexity in the context of waves, encompasses many scales which are conveniently referenced to the relevant wavelengths =2c/ in the interrogating wave signal, where c is the wavespeed in a reference ambient medium and  is the radian frequency. • These relative scales si /  can be associated with physical dimensions si di; wavelengths si i in various materials; temporal widths si Ti of the signal spectra; sampling window widths si Wi in the processing data, etc. • The wave modeler must decide how to parameterize a complex physical problem so as to take best advantage of the wave-based and computational tools at his disposal. It is natural to employ an architecture which decomposes the overall complex problem domain into simpler more tracktable interacting subdomain (SD) problems.

4. Complexity Architecture • A general framework for decomposing an overall complex problem space into interacting subdomains

5. Guided waves and Reduction • The steady-state EM vector fields excited by a specified electric and magnetic current distributions, J and M, respectively, are defined by Maxwell’s field equations: • on the perfectly conducting boundary of the uniform waveguide, the tangential component of the electric field must vanish:

6. x z ZL Z0,  =  + j Load Source L Complexity Architecture • Transmission line model

7. S21 Incident Transmitted a1 b2 S22 S11 [S] Port 2 Port 1 Reflected Reflected b1 a2 Incident S12 Transmitted Complexity Architecture • Scattered (S) parameter model

8. Maxwell’s Equations and Decomposition Full, Source-excited Maxwell’s equations + BC Longitudinal Fields (dependent) Transverse Source-excited Maxwell’s equations + BC Eigensolutions (source-free) Modal Amplitudes (scalar) Source-free Transverse Field equations + BC TE Type (Dirichlet) Transmission Line Equations Transverse Vector Eigenfunctions (scalarization) TM Type (Neumann) Mixed (Cauchy)

9. Guided waves and Reduction • EIGENVALUE PROBLEM (Dirichlet type) • TRANSMISSION LINE PROBLEM

10. Sturm-Liouville Equation (SL) • Each 1D problem in a coordinate-separable reduced 2D or 3D problem is parameterized by SL theory which deals with eigenspectra of 1D linear second-order diferential operators in bounded and unbounded domains. with suitable boundary condititons at z=ZL (left boundary) and z=ZR (right boundary) represents our EM equations. Depending on the p(z), q(z) and (z) this may be Helmholtz wave equation, Laplace equation, diffusion equation, etc. • This represents a Transmission Line problem if  is a fixed parameter. • It is an Eigenvalue problem if  is a free parameter that does not equal to an eigenvalue (Since eigenvalues are real, choosing Im {}0 guarantees uniqueness).

11. Alternative representations • Phase space wave dynamics can be organized around two complementary phenomenologies - progressing and oscillatory - which are closely related to local vs. global descriptions. • Progressing wave objects (rays or wavefronts) are responsible for point-to-point propagation and sample the physical environmentlocally along their trajectories. • Oscillatory wave objects (modes or resonances) form standing waves over extended (global) portions of the physical environment. • An alternative approach to the local-global parameterization of wave phenomena is through the use of Poisson summation, which converts an infinite or truncated n-sum (poorly convergent) series into an m-sum (good convergent ) series in Fourier domain.

12. Sturm-Liouville Equation • A generic problem for transmission lines is the Green’s function problem. Solving Sturm-Liouville equation when  is a fixed parameter yields nothing but the Green’s function. • The Green’s function is the response of a linear system to a point source of unit strength. A source of unit strength at a position “a” along, for example, z coordinate can be represented by the Dirac delta function. • Green’s function solution may also be built from eigenfunction solutions. • Alternatively, eigenfunctions may also be derived from the Green’s function solution. • This illustrates that a guided wave problem may be handled as either an eigenvalue problem or transmission line problem, since the former also yields the latter and vice versa.

13. Normal Mode Solution • NM is the solution of source-free wave equation in a coordinate system that yields separation of variables • NM satisfy transverse boundary conditions on each transverse cross-section • NM confinement may be due to fixed transverse boundaries and/or mode dependent virtual boundaries (refractive confinement) • NM propagate in longitudinal direction with distinct propagation constants • Each NM carries finite energy (can be normalized) and is independent of every other NM (orthogonality)

14. Example : Parallel plate waveguide x PEC z PEC • Start with Maxwell’ equations and obtain two sets of equations as

15. Example : Parallel plate waveguide • Set 1: TMz Set 1 and Set 2 are decoupled • Set 2: TEz

16. Example : Parallel plate waveguide • Any of Jx, My or Jz excite SET 1 (TMz) • Any of Jy, Mx or Mz excite SET 2 (TMz) • Lets look at TEz with Jy source only and start with second order decoupled, Maxwell’s (wave) equations. Boundary conditions + wavenumber

17. Example : Parallel plate waveguide • The Green’s function associated with this problem is: + Normalized Eigenfunciton Z dependent coefficient

18. Example : Parallel plate waveguide • Using the identities one obtains • Defining and yields

19. Example : Parallel plate waveguide • Z dependent solution is • and, finally the Green’s function is obtained as • Ey may directly be obtained from the Green’s function as

20. x PEC z PEC Example : Parallel plate waveguide • For a line source at (x’,z’) the solution will then be

21. Mode 1 Mode 5 Mode 25 f=300 MHz Modal Caustics Height [m] a0 = -410-5 Normalized Mode Amplitude Example #2: Open surface waveguide

22. f=300 MHz 15 km Height [m] 100 NM 10-6 contribution 0 km 12 km Normalized Field Strength Software Calibration: SSPE vs. NM

23. Conclusions and Discussions • Field and network theories are two fundamental approaches in ECE. • Any ECE problem can be postulated via one of these two approaches, the solution can be derived, and may then be transformed into the other. • The problem at hand, the parameter regime, or the geometry give clues to choose the best suitable approach.