Prof. David R. Jackson ECE Dept. Spring 2016 Notes 42 ECE 6341 Notes 44 1

Overview In this set of notes we derive the SDI formulation using a more mathematical, but more general, approach (we directly Fourier transform Maxwell’s equations). 2  This allows for all possible types of sources (horizontal, vertical, electric, and magnetic) to be treated in one derivation.

General SDI Method where Start with Ampere’s law: Assume a 2D spatial transform: 3

Hence we have Note that Take the components of the transformed Ampere’s equation: Next, represent the field as General SDI Method (cont.) 4

Examine the TM z field: (Ignore equation) (2) (1) General SDI Method (cont.) 5

TM z Fields We wish to eliminate from Eq. (1). To do this, use Faraday’s law: Take the component of the transformed Faraday’s Law: (3) 6 Recall:

Substitute from (1) into (3) to eliminate or TM z Fields (cont.) (1) (3) 7

Hence (4) TM z Fields (cont.) Note that 8

Equations (2) and (4) are the final TM z field modeling equations: TM z Fields (cont.) 9

Define TEN modeling equations: We then have TM z Fields (cont.) 10

Telegrapher’s Equation Allow for distributed sources so v+v+ v zz i+i+ i-i- 11

Telegrapher’s Equation (cont.) Also, Hence, in the phasor domain, Hence, we have so 12

Telegrapher’s Equation (cont.) Compare field equations for TM z fields with TL equations: 13

Telegrapher’s Equation (cont.) We then make the following identifications: Hence so 14

Sources: TM z For the sources we have, for the TM z case: 15

Assume Special case: planar horizontal surface current sources: Sources: TM z (cont.) These correspond to lumped current and voltage sources: lumped parallel current source lumped series voltage source Then we have 16

Assume For a vertical electric current: Sources: TM z (cont.) “planar vertical current distribution” z = 0 17

This corresponds to a lumped series voltage source: Sources: TM z (cont.) 18

Special case: vertical electric dipole Sources: TM z (cont.) Hence 19

TE z Fields Use duality: TM z TE z 20

TE z (cont.) Define: We then identify 21

For the sources, we have Special case of planar horizontal surface currents: TE z (cont.) lumped series voltage source lumped parallel current source 22

Assume For a vertical magnetic current: Then we have Sources: TE z (cont.) This corresponds to a lumped parallel current source: 23

Sources: TE z (cont.) Special case: vertical magnetic dipole Hence 24

Summary 25 Horizontal Vertical Results for 3D (volumetric) sources Distributed sources: either parallel current sources or series voltage sources

Summary Lumped sources: either parallel current sources or series voltage sources 26 Results for 2D (planar) sources Horizontal Vertical

Summary (cont.) TEN Models

Summary (cont.) Michalski functions