8 Rect. Coord. Curl is a measure of fields ability to produce torque 9 8. Laplacian Rect. Coord. 10 9. Helmholtz Theorem A vector is completely defined if its DIV Curl are specified if the field vanished as rh (or if normal component is specified over some closed surface) Note we specified that DIV Curl are ROCs are related to sources of the field. Thus a vector field is specified by its vector scalar source. 11 10. Divergence Theorem 12 10. Stokes Theorem 13 B. Maxwells Equations 1. Gauss Law 14 Line of Charge For a Closed Surface 15 Apply Divergence Theorem Maxwell Eqn. (Gauss Law for Points) 16 2. Amperes Law 17 For a very long straight wire lines encircle conductor In this case 18 3. Potential 19 Conservative Field And Grad. of a scalar From To find V from a collection of charges
20 For general situations where Now you can get from 21 4. Equation of Continuity Current Closed Surface Around One Plate Current Entering Closed Surface 22 For I out of closed surface By Divergence Theorem Equation of Continuity 23 Returning to Amperes Law In General Maxwell Equation Current density is composed of two parts 24 5. Gauss Law for Magnetics 25 6. Faradays Law Apply Stokes Thm 26 7. M.E. and Sources of Fields 27 8. Magnetic Vector Potential 28 (No Transcript) 29 9. Poynting Vector and Power Outward Power Flow From Stored Magnetic Electric Energies 30 Instantaneous Poynting Vector 31 C. Radiation (Assume sinusoidal time variation) 1. Total the gradient of a scalar Scalar Potential from Charges 32 2. Wave Equations From Get And (sometimes called ß) 33 Looks messy but note we still have not specified yet. If we choose Lorentz Gauge Condition (which is really just the equation of continuity in disguise!) we get 34 And for single source notation 35 Solutions to Wave Eqns. Are Found by Considering a Point Source (called a free-space Greens Function observation pt. source pt. is the wave eqn. for G (this you can show with lots of effort) 36 We then deduce convolution integral Potential integrals are solutions to wave equation!! 37 3. Elementary Source We will now solve a most simple antenna and use it as a building block (in our computer modeling) to solve complicated antennas. Short Electric Dipole Constant (uniform) I How short is short Why is it called a Dipole How can you achieve uniform I 38 dipole moment 39 Find from using sph. coordinates.
40 1/r 1/r2 1/r3 Far Out Close In Real Close In 41 3 Regions 42 Far Field In-phase terms Ratio of 1/r2 power variation Magnitude of plane wave Plane wave Spherical wave! 43 Radially outward and variation No loss of power just spreads out N-S E-W Strongest at Equator Zero at Poles Up close looks like a plane wave 44 Near and Fresnel Usually lumped together into near field 90o out of phase with Energy oscillates in out and is capacitive (Near fields tied to J P. Far fields tied to displacement current) 45 (No Transcript) 46 Near Antenna E H out of phase H in phase with I E in phase with Q on ends Far from Antenna Never returns E in phase with H P 47 Einduction Eradiation are 180o out of phase Hinduction Hradiation are 90o out of phase Why 48
D. Antenna Concepts
1. Antennas are field warpers which transform transmission line fields into spherical waves with prefered directions and vice versa. Antennas made of wires (linear antennas) have currents and charges on them and if you know or can guess I Q you can find the far out stuff via potential integrals (solutions to the wave equation). 2. 49 If you dont know I or Q you have lots of hard work to do because all you have is a source function and conducting surface where E is perpendicular to surface and Etangential is zero. You then have to write a boundary value equation to force Etan 0 at the conductors by writing Etan in terms of where K is a Greens function. The problem then is to find I which is unfortunately inside the . 3. 50 Once you have E H you can use S to get the total power radiated over a sphere and define 4. 51 (No Transcript) 52 Patterns are usually plotted as a cross section of the 3D surface in a plane. Often we choose the planes as perpendicular cuts through the main beam (maximum lobe). E Plane Contains axis of main beam and vector H Plane Contains axis of main beam and vector 53 Toroidal Shaped Could plot fields too 54 Gain and Directivity are dependent on . A maximum (constant) value is often specified 6. 55 In general for measurements Antenna (same r) 56 (No Transcript) 57 (No Transcript) 58 (No Transcript) 59 Since 60 Far field GT GR are often off-frequency What about near field coupling Tough! Got to use network concepts. 61 Note how super-simple idealized the short dipole is how complex the solution was. We could do the same set-up for the next simplest antenna a /2 dipole but would find we would have to integrate the assumed current to find . Anything more complex than that we would have to know the current and be able to integrate it. There is another possibility We could treat complex structures as a collection of short dipoles of different but uniform currents! 62 Looks simpler and it is but you must know or guess I( l). IF you cant do this you must develop an integral equation for I. 63
D. Integral Equation Formulation
The typical linear antenna is a cylindrical dipole L/2 2a Z0 Gap is fed by -L/2 field or Vo v. generator 64 (No Transcript) 65 Apply surface boundary condition With Greens Function Notation with 66 Reduce volume to surface by assuming perfect conductor applying axial symmetry it is OK to consider just one value of Pocklingtons Eqn. 67 Look at Kernel It has a singularity of third order at and the integration is over a surface thus the integral diverges and Pocklingtons Eqn. cannot be used for numerical analysis. This does not make the integral equation meaningless. If the THIN WIRE APPROXIMATION is applied 68 And observation points are taken along a line on the cylinder surface so that We get Pocklingtons Thin Wire Equation
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