Antennas & Propagation

Transcription

1 Antennas & Popagation 1 Oveview of Lectue II -Wave Equation -Example -Antenna Radiation -Retaded potential

2 THE KEY TO ANY OPERATING ANTENNA ot H = J +... Suppose: 1. Thee does exist an electic medium, which povides a cuent I and thus a cuent density J.. This causes location vaying magnetic field H 3. This causes location vaying magnetic flux B, but no time vaying magnetic flux. Thus no ot E, thus no time vaying electic flux. Thus no wave! 3 ot E = B ot H = J D + Suppose: 1. Thee is a time vaying cuent density J.. This causes location and time vaying magnetic field H 3. This causes location and time vaying magnetic flux B. 4. This causes location and time vaying electic field E. 5. This causes location and time vaying electic flux D. 6. This causes location and time vaying magnetic field H, even if without cuent density J. 4

3 The Tansmitting Antenna ot E = µµ 0 H E ot H = J + εε 0 Thus, we only need a medium, which is capable of caying a time-vaiant cuent. We will call this medium: Antenna. Outside the Antenna the electomagnetic field can popagate on its own without the souce J, since both fields ae coupled though the fomulas! 5 The Receiving Antenna ot E = µµ 0 H E ot H = J + εε 0 We only need a medium, which has fee electons to geneate a cuent out of a timevaying electomagnetic field. 6

4 Antenna Radiation An Antenna is an efficient way of conveting a guided wave into a adiating wave o vice vesa. wave guide, mico stip, tansmission line fee space taveling wave 7 Antenna Radiation Tansmission Line Cuent Distibution V Mutual Cancellation (Half-wave) Dipole Radiation V 8

5 Antenna Radiation 9 Radiation By Cuents and Chages An obseve at some distance fom the vaying chage distibution would sense tempoally vaying electic and magnetic fields. These fields ae known as adiation fields and induction fields. In ode to elate these adiation fields to thei souces, we will conside the case whee ρ and J ae not zeo, esulting in an inhomogeneous wave equation (Helmholtz Equations). 10

6 Stating fom Maxwell s equations div D = ρ div B = 0 (1) () D = ε ε E 0 B = µ µ H 0 (5) B ot E = (3) (4) ot H = J + D 11 The magnetic flux density can be elated to a vecto magnetic potential by B = A Substituting it into the diffeential fom of Faaday s law A ( E + ) = 0 t The tem between paentheses can then be expessed as the gadient of a scala potential V E = V A 1

7 Taking the cul of A Using the vecto identity of and substituting in the diffeential fom of Ampee s law, we have We have µ H A = ( A) A V µε = µ J + ( A + µε ) A E A A = µ J + µε ( V ) A Since we ae fee to choose the divegence of, A, we let V A = µε 13 Hence, the nonhomogeneous vecto wave equation fo vecto potential ( the nonhomogeneous vecto Helmholtz equation ) is : A µε A = µ J The coesponding nonhomogeneous scala wave equation fo scala potential V is given by V V µε ρv = ε 14

8 The solutions fo the two equation ae espectively V A (, t ) (, t ) = = 1 4πε µ 4π V ' V ' J ρ ( ', t ' u) ' ( ', t ' u) ' dv' dv' The vecto magnetic potential and the electic scala potential at distance fom the souce depend espectively on the value of the chage density and the electic cuent density at ealie time ( t - - /u). 15 Retaded Potential The vecto electic potential expession epesents the supeposition of potentials due to vaious cuent elements (I dl), at distant point P ( at a distance of ). The effect eaching a distant point P fom a given element at an instant t, due to a cuent value which is followed at an ealie time. This time, of couse, depends on the distance taveled fom dl to P. Hence, etadation time must be taken into account. 16

9 Let the instantaneous cuent in a shot wie be a sinusoidal function of time I = Im cos( ωt) Taking into account the etadation effect, the instantaneous cuent becomes I = I m cosω(t c ) 17 With efeence to the Figue The geneal expession fo magnetic vecto potential is given by µ J(t / c) A(, t) = 4π V' dv' Using I = Jds and dv' = ds dl The equations fo the etaded vecto magnetic potential can be witten as A z µ I lcos( ωt β) = m 4π 18

10 Pocedue to solve adiation poblems 1. Repesent signal to be tansmitted though cuent density J.. Resolve J into its hamonics. 3. Find the hamonic magnetic vecto potential A. 4. To find the magnetic field H, solve : H = A / µ 5. To find the electic field E, solve : E = H / jωε 19