Plane Wave: Introduction

Transcription

1 Plane Wave: Introduction According to Mawell s equations a timevarying electric field produces a time-varying magnetic field and conversely a time-varying magnetic field produces an electric field ( i.e. ) Called electromagnetic waves Wireless applications are possible because electromagnetic waves can propagate in free space without any guiding structures. 1

2 Spherical Wave and Plane Wave When energy is emitted by a source such as an antenna it epands outwardly from the source in the form of spherical waves. Far away from the source the spherical wave could be considered as a plane wave.

3 3 Time-armonic Fields In last chapter both electric and magnetic fields are epressed as functions of time and position. In real applications time signals can be epressed as sum of sinusoidal waveforms. So it is convenient to use the phasor notation to epress fields in the frequency domain. { } t e y t y ω ) ( Re ) ( { } t t e y t e y t t y ω ω ω ) ( Re ) ( Re ) (

4 4 Time-armonic Fields Therefore the differential form of Mawell s equations becomes these the differential form time-harmonic Mawell s equations B B D D D J D J B B v v t t ρ ρ ω ω Attention: A lot of equations DBJρ v are functions of yt DBJρ v are functions of y

5 5 Time-armonic Fields Using the constitutive relations and the equations becomes / B D J D J B ε ρ ρ ωε ω ωµ ω v v D B ε µ

6 Wave equations in Source-Free Media In a source-free media (i.e.charge density ρ v ) Mawell s equations become: ωµ ( σ ωε ) Q J σ The equation ( σ ωε ) can be written as ωε c where ε c ε ' ε '' ε σ / ω and it is called comple permittivity. 6

7 Wave equations in Source-Free Media We will derive a differential equation involving or alone. First take the curl of both sides of the 1st equation: ωµ ωµ ( ωε ) c using the vector identity A A A. y and called Laplacian ωµ ωε 7 c

8 Wave equations in Source-Free Media ωµ ( ωε ) ω µε c where ω µε c and is called propagation constant The equation is called the homogeneous wave equation for. c Similarly we can obtain (Try yourself) ωµ ωε 8 c

9 Plane wave in lossless media If the medium is nonconducting (σ) ε ε σ / ω c ε and then ω µε or ω µε or β In the tetbook the wavenumber k is used which is defined as k ω µε 9

10 1 Plane wave in lossless media In rectangular cocodinates can be separated into three equations. Suppose is a function of only and y we have y i y i i

11 11 Comparison Wave equation Transmission Line equations In rectangular co-ordinates If is a function of only and y I I V V i i y i

12 1 Solution This second order linear differential equation has two independent solutions so the solution of is o o o o e e or e e β β ) ( ) ( The 1st term is a wave travelling in the direction and the nd term is a wave travelling in the - direction.

13 Parameters Let u p be the phase velocity with which either the forward or backward wave is travelling and λ is the wavelength. Since β ω µε therefore u p ω β 1 µε λ π β In free space (vacuum) ε ε o µ µ o and the phase velocity is the same as the velocity of light in vacuum (3 1 8 m/s). 13

14 Magnetic field Similarly for the magnetic field can be found by solving ωµ under the conditions y We obtain 1 ωµ [ β β ] oe oe a y [ β β ] e e a y β o o ωµ 14

15 Magnetic field We notice that the magnetic field is in the y-direction i.e. perpendicular to the electric field and the ratio of magnitudes of electric and magnetic fields is: o o η ωµ β o o η η is called the intrinsic (or wave) impedance of the medium. In free space η has a value approimately equal to 377Ω. µ ε 15

16 Summary Plane wave is a particular solution of the Mawell s equation where both the electric field and magnetic field are perpendicular to the propagation direction of the wave. -and -field vectors for a plane wave propagating in -direction ample: M7.3 16

17 ample ample: A uniform plane wave with a propagates in a lossless medium (ε r 4 µ r 1 σ) in the -direction. If has a frequency of 1M and has a maimum value of 1-4 (V/m) at t and 1/8 (m) a) write the instantaneous epression for and b) determine the locations where is a positive maimum when t1-8 s. Solution: ω µ µ ε ε since velocity of light c o r o r ω c 1 µ ε o o µ r ε r π π 3 17

18 ample 4 8 4π (a) a 1 cos(π 1 t φ) 3 Setting 1-4 at t and 1/8 π φ k 6 The -field: a y y ay η η µ rµ o ε ε r o 6π (b) At t1-8 for to be maimum π1 m 8 ( ± 8 3 ) n 4π 3 m 1 8 ± nπ n