Lecture 3 Date: 4.4.16 Plane Wave in Free Space and Good Conductor Power and Poynting Vector
Wave Propagation in Loy Dielectric Wave propagating in z-direction and having only x-component i given by: E ( z) E e E e x z ( j ) z Where: 1 1 1 1 Inerting the time factor yield: j t ( ) ˆ z j t z ˆ x x x E( z, t) Re E ( z) e a Re E e e a The olution for magnetic field i: z j( t z) H ( z, t) ReH ˆ e e a y Where: H E η i a complex quantity known a the intrinic impedance of the medium. j e j j / 1 1/4 tan θ η 45
Wave Propagation in Loy Dielectric (contd.) In term of β, the wave velocity u and wavelength λ are: u Furthermore, the ratio of the magnitude of conduction current denity J c to that of the diplacement current denity J d i: J c E J d j E tan tanθ i known a the lo tangent and θ i the lo angle of the medium. A medium i good (lole or perfect) dielectric if tanθ i very mall (σ ωε) or a good conductor if tanθ i large (σ ωε)
Plane Wave in Lole Dielectric In a lole dielectric, σ ωε. In uch a cenario: σ, ε = ε ε r, μ = μ μ r. Therefore: u 1 Thu E and H are in time phae with each other.
Plane Wave in Free Space In thi cae: σ =, ε = ε, μ = μ. Therefore: c u 1 c c The fact that EM wave travel in free pace with the peed of light i ignificant. It provide evidence that light i the manifetation of an EM wave. We have: The plot of E and H are hown below. 1 377 Furthermore: E E t z a ˆ co( ) x E = E + co( βz) a x E H co( t z) aˆ y H = H + co( βz) a y
Plane Wave in Free Space (contd.) In general, if a E, a H and a k are unit vector along E, H and the direction of propagation, then: aˆ aˆ aˆ aˆ aˆ aˆ aˆ aˆ aˆ k E H k H E E H k Both E and H field are everywhere normal to the direction of wave propagation. It mean that the field lie in a plane that i tranvere or orthogonal to the direction of propagation. They form an EM wave that ha no electric or magnetic field component along the direction of propagation uch a wave i called tranvere electromagnetic (TEM) wave. A combination of E and H i called a uniform plane wave becaue field have ame magnitude throughout any tranvere plane. The direction in which the electric field point i the polarization of a TEM wave Eentially, polarization of a uniform plane wave decribe the locu traced by the tip of the E vector (in the plane orthogonal to the direction of propagation) at a given point in pace a a function of time.
Plane Wave in Free Space (contd.) E( z, t) E e co( t z) a x It i polarized in x-direction z ˆ E = E + co(ωt) a x It illutrate a uniform plane wave H = H + co(ωt) a y In practice, a uniform plane wave can t exit becaue it tretche to infinity and would repreent an infinite energy however thee wave are characteritically imple and fundamentally important. Thee erve a approximation for practical wave uch a thoe from radio antenna at ditance ufficiently far from radiating ource. The on-going dicuion are applicable for any other iotropic medium.
Plane Wave in Good Conductor In a good conductor, diplacement current i negligible in comparion to conduction current (J conduction J diplacement ) Becaue, for a perfect or good conductor, σ ωε. Although thi inequality i frequency dependent, mot good conductor (uch a copper and aluminum) have conductivitie on the order of 1 7 mho/m and negligible polarization uch that we never encounter the frequencie at which the diplacement current become comparable to the conduction current. For a good conductor: σ =, ε = ε, μ = μ μ r. Therefore: f u
Plane Wave in Good Conductor (contd.) j Furthermore: 45 If: E E e t z a x Thu E lead H by 45 z ˆ co( ) Then: E H e t z a z co( 45 ) ˆ y The amplitude of E or H i attenuated by the factor e αz a it travel along the medium. The rate of attenuation in a good conductor i characterized by ditance called kin depth δ a ditance over which plane wave i attenuated by a factor e 1 (about 37% of the original value) in a good conductor. E(x) E +
Plane Wave in Good Conductor (contd.) kin depth i a meaure of the depth to which an EM wave can penetrate the medium. E e E e 1 1 Valid for any material medium 1 1 f For a partially conducting medium, the kin depth can be coniderably large. 1 1 j j /4 For a good conductor: e For good conductor, α = β = 1, therefore: z/ δ z E E e t a ˆ co( ) x
Plane Wave in Good Conductor (contd.) z E E e t a z/ ˆ co( ) x It how that kin depth (δ) i the meaure of exponential damping the wave experience a it travel through the conductor. E + 66.1/ f (mm) For Copper!.368E + It demontrate that the field dampen and will hardly propagate through good conductor
Example 1 Uniform plane wave (f = 1MHz) at an air/copper interface. Determine α 1, α, β 1, β, u 1, u, λ 1, and λ. α 1 =, β 1 = ω c α = β = 1 δ
Example 1 (contd.) In the air, c 8 31 / m 6 1 1.9 rad / 8 c 31 m 8 c 31 1 3m 6 f 1 In the copper, 1 1.66 f f f 4 1 7 5.81 7 at 1 MHZ:.66mm 1 3 15.1 Np / m.415mm u f 415 m /
Example A plane wave E = E co(ωt βz) a x i incident on a good conductor at z. Find the current denity in the conductor. Since, J = σe, the wave equation change to: J J Furthermore, the incident E ha only an x-component that varie with z. Therefore, J = J x z, t a x and: d J x J x dz The olution i: z z J x Ae Be
Example (contd.) B i zero conidering that wave i propagating in +z direction. Furthermore, in a good conductor σ ωε o that α = β = 1. Therefore, δ z(1 j)/ Therefore: J Ae x j 1 j z(1 j)/ J x J x e 1 j J x () Where, J x () i the current denity on the conductor urface. J x Thi depict the cenario
Example 3 Indrapratha Intitute of Given the current denity of previou problem J x = J x ()e z(1+j)/δ, find the magnitude of total current through a trip of the conductor of infinite depth along z direction and width w along y direction. I w J dydz x w () z(1 j)/ I J x dye dz J x () δ I Jx() w 1 j I J x () w It actually reemble a uniform current denity J x () flowing through a thin urface width w and depth δ. A J x decay exponentially with depth z, a conductor of finite thickne d can be conidered electrically equivalent to one of infinite depth a long a d exceed a few kin depth (δ).
Example 4 In the previou example, what i the voltage acro a length l at the urface. What i the impedance of the conductor in conideration? J x () δ J () x V El l Z V J x() 1 j l I J () w x Z V 1 j l l Z Z I w w Z i urface impedance and the real part of thi i called ac reitance.
Plane Wave in Good Conductor (contd.) Electromagnetic Shielding The previou example how that we may encloe a volume with a thin layer of good conductor to act a an electromagnetic hield. Depending on the application, the electromagnetic hield may be neceary to prevent wave radiating out of the hielded volume or to prevent wave from penetrating into the hielded volume.
Plane Wave in Good Conductor (contd.) Given a plane wave incident on a highly-conducting urface, the electric field (and thu the current denity) i found to be concentrated at the urface of the conductor. The ame phenomenon occur for a current carrying conductor uch a a wire. The effect i frequency dependent, jut a it i in the incident plane wave example. Thi phenomenon i known a the kin effect. Therefore, one can ay, The proce whereby the field intenity in a conductor rapidly decreae i called kin effect. kin effect i the tendency of the charge to migrate from the bulk of the conducting material to the urface, reulting in higher reitance (for ac!) The field and aociated current are confined to a very thin layer (the kin) of the conductor urface.
Plane Wave in Good Conductor (contd.) For a wire of radiu a, it i a good approximation at high frequencie to aume that all of the current flow in the circular ring of thickne δ. kin effect i ued to advantage in many application. For example, becaue the kin depth in ilver i very mall, the difference in performance between a pure ilver and ilver-plated bra component i negligible, o ilver plating i often ued to reduce the material cot of waveguide component. Furthermore, hollow tubular conductor are ued intead of olid conductor in outdoor televiion antenna.
Plane Wave in Good Conductor (contd.) The kin depth i ueful in calculating the ac reitance. The reitance R = l σs The kin reitance R i the real part of η. i called the dc reitance R dc. R 1 f Reitance of a unit width and unit length of the conductor having croectional area 1 δ. Therefore, for a given width w and length l, the ac reitance i: For a conductor wire of radiu a: R R ac dc l l w a a l l S a R ac l Rl w w Since, δ a at high frequencie, R ac i far greater than R dc. In general, the ratio of the ac and dc reitance tart at 1. for dc and very low frequencie and increae a the frequency increae.
Power and Poynting Vector For any wave with an electric field E and magnetic field H, the direction of wave propagation i alo the direction of power per unit area (or power denity) carried by the wave. It i repreented by Poynting Vector S. S E H W/m Intantaneou Poynting Vector direction and denity of power flow at a point S a n The total power flowing through thi aperture i: a k P = A S. a n da = SAcoθ
Power and Poynting Vector (contd.) Except for the fact that unit of S are per unit area, the Poynting Vector i the vector analogue of the calar expreion for the intantaneou power P(z, t) flowing through a tranmiion line: P( z, t) v( z, t) i( z, t) 1 * Pav( z) Re V ( z) I ( z) From LC we can recall In a imilar manner, power denity (W/m ) aociated with a time-harmonic EM field in term of E and H phaor i: 1 P Re E H * ave
Example 5 Determine the expreion for the time-average power denity for an EM plane wave in term of electric field only and magnetic field only; given (a) a loy medium, (b) a lole medium. (a) 1 P Re E H * ave P 1 ReE H aˆ * ave k P aˆ Re E H k * ave E H j e H * E E e * * * j P ave * aˆ k E Re E j e P ave * k EE E Re co j e aˆ aˆ k
Example 5 (contd.) aˆ P Re H e H k j * ave P ave * k EE H Re co j e aˆ aˆ k (b) Lole Medium η real, θ η = P ave E aˆ k P ave H aˆ k