ECE 6340 Intermediate EM Waves. Fall Prof. David R. Jackson Dept. of ECE. Notes 4

Transcription

1 ECE 6340 Intemediate EM Waves Fall 016 Pof. David R. Jackson Dept. of ECE Notes 4 1

2 Debye Model This model explains molecula effects. y We conside an electic field applied in the x diection. Molecule: q -q x Molecule at est p x = 0 The dipole moment p x of this single molecule epesents the aveage dipole moment in the x diection fo all of the dipoles in a little volume. A zeo dipole moment p x coesponds to a andom dipole alignment in the actual mateial. E x θ Molecule with applied field E x p x > 0

3 Debye Model (cont.) Toque on dipole due to electic field: E ( E) ( E) + T = q + q ( + ) = qe = p E y θ x E x 3

4 Debye Model (cont.) T = d θ Note: T = -T z y T E I dt T = T + T + T E S F π = p E = zˆ pexsin θ = qd E cosθ x E x θ x T E = qd E x cosθ T T S F = sθ dθ = c dt s = sping constant c = fiction constant 4

5 Debye Model (cont.) Hence dθ qdex cosθ sθ c = I dt d θ dt Assume θ << 1, cosθ 1 (small fields) dθ qdex sθ + c + I dt d θ dt 5

6 Debye Model (cont.) dθ qdex sθ + c + I dt d θ dt θ Note that px = qdsinθ qd ( ) θ E x p Hence: θ x qd Inset this into the top equation. 6

7 Debye Model (cont.) Then we have: p d p d p qdex s + c + I qd qd dt qd dt x 1 x 1 x o d px d px spx + c + I = qd dt dt ( ) E x Assume sinusoidal steady state: sp + jωcp ω I p = q d E ( ) x x x x 7

8 Debye Model (cont.) Hence, we have p x = E x qd ( ) ( ω ) s I + jωc Denote N m = # molecules 3 m The tem P denotes the total dipole moment pe unit volume. Then we have P = N p M x m x The M supescipt eminds us that we ae talking about molecules. 8

9 Debye Model (cont.) Also, fo a linea mateial, P = χ E M M x 0 e x Hence χ P N p M M x m x e = = 0Ex 0Ex χ M m Theefoe ( ) e N 1 = qd s ω I + jωc ( ) 0 Assume ω I << s (The fequency is faily low elative to molecula esonance fequencies. That is, the fequency is at millimete wave fequency and below.) 9

10 Debye Model (cont.) χ M e N m ( ) 1 1 qd c 0 s 1+ jω s Denote the time constant as: Denote the zeo-fequency value as: τ = c s χ M e N 1 = 0 s m ( 0) ( qd) Then we have χ M e χe 1 + M ( 0) jωτ (eal constant) 10

11 Debye Model (cont.) χ M e χe 1 + M ( 0) jωτ This would imply that ( 0) M χe = jωτ At high fequency the molecules cannot espond to the field, so the elative pemittivity due to the molecules tends to unity. This equation gives the wong esult at high fequency, whee atomic effects become impotant. 11

12 Debye Model (cont.) Include BOTH molecule and atomic effects: P = P + P M A x x x = χ E + χ E M A 0 e x 0 e x = χe 0 e x Molecule effects: Atomic effects: A χ e χ M e χe = 1 + M ( 0) jωτ = constant (eal) Note: Atoms can espond much faste to the field than molecules, so the atomic susceptibility is almost constant (unless the fequency is vey high, e.g., at THz fequencies and above). 1

13 Debye Model (cont.) We then have that χ = χ + χ M A e e e Hence: χ e M χ ( 0) e = + χe 1+ jωτ A 13

14 Debye Model (cont.) Pemittivity fomula: 1 = + χ e Hence, we have A χe = 1+ χe a = M a jωτ 0 ( ) jωτ whee a a 1 = 1+ χ = χ M e A e ( 0) (a 1 and a ae eal constants.) 14

15 Debye Model (cont.) Note that: 0 ( ) = ( ) = a + a a 1 1 = a a jωτ so a a 1 = ( ) 0 = ( ) ( ) Hence: = + ( ) 0 ( ) ( ) 1+ jωτ 15

16 Debye Model (cont.) = ( ) + 1 ( 0) ( ) + ( ωτ ) ( 0) ( ) + ( ωτ ) = ωτ 1 ( 0) ( ) 1/τ ω 16

17 Debye Model (cont.) Fequency fo maximum loss: Let x = ωτ x = 0 1+ x ( ) ( ) ( ) A maximum occus at x = 1 o ω = 1 τ 17

18 Debye Model (cont.) Complex elative pemittivity fo pue (distilled) wate Wate obeys the Debye model quite well. Wate obeys the Debye model quite well. 18

19 Example Wate obeys the Debye model quite well. Calculate the complex elative pemittivity c fo ocean wate at 10.0 GHz. Ocean wate: σ = 4 [S/m] 1 c j σ = = = j σ ω ω c c = ( 60 j35) j π ( )( ) fom pevious plot fo distilled wate Hence ( 60 35) ( 7.19) c = j j o = 60 j4.19 c 19

20 Cole-Cole Model This is a modification of the Debye model. 0 ( ) ( ) ( ) = + 1 ( j ) 1 α + ωτ When α = 0, the model educes to the Debye model. This model has often been used to descibe the pemittivity of some polymes, as well as biological tissues. 0

21 Cole-Cole Model (Cont.) Paametes fo Some Biological Tissues 1

22 Haviliak Negami Model This is anothe modification of the Debye model. = + ( ) 0 ( ) ( ) ( 1 ( ) α + jωτ ) β When α = 1 and β = 1, the model educes to the Debye model. This has been used to descibe the pemittivity of some polymes.

23 Loentz Model Explains atom and electon esonance effects (usually obseved at high fequencies, such as THz fequencies and optical fequencies, espectively). Electons Atom: E x = 0 q e q n Dipole effect: q n = - q e E x 3

24 Loentz Model (cont.) Model: q e m q n E x x The heavy positive nucleus is fixed. Equation of motion fo electons in atom: F x = d x m dt F = F + F + F E S F x x x x = dx qeex sx c dt 4

25 Loentz Model (cont.) so dx qeex sx c = m dt d x dt Fo a single atom, p = q x = qx o x= p / ( ) x n e x q e Hence d p x qe Ex = s px + c + m dt d dt p x Sinusoidal Steady State: q E = sp + jωcp ω mp e x x x x 5

26 Loentz Model (cont.) p x = E x q e s m + jωc ( ω ) Denote P = N p A x a x = χ 0 Theefoe, N a A e # atoms = 3 m E x χ N p = A a x e 0 Ex The tem P denotes the total dipole moment pe unit volume. The A supescipt eminds us that we ae talking about atoms. 6

27 Loentz Model (cont.) Hence χ N q ( s ω m) + jωc A a e e = 0 Na q e 1 = s 0m c ω + jω m m Denote: A N q s c m m m a e = ω0 = cf = 0 (eal constants) 7

28 Loentz Model (cont.) We then have χ A e = A ( ) 0 ω ω + jωc f A Pemittivity fomula: 1 = + χ A e Hence A = 1+ A ( ) 0 ω ω + jωc f Real and imaginay pats: A ( ω0 ω ) A A ω cf A ( ω0 ω ) + ( ωcf ) ( ω0 ω ) + ( ωcf ) = 1+ = 8

29 Loentz Model (cont.) A 1 + A / ω 0 A A 1 ω 0 ω Low fequency value fo A χ e (This is still "high" fequency in the Debye model.) ω = ω A shap esonance occus at 0 9

30 Low fequency: c ( 0) j σ ω Total Response Response of a hypothetical mateial MW THz Vis UV Fequency (Hz) Fom the dielectic spectoscopy page of the eseach goup of D. Kenneth A. Mauitz. 30

31 Atmospheic Attenuation 60 GHz 90 GHz 31

32 Atmospheic Attenuation (cont.) Atmospheic absoption (% powe absobed) fo millimete-wave fequencies ove a 1-km path 90 GHz % powe absobed: P = ( e α ) % (1000) abs Attenuation in db/km: db / km = 10log = 10log P P out in = α (1000) ( e ) P % abs 10log Teabeam Document Numbe: Revision: A / Release Date: Copyight 00 Teabeam Copoation. All Rights Reseved. 3

33 Plasma Electically neutal plasma medium (positive ions and electons): Electon (-) Ion (+) We assume that only the electons ae fee to move when an electic field is applied. This causes a cuent to flow. 33

34 Plasma (cont.) Equation of motion fo aveage electon: F dv = m = e E mv dt υ ( ) ( ) Thee is no sping foce now. Foce due to electic field Foce due to collisions with ions (loss of momentum) υ = collision fequency (ate of collisions pe second of aveage electon) Notes: (1) The last tem assumes pefect inelastic collisions (loss mechanism). () We neglect the foce due to the magnetic field. 34

35 Plasma (cont.) Sinusoidal steady state: m jωv = e E mvυ ( ) ( ) ( e) ( + ) v = E m jω υ Cuent: J ρvev E m j ve ( e) ( ω+ υ) ρ = = 35

36 Plasma (cont.) Ampees law: H = J + jω E c 0 ve ( e) ( ω+ υ) ρ = E + jω 0E m j ρ = jω 0 + m j = jω E J vev E m j ve ( e) ( ω+ υ) E ve ( e) ( ω+ υ) ρ = ρ = We assume that thee is no polaization cuent only conduction cuent. Hence we use 0. 36

37 Plasma (cont.) Hence: jω c ve ( e) ( ω+ υ) ρ = jω 0 + m j so c ( e) ( + ) 1 ρ ve = 0 + jω m jω υ o c = 0 ρ ve ( e) m 1 ω ω ( jυ) 37

38 Plasma (cont.) c = 0 ρ ve ( e) m 1 ω ω ( jυ) Define: ω 0 p ρ ve ( e) m (ω p plasma fequency) We then have c 1 ω p = 0 ω ω ( jυ) 38

39 Plasma (cont.) Lossless plasma: υ = 0 c 1 ω p = 0 ω (Dude equation) Plane wave in lossless plasma: ω> ω : = > 0, k = ω µ = β ( popagation) p c c 0 c ω< ω : = < 0, k = ω µ = jα ( attenuation) p c c 0 c 39

40 Plasma (cont.) The Dude model is an appoximate model fo how metals behave at optical fequencies. Plasmonic behavio Relative pemittivity f = p [Hz] Visible Measued complex elative pemittivity of silve at optical fequencies 40

41 Plasma (cont.) At micowave fequencies, a plasma-like medium can be simulated by using a wie medium. a ω p = c d 1 π ln d π a d d y x c = 1 µ

42 Plasma (cont.) An example of a diective antenna using a wie medium: ω > ω, ω ω 0< << 1 p p x z θ y h hs << 1 y x Geometical optics (GO): Refaction towads boadside 4