ECE 598 JS Lecture 07 Nonideal Conductors and Dielectrics

Transcription

1 ECE 598 JS Lecture 07 Nonideal Conductors and Dielectrics Spring 2012 Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois 1

2 μh E = t ε E H = J + t Material Medium J = σ E E = jωμ H H = J + jωε E σ: conductivity of material medium (Ω 1 m 1 ) since σ H = σ E + jωε E = E( σ + jωε ) = jωε 1+ E jωε ε ε 1+ σ jωε or then E ωμε = + σ E jωε 2

3 Wave in Material Medium 2 2 σ 2 E = ω με 1+ E = γ E jωε 2 2 σ γ = ω με 1+ jωε γ is complex propagation constant σ γ = jω με 1+ = α + jβ jωε α: associated with attenuation of wave β: associated with propagation of wave 3

4 Wave in Material Medium Solution: E = xe ˆ e = xe ˆ e e o γ z α z j β z o decaying exponential 2 με σ α = ω ωε 1/2 2 με σ β = ω ωε Magnetic field E H yˆ o α = e e η z jβ z 1/2 Complex intrinsic impedance η = jωμ σ + jωε 4

5 Skin Depth E = xe ˆ e = xe ˆ e e o γ z α z j β z o Wave decay The decay of electromagnetic wave propagating into a conductor is measured in terms of the skin depth Definition: skin depth δ is distance over which amplitude of wave drops by 1/e. δ = 1 α For good conductors: δ = 2 ωμσ 5

6 Skin Depth e -1 Wave motion I δ = z C V t For perfect conductor, δ = 0 and current only flows on the surface 6

7 DC Resistance R dc l σ wt = l: conductor length σ: conductivity 7

8 AC Resistance R ac l l l πμ f = = = σ wδ σw 2/ ωμσ w σ l: conductor length σ: conductivity f: frequency 8

9 Frequency-Dependent Resistance Jδ =1 α z J J e dz J 0 = = α δ Approximation is to assume that all the current is flowing uniformly within a skin depth 9

10 Frequency-Dependent Resistance Resistance changes with I z I = CL t R ac R dc Resistance is ~ constant when δ >t f 1 π f μ = w σ 1 = σ wt 10

11 Reference Plane Current R ac, ground l 6h πμ f σ 11

12 Skin Effect in Microstrip ε r H. A. Wheeler, "Formulas for the skin effect," Proc. IRE, vol. 30, pp ,

13 Skin Effect in Microstrip Current density varies as J = J e e o y/ δ jy/ δ Note that the phase of the current density varies as a function of y y/ δ jy/ δ Jw o δ o 1 j 0 I = J we e dy = + Jo σ Eo = Jo Eo = σ The voltage measured over a section of conductor of length D is: V = E D= o JD o σ 13

14 The skin effect impedance is where with V JD ( 1+ j) ( 1 ) o Zskin = = = + j π f I σ Jowδ w ρ = Skin Effect in Microstrip 1 σ D D R = skin X = skin f w π μσ μρ is the bulk resistivity of the conductor Zskin = Rskin + jxskin Skin effect has reactive (inductive) component 14

15 Internal Inductance The internal inductance can be calculated directly from the ac resistance L internal R ac = = ω R skin ω Skin effect resistance goes up with frequency Skin effect inductance goes down with frequency 15

16 Surface Roughness Copper surfaces are rough to facilitate adhesion to dielectric during PCB manufacturing When the tooth height is comparable to the skin depth, roughness effects cannot be ignored Surface roughness will increase ohmic losses 16

17 Hammerstad Model R H ( f ) KHRs f when δ < t = Rdc when δ t L H ( f ) L external ( f ) = RH ( fδ = t) + L external RH + when δ < t 2π f 2π f δ = t when δ t 17

18 Hammerstad Model K H 2 2 hrms = 1+ arctan 1.4 π δ h RMS : root mean square value of surface roughness height δ: skin depth f δ=t : frequency where the skin depth is equal to the thickness of the conductor 18

19 Hemispherical Model R L hemi hemi ( f ) ( f ) KhemiRs f when δ < t = Rdc when δ t L external ( f ) = Rhemi ( fδ = t ) + L external Rhemi + when δ < t 2π f 2π f δ = t when δ t 19

20 Hemispherical Model K hemi 1 when Ks 1 = Ks when Ks > 1 K s = 2 ( ) () () ( )( ) ( )( ) () 2 j 3 1 δ / r 1+ j α 1 = ( kr) δ /2r 1+ j ( + ) + ( o )( tile base ) ( μωδ/4) A Re η 3 π / 4k α 1 β 1 μ ωδ / 4 A A 2 () 2 j j/ k rδ 1/ 1 β 1 = ( kr) j/ k rδ 1/ 1 o tile ( )( ( j) ) ( )( ( j) ) 20

21 Huray Model R Huray ( f ) KHurayRs f when δ < t = Rdc when δ t L Huray ( f ) L external ( f ) = RHuray ( fδ = t ) + L external RHuray + when δ < t 2π f 2π f δ = t when δ t 21

22 Huray Model K P Huray = = P + P flat N _ spheres P ( μ ωδ /4) flat N 1 2 3π () () PN _ spheres = Re η Ho α 1 + β k n= 1 flat o tile ( )( ) ( )( ) () 2 j 3 1 δ / r 1+ j α 1 = ( kr) δ /2r 1+ j A η = μ / ε ε' ( )( ( j) ) ( )( ( j) ) 2 () 2 j j/ k rδ 1/ 1 β 1 = ( kr) j/ k rδ 1/ 1 o o n H o : magnitude of applied H field. 22

23 Dielectrics and Polarization No field Applied field Field causes the formation of dipoles polarization Bound surface charge density q sp on upper surface and +q sp on lower surface of the slab. 23

24 Dielectrics and Polarization D= ε E + P= ε E + ε χ E = ε + χ E = ε E ( 1 ) o a o a o e a o e a s a P : polarization vector D : electric flux density E : applied electric field a χ e ε o ε s : electric susceptibility : free space permittivity : static permittivity 24

25 Dielectric Constant Material Air Styrofoam Paraffin Teflon Plywood RT/duroid 5880 Polyethylene RT/duroid 5870 Glass-reinforced teflon (microfiber) Teflon quartz (woven) Glass-reinforced teflon (woven) Cross-linked polystyrene (unreinforced) Polyphenelene oxide (PPO) Glass-reinf orced polystyrene Amber Rubber Plexiglas ε r

26 Dielectric Constants Material Lucite Fused silica Nylon (solid) Quartz Bakelite Formica Lead glass Mica Beryllium oxide (BeO) Marble Flint glass Ferrite (FqO,) Silicon (Si) Gallium arsenide (GaAs) Ammonia (liquid) Glycerin Water ε r

27 AC Variations When a material is subjected to an applied electric field, the centroids of the positive and negative charges are displaced relative to each other forming a linear dipole. When the applied fields begin to alternate in polarity, the permittivities are affected and become functions of the frequency of the alternating fields. 27

28 AC Variations Reverses in polarity cause incremental changes in the static conductivity σ s heating of materials using microwaves (e.g. food cooking) When an electric field is applied, it is assumed that the positive charge remains stationary and the negative charge moves relative to the positive along a platform that exhibits a friction (damping) coefficient d. 28

29 Complex Permittivity N e ε = 1+ Q ε o ' r NQ e ε m o 2 ( 2 2 ω ) o ω 2 d m 2 ( 2 2) o ωo ω + ω ( 2 2) o : dipole density ω = : dipole charge : free space permittivity s : spring (tension) factor o ε " r s m = m ω o NQ e ε m 2 : mass d ω m 2 d ω ω + ω m d : damping coefficient : natural frequency ω: applied frequency 2 29

30 Complex Permittivity H = J i + J c + jωε E = J i + σ se + jω ( ε ' jε ") E H = J + σ + ωε " E+ jωε ' E = J + σ E+ jωε ' E ( ) i s i e σ = equivalent conductivity = σ + ωε" = σ + σ e s s a σ = a alternating field conductivity = ωε " σ s = static field conductivity σ e : total conductivity composed of the static portion σ s and the alternative part σ a caused by the rotation of the dipoles 30

31 Complex Permittivity J = J + J + J = J + σ E+ jωε ' E t i ce de i e J i J t : total electric current density : impressed (source) electric current density J ce J : effective electric conduction current density de : effective displacement electric current density σ e Jt = Ji + σee+ jωε' E = Ji + jωε' 1 j E = Ji + jωε' ( 1 jtanδe) E ωε ' σ e σs + σa σs σa tanδe = effective electric loss tangent = = = + ωε' ωε' ωε' ωε' 31

32 Complex Permittivity σ s ε " ε tanδe = + = tanδs + tanδa = ωε' ε' ε " e ' e tanδ a tanδ s = static electric loss tangent = σ s ωε ' σ a = alternating electric loss tangent = = ωε' " ε ' ε σ e Jcd = Jce + Jde = σee+ jωε' E = jωε' 1 j E = jωε' ( 1 jtanδe ) E ωε ' 32

33 Dielectric Properties Good Dielectrics: σ e 1 ωε ' σ e Jcd = jωε'1 j E jωε' E ωε ' Good Conductors: σ e 1 ωε ' σ e Jcd = jωε'1 j E σee ωε ' 33

34 Dielectric Properties Good Dielectrics: σ e 1 ωε ' σ e Jcd = jωε'1 j E jωε' E ωε ' Good Conductors: σ e 1 ωε ' σ e Jcd = jωε'1 j E σee ωε ' 34

35 Kramers-Kronig Relations There is a relation between the real and imaginary parts of the complex permittivity: 0 ( ') ( ω ') " ' 2 ' r ε r ( ω) = 1 + ω ε dω' 2 2 π ω ω 0 ( ') ( ω ) ' " 2ω 1 ε r ' ε r ( ω) = dω' 2 2 π ω ω Debye Equation ' ' " ' ε rs εr( ω) = εr( ω) jεr( ω) = εr ε ' r jωτ e 35

36 Kramers-Kronig Relations τ e is a relaxation time constant: τ e ε = τ ε ' rs ' r ' r ( ) ε ω ε ε ' ' r = r ε ' r 2 ( ωτ ) e ε " r ( ω) = ( ' ' ) ε ε ωτ r r e 1+ ( ωτ ) e 2 36

37 Dielectric Materials Material ε r tanδ Air Alcohol (ethyl) Aluminum oxide Bakelite Carbon dioxide Germanium Glass Ice Mica Nylon Paper Plexiglas Polystyrene Porcelain * x x x x10-4 2x x x x x

38 Dielectric Materials Material ε r tanδ Pyrex glass Quartz (fused) Rubber Silica (fused) Silicon Snow Sodium chloride Soil (dry) Styrofoam Teflon Titanium dioxide Water (distilled) Water (sea) Water (dehydrated) Wood (dry) x x x x x x x10-4 3x x x x