Lecture 2 Review of Maxwell s Equations and the EM Constitutive Parameters

Transcription

1 Lecture 2 Review of Maxwell s Equations and the EM Constitutive Parameters Optional Reading: Steer Appendix D, or Pozar Section 1.2,1.6, or any text on Engineering Electromagnetics (e.g., Hayt/Buck)

2 Time-domain Maxwell s Equations: Faraday s Law Faraday s law in integral form V electromotive force (EMF) dφ = E = dt d E d l= d dt B s C S C Faraday s law in differential (point-wise) form ( E) ds= B ds B E = t t S C S C ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 2

3 The Time-domain Faraday Law: Hi-Sci Bonus Question Are the two expressions below equivalent? If not, can you state what the relationship is between the two? d? B ds= B ds dt t SC SC Hint: Consider both the transformer EMF and the motional EMF. See Hayt/Buck, 8 th ed., section 9.1, eq. (14). ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 3

4 Time-domain Maxwell s Equations: Ampère s Law Ampère s law in integral form D H d l = J + d = Ie + Iσ + I t s C S C excitation (source) current conduction current D displacement current Ampère s law in differential (point-wise) form J σ H = J + σ E + e J D t J D Maxwell s correction: displacement current ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 4

5 The Time-domain Ampère Law: Hi-Sci Bonus Question Show that Ampère s law of magnetostatics does not hold for timevarying EM fields (it is inconsistent with the conservation of charge).? ρv H = J J = t Show that Ampère s law with Maxwell s correction observes the conservation of charge for time-varying fields. ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 5

6 All Four Time-domain Maxwell s Equations Faraday s Law the curl MEs Ampère s Law Gauss Law of Electricity the div MEs Gauss Law of Magnetism E d l = B d t s C S C D H d l = J + d t s C S C D d s = ρvdv = Q S integral form B S V S d s = 0 free differential form B E = t D H = + σ E+ Je t D = ρ v B =0 J Gauss laws follow from the conservation of charge and the curl MEs ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 6

7 Time-domain Maxwell s Equations: Hi-Sci Bonus Questions Prove that Gauss law of electricity follows from Ampère s law and the conservation of electrical charge. Prove that Gauss law of magnetism follows from Faraday s law. Reminder: Conservation of electrical charge is a fundamental law of physics expressed as J = ρ v / t ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 7

8 Constitutive Relations Maxwell s equations are 4 but only 2 of them are independent (the two curl equations) there are 4 unknown vectors in the 2 curl Maxwell equations we need 2 more independent vector equations for a solution constitutive equations the constitutive EM equations are essential in describing the EM field interaction with matter in vacuum (SI): D= ε 0E, J = 0, B= µ 0H 7 µ 0 = 4π 10 H/m 12 ε F/m 1 8 ε0 = 2, c = m/s c µ in matter: D= F ( EH, ), J= F ( EH, ), B= F ( EH, ) 0 p c m ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 8

9 Constitutive Relations (2) we often assume that materials are isotropic, linear and dispersion-free: D= ε0e+ P= εε 0 re, εr = 1+ χe Jσ = σ E B= µ H+ M= µµ H, µ = 1+ χ 0 0 r r m the above assumption does not hold for many materials: water and living tissue plasma magnetized plasma ferromagnetic materials and ferrites piezoelectric crystals ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 9

10 Constitutive Relations: Hi-Sci Bonus Questions Define what is meant by a material being electrically or magnetically heterogeneous nonlinear anisotropic bi-anisotropic dispersive In each case, describe how the constitutive relations appear as functions of either space, or field strength, or field polarization, or frequency, etc. Is human tissue electrically heterogeneous? Give examples. Under normal conditions, is it magnetically homogenous or heterogeneous? ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 10

11 Time-harmonic EM Analysis: Field Phasors time-domain field vector { jωt E xyze } E( xyzt,,, ) = Re (,, ) { jωt H xyze } H( xyzt,,, ) = Re (,, ) E( xyzt,,, ) = xˆ E( xyz,, )cos[ ωt+ ϕ ( xyz,, )] + x yˆ E( xyz,, )cos[ ωt+ ϕ ( xyz,, )] + y zˆ E( xyz,, )cos[ ωt+ ϕ ( xyz,, )] z frequency-domain field vector (vector-field phasor) jϕ x ( xyz,, ) E = xˆ E( xyz,, ) e + E = xˆ E ( xyz,, ) + x jϕ ( xyz,, ) y yˆ E( xyz,, ) e + y zˆ E( xyz,, ) e z jϕ ( xyz,, ) z x z y ( xyz,, ) ye x y ( xyz,, ) + zˆ E ( xyz,, ) z ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 11

12 Maxwell Equations in Phasor Form phasor of a function s time derivative f f( xyzt,,, ) Fxyz (,, ) ( xyzt,,,) t f F spatial derivatives, ξ = xyz,, ξ ξ jωfxyz (,, ) time domain frequency domain (phasor) form B E = t E = jωb D H = + σ E+ J H = jωd + σe + J e e t usually given as jωd only D = ρ v D = ρ v B =0 B =0 J = ρ v / t J = jωρ ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 12 v

13 Maxwell Equations in Phasor Form: Example The following E-field exists in vacuum: E( z, t) = xˆe 0cos( ωt kz) + yˆe 0cos( ωt kz + π / 2) where E 0 = 5 V/m, ω is the radian frequency and k is the wave number. 1)Convert E(z,t) into a vector phasor E ( z). 2)Find the H-field phasor H ( z) using the frequency-domain ME. 3)Find k in terms of ω, ε 0 and μ 0 (k 0). 4)Convert the H-field phasor back into the time domain: H(z,t). 5)Prove that the above E-field satisfies Gauss law of electricity. ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 13

14 Maxwell Equations in Phasor Form: Example ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 14

15 Constitutive Relations in the Frequency Domain dielectric polarization and complex permittivity D = εe, ε = ε jε or ε = ε0( εr jεr ) ε r 0 ε = ε (1 j tan δ ), tan δ = ε / ε d E( xyzt,,, ) = eˆ Exyz (,, )cos( ωt) D( xyzt,,, ) = dˆ ε Exyz (,, )cos( ωt δ ) d d Im ε E E δ d jε E D ε 0 Re magnetization and complex permeability B = µ H, µ = µ jµ or µ = µ 0( µ r jµ r ) µ = µ (1 j tan δ ), tan δ = µ / µ m m µ r Im µ H H δ m jµ H B µ 0 ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 15 0 Re

16 Complex Permittivity: Hi-Sci Bonus Question Prove that the equation H = jωε E + σe + J where the permittivity may be a complex number such that ε = ε jε (εꞌ and εꞌꞌ being positive real) can also be written as H = jωε E + J e where σ σ ε = ε j = ε j ε + ω ω D one can replace ε" with an equivalent (or effective) conductivity σ eff, and, vice versa, σ can be replaced by an effective imaginary permittivity σ = ωε = ωε ε = ωε ε δ εeff = εε 0,eff = σ/ ω eff 0 r 0 r tan d ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 16 e r

17 Constitutive Relations in the Frequency Domain: Example 1 High-frequency voltage of magnitude V 0 = 10 V is applied to a parallel-plate capacitor, the plates of which are d = 1 mm apart. The dielectric has complex relative permittivity ε = 10 j10. 1) Find the E-field phasor inside the capacitor. 2) Find the D phasor inside the capacitor. 3) Find the phasor of the surface charge density ρ s at the plates. 4) Sketch the E and D phasor vectors in the complex plane. 5) What is the dielectric loss tangent tanδ d? 6) What is the equivalent (or effective) conductivity σ eff provided that the frequency is 3 GHz? r ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 17

18 Constitutive Relations in the Frequency Domain: Example 1 ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 18

19 Constitutive Relations in the Frequency Domain: Example 2 The following parameters of a conducting medium (cold plasma) are given: N number density of charged particles [m 3 ] q charge of a single charged particle [C] τ relaxation constant (aka average momentum relaxation time) [s] Find the complex relative permittivity εr = εr jεr of the medium at the frequency f = 1.3 GHz if: N = 3x10 17 m 3, q 1.602x10 19 C, m 9.109x10 31 kg, τ = 7x10 10 s. ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 19

20 Constitutive Relations in the Frequency Domain: Example 2 Heads up: In this example, we consider the case of a material (e.g. plasma) where the loss is due to conduction. In a Tutorial, we will show how to obtain the complex relative permittivity of a dielectric material where the loss is due to high-frequency dipole polarization (dipole relaxation). ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 20

21 Tech Brief: Constitutive Parameters of Human Tissues the permittivity of human tissues in the RF/μwave bands is dictated by the tissue water content water has very high permittivity and loss, the loss being much dependent on solutes (salts, proteins, nucleic acids, etc.) the higher the water content, the higher the permittivity and conductivity (e.g., blood has much higher permittivity than bone) strong frequency dependence is observed (frequency dispersion) since common RF/μwave sources emit in air, their radiation is strongly rejected by the human body ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 21

22 Tech Brief: Constitutive Parameters of Human Tissues 2 complex permittivity of water vs. frequency [Andryieuski et al., Scientific Reports, 2015] ε r ε r ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 22

23 Tech Brief: Constitutive Parameters of Human Tissues 3 measured complex permittivity of water in the low-ghz range [Burdette et al., In-situ tissue permittivity at microwave frequencies: perspective, techniques, results, 1986] ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 23

24 ε r Tech Brief: Constitutive Parameters of Human Tissues 3 permittivity and conductivity of blood [Gabriel et al., Phys. Med. Biol. 41 (1996)] frequency (Hz) σ [S/m] frequency (Hz) ElecEng 4FJ4 Nikolova 24

25 Tech Brief: Constitutive Parameters of Human Tissues 4 [Gabriel et al., Phys. Med. Biol. 41 (1996)] ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 25

26 Tech Brief: Constitutive Parameters of Human Tissues (FAT) [Gabriel et al., Phys. Med. Biol. 41 (1996)] ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 26

27 Summary the 4 Maxwell equations (2 curl and 2 divergence) form the basis of microwave and antenna engineering the 2 div equations (the Gauss laws) follow from the curl equations and the continuity of charge phasors are used in harmonic (single-frequency) field analysis the imaginary parts of the complex permittivity and permeability must be negative (or zero) indicating loss (no loss) (a positive imaginary part would indicate a gain material!) the dielectric properties of human tissues differ significantly from those of air/vacuum and are dictated by the amount of water high water content leads to high real permittivity and high loss ElecEng 4FJ4 Nikolova LECTURE 02: MAXWELL'S EQUATIONS AND CONSTITUTIVE PARAMETERS 27