UNIVERSITÀ DI PISA. Plane waves 07/10/2011 1

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1 UNIVERSITÀ DI PISA Electromagnetic Radiations and Biological i l Interactions Laurea Magistrale in Biomedical Engineering First semester (6 credits), academic ear 11/1 Prof. Paolo Nepa p.nepa@iet.unipi.it Plane waves Edited b Dr. Anda Guraliuc 7/1/11 1

2 Lecture Content Plane waves Plane Waves in Time Domain Polariation Plane Waves in Frequenc Domain (in dispersive i and loss media) 7/1/11

3 Introduction One of the most important consequences of Mawell s equations is the eistence of electric and magnetic field perturbations that travels with a finite velocit (in a material or even in free space). A particular simple solution of Mawell s equations is the plane wave solution; it allows introducing the fundamental parameters of electromagnetic wave propagation: a. propagation constant, phase and attenuation constants b. wavelength c. phase velocit d. medium characteristic impedance e. polariation Plane waves are particularl important in applications where the electromagnetic field distribution, sufficientl far awa from the source, can be effectivel approimated b a local plane wave. 7/1/11 3

4 Plane Waves (time domain) Assumptions: free space medium (linear, isotropic, homogeneous and non dispersive) with no charges (ρ=) and no currents (j=). A non vanishing solution can be obtained due to the presence of sources located outside the volume where Mawell s equations are going to be solved. 9 = 1/ 36π 1 F / m = F / m µ π 7 = 4 1 H / m drt (,) = ert (,) & brt (,) = µ hrt (,) ert (, ) = µ hrt (, ) hrt (,) = ert (,) [ ert (, )] = [ µ hrt (, )] = A further assumption: electric and magnetic fields are independent of and coordinates (looking for a solution onl dependent on : PLANE WAVE SOLUTION) i i i et (,)& ht (,) e e e= = i + i e e e A A A A A A A = i + i + i 7/1/11 4

5 Plane Waves (time domain) et (, ) = µ ht (, ) ht (,) = et (,) e e i i µ h i h i hi ( ) + = + + h h ( ) i + i = e i + e i + e i TWO vector equations e h = µ h e = & e h = µ h e = & e = h = SIX scalar equations e = const & h = const (static components) we will assume e = & h = (since looking for a dnamic solution) 7/1/11 5

6 Wave equation (D Alambert s equation) e h = µ h e = e h = µ h e = t e e = µ e 1 e = c c = 1 µ e h = µ h e = e h = µ h e = h h = µ h 1 h = c e 1 e = c h 1 h = c D Alambert s equation 7/1/11 6

7 Wave equation (D Alambert s equation) e 1 e = c h 1 h = c ( ) ( ) ( ) ( ) e (,) t = f ct + f 1 + ct h (, t) = f ct + f ct Forward wave (>) (>) Backward wave (<) If considering onl the forward wave which is function of u=-ct h e h u e u = = u u u u = 1& = c h u e = c u h e = u µ h u e u const wrt u ( ) ( ) =.... µ Due to the finite propagation velocit, the electromagnetic field will be ero at t=t in an interval [ 1, ] far form the source (if the latter is ecited at t=t ); then when u is in the interval [u 1,u ]= [ 1 -ct, -ct ] field components are ero and above constant must be ero for an value of u. = = µ h e const µ µ e ( ct) = h ( ct) = ζ h ( ct) ζ = = 1πΩ 377Ω Free space characteristic impedance 7/1/11 7

8 Plane Waves properties (forward wave) e h = µ h e = & > (forward wave) e ct µ h ct h ct ( ) = ( ) = ζ ( ) e h = µ h e = & > (forward wave) e ct µ h ct ζ h ct ( ) = ( ) = ( ) ζ h e ζ h = e e e ( ct) e ( ct) = = ζ h ( ct) h ( ct) ζ h = e e The instantaneous electric field can var arbitraril in time while magnetic field vector (amplitude and direction) will satisf above conditions at an time and for an observation point 7/1/11 8

9 Plane Waves properties (backward wave) e h = µ h e = & < (backward wave) e + ct = µ h + ct = ζ h + ct ( ) ( ) ( ) e h = µ h e = & < (backward wave) e + ct = µ h + ct = ζ h + ct ( ) ( ) ( ) e e ζ h = e e e e ( + ct) e ( + ct) = = ζ h ( + ct) h ( + ct) ζ h = e ζ h The instantaneous electric field can var arbitraril in time while magnetic field vector (amplitude and direction) will satisf above conditions at an time and for an observation point 7/1/11 9

10 Plane Waves properties Electric field and magnetic field are constant on an plane perpendicular p to the plane wave propagation direction. The (finite) perturbation propagation velocit is c = 1/ µ The electric and magnetic fields are related b the medium characteristic impedance 1. e h, (, t) For each plane wave e (forward or backward) = ζ, ( t, ) h ( e h= eh + eh = ζ hh ζ hh = ). ( ) e = e + e = ζ e + ζ e = ζ h ζ 3. 1 > ( forward wave ) h = i e e = ζ h i ζ 1 < ( backward wave) h = ( i ) e e = ζ h ( i ) ζ The phsical properties of a plane wave are independent of the coordinate sstem and propagation direction. For an propagation direction denoted b the versor i: ( rt, ) : er ( i ct )& h ( r i ct), ei = 1 h= i e ζ or equivalentl hi = e= ζ h i 7/1/11 1

11 Plane Waves eample: sinusoidal signals e λ c ζ h π e bcos ct bcos π π π = = t = bcos t e = bcos ( ct) λ ( ) ( β ) λ λ T T = λ / c time period e = ; e = = π / T = π f ππ 1 β = phase constant c = λ µ e bcos( t β ) = e = bcos t For magnetic field: h = ; h = ζ e ; h = = ( ) λ π 1 = = e = bcos t π/ = bsin t 4 β 4 ( ) ( ) π c λ = = Electromagnetic wavelength = spatial period β f 7/1/11 11

12 Linear Polariation Polariation=determines the orientation of the electric field in a fied spatial plane orthogonal to the direction of the propagation. Assume = e = acos( t β) = acos( t) e = bcos( t β + δ ) = b cos( t + δ ) e= e i + e i = acos( t) i + bcos( t+ δ) i δ = Linear Polariation If e and e are in phase e is linearl polaried along a direction given b the angle: e α = tg = tg e 1 1 b a b et= ( ) e () t e () t et () a T et= ( ) 7/1/11 1

13 Circular Polariation a= b& δ =± π / Circular Polariation π e= acos( t) i + acos( t± ) i = acos( t) i ± asin( t) i e = a t e -a a e () t α et () e () t et= ( ) 1 1 sint α = tg = tg m = m t e cost δ = π / δ = π / (LHCP) (RHCP) Light Hand Circular Polariation Right Hand Circular Polariation 7/1/11 13

14 Consider e = acos( t) e = bcos( t+ δ ) Elliptical Polariation a b& δ ± π / Elliptical Polariation e e + = cos + cos + a b ( t ) t δ cost e e e e + cosδ sin δ a = b a b e = a e = bcosδ e = a e = bcosδ If: e = b e = acosδ e = b e = acosδ e < a& e < b bcosδ a cos δ b a cosδ bcosδ ab e If the sstem is rotated b a θ angle: tgθ = cosδ e ' + a b a' = b' ' ' e b cos( t + δ ) α = θ ( e, ); tgα = = e acos( t) dα absinδ Angular velocit: () t = = dt e () t a ' 1 Ellipse equation 7/1/11 14

15 Plane Wave Polariation a<b δ = π /4 δ = π / δ = 3 π /4 δ = b b b b a a a a α Elliptical polariation (LHEP) Elliptical polariation (LHEP) Elliptical polariation (LHEP) Linear polariation (ellipse becomes a line) δ = π /4 δ = π / δ = 3 π /4 δ = π b b b b a a a a α Elliptical polariation (RHEP) Elliptical polariation (RHEP) Elliptical polariation (RHEP) Linear polariation 7/1/11 15 (ellipse becomes a line)

16 Plane Wave Polariation Linear Circular Elliptical Polariation Animation 7/1/11 16

17 Plane Waves Frequenc Domain solution Frequenc domain analsis allows to stud EM propagation in dissipative and dispersive medium Medium: linear, homogeneous, isotropic, Dr (, ) = ( ) Er (, ) dispersive in time and with Br (, ) = µ ( ) Hr (, ) losses In time domain it corresponds to a temporal convolution ( ) = ( ) j ( ) µ ( ) = µ ( ) j µ ( ) Dependence on accounts for time dispersion, while the imaginar part is related to losses Frequenc domain Mawell s equations Er (, ) = jbr (, ) H (, r ) = jd(, r ) + J(, r ) ( Dr (, )) = ρ( r, ) ( Br (, )) = Er (, ) = j µ ( ) Hr (, ) H (, r ) = j ( ) E(, r ) + J(, r ) ( ( ) Er (, )) = ρ( r, ) ( µ ( ) H ( r, )) = ( ) Er (, ) = Er (, ) = Medium homogeneit µ ( ) H( r, ) = H(, r ) = 7/1/11 17

18 Plane Waves Frequenc Domain solution Assumption: Er (, ) = jµ ( ) Hr (, ) J(, r ) = & ρ(, r ) = Hr (, ) = j ( ) Er (, ) Er (, ) = H (, r ) = ( ( ) ( )) A = A A 1 Er (, ) = Hr (, ) jµ ( ) 1 ( Er (, ) ) = j ( ) Er (, ) jµ ( ) ( ) Er (, ) + Er (, ) = ( ) µ ( ) Er (, ) = Er (, ) = Er + µ Er = (, ) ( ) ( ) (, ) 7/1/11 18

19 Plane Waves Frequenc Domain solution Er (, ) = jµ ( ) Hr (, ) Hr (, ) = j ( ) Er (, ) Er (, ) = Hr (, ) = ( ( ) ( )) A = A A 1 H ( r, ) = E ( r, ) j ( ) 1 ( H (, r ) ) = jµ ( ) H(, r ) j ( ) ( ) H ( r, ) + H( r, ) = ( ) µ ( ) H( r, ) = H ( r, ) = Hr + µ Hr = (, ) ( ) ( ) (, ) Er (, ) + ker (, ) = Homogeneous Vector k = µ ( ) ( ) Helmholt s Equations H(, r ) + k H(, r ) = Propagation constant 7/1/11 19

20 Helmholt s equation solution Er + ker = (, ) (, ) In a rectangular coordinate sstem: ( ( ) ( )) A = A A Er (, ) = E( r, ) i+ E( r, ) i + E( r, ) i Consider onl the component along : E r + k E r = φ φ φ (, ) (, ) φ = + + E (, r ) E (, r ) E (, r ) ke(, r) = Assumption : Er (, ) = E (, ) (looking for a solution onl dependent on : PLANE WAVE SOLUTION) E (, ) + (, ) = ke In a similar wa, it can be shown that: + jk E (, ) = E e + E e + jk E (, ) = E e + E e jk jk + jk H (, ) = H e + H e + jk H (, ) = H e + H e 7/1/11 jk jk

21 E (, ) = j µ ( ) H (, ) H(, ) = j ( ) E(, ) E (, ) = H (, ) = Helmholt s equation solution i i i E (, ) = E E E = i+ i = jµ ( ) H (, ) E E E i i i H (, ) = H H = i + i = j ( ) E(, ) H H H E = H = Field components along the propagation p direction must vanish 7/1/11 1

22 Helmholt s equation solution E (, ) = j µ H (, ) H (, ) = je (, ) E E = ζ H + + = ζ H + jk jk E (, ) = E e + E e + E jk E H (, ) = e e ζ ζ jk E (, ) = H (, ) = jµ H (, ) je (, ) E E = ζ H + + = ζ H + jk E (, ) = E e + E e + E jk E H (, ) = e e ζ ζ jk jk ζ = µ / = R+ jx Medium characteristic impedance k = µ ( ) ( ) = β jα β Phase constant α If α = (lossless medium, µ and real): Attenuation constant k = µ = β e = e = e e jk j ( β j α ) j β α 7/1/11

23 Plane Waves phase velocit + Back to jφ { } { } time domain: + e (, t) Re E e jk e jt + Re E e α e jt e jβ + e E e α + = = = cos( t β+ φ ) φ(,) t = t β+ φ + (phase of e (,) t ) Consider φ at (,t) and (+Δ,t+Δt): φ = [ ( t + t ) β ( + ) ] ( t β ) = t β φ = = t β e (,) t v f = = tt β λ t t + t Phase velocit v f = = β Re{ k} = v t f π v f λ = = Electromagnetic wavelength = spatial period β f 7/1/11 3

24 Plane Waves phase velocit + Back to jφ { } { } time domain: jk j t j t j e (, t) = Re E + e e = Re E + e α e e β e = E + e α cos( t β + φ + ) e (,) t = T φ(,) t = t β+ φ + (phase of e (,) t ) = e = b cos( t) 1 = λ π e bcos t / bsin t 4 = β 4 = = ( π ) ( ) = + λ /4 t Free space (, µ ) 1 8 v = = c 31 m/sec f µ If α = (lossless medium, µ and are real): v f 1 = µ ( ) ( ) µ ( ) ( ) 7/1/11 4

25 Plane Waves properties + jk jk + jk E (, ) = E e + E e E (, ) = E e + E e E + ζ H + = E + = ζ H + E = ζ H E = ζ H (, ) / jk jk H E + ζe E jk = / ζe H (, ) = E + / ζe + E / ζe jk jk k jk E = H = ζ = µ / = R + jx E H = E = ζ H k = µ ( ) ( ) = β jα β Phase constant α 1 > ( forward wave) H = i E E = ζ H i ζ 1 < ( backward wave ) H = ( i ) E E = ζ H ( i ) ζ β µ Dielectric medium r with µ and real: π c λ λ = = = β f jk Attenuation constant = f ( ) ζ = µ / = ζ / v = [rad/m] c [m 1 ] or [Neper/m] 7/1/11 r r 5 r r r

26 Plane waves in a conductor Dielectric: σ= Conductor: σ H = j E σ H = j E+ σ E = j + E = j eff E j eff = 1+ σ j + jk jk + jk E (, ) = E e + E e E (, ) = E e + E e E + ζ H + E + = ζ H + = E = ζ H E = ζ H / jk jk H E + e E jk (, ) = ζ / ζ e H (, ) = E + / ζe + E / ζe jk jk k E = H = σ ζ = µ / = µ / 1 R jx eff + = + j σ β Ph = µ = µ 1+ = β jα j α Phase constant Attenuation constant 7/1/11 6

27 Conductor propagation constant µ µ characteristic impedance ζ = = eff σ 1 j r r σ propagation constant k = µ = µ 1 j β j α eff r = r ( ) Re( ) + Re = j µ σ r β = r π c/ f λ = = β σ r r µ σ r α = 1+ 1 r δ 1/ = α = r c σ 1+ 1 r 1 ( δ ) α e = = e = 1/ e (about.37 or 8.69dB) 7/1/11 7

28 Good conductor σ 1 >> r propagation constant: characteristic impedance: k j σ j σµ σµ = µ 1 j r = r µ µ µ ζ = = j σ jσ σ 1 j r r ( 1 j) = j & ( j) 1+ = j σµ k (1 j) = β jα β = α = µ 1 ζ (1 j) (1 j) R (1 j) s σ + = σδ + = + σµ δ = Good conductor 1 Good conductor µσ R penetration depth s = σδ surface resistivit 7/1/11 8

29 Low losses material σ r << 1 π c / f λ = β r k jσ σ µ = µ 1 r j r c r r 1 r δ = α σ µ µ µ σ = + ζ 1 j ζ / r σ r r 1 r j r 7/1/11 9

30 Penetration Depth: eamples Material Frequenc Conductivit [S/m] Depth penetration [mm] Aluminum 1H 3.54*1^ GH 3.54*1^7.85*1^ 3 Blood 9MH GH Fat 9MH GH Sea water 3H 5 13 Penetration depth eplains the skin effect: while the frequenc increases, the penetration depth decreases, and the currents onl flow on the conductor surface. 7/1/11 3