5. Plane Electromagnetic Waves

Transcription

1 5. Plane lectomagnetic Waves Pof. Rakhesh Singh 1

2 5.1 Intoduction lectomagnetic Waves Plane waves Poynting vecto Plane waves in vaious media Polaiation Lossless medium Lossy conducting medium Good conducto Good dielectic Fig. 5.1 Plane Waves

3 5. Plane waves 5..1 What ae plane waves? Spheical waves become plane waves in the fa field behaves like plane waves In plane waves wavefonts ae pependicula to the wave popagation vecto Mathematically, fields in plane waves assumes the following fom F, t ( ) = F e j k ( ωt) 3

4 5. Plane waves In ectangula o Catesian coodinate system ) ) ) k = k x x + k y y + k = xx ) + yy ) + ) k = k k = k + k + k = ω µε ( ) ( ) ( ) x y Note that the constant phase suface fo such waves ) ) ) ) k = k x + k y + k xx + yy + = k x + k y + k = con t ) ) ( x y ) ( ) x y tan 4

5 5. Plane waves Since the field stength is unifom eveywhee it is also known as unifom plane waves Fo plane waves fom the Maxwell s equations, the following elations could be deived k = H; k H = ; k = ; k H = ωµ ωε lectic and magnetic field ae pependicula to each othe lectic field, magnetic field and wave popagation vecto ae pependicula to each othe 5

6 6 5. Plane waves Popeties of a unifom plane wave: No electic o magnetic field in the diection of popagation Tansvese electomagnetic wave: TM wave (electic and magnetic field pependicula to wave popagation vecto) The value of the magnetic field is equal to the magnitude of the electic field divided by η (~377 Ohm) at evey instant magnetic field amplitude is much smalle than the electic field amplitude

7 5. Plane waves The diection of popagation is in the same diection as Poynting vecto * 1 S * ins tant = H ; Sins tan t = Re H { } The instantaneous value of the Poynting vecto is given by /η, o H η The aveage value of the Poynting vecto is given by /η, o H η / 7

8 5. Plane waves 5.. Wave polaiation Polaiation of plane wave efes to the oientation of electic field vecto (o magnetic field vecto), which may be in fixed diection o may change with time Polaiation is the cuve taced out by the tip of the aow epesenting the instantaneous electic field The electic field must be obseved along the diection of popagation 8

9 5. Plane waves Types of polaiation Linea polaied Ciculaly lliptically (LP) polaied (CP) polaied (P) LHCP RHCP RHP LHP 9

10 5. Plane waves If the vecto that descibes the electic field at a point in space vaies as function of time and is always diected along a line which is nomal to the diection of popagation the field is said to be linealy polaied If the figue that electic field tace is a cicle (o ellipse), then, the field is said to be ciculaly (o elliptically) polaied 1

11 5. Plane waves Besides, the figue that electic field taces is cicle and anticlockwise (o clockwise) diection, then, electic field is also said to be ight-hand (o lefthand) ciculaly polaied wave (RHCP/LHCP) Besides, the figue that electic field taces is ellipse and anticlockwise (o clockwise) diection, then, electic field is also said to be ight-hand (o lefthand) elliptically polaied (RHP/LHP) 11

12 5. Plane waves Fo pope analysis of polaiation, let us conside the supeposition of an x-linealy polaied wave with complex amplitude x and a y-linealy polaied wave with complex amplitude y, both popagating in the positive -diection Note that since the electic field is vaying with both space and time It is a function of both space and time 1

13 5. Plane waves It is easie to analye at a paticula instant of time fist And add the time dependence late The total electic field can be witten as ( ) ( ) ( ) j j j j e y e x e e y x y x β φ φ β + = + = ˆ ˆ ˆ ˆ 13 Note x and y may be complex numbes and x and y ae the amplitudes of x and y and ae the phases of x and y ( ) ( ) ( ) j j y j x j y x e y e x e e y x y x β φ φ β + = + = ˆ ˆ ˆ ˆ x φ y φ

14 5. Plane waves Putting in the time dependence and taking the eal pat, we have, ( t) = cos( ω t β + φ ) xˆ + cos( ωt β + )yˆ, x x y φy Let us conside a numbe of possibilities: Linealy polaied (LP) wave: If both x and y ae eal (say x = ox and y = oy ), then, LP ( ) ( ) jβ ( ) jβ = x + yˆ e = xˆ + yˆ e ˆ x y x y 14

15 5. Plane waves Putting in the time dependence and taking the eal pat, we have, LP ( t) = cos( ωt β) xˆ + cos( ωt )yˆ, x y β The amplitude of the electic field vecto is given by LP ( ) ( ) ( ), t = + cos( ωt β) x which is a staight line diected at all times along a line that makes an angle θ with the x-axis given by the following elation = = y 1 y 1 y θlp tan tan x x 15

16 5. Plane waves If x and y =, we have a linealy polaied plane wave in x- diection LP (, t) = cos( ωt β)xˆ ox If x = and y, we have a linealy polaied plane wave in y- diection LP (, t) = cos( ωt β)yˆ oy 16

17 5. Plane waves asie to fix space to see the polaiation Fo a fixed point in space (say =), LP (, t) = cos( ωt)xˆ = Fo all times, electic field will be diected along x-axis hence, the field is said to be linealy polaied along the x-diection (efe to figue in next slide) oy 17

18 5. Plane waves Fig. 5. (a) LP wave x y 18

19 5. Plane waves Ciculaly polaied (CP) wave: Now conside the case x = j y = o, whee o is eal so that π j j y x = e ; = e ; j = ( xˆ jyˆ ) e β RHCP o 19 The time domain fom of this field is (putting in the time dependence and taking the eal pat) π (, ) [ ˆ cos( ) ˆ RHCP t = o x ωt β + y cos( ωt β )]

20 5. Plane waves Note that x- and y-components of the electic field have the same amplitude but ae 9 out of phase In ode to obseve the behavio of the RHCP wave, assume that we ae obseving the wave at a fixed position Let us choose a fixed position (say =), then, RHCP ( =, t) = cos( ω t) xˆ + sin( ωt)yˆ o o

21 5. Plane waves The diection of the tip of the electic field vecto is sinωt ( t) cosωt 1 1 θrhcp = tan = tan tan ω = ωt which shows that the polaiation otates with unifom angula velocity ω in anticlockwise diection fo popagation along positive -axis 1

22 5. Plane waves An obseve standing at = will see the electic field otating in a cicle and the field neve goes to eo Since the finges of ight hand point in the diection of otation of the tip of the electic field vecto when the thumb points in the diection of popagation, this type of wave is efeed to as ight hand ciculaly polaied wave (RHCP wave)

23 5. Plane waves Fig. 5. (b) RHCP wave x y 3

24 5. Plane waves lliptically polaied (P) wave: Now, conside a moe geneal case of P wave, when the amplitude of the electic field in the x- and y- diections ae not equal in amplitude and phase unlike CP wave, so that, ( ) ( jφ ) jβ P = xˆ + Ae yˆ e Putting in the time dependence and taking the eal pat, we have, (, t) = cos( ω t β) xˆ + Acos( ωt β + φ )y P ˆ 4

25 5. Plane waves LP A = numbe, φ =,π LHCP: A = 1, φ = RHCP: π A = 1, φ = π 5

26 5. Plane waves How do we know whethe it is LHP o RHP? A can take any value fo P wave If φ is in the uppe half of the complex plane then the wave is LHP wheeas φ is in the lowe half of the complex plane, then the wave is RHP Let us choose a fixed position (say =) like in the CP case, then, = cos t x + A t + yˆ P = ( ω ) ˆ cos( ω φ ) 6

27 5. Plane waves Some paticula cases: ( a) A = 1, φ = ; = cos t x + y LP = ( b) A = 1, φ = π ; = cos ωt x y LP = ( ω )( ˆ ˆ ) ( ) ( )( ˆ ˆ ) ( ) π ( c) A = 1, φ = ; = { cos( ωt) xˆ yˆ sin ( ωt) } ( LHCP) = π ( d) A = 1, φ = ; = cos ωt x + y sin ωt RHCP = { ( ) ˆ ˆ ( )} ( ) 7

28 5. Plane waves ( e) π A = 3, φ = ; = { cos( ωt) xˆ yˆ 3sin ( ωt) } = ( LHP) ( f ) π A =.5, φ = ; = { cos( ωt) xˆ + yˆ.5sin ( ωt) } = ( RHP) ( g ) π π A = 1, φ = ; = ( ) cos ωt xˆ + yˆ cos ω t 4 = + 4 ( LHP ) ( h) π π A = 1, φ = 3 ; = ˆ ˆ cos ωt x + y cos ωt 3 4 = 4 RHP ( ) ( ) 8

29 5. Plane waves Fig. 5. (c) LHP wave Diection of popagation lectic field x y Magnetic field at each point is othogonal to the electic field 9

30 5. Plane waves Depending on how we obseve the wave, the polaiation (LHCP, RHCP, LHP, RHP) will be diffeent In ou case, we ae obseving the wave standing opposite facing the wave popagating If we stand in diection following the diection of the wave, then the polaiation (LHCP, RHCP, LHP, RHP) will be diffeent 3

31 5.3 Poynting vecto & powe flow in M fields The ate of enegy flow pe unit aea in a plane wave is descibed by a vecto temed as Poynting vecto which is basically cul of electic field intensity vecto and magnetic field intensity vecto * S = H The magnitude of Poynting vecto is the powe flow pe unit aea and it points along the diection of wave popagation vecto 31

32 5.3 Poynting vecto & powe flow in M fields The aveage powe pe unit aea is often called the intensity of M waves and it is given by 1 Savg = Re H ( * ) Let us ty to deive the point fom of Poynting theoem fom two Maxwell s cul equations fo time hamonic fields = jωµ H H = jωε + J 3

33 5.3 Poynting vecto & powe flow in M fields Fom vecto analysis, ) ( ) ( ) ( ) ( ) ( * * * * * * J j H j H H H H + = = ωε ωµ 33 We can futhe simplify ( ) ( ) * * * * ) ( J j H H j H = ωε ωµ

34 5.3 Poynting vecto & powe flow in M fields It is basically a point elation It should be valid at evey point in space at evey instant of time The powe is given by the integal of this elation of Poynting vecto ove a volume as follows V ( ) ( ) * * H dv = H ds = S ds = S jωµ V * ( H H ) dv jωε This is the integal fom of instantenous Poynting vecto and powe flow in M fields S V * ( ) dv V * J dv 34

35 5.3 Poynting vecto & powe flow in M fields 1 Re Fo time aveage powe flow, we have, S S ds = Re jωµ H dv Re jωε dv Re σ V V V In ode to take into account the losses (loss tangent (ϵ) fo PCB FR4 boad is.) ' '' ' = ε jε, µ = µ j Hence, '' ε ' ε '' ε µ, tanδε =, tanδ µ = '' µ ' µ dv 1 Re S ω '' ω '' 1 S ds = µ H dv ε dv σ V V V dv 35

36 5.3 Poynting vecto & powe flow in M fields Poynting theoem states that the powe coming out of the closed volume is equal to the total decease in M enegy pe unit time i.e. time aveage powe dissipation fom the volume which constitutes of magnetic losses dielectic losses conductivity in the volume 36

37 5.3 Poynting vecto & powe flow in M fields xample: Let us assume a plane wave popagating in the + diection in fee space, then ; jk j j e e H e β β η = = = ) 37 The instantaneous value of the Poynting vecto: ( ) ( ) ( ) ( ) ( ) ( ) ˆ ˆ ˆ ˆ ˆ 1 ˆ η η η η η β β e e H S j j = = = = = =

38 5.3 Poynting vecto & powe flow in M fields 38 o Note that the diection of Poynting vecto is also in the - diection same as that of the wave vecto o The aveage value of the Poynting vecto: 1 1 S ( ) avg = Re H = Re η = η 1 we = ε o Stoed lectic negy: o Stoed Magnetic negy: ˆ ˆ wm = µ H = µ = µ ε = ε = w η µ e

39 5.3 Poynting vecto & powe flow in M fields Let us summaie fo plane waves: The diection of popagation is in the same diection as of Poynting vecto The instantaneous value of the Poynting vecto is given by /η, o H η The aveage value of the Poynting vecto is given by /(η ), o H η / The stoed electic enegy is equal to the stoed magnetic enegy at any instant 39

40 5.4 Plane waves in vaious media What kinds of mateials we have in class oom? Desk made of woods Walls made of bicks Glass in windows Fans made up metals Pape of books/notebooks How do we chaacteie the behaviou of plane waves in diffeent media made up of diffeent mateials? Fo that matte, What paametes of the media that influence the plane M waves? 4

41 5.4 Plane waves in vaious media A medium in electomagnetics is chaacteied by thee paametes: lectic pemittivity ε= ε ε, Magnetic pemeability µ= µ µ and Conductivity σ Dielectic mateial (ε 1, µ =1, σ =) Conductos (ε =1, µ =1, σ ~1 7 ) such as Silve, Coppe Magnetic mateials (ε =1, µ 1, σ ~1 7 such as Ion, Nickel) 41

42 5.4 Plane waves in vaious media Relative electic pemittivity ε ae dy wood (1.4-.9), pape (1.5-3) bick (3.58), glass (5-1), wate (8.4), Based on these values we may pedict the behavio of plane waves Lossless medium In a lossless medium like fee space o lossless dielectic, εand µ ae eal, σ=, ( ) ( ) Qγ ωµσ ωε α β = j + j = + j 4

43 5.4 Plane waves in vaious media ( jβ ) β ω µε γ = j ω µε = = so β is eal Assume x-polaied plane wave with the electic field of only x- component, no vaiation along x- and y-axis and popagation along -axis, i.e., x = = y 43

44 5.4 Plane waves in vaious media Helmholt wave equation educes to x + β x = whose solution gives waves in one dimension as follows = + e x + e j β + j β whee + and - ae abitay constants 44

45 5.4 Plane waves in vaious media Putting in the time dependence and taking eal pat, we get, x (, t) = cos( ω t β) + cos( ωt + β) Fo constant phase, ωt-β=constant=b(say) Since phase velocity, + v p d d ωt b) ω 1 1 = = ( ) = = = dt dt β β µε µ µ ε ε Qβ = ω µε 45

46 5.4 Plane waves in vaious media Fo fee space, v p = 1 = c = m / µ ε which is the speed of light in fee space This emegence of speed of light fom electomagnetic consideations is one of the main contibutions fom Maxwell s theoy The magnetic field can be obtained fom the souce fee Maxwell s cul equation s 46

47 5.4 Plane waves in vaious media How do we find the magnetic fields? = jωµ H xˆ yˆ ˆ j j j H = = = = yˆ + e jωµ ωµ ωµ x y ωµ + jβ + jβ e + e + jβ + jβ { e e } + jβ + jβ ( e ) + jβ + jβ {( ) ( )} ( ) + jβ + jβ jβ ( e ) + ( e ) jβ jβ e e H = ( j) yˆ = j yˆ ωµ ωµ β ( ) ( ) 1 + jβ + jβ = yˆ = [ e e ] yˆ ωµ η 47

48 5.4 Plane waves in vaious media η is the wave impedance of the plane wave η = ωµ = β Fo fee space, µ = ε x Hy µ o ηo = = 1π = 377 ε o 5.4. Lossy conducting medium If the medium is conductive with a conductivity σ, then the Maxwell s cul equations can be witten as Ω 48

49 5.4 Plane waves in vaious media = jωµ H; H = jω ε + σ = ( jω ε + σ ) = jω ε ; σ jσ jσ ε eff ( ω) = ε + = ε = ε 1 jωω ω ωε It is like you ae accounting losses due to conductivity into account by intoducing imaginay pat in the ϵ The effect of the conductivity has been absobed in the complex fequency dependent effective pemittivity eff 49

50 5.4 Plane waves in vaious media + ω µε ω = + jγ = ( ) ( ) eff We can define a complex popagation constant ( ) γ = j ω µε ω = α + j β eff whee α is the attenuation constant and β is the phase constant 5

51 5.4 Plane waves in vaious media What is implication of complex wave vecto? The wave is exponentially decaying The dispesion elation fo a conducto (usually nonmagnetic) is ε eff ( ω) ω j ( ) j j n ( ) j n ( ) γ = ω µε ω = ω µ ε = ω µ ε ω = ω eff eff eff ε c whee n eff is the complex efactive index 51

52 5.4 Plane waves in vaious media 1-D wave equation fo geneal lossy medium becomes whose solution is 1-D plane waves as follows = x x γ 5 j j x e e e e e e β α β α γ γ = + = ) (

53 5.4 Plane waves in vaious media Putting the time dependence and taking eal pat, we get, x (, t) = + e α cos( ωt β) + e α cos( ωt + β) The magnetic field can be found out fom Maxwell s equations as in the pevious section 1 + γ γ H y ( ) = [ e e ] η eff 53

54 5.4 Plane waves in vaious media whee useful expession fo intinsic impedance is η eff jωµ jωµ µ = = = γ jω µ ε ω ε ω eff ( ) eff ( ) The electic field and magnetic field ae no longe in phase as ε eff is complex Poynting vecto o powe flow fo this wave inside the lossy conducting medium is 54

55 5.4 Plane waves in vaious media * α jβ α jβ + + * j e e + + α β + α jβ e e α * * η eff η η S = H = e e xˆ yˆ = e e ˆ = e ˆ eff eff it is decaying in tems of squae of an exponential function Good dielectic/conducto Note that σ/ωε is defined as loss tangent of a medium A medium with σ/ωε <.1 is said to be a good insulato wheeas a medium with σ/ωε >1 is said to be a good conducto 55

56 5.4 Plane waves in vaious media Fo good dielectic, jσ Qσ << wε γ = jω µε ( 1 ) ωε can be appoximated using Taylo s seies expansion obtain α and β as follows: jσ γ = j ω µε 1 = α + ωε γ = jω σ µε + σ µ α = β = ω µε ε µ = α + ε j β jβ 56

57 5.4 Plane waves in vaious media What is the phase velocity of wave in vaious media? v p d d ωt b) ω 1 1 = ( ) dt = dt β = β = = µε µ µ ε ε Most dielectics have ε 1, hence the phase velocity is usually lesse than the speed of light 57

58 5.4 Plane waves in vaious media How about the wave impedance and wavelength? η = ωµ = β µ = ε x Hy π ; λ = = β π ω µε Wave impedance and wavelength fo M waves in dielectic is usually lowe than that of the fee space 58

59 5.4 Plane waves in vaious media Fo a good conducto, σ >> ωε ; γ = jω µε 1 Theefoe, γ = j σ jω µε = α + ωε γ = jωµσ = α + jβ jσ ωε jβ γ (1 + ωµσ j ) α = β = wµσ 59

60 5.4 Plane waves in vaious media Skin effect The fields do attenuate as they popagate in a good dielectic medium α in a good dielectic is vey small in compaison to that of a good conducto As the amplitude of the wave vaies with e -α, the wave amplitude educes its value by 1/e o 37% times ove a distance of skin depth (α=1) δ = 1 α 1 1 = = = = β ωµσ π f µσ π f µσ 6

61 5.4 Plane waves in vaious media α = π fµσ o Fo example, ofo coppe at 1 GH, oattenuation constant is 1.4e5 o Fo wave incident with amplitude 1, o at a distance of 1mm, the wave amplitude educes to e-6 o Skin depth is e-3 mm 61

62 5.4 Plane waves in vaious media 6 δ = This means that in a good conducto α (a) highe the fequency, lowe is the skin depth (it means cuent flows at the suface of the conducto) (b) highe is the conductivity, lowe is the skin depth and (c) highe is the pemeability, lowe is the skin depth Let us assume an M wave which has x-component and popagating along the -axis Then, it can be expessed as x α j( β ωt ) (, t) e e = 1

63 5.4 Plane waves in vaious media Taking the eal pat, we have, x α ( t) = e cos( ωt β), Substituting the values of α and β fo good conductos, we have, ( t) e ( t f ) π fµσ = cos ω π x, µσ Now using the point fom of Ohm s law fo conductos, we can wite ( t) e ( t f ) π fµσ = σ cos ω π J x = σx µσ, 63

64 5.4 Plane waves in vaious media What is the phase velocity and wavelength inside a good conducto? v p = ω π ωδ ; λ πδ β = = β = Skin depth fo coppe at 1 GH is e-3 mm Hence, phase velocity is quite small in compaison to speed of light. 64

65 5.5 Summay Plane waves ) ) ) ) k = k x + k y + k xx + yy + = k x + k y + k = con t ) ) ( x y ) ( ) x y tan lectomagnetic Waves Plane waves in vaious media Polaiation LP = xxˆ + y yˆ e j ( ˆ ˆ RHCP = o x jy) e β jφ jβ = xˆ + Ae yˆ e P 1 Re 65 jβ ( ) ( ) ( ) ( ) S Poynting vecto Lossless medium β = ω µε ω 1 = = β µε v p Lossy conducting medium ε jσ = ωε ( ω) ε eff 1 ωµ µ η = = ωµ β ε ηeff = γ ω '' ω '' 1 S ds = µ H dv ε dv σ dv V V V eff ( ) γ = jω µε ω = α + jβ j = µ ε eff ( ω) v p Good conducto α = β = 1 1 δ = = = α β ωµσ π fµσ = ω π ωδ ; λ πδ β = = β = Fig. 5.3 Plane waves in a nutshell 1 Good dielectic σ α = β = ω µ ε µε π fµσ ( t) = σ e cos( ωt πf ) J x = σx, µσ