EELE 3332 Electromagnetic II Chapter 10
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1 EELE 333 Electromagnetic II Chapter 10 Electromagnetic Wave Propagation Ilamic Univerity of Gaza Electrical Engineering Department Dr. Talal Skaik 01 1
2 Electromagnetic wave propagation A changing magnetic field produce an electric field, and a changing electric field produce a magnetic field. Accelerating charge produce electromagnetic wave. The exitence of EM wave, predicted by Maxwell equation, wa firt invetigated by Heinrich Hertz (ometime called Hertzian wave). In general, wave are mean of tranporting energy or information. Typical example of EM wave include radio wave, TV ignal, radar,
3 WAVES Mechanical wave Electromagnetic wave Tranvere wave Longitudinal wave Tranvere wave Longitudinal Wave: Vibration i parallel to the direction of propagation. Sound and preure wave are longitudinal. Tranvere Wave: the motion of the matter particle i perpendicular to the direction of propagation of the wave itelf. 3
4 Mechanical Wave A material medium i neceary for the tranmiion for mechanical wave. Mechanical wave cannot travel through vacuum. Diturbance i tranmitted from one layer to the next through the medium. Tranvere Wave (in tring) Longitudinal Wave (in pring) 4
5 Electromagnetic Wave Material medium i not eential for propagation. Em wave travel through vacuum. Diturbance of electric and magnetic field travelling through pace. All electromagnetic wave are tranvere wave. Electromagnetic wave travel through empty pace!! All electromagnetic wave travel at c = m/ in vacuum (peed of light). Electric field, magnetic field, and direction of travel are mutually perpendicular. 5
6 Electromagnetic wave Goal in thi chapter: Solve Maxwell equation and decribe EM wave motion in the following media: 1. Free Space ( =0, =, = ) 0 0. Lole Dielectric ( =0, =, = or ) 0 r 0 3. Loy Dielectric ( 0, =, = ) 0 r 0 4. Good Conductor (, =, = or ) r r 0 r 0 where i the angular frequency of the wave 6
7 Electromagnetic wave The wave E Ain( t z) ha the following characteritic: It varie with both time and pace. It i time harmonic. The amplitude of the wave i A. The phae of the wave (in radian) i the term (ωt-βz), depend on time t and pace variable z. ω i the angular frequency in (radian/econd). β i the phae contant, or wave number in (radian per meter). 7
8 E Ain( t z) Wave take ditance λ to repeat itelf. λ i called the wavelength (in meter) Wave take time T to repeat itelf. T i called the Period (in econd) 8
9 Since it take time T for the wave to travel ditance at the peed u, ut 1 ( u peed of the wave, depend on the medium u ) But T 1/ f, where f i the frequency of the wave in Hertz (Hz). u f Since f, and = u f f (rad/m) Thi how that for every wavelength of ditance traveled, a wave undergoe phae change of radian. 9
10 Conider a fixed point P on the wave. Sketch E=A in(ωt-βz) at time t=0, T/4, and T/. It i evident that a the wave advance with time, point P move along +z direction. So, the wave E=A in(ωt-βz) i travelling with a velocity u in the +z direction. 10
11 Point P i a point of contant phae, therefore tz contant dz z u dt Note: A in(ωt - βz) i wave propagating in +z direction (forward travelling, or poitive-going wave) A in(ωt + βz) i wave propagating in -z direction (backward travelling, or negative going wave) 11
12 1
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14 Example 10.1 An Electric field in free pace i given by 8 E 50co(10 t x ) a V/m (a) Find the direction of wave propagation. (b) Calculate β and the time it take to travel a ditance of λ/. (c) Sketch the wave at t=0, T/4, and T/. y ( a) The wave i propagating along - a x direction (b) in free pace u c, = rad/m 8 u c If T i the period of the wave, it take T econd to travel a ditance at peed c. Hence to travel a ditance / will take: T 1 t n. 8 f 10 14
15 8 Example co(10 E t x) a V/m ( c) At t=0, E 50co x T 1 At t= = =, 4 4 f 4 E y T 1 At t= = =, f E y y 50co. x 4 50co x / 50in x 50co. x 50co x 50co x y Notice that the wave travel along a x 15
16 Conider a linear, iotropic, homogeneou, LOSSY dielectric medium that i charge free ( =0). Maxwell' equation in phaor form are: E 0 (1) H 0 v () E jh (3) H j E (4) 10.3 Wave Propagation in Loy Dielectric A loy dielectric i a medium in which an EM wave, a it propagate, loe power owing to imperfect dielectric. (partially conducting medium with σ 0) Taking the curl of both ide of equation (3) give: E j H (3) Since A ( A) A ( E ) E j j E E E 0, where j j 16
17 E E 0 where j j i called the propagation contant of the medium. By imilar procedure, it can be hown that for the H-Field, H H 0 E E 0 H H 0 Sinc e i a complex quantity, let j j j j j j Wave Propagation in Loy Dielectric j j, (Vector Wave Equation) (olve for and )
18 Wave Propagation in Loy Dielectric α i attenuation contant (Np/m): define the rate of decay of the wave in the medium. meaured in Neper per meter (Np/m). α=0 for lole medium (ζ=0) An attenuation of 1 neper indicate a reduction of e-1 of the original value. (1 Np=0 log10e=8.686 db). β i phae contant (rad/m) : i a meaure of the phae hift per unit length in radian per meter. (alo called wave number) u 18
19 Aume the wave propagate along a and E ha only x component, then E E ( z) a Since E E 0 ( E Vector Laplacian ) Hence E x x Wave Propagation in Loy Dielectric x ( z) x E x y ( z) z ' z ' x o 0 o 0 z ( z) x E x z ( ) 0 Ex ( z) or E ( ) 0 x z z Thi i a calar wave equation, a differential equation with olution: E ( z) E e E e (where E and E are contant) ( Second part i zero ince we aumed wave traveling along +a ). E z z jt z j( t z) E( z, t) Re Ex ( z) e a x Re E0e e a x z E( z, t) E e co( t z) a 19 0 x
20 Wave Propagation in Loy Dielectric E( z, t) E e z co( t z) a 0 A Sketch of E at time t=0 and t=δt i hown x Notice E ha only x-component and it i travelling in the +z direction. 0
21 H( zt, ) can be obtained a: H( z, t) Re H e e a, H z j( t z) 0 0 y 0 where i complex quantity known a intrinic impedance, in ohm of the medium. j e j E j with / 1 1/4, tan E 0 z j( t z) H= Re e e a j y e E or z t e t z 0 z H(, ) co( ) a y 1
22 Notice that E and H are out of phae by Ѳη at any intant of time. Thu, E lead H (or H lag E) by Ѳη. The ratio of the magnitude of the conduction current denity Jc to that of the diplacement current denity Jd in a loy medium i J J Where tanѳ i known a the lo tangent and Ѳ i the lo angle of the medium. c d z E( z, t) E e co( t z) a E 0 0 z H( z, t) e co( t z ) a y E j E or tan tan x
23 tanѳ i ued to determine how loy the medium i: Good (lole or perfect) dielectric if tanѳ i vary mall (ζ<<ωε) Good conductor if tanѳ i very large (ζ>>ωε) Behaviour of a medium depend not only on parameter ζ, ε, and µ, but alo on the frequency. A medium regarded a a good conductor at low frequencie may be a good dielectric at high frequencie. Since tan and tan j Since H j E j 1 E = jce where, c 1 j j, c ' j '' with ', '' c i calle dthe complex permittivity of the medium. Notice that the ratio of ''to ' i the lo tangent of the medium. '' tan = ' 3
24 10.4 Plane wave in lole dielectric In a lole dielectric, ζ<<ωε (pecial cae of ection 10.3), except that 0, =, = 0 r 0 j j, Since 0 j j 0, r u 1, = j Alo, Since and 0 j 0 and thu E and H are in time phae with each other. o 4
25 10.5 Plane wave in free pace In free pace 0, = 0, = 0 Thi may be regarded a pecial cae of ection , u where c=310 m/, i the peed of light in a vaccum. 0 0 c 1 c, = 0 0 o i called the intrinic impedance of free pace. 5
26 Plane wave in free pace If E= E co( t-z) a then H= H H 0 x 0 co( t-z) a E0 co( t-z) a In general, if a, a, and a are unit vector along the E field, H field, and the direction of ave propagation: a k a E y y E H k a a a a a a a H k H E E H k Plot of E and H (a) a function of z at t 0; and (b) at z 0. The arrow indicate intantaneou value. 6
27 Wave repreentation Plane Wave and it repreentation Circular Wave and it repreentation 7
28 Plane wave in free pace 8
29 Plane wave in free pace Both E and H field (or EM wave) are everywhere normal to the direction of wave propagation, ak. They form an EM wave and have no electric or magnetic field component along the direction of propagation. Such a wave i called a tranvere electromagnetic (TEM) wave. A combination of E and H i called a uniform plane wave becaue E (or H) ha the ame magnitude throughout any tranvere plane, defined by z=contant. Uniform plane wave erve a approximation to practical wave uch a thoe from a radio antenna a ditance ufficiently far from radiating ource. 9
30 Plane Wave in Good Conductor A perfect or good conductor i one in which ζ>>ωε, = 0, = 0 r = 1 j j j j j = (1 j) f u, = Alo, Since j j and j o o 45 Thu E lead H by
31 Plane Wave in Good Conductor If E E e co( t z) a z 0 x then E e t z 0 z o H co( 45 ) a y Therefore, a the wave travel in a conducting medium, it amplitude i attenuated by a factor e -αz. The ditance δ, through which the wave amplitude decreae to a factor e -1 (about 37% of the original value) i called kin depth or penetration depth of the medium. E e E e (general) The kin depth i a meaure of the depth to which an EM wave can penetrate the medium. 31
32 Plane Wave in Good Conductor Skin depth illutration Since for good conductor = f 1 1 f 1 z E e t a z/ Since = E 0 co( ) x 3
33 Plane Wave in Good Conductor Skin Depth in Copper Frequency (Hz) Skin depth (mm) x x 10-4 For copper, σ=5.8x10 7 S/m, µ= µ 0, 66.1/ f (in mm) The kin depth decreae with increaing frequency. Thu, E and H can hardly propagate through good conductor. The field and current are confined to a very thin layer (the kin) of the conductor urface. For a wire of radiu a, it i a good approximation at high frequencie to aume that all of the current flow in the circular ring of thickne δ. (a hown) << a 33
34 34
35 Plane Wave in Good Conductor Skin effect : i the tendency for high-frequency current to flow on the urface of a conductor. * The effective conductor cro ection decreae and the conductor reitance increae. It i ued to advantage in many application: Since the kin depth in ilver i very mall, ilver plating i often ued to reduce the material cot of waveguide component. (e.g ilver-plating on bra). Hollow conductor are ued intead of olid conductor in outdoor televiion antenna, and thu aving weight and cot. 35
36 Plane Wave in Good Conductor The dc reitance i given by: The urface or kin reitance R (in ) i given by: R R dc l S 1 f (real part of for a good conductor) The ac reitance R ac i calculated by uing the dc formula with S w, where w i the width: For a conductor of radiu a, w a. So, R ac l Rl w w l Rac a a a f R l dc a At high frequencie, R i far greater than R ac dc 36
37 Example 10. A loy dielectric ha an intrinic impedance of at a particular radian frequency. If, at that frequency, the plane wave propagating through the dielectric ha the magnetic field component: x 1 H10e co t x ay A/m find E and. Determine the kin depth and wave polarization. 37
38 The given wave travel along a a ; a a, o k x H y a a a a a a a E E k H x y z a z o that Eo Ho 10,o e E 000e H o Example Solution a x o j/6 j/6 o Except for the amplitude and phae difference, E and H alway have the ame form. Hence E Re 000e e e j /6 x jt x x E e co t az kv/m 6 a E 38
39 Knowing that =1/, we need to determine. Since 1 1 and 1 1 But o tan =tan Hence 1/ Np/m = 3 The wave ha an E component; hence it i polarized along the z-direction. 39 z Example Solution 1/
40 Example 10.3 In a lole medium for which = 60, =1 and H = 0.1 co ( t z) a in ( t z) a A/m, calculate,, and E. r Solution In thi cae, =0, =0, and =1, o x o r = / or r r 4 60 o r r oo r r 4 c or 8 c c rad/ r y 40
41 Example Solution From the given H field, E can be calculated in two way: uing the technique (baed on Maxwell' equation) developed in thi chapter or directly uing Maxwell' equation a in Chapter 9. Method 1: To ue the technique developed in thi chapter, we let where H = H + H 1 H = 0.1 co ( t z) a and H = 0.5 in ( t z) a 1 x y and the correponding electric field where E = E + E 1 E =E co ( t z) a and E = E in ( t z) a 1 1o E o E 1 41
42 Notice that although H ha component along a and a, it ha no component along the direction of propagation; it i therefore a TEM wave. For E Hence For E Hence 1 1 a a a a a a E k H z x y 1 1 E 1o H 60 (0.1) 6 1o E = 6 co t z 1 y a a a a a a E k H z y x Adding E and E give E ; that i, E Example Solution o H 60 (0.5) 30 o a E = 30 in t z x 8 =94.5 in ( t z) ax co (1.5 E 8 10 t z) a V/m x a y y 4
43 Method : We may apply Maxwell' equation directly. E 1 H = E E H dt t where 0. But Example Solution H y H x H x x z ax a z z H ( z) H ( z) 0 x x y Hence = 0. 5 co ( t z) a 0.1 in ( t z) x E Hdt in ( t z) ax in ( t z) a = 94.5 in ( t z) a co ( t z) a V/m x a y y y 43
44 Example 10.4 A uniform plane wave propagating in a medium ha z 8 E e in(10 t z) ay V/m. If the medium i characterized by = 1, = 0, and = 3 mho/m, find,, and H. r r Solution We need to determine the lo tangent to be able to tell whether the medium i a loy dielectric or a good conductor howing that the medium may be regarded a a good conductor at the frequency (10 )(3) of operation. Hence, = = Np/m, 61.4 rad/m 1/ 44
45 7 8 1/ (10 ) 800 Alo, 3 o tan Hence z H Hoe in t z a 4 a a a a a a H k E z y x and H o Example Solution Eo Thu in z 8 H e t 61.4 z x ma/m H 4 a 45
46 A plane wave E = E co ( t z) a i incident on a good o Example 10.5 conductor at z = 0. Find the current denity in the conductor. Solution Since the current denity J = E, we expect J to atify the wave equation ( E E 0), that i, J J Alo the incident E ha only an x-component and varie with z. Hence J = J (z, t) a and x x d J x J x 0 dz which i an ordinary differential equation with olution z J x Ae Be 0 x z 46
47 Example Solution The contant B mut be zero becaue J i finite a z. But in a good conductor, >> o that =1/. Hence = j 1 and J or where J x x Ae x j z(1 j)/ J J (0) Ae x x z(1 j)/ 1 (0) i the current denity on the conductor urface. j 47
48 Example 10.6 For the copper coaxial cable hown, let a = mm, b = 6 mm, and t = 1 mm. Calculate the reitance of m length of the cable at dc and at 100 MHz. Solution : Let where R and R are the reitance of the inner and outer conductor. At dc, R R i o o i R = R + R o.744 m 7 3 S a ( 10 ) S [ b t] b t bt m Hence R m dc i 48
49 At f=100 MHz, f Ri S w a a ( S w) Since = 6.6 m << t = 1 mm, = b for the outer conductor. Hence, R R o ac Example Solution 8 7 f b which i about 150 time greater than R. Thu, for the ame effective current i, the ohmic lo ( ir the dc power lo by a factor of 150. dc ) of the cable at 100 MHz i far greater than 49
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