8 Plane Electromagnetic Waves - National Chiao Tung ocw.nctu.edu.tw/course/emii031/CH08.pdfUEE 3201...

Click here to load reader

  • date post

    13-May-2018
  • Category

    Documents

  • view

    223
  • download

    3

Embed Size (px)

Transcript of 8 Plane Electromagnetic Waves - National Chiao Tung ocw.nctu.edu.tw/course/emii031/CH08.pdfUEE 3201...

  • UEE 3201 Electromagnetics II Fall, 2014

    8 Plane Electromagnetic Waves

    8.2 Plane Waves in Lossless Media

    1. Homogeneous wave equation in free space

    2E 1c2

    Et2

    = 0

    For a time harmonic (sinusoidal) field,

    2E+ k20E = 0, (E = phasor)

    where k0 = 00 = /c = free space wave number.

    In Cartesian corrdinate,

    ( 2

    x2+

    2

    y2+

    2

    z2+ k20)Ex,y,z = 0

    2. wavefront: set of points/positions in space with same phase

    plane wave: wave with planar wavefront perpendicular to the direction

    of propagation

    spherical wave: wave with spherical wavefront

    uniform wave: wave with constant E and H field magnitude across the

    wavefront

    3. Now consider a uniform plane wave with constant Ex on the wavefront

    xy plane,

    General solution in phasor form is

    where E+0 and E0 are complex constant to be determined from bound-

    ary or initial conditions.

    4. The real-time solution represented by the first phasor term is

    = wave traveling in the +z direction

    cYi Chiu, ECE Dept., NCTU 8-1

  • UEE 3201 Electromagnetics II Fall, 2014

    wave velocity (phase velocity up)

    =

    5. Real-time solution represented by the second phasor term is

    Ex (z, t) = [E0 ej(t+k0z)]

    = E0 cos(t+ k0z)

    = wave traveling in z direction with phase velocity up = c

    - Sign of up (or k0) represents direction of wave propagation.

    - If there is only one wave in the +z direction, then E0 = 0.

    6. H field can be found from E and Maxwells equations:

    For +z wave,

    Similarly for z wave,

    Hy (z) = 1

    0Ex (z)

    cYi Chiu, ECE Dept., NCTU 8-2

  • UEE 3201 Electromagnetics II Fall, 2014

    7. 0 =E+x (z)H+y (z)

    = intrinsic impedance of the free space

    If 0 = real E+x (z) and H+y (z) are in phase

    8. Real time solution of the H field

    (directions of wave propagation (az), electric field (ax) and magnetic

    field (ay) are mutually perpendicular.)

    Ex 8-1: A uniform plane wave E = axEx propagates in +z direction,

    r = 4, r = 1, = 0, f = 100MHz

    (a) instantaneous expression of E

    (b) instantaneous expression of H

    cYi Chiu, ECE Dept., NCTU 8-3

  • UEE 3201 Electromagnetics II Fall, 2014

    (c) positions for postitive peaks Ex at t = 108 sec

    - E and H are perpendicular to each other and both are transverse

    (perpendicular) to propagation direction

    transverse electromagnetic wave (TEM wave)

    9. Doppler effect (see the textbook)

    10. For a uniform plane wave propagating in an arbitrary direction in a loss-

    less medium, the solution to the wave equation 2E+k2E = 0 has the

    general form (from separation of variables, let E = E0X(x)Y (y)Z(z))

    E(x, y, z) = E0ejkxxejkyyejkzz

    = E0ej(kxx+kyy+kzz)

    = E0ejkR

    where R = axx+ ayy + azz = position vector

    k = axkx + ayky + azkz = wave vector

    |k|2 = k2x + k2y + k2z = k2 = 2

    cYi Chiu, ECE Dept., NCTU 8-4

  • UEE 3201 Electromagnetics II Fall, 2014

    Wavefront of the wave E(x, y, z) = set of points with same phase

    k = ank = wave vector

    an = propagation direction

    k = |k| = = 2/ = wave number

    11. For a uniform plane wave in a charge free region,

    E0 is transverse to the propagation direction

    12. H(R) =

    where = /k =/ () = intrinsic impedance of the medium,

    H(R) = 1 (an E0) ejkR

    H an and E, an is in the direction of EH (right hand rule)

    TEM wave in the an (or k) direction

    Ex 8-2: Find E(R) in terms of H(R)

    ()

    E(R) = an H(R)

    cYi Chiu, ECE Dept., NCTU 8-5

  • UEE 3201 Electromagnetics II Fall, 2014

    13. Polarization of plane waves (, )

    Polarization is the time varying behavior of the orientation of the E

    field at a given point in space. For example, in Ex 8-1,

    The tip of E field oscillates only along the x axis

    linear polarization (linaerly polarized)

    14. Now consider another wave in the +z direction,

    = superposition of two orthogonal linearly polarized wave with

    y polarization lagging x polarization by /2.

    The instantaneous E field is

    At a given position z = 0,

    Locus of tip of E(0, t) is(E1(0, t)E10

    )2+

    (E2(0, t)E20

    )2= cos2 t+ sin2 t = 1

    ellipse in the counterclockwise sense

    elliptically polarized if E10 = E20circularly polarized if E10 = E20

    right-hand or positive circularly polarized wave

    cYi Chiu, ECE Dept., NCTU 8-6

  • UEE 3201 Electromagnetics II Fall, 2014

    15. If E2(z) leads E1(z) by 90 (/2),

    E(z) = axE10ejkz + ayjE20e

    jkz

    E(0, t) = axE10 cost ayjE20 sint

    elliptically or circularly polarized wave

    = tan1 E2(0, t)E1(0, t)

    = t for E20 = E10 left-hand or negative circularly polarized wave

    16. If E1(z) and E2(z) are in time phase,

    linearly polarized

    17. In general,

    elliptically polarized

    Ex 8-3: Linear and circular waves

    Consider a linearly polarized wave propagating in +z direction,

    = superposition of a right-hand and a left-hand circular waves

    Note: - AM wave: linearly polarized with E field perpendicular to ground

    - TV wave: linearly polarized in the horizontal direction

    - FM wave: circularly polarized

    orientation of receiving antenna should be adjusted accordingly

    cYi Chiu, ECE Dept., NCTU 8-7

  • UEE 3201 Electromagnetics II Fall, 2014

    8.3 Plane Waves in Lossy Media

    1. wave equation in lossy media

    - kc = c = complex wave number (assume = real)

    - plane wave ejkz in lossless medium becomes ejkcz in lossy medium

    We can define a propagation constant

    * For lossless media, = 0, = 0 = 0, = k =

    The Helmholtz wave equation becomes

    For a linearly polarized uniform plane wave in +z direction,

    ez: attenuation factor

    : attenuation constant (Np/m, 1/m)

    (wave amplitude decresed by a factor of e1 after traveling a dis-

    tance of 1/ meters)

    ejz: phase factor

    : phase constant (rad/m)

    (similar to the wave number in lossless media)

    8.3.1 Low-Loss Dielectrics

    1. propagation constant

    cYi Chiu, ECE Dept., NCTU 8-8

  • UEE 3201 Electromagnetics II Fall, 2014

    For a good but imperfect insulator, , or / 1,

    = + j = jc = j(1 j/)1/2

    attenuation constant

    2

    phase constant

    [1 + 18

    (

    )2]2. intrinsic impedance

    c =

    (1 j

    )1/2

    Ex and Hy are not in time phase

    3. phase veolcity

    up =

    1

    (1 18

    (

    )2)8.3.2 Good Conductor

    1. propagation constant

    = + j = j

    (1 + j

    )1/2

    2. intrinsic impedance

    3. phase velocity

    up =

    2

    cYi Chiu, ECE Dept., NCTU 8-9

  • UEE 3201 Electromagnetics II Fall, 2014

    4. example: copper

    after a distance of = 1/ = 0.038 mm, the wave amplitude will be

    attenuated by a factor of e1 = 0.368.

    (At f = 10 GHz, = 0.66m high frequency EM wave is attenuated

    very rapidly in a good conductor.)

    5. skin depth or depth of penetration

    = distance over which the wave amplitude is attenuated by a factor

    of 1/e

    =

    =

    =

    E,J are distributed near the surface of a conductor at high

    frequency.

    (See Table 8-1)

    Ex 8-4: Seawater (Summary)

    cYi Chiu, ECE Dept., NCTU 8-10

  • UEE 3201 Electromagnetics II Fall, 2014

    At f = 5 MHz,

    / = 200 1 good conductor,

    attenuation constant = phase constant =f = 8.89 (Np/m,

    rad/m),

    wavelength in water = 2/ = 0.707 m,

    skin depth = 0.112 m

    (a) At 5 MHz, the wavelength in air is 60 m. It is very difficult to

    communicate with a submarine due to the small penetration depth

    in seawater and long wavelength in air (difficult to build efficient

    tranmission antenna).

    (b) After the real-time electric field Ex(z, t) is calculated, it is in-

    correct to calculate the real-time magnetic field Hy(z, t) from

    Hy(z, t) = Ex(z, t)/c. Since the complex impedance is defined

    in the phasor form (frequency domain), the correct formula are

    Hy(z) =Ex(z)

    c,

    Hy(z, t) = [Ex(z)

    cejt

    ]8.3.3 Ionized Gas (plasma)

    Plasma - assembly of equal numbers of positive ions and negative electrons

    and possibly other neutral species

    - electrons are much lighter than ions

    - motion of ions can be neglected and electrons can be viewed as a

    free electron gas (collision is neglected)

    - neutral species, if any, are not affected by electric field

    cYi Chiu, ECE Dept., NCTU 8-11

  • UEE 3201 Electromagnetics II Fall, 2014

    1. From Newtons force law, force on an electron in a time harmonic E

    field is

    In phasor form,

    Displacement x from the background positive ions gives rise to a dipole

    moment

    If there are N electrons per unit volume, the polarization vector is

    Equivalent permittivity of a plasma is

    - propagation constant in a plasma

    - intrinsic impedance

    cYi Chiu, ECE Dept., NCTU 8-12

  • UEE 3201 Electromagnetics II Fall, 2014

    (a) f < fp , = pure real

    pure attenuation of wave in plasma

    p = E/H = pure imaginary

    plasma is a reactive load

    no power transmission (electrons can move fast enough to

    screen the EM fields)

    (fp 9N 0.9 9 MHz for ionosphere)

    (b) f > fp , = pure imaginary

    no attenuation in plasma

    Ex 8-5: Communication with space ship

    cYi Chiu, ECE Dept., NCTU 8-13

  • UEE 3201 Electromagnetics II Fall, 2014