Dr' Hugh Blanton

1
ENTC 3331 RF Fundamentals

• Dr. Hugh Blanton
• ENTC 3331

2
Plane-Wave Propagation
3

• Electric Magnetic fields that vary harmonically
  with time are called electromagnetic waves

4

• In order to simplify the mathematical treatment,
  treat all fields as complex numbers.

5

• The mathematical form of the Maxwell equations
  remains the same, however, all quantities (apart
  from x,y,z,t) are now complex.

6

• For
• It follows that

7

• The Maxwell equations (in differential form) can
  thus be expressed as
• In a vacuum (space)
• In air (atmosphere)

8

• Thus, the Maxwell equations (in differential
  form) and in air can be expressed as
• The Maxwell equations are fundamental and of
  general validity which implies
• It should be possible to derive a pair of
  equations, which describe the propagation of
  electromagnetic waves.

9

• We expect solutions like
• How do we get from
• to

10

• Recall that
• and apply to both sides of
• but

11
0
12
wave number k2
wave equation
13
wave equation

• The previous two equations are called wave
  equations because their solutions describe the
  propagation of electromagnetic waves

14

• In one dimension
• If this describes an electromagnetic wave, it may
  also hold for a single photon.

15

• For a photon, is significant at the current
  location of the photon.
• The probability of finding a photon at location x
  is .
• This implies

Schrodingers equation
16
strict derivation
heuristic analogy
Schrodingers Equation (Postulates of Quantum
Mechanics
physics of the macroscopic world
Maxwells equations (Newtons laws)
physics of the microscopic world
Wave Equation
particles and waves
particles-wave duality
17

• What are the solutions of the electromagnetic
  wave equations?

18

• Perform the Laplacian

19

• That is

20

• Consider a uniform plane wave that is
  characterized by electric and magnetic fields
  that have uniform properties at all points across
  an infinite plane.

21
no component in the z-direction
x
y up
wave crescents
z
22

• Consequently,
• simplifies to

23

• The most general solutions of
• are
• where and are constants determined
  by boundary conditions.

24

• For mathematical simplification rotate the
  Cartesian coordinate system about the z-axis
  until
• The plane wave is
• The first term represents a wave with amplitude
  traveling in the z-direction, and
• the second term represents a wave with amplitude
  traveling in the z direction.

25

• Let us assume that consists of a wave
  traveling in the z-direction only

26

• Magnetic field, ?
• We must fulfill the Maxwell equation
• But

27
(No Transcript)
28
(No Transcript)
29
(No Transcript)
30
(No Transcript)
31

• Recall

32

• This is possible if
• Electric and magnetic field vectors are
  perpendicular!

33
Transversal electromagnetic wave (TEM)
34

• Electromagnetic Plane Wave in Air
• The electric field of a 1-MHz electromagnetic
  plane wave points in the x-direction.
• The peak value of is 1.2p (mV/m) and for t 0,
  z 50 m.
• Obtain the expression for and
  .

35

• The field is maximum when the argument of the
  cosine function equals zero or multiples of 2p.
• At t 0 and z 50 m

36
(No Transcript)
37
(No Transcript)
38
PLANE WAVE PROPAGATION

• POLARIZATION

39
Wave Polarization

• Wave polarization describes the shape and locus
  of tip of the vector at a given point in space
  as a function of time.
• The direction of wave propagation is in the
  z-direction.

40
Wave Polarization

• The locus of , may have three different
  polarization states depending on conditions
• Linear
• Circular
• Elliptical

41
Polarization

• A uniform plane wave traveling in the z
  direction may have x- and y- components.
• where

42
Polarization

• and are the complex amplitudes of
  and , respectively.
• Note that
• the wave is traveling in the positive
  z-direction, and
• the two amplitudes and are in
  general complex quantities.

43
Polarization

• The phase of a wave is defined relative to a
  reference condition, such as z 0 and t 0
  or any other combination of z and t.
• We will choose the phase of as our
  reference, and will denote the phase of
  relative to that of , as d.
• Thus, d is the phase-difference between the
  y-component of and its x-component.

where ax and ay are the magnitudes of Ex0 and
Ey0
44
Polarization

• The total electric field phasor is
• and the corresponding instantaneous field is

45
Intensity and Inclination Angle

• The intensity of is given by
• The inclination angle ?

46
Linear Polarization

• A wave is said to be linearly polarized if
  Ex(z,t) and Ey(z,t) are in phase (i.e., d 0) or
  out of phase (d p).
• At z 0 and d 0 or p,

47
Linear Polarization (out of phase)

• For the out of phase case
• w t 0 and
• That is, extends from the origin to the point
  (ax ,?ay) in the fourth quadrant.

48
Linear Polarization (out of phase)

• For the in phase case
• w t 0 and
• That is, extends from the origin to the point
  (ax , ay) in the first quadrant.

y
x
49

• The inclination is
• If ay 0, y 0? or 180?, the wave becomes
  x-polarized, and if ax 0, y 90 ? or -90 ?,
  and the wave becomes y-polarized.

50
Linear Polarization

• For a z-propagating wave, there are two possible
  directions of .
• Direction of is called polarization
• There are two independent solution for the wave
  equation

51
Linear Polarization
E
z
B
Can make any angle from the horizontal and
vertical waves
52
Linear Polarization
Looking up from z
x-polarized or horizontal polarized ay0
?0 or 180
y-polarized or vertical polarized ax0
?90 or -90
53
Circular Polarization

• For circular polarization, ax ay.
• For left-hand circular polarization, d p/2 .
• For right-hand circular polarization, d -p/2 .

54
Left-Hand Polarization

• For ax ay a, and d p/2,
• and the modulus or intensity is

55

• The angle of inclination is

56

• At a fixed z, for instance z 0, y -wt.
• The negative sign means that the inclination
  angle is in the clockwise direction.

57
Right-Hand Circular

• For ax ay a, and d p/2,
• ,
• The positive sign means that the inclination
  angle is in the counter clockwise direction.

58

• A RHC polarized plane wave with electric field
  modulus of 3 (mV/m) is traveling in the
  y-direction in a dielectric medium with e 4eo,
  m mo, and s 0.
• The wave frequency in 100 MHz.
• What are
• and

59

• Since the wave is traveling along the y-axis, its
  field components must be along the z-axis and
  x-axis.

w
60
(No Transcript)
61
(No Transcript)
62
(No Transcript)
63
Elliptical Polarization

• In general,
• ax ? 0,
• ay ? 0, and
• d ? 0.
• The tip of traces an ellipse in the x-y
  plane.
• The wave is said to be elliptically polarized.
• The shape of the ellipse and its handedness
  (left-hand or right-hand rotation) are determined
  by the values of the ratio and the
  polarization phase difference, d.

64
Elliptical Polarization

• The polarization ellipse has a major axis, ax
  along the x-direction and a minor axis ah along
  the h-direction.

65
Elliptical Polarization

• The rotation angle g is defined as the angle
  between the major axis of the ellipse and a
  reference direction, chosen below to be the
  x-axis.

66
Elliptical Polarization

• g is bounded within the range

67
Elliptical Polarization

• The shape and the handedness are characterized by
  the ellipticity angle, c.

implies LH rotation - implies RH rotation
68
Elliptical Polarization
is called the axial ratio and varies between 1
for circular polarization and ? for linear
polarization
69
Elliptical Polarization
70
Elliptical Polarization
Positive values of c (sind gt 0) ? LH
Rotation Negative values of c (sind lt 0) ? RH
Rotation
Also
71
Example 7-3

• Find the polarization state of a plane wave
• Change to a cosine reference

72
Example 7-3

• Find the corresponding phasor
• Find the phase angles
• Phase difference
• Auxiliary

73

• can have two solutions
• 
  or
• Since cosd lt 0, the correct value of g is -69.2?.

74

• Since the angle of c is positive and less than
  45?,
• The wave is elliptically polarized and
• The rotation of the wave is left-handed.

75
Polarization States
The wave is traveling out of the slide!