Dr. Hugh Blanton ENTC 3331

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ENTC 3331 RF Fundamentals. Dr. Hugh Blanton ENTC 3331. Plane-Wave Propagation. Electric &amp; Magnetic fields that vary harmonically with time are called electromagnetic waves:. In order to simplify the mathematical treatment, treat all fields as complex numbers.

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ENTC 3331

RF Fundamentals

Dr. Hugh Blanton

ENTC 3331

### Plane-Wave Propagation

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

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

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

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For
• It follows that

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• In a vacuum (space)
• In air (atmosphere)

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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.

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

Dr. Blanton - ENTC 3331 - Wave Propagation 9

Recall that
• and apply to both sides of
• but

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0

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wave number

=k2

wave equation

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

wave equation

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

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• The probability of finding a photon at location x is .
• This implies:

Schrodinger’s equation

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strict derivation

heuristic analogy

Schrodinger’s Equation (Postulates of Quantum Mechanics

physics of the macroscopic world

Maxwell’s equations (Newtons laws)

physics of the microscopic world

Wave Equation

particles and waves

particles-wave duality

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Perform the Laplacian

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That is:

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

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no component in the z-direction

x

y “up”

wave crescents

z

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Consequently,
• simplifies to

Dr. Blanton - ENTC 3331 - Wave Propagation 22

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

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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.

Dr. Blanton - ENTC 3331 - Wave Propagation 24

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

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

Dr. Blanton - ENTC 3331 - Wave Propagation 26

Recall

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x

z

y

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

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Transversal electromagnetic wave (TEM)

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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 .

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

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### PLANE WAVE PROPAGATION

POLARIZATION

y

x

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.

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

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

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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.

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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 axand ayare the magnitudesof Ex0and Ey0

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

Dr. Blanton - ENTC 3331 - Wave Propagation 44

Intensity and Inclination Angle
• The intensity of is given by:
• The inclination angleψ

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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,

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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.

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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

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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.

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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

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Linear Polarization

E

+z

B

Can make any angle from the horizontal and vertical waves

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Linear Polarization

Looking up from +z

x-polarized or horizontal polarized

ay=0 ψ=0° or 180°

y-polarized or vertical polarized

ax=0 ψ=90° or -90°

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

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Left-Hand Polarization
• For ax = ay = a, and d = p/2,
• and the modulus or intensity is

Dr. Blanton - ENTC 3331 - Wave Propagation 54

The angle of inclination is:

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

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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.

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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

Dr. Blanton - ENTC 3331 - Wave Propagation 58

x

z

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

w

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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.

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

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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.

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Elliptical Polarization
• g is bounded within the range:

Dr. Blanton - ENTC 3331 - Wave Propagation 66

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

+ implies LH rotation

- implies RH rotation

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Elliptical Polarization

is called the axial ratio and varies between 1 for circular polarization and  for linear polarization

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Elliptical Polarization

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Elliptical Polarization

Positive values of c (sind > 0) LH Rotation

Negative values of c (sind< 0) RH Rotation

Also

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Example 7-3
• Find the polarization state of a plane wave
• Change to a cosine reference:

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Example 7-3
• Find the corresponding phasor:
• Find the phase angles:
• Phase difference:
• Auxiliary:

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can have two solutions:
• or
• Since cosd < 0, the correct value of g is -69.2.

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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.

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Polarization States

The wave is traveling out of the slide!

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