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ECE 480 Wireless Systems Lecture 3 Propagation and Modulation of RF Waves. Circular Polarization. Magnitudes of the x – and y – components of are equal Phase difference is. Left – Hand Circular (LHC) Polarization. Right – Hand Circular (RHC) Polarization.
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ECE 480 Wireless Systems Lecture 3 Propagation and Modulation of RF Waves
Circular Polarization Magnitudes of the x – and y – components of are equal Phase difference is Left – Hand Circular (LHC) Polarization Right – Hand Circular (RHC) Polarization
Linear polarization = f (z, t) f (z, t) Circular polarization f (z, t) = f (z, t)
Back to LHC Polarization Consider the LHC wave at z = 0 Inclination angle decreases with time
The direction of polarization is defined in terms of the rotation of as a function of time in a fixed plane orthogonal to the direction of propagation
Example RHC Polarized Wave An RHC polarized plane wave with electric field modulus of 3 mV/m is traveling in the + y direction in a dielectric medium with f = 100 MHz Obtain expressions for
Solution The wave is traveling in the + y direction. Therefore, the field components are in the x and z directions. direction of propagation
Assign a phase angle of 0 o to the z component of (arbitrary) The x component of will have a phase shift Both components have a magnitude of a = 3
Elliptical Polarization Most general case The tip of traces an ellipse in the x – y plane Can be left – handed or right - handed Major axis: Minor axis: Rotation Angle Ellipticity Angle
Rotation angle The shape and rotation are defined by the ellipticity angle R = 1 Circular R = Linear axial ratio
Circular Linear
How is the type of polarization determined? Positive values of corresponding to sin > 0 define left – handed rotation Negative values of corresponding to sin < 0 define right – handed rotation a x and a y are, by definition, > 0 Two possible values of in this range
Example: Polarization State Determine the polarization of a plane wave with an electric field given by Solution Convert the sin term to a cos term by subtracting 90 o Convert the – cos term to a + cos term by adding 180 o
There are two possible solutions for since the tan function is positive in both the first and third quadrants Which is correct?
By a similar analysis, The wave is elliptically polarized and the rotation is left - handed
Plane – Wave Propagation in Lossy Media can be written as = attenuation constant = phase constant
Equate the real and imaginary parts Solve for and
For a uniform plane wave traveling in the + z direction with an electric field the wave equation becomes The solution is
The magnitude of is Decreases exponentially with e - z also decreases exponentially with e - z Define: Skin Depth, s Distance that a wave must travel before it is attenuated by
In a perfect dielectric In a perfect conductor
Expressions are valid for any linear, isotropic, homogeneous medium Low – Loss Dielectric Quasi – Conductor (Semiconductor) Good Conductor
Low – Loss Dielectric Consider For Divide into real and imaginary parts - Same as for lossless medium
Example – Plane Wave in Seawater A uniform plane wave is traveling downward in the + z direction in seawater, with the x – y plane denoting the sea surface and z = 0 denoting a point just below the surface. The constitutive parameters of seawater are: The magnetic field intensity at z = 0 is given by • Determine expressions for • The depth at which the amplitude of E is 1% of its value at z = 0
Solution a. The general expressions for the phasor fields are Seawater is a good conductor at 1 KHz
at z = 0: Compare with original expression
Note that they are no longer in phase. The electric field always leads the magnetic field by 45 o. b. Set the amplitude to 0.01
Electromagnetic Power Density Define: Poynting Vector Direction of S is in the direction of propagation, k unit vector normal to the surface Power through a surface, A
Plane Wave in a Lossless Medium Consider a plane wave traveling in the + z direction Want to find the power density vector, S
Phasor – Domain Approach is valid for any media
Homework The electric field of a plane wave is given by Identify the polarization state, determine the polarization angles (, ), and sketch the locus of E (0, t) for each of the following cases
Homework In a medium characterized by Determine the phase angle by which the magnetic field leads the electric field
Radiation and Antennas • An antenna may be considered as a transducer that converts a guided EM wave to a transmitted wave or an incident wave to a guided EM wave • Antenna dimensions are generally referred to in wavelength units
Reciprocity • Antenna radiation pattern: The directional function that characterizes the distribution pattern radiated by an antenna • Isotropic antenna: A hypothetical antenna that radiates equally in all directions • Used as a reference radiator to compare antennas • Reciprocal antennas: Antennas that have the same radiation patterns for transmission as for reception
Two aspects of antenna performance • 1. Radiation Properties • Direction of the radiation pattern • Polarization state of the radiated wave in the TX mode (Antenna Polarization) • In the RX mode, the antenna can extract only that component of the wave whose E – field is parallel to that of the antennas polarization direction • 2. Antenna Impedance • Pertains to the impedance match between the antenna and the generator