Wave Propagation Wave; in general Wave in free space Wave in perfect and lossy dielectric Power calculation
Wave; in general • •
Generally, wave is a function of space and time, defines the spatial and time varying nature of any phenomena like energy, potential or power.. Continuity equation is a best example for “First order wave equation”, from there the second order equation been derived for many cases, in general they used to call as “Wave equation”. While solving that we can understand the linear and non-linear properties of a wave.
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Left hand side is zero, since it was assumed J=o and ρv=0 Consider the following two functions,
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Define the wave propagation in positive and negative direction. They can be exponential functions, which indicated they are exhibiting the property periodically with time or frequency, by then they will be called “Time-harmonic or Frequency harmonic”
Inside the wave •
Lets consider a wave
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Each terms are defining the property of wave,
Propagation • •
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As definition, the wave is function of time and space This picture is clearly indicating the meaning of propagation At every time, t=0, t=T/4, t= T/2, the amplitude is moving and will be equal to “βz” Which mathematically explain the propagation of spatial waves in time.
From sadiku
Example
Wave propagation in Lossy medium •
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Analysis in Lossy medium is more generalized one, lossless or free space analysis are special case of Lossy case Lossy means the poor conductance property of medium, thus the amplitude or phase shall not be retain after a time/distance of travel Perfect conductor σ=∞, and lossy shall have a finite value. Consider a EM field is expressed by unit amplitude with wave function as ejωt Then, the εaxwell’s equations can be derived as,
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Taking curl for last equation, We studied few identities in the first chapter, use it
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is defining the changes that happening in wave due to medium, thus it reveals the information of the medium intern. Since it derived from it’s square, it usually defined by complex variables
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Attenuation consant α=0, it become lossless medium, else it’s a lossy one, this is giving changes in amplitude and phase, later we will study as “Skin effect” is propagation constant, defines the wave’s nature.
Wave impedance •
The impedance experienced by the wave is called Wave impedance, this can be estimated from E and H field component. First we shall attempt to derive the wave equation of fields in the lossy medium
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Left hand side of wave equation
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Since is complex one, then the solution for this equation will be like
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The solution has interesting meaning,
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z means the flow, there the positive sign denotes the negative direction, vice versa Exs(z) is the magnitude of wave, then, the wave can be rewritten as
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Or
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Similarly,
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From this equation only all the wave propagation concepts been analyzed, especially, the antennas. Here while looking close unto the e-αz, it says at αz=1, the wave amplitude reduces by e-1amount, where e-1 is the attenuation unit (measured in Neper/meter).
When σ=0, then α=0, is the case of “δossless”, will have only phase constant
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Any lossy medium will have conductance, lead to conduction current and displacement current.
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Is defining the phase shift created by the medium, named as “δoss tangent” A dielectric medium is said be “Perfect or δossless”, when σ<<ω and good conductor when σ>>ω Loss tangent can be arrived in another way
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From wave equation
Special case: Lossless • •
Lossless dielectric has
Special case: Free space
Free space Wave impedance
EM Waves • •
Consider in free space, the Electric field Then, the Magnetic field
Wave Polarization •
While looking from the direction of propagation, ak, if one can able to draw the variations of amplitude, that shows the polarization of wave. Each H or E field is called as “Plane wave”.
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As shown in figure, if the E or H field are not in the direction of propagation, then that is called Transverse Electromagnetic field (TEM) wave. Here the spotting the movement of amplitude, that’s called “Polarization” Linear Circular Elliptical
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Plane waves in Good conductor •
Perfect or Good conductor shall be defined by
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So the other constants
Skin Effect and Skin Depth • • •
Amplitude of wave is decreasing rapidly along the distance, called “Skin efffect” The attenuation of E field (or H) is defined by the term Here, at a distance of travel, , the amplitude is reduced by the amount of e-1, is called Skin depth. e-1 is about 0.368 (or 37%)
