I need help with pre lab on ( Wave Propagation and Reflection Visualization in 3D ). All information is attached

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Znex56

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Engineering

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Please solve the Pre lab problems, which is about (Wave Propagation and Reflection Visualization in 3D).

In some of the questions in the Pre Lab, you will be asked to use a group number. Please use number 1 as the group number.

Please solve the problems in a separate sheet and label each answer correspondingly.

Please show all the steps and work taken to get to the final answer.

The attached is the Pre lab sheet along with lecture slides, which may help in solving the questions.

Thank you ,,

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Laboratory #9 EE Science II Laboratory #9 Wave Propagation and Reflection Visualization in 3D Pre-Laboratory Assignment Your name and group number: _______________ NOTATION: Vectors denoted by ~ and → are in the phasor domain and time domain respectively. 1. An electric field Ex is defined in the phasor domain by the expression in Eqn. (1), where the terms E+ and k are functions of your group number. Express the same field in the time domain. 𝐸𝐸�𝑥𝑥 (𝑧𝑧) = (𝐸𝐸 + )𝑒𝑒 −𝑗𝑗𝑗𝑗𝑗𝑗 where, �𝑘𝑘 = 𝜆𝜆 = 2𝜋𝜋 𝜆𝜆 𝐺𝐺 100 (1) � , [𝜔𝜔 = 𝑘𝑘𝑘𝑘] [𝑚𝑚] and G is your group number 𝑚𝑚 𝑐𝑐 = 299 792 458 � � 𝑉𝑉 𝐸𝐸 + = 𝐺𝐺 � � 𝑠𝑠 𝑚𝑚 Time domain representation of the electric field for your group (show work for full credit) = _______________________________ (1 Point) Evaluate your answer for different values of z and t by completing Table I. (1 Point) (a) What happens to the electric field as time increases? (1 Point) (b) OPTIONAL: If it is useful for you, use the x-z coordinate grid attached to the end of this document to plot the E field as a function of z for (1) t = 0 and (2) t = λ/(4c). Note: for each z, the electric field is a vector in the x direction. Table 1: Evaluate your equation and complete the following table z 0 𝒕𝒕 = 𝟎𝟎 𝒕𝒕 = 𝝀𝝀 𝟒𝟒𝟒𝟒 𝒕𝒕 = 𝟐𝟐𝟐𝟐 𝟒𝟒𝟒𝟒 𝒕𝒕 = (0.2)𝜆𝜆 (0.4)𝜆𝜆 (0.6)𝜆𝜆 (0.8)𝜆𝜆 𝜆𝜆  University of South Florida 1 EE209-pre-18_Fall.docx 𝟑𝟑𝟑𝟑 𝟒𝟒𝟒𝟒 Laboratory #9 2. Consider the following equation for the electric field of an electromagnetic plane wave propagating in free space: 𝐸𝐸�𝑥𝑥 (𝑧𝑧) = (𝐸𝐸 + )𝑒𝑒 −𝑗𝑗𝑗𝑗𝑗𝑗 + (𝐸𝐸 − )𝑒𝑒 𝑗𝑗𝑗𝑗𝑗𝑗 Calculate the magnetic field of the plane wave and express it in the time domain. (2 Points) Answer: ��⃗ 𝑯𝑯 = __________________________________ Hint: You can calculate the phasor representation of magnetic field using either of the following two equations and then express your answer in the time domain: � (1) 𝛻𝛻 × 𝐸𝐸� = − 𝑗𝑗𝑗𝑗𝑗𝑗𝐻𝐻 � = 1 𝑘𝑘� × 𝐸𝐸� where 𝑘𝑘� is the direction of propagation and η is the impedance of free space (2) 𝐻𝐻 𝜂𝜂 3. The electric field ��⃗ 𝐸𝐸𝚤𝚤 (𝑧𝑧) of a uniform electromagnetic plane wave propagating in medium 1 (shown in Figure 1) is defined by the expression: 𝐸𝐸�⃗𝑖𝑖 = 𝑥𝑥� 𝐸𝐸0 cos(𝜔𝜔𝜔𝜔 − 𝑘𝑘1 𝑧𝑧) = 𝑥𝑥� 𝐸𝐸0 cos �� 2𝜋𝜋 � �𝑢𝑢𝑝𝑝1 𝑡𝑡 − 𝑧𝑧�� 𝜆𝜆1 where , E0 is the amplitude of the electric field and is equal to your group number, ω is the angular frequency, k1 is the wavenumber in medium 1, up1 is the phase velocity in medium 1, and λ1 is the wavelength in medium 1. The wave incidents a dielectric boundary at z = 0 and a part of the wave transmits through medium 2 with a transmission coefficient τ while the rest is reflected back into medium 1 with a reflection coefficient Γ. The ��⃗𝑡𝑡 (𝑧𝑧) and the reflected wave ��⃗ 𝐸𝐸𝑟𝑟 (𝑧𝑧) can be expressed as: transmitted wave 𝐸𝐸 𝐸𝐸�⃗𝑡𝑡 = 𝑥𝑥� 𝜏𝜏𝐸𝐸0 cos �� 2𝜋𝜋 � �𝑢𝑢𝑝𝑝2 𝑡𝑡 − 𝑧𝑧�� 𝜆𝜆2 𝐸𝐸�⃗𝑟𝑟 = 𝑥𝑥� Γ𝐸𝐸0 cos �� 2𝜋𝜋 � �𝑢𝑢𝑝𝑝1 𝑡𝑡 + 𝑧𝑧�� 𝜆𝜆1 where, up2 and λ2 are the phase velocity and wavelength in medium 2 respectively. Let medium 1 have a permittivity equal to εo and medium 2 have a permittivity equal to 3.88 εo. Answer the following questions. Show work for full credit. 2𝜋𝜋 (1) Show mathematically why 𝜔𝜔𝜔𝜔 − 𝑘𝑘1 𝑧𝑧 = � � �𝑢𝑢𝑝𝑝1 𝑡𝑡 − 𝑧𝑧�. (1 Point) 𝜆𝜆1 (2) What is the reflection coefficient, Γ at the dielectric boundary? (0.5 Points) Γ = ______________ (3) What is the transmission coefficient, τ at the dielectric boundary? (0.5 Points) τ = ______________ (4) Calculate the wavelength and phase velocity in both medium 1 and medium 2. (1 Point) λ1 = _____________ λ2 = ______________ up1 = ______________ up2 = _______________ 𝐸𝐸𝑟𝑟 (𝑧𝑧, 𝑡𝑡) by substituting (5) Let E0 be your group number. Complete the expressions for ��⃗ 𝐸𝐸𝚤𝚤 (𝑧𝑧, 𝑡𝑡), ��⃗ 𝐸𝐸𝑡𝑡 (𝑧𝑧, 𝑡𝑡), and ��⃗ the values calculated above: (1 Point) ��⃗ 𝐸𝐸𝚤𝚤 (𝑧𝑧) = _________________________________ ��⃗ 𝐸𝐸𝑡𝑡 (𝑧𝑧) = __________________________________  University of South Florida 2 EE209-pre-18_Fall.docx Laboratory #9 ��⃗ 𝐸𝐸𝑟𝑟 (𝑧𝑧) = ___________________________________ �⃗𝑖𝑖 (𝑧𝑧, 𝑡𝑡), 𝐻𝐻 �⃗𝑡𝑡 (𝑧𝑧, 𝑡𝑡), and 𝐻𝐻 �⃗𝑟𝑟 (𝑧𝑧, 𝑡𝑡) (1 Point) (6) Write down the expressions for the magnetic fields 𝐻𝐻 Figure 1: A uniform plane wave with electric field ��⃗ 𝐸𝐸𝚤𝚤 (𝑧𝑧, 𝑡𝑡) is incident on a dielectric boundary at z = 0. A part of the wave is transmitted as ��⃗ 𝐸𝐸𝑡𝑡 (𝑧𝑧, 𝑡𝑡) while the rest is reflected as ��⃗ 𝐸𝐸𝑟𝑟 (𝑧𝑧, 𝑡𝑡)  University of South Florida 3 EE209-pre-18_Fall.docx Laboratory #9 x-z Coordinates Grid x z  University of South Florida 4 EE209-pre-18_Fall.docx Laboratory #9 EE Science II Laboratory #9 Wave Propagation and Reflection Visualization in 3D Summary In this laboratory, you will explore plane waves and wave propagation characteristics in 3-dimensional space. The lab guides you through concepts such as plane waves in unbounded media, properties of plane waves in both insulating materials, reflection/transmission of waves at planar boundaries between different materials, and wave progration in a bounded media such as a transmission line. Understanding wave propagation in bounded and unbounded media is crucial to understanding wireless systems such as mobile communication systems, Wi-Fi, radars, etc. In the prelab, you will review concepts learned in coursework by calculating the electric field of a plane wave in free space and at the boundary of a dielectric material. In the lab experience, you will use MATLAB as a software tool for solving electromagnetic-wave based problems. You will use MATLAB to simulate the questions in the prelab and compare the solutions with your hand-calculated results. Finally, you will be presented with the corresponding visualization in 3D space of the electric and magnetic field for a plane wave in free space, in the presence of a dielectric material (i.e. transmission and reflection) and inside a transmission line (i.e. coaxial cable). During this lab, you will: a) Understand the concepts of plane wave and wave propagation. b) Study plane wave propagation using MATLAB and compute the electric field of a plane wave in free space for various time instants. c) Improve your understanding of how dielectric boundary conditions affect wave propagation through a specific example of the electric field of a plane wave in free space in the presence of a dielectric plane perpendicular to the propagation of the wave (i.e. Propagation and reflection) for various time instants. d) Develop necessary skills in the use of MATLAB to calculate and plot a time-varying polarized field and learn to interpret the physical meaning of said plot. e) Gain knowledge about transmission lines and relevant applications in Electrical Engineering. By the end of this experience, you should be able to understand the time variant graphical representation of electromagnetic waves in the time domain, comprehend the concept of a plane wave in three-dimensional spaces, get a sense of how plane waves propagate in transmission lines and be able to compare this with waves in free space. Objectives • Gain an understanding of electric and magnetic field components of a plane wave and their relationship • Gain experience using MATLAB as a tool to solve plane wave problems • Visualize time-varying electromagnetic waves in 3D at the Advanced Visualization Center and through MATLAB simulations. • Link the outcome of MATLAB operations to 3D displays in the Advanced Visualization Center (AVC). Equipment and Software * 3D Visualization Center * MATLAB  University of South Florida 1 EE209-sum-18_Fall.docx Plane-Wave Propagation – Part 4 Electrical Engineering Science II – Electromagnetics Video Presentation 10/29/17 EE Science II - Electromagnetics 1 Outline • Plane wave propagation in lossy media • Electromagnetic power density 10/29/17 EE Science II - Electromagnetics 2 Plane Wave propagation in lossy media "$ $ ! #−& # =0 " &: Propagation Constant " " & = −) * + − , ) 10/29/17 EE Science II - Electromagnetics +. = + − , ) 3 Attenuation of Plane Waves in lossy medium & = / + ,1 2 #$3 4 "$ − & # 4 = 0 3 " 24 " 10/29/17 EE Science II - Electromagnetics 4 Low and high frequency approximations 10/29/17 EE Science II - Electromagnetics 5 Electromagnetic Energy Density Poynting vector: Total power intercepted by A: Average power transmitted by a plane wave: " 5678 #$ = 4̂ 2; " #$ 0 ?"@A 5678 (4) = 4̂ > cos EF 2 ;. EE Science II - Electromagnetics 10/29/17 ;. = ;. > GHI 6
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Explanation & Answer

attached is my answer

Question 1:
With the given values, the expression for the electric field in the x-direction can be given as
follow:

E ( z , t ) = E + e − jkz e − jt ax + E − e − jkz e − jt a y
with ax  a y = az . Therefore,

Ex ( z ) = E + e− jkz e− jt ax
 Ex ( z ) = Ge− j ( kz +t ) ax
It is obvious that the forward wave is decaying as the time increases, which means

lim e− jt = 1 and lim e− jt = 0
t →0

t →

Therefore, it follows that the electric field decreases as time increases.
Consider the following diagram

where

Et ( z ) = Ge

− J ( t + kz )

 Et ( z ) = Ge
 Et ( z ) = Ge
(i) For z = 0 :
At t = 0 , we have

− J ( kct + kz )
2 
 2
−J
ct +
z
 
 

2 
 2
Ex ( z ) = Ge − j 
c0 +
0
 
 
 Ex ( z ) = Ge − j ( 0) = G
At t =


4c

, we have

 2  
 2
Ex ( z ) = Ge − j 
c +
 
4  4
 
 Ex ( z ) = Ge

 
− j 
2




 Ex ( z ) = G  cos − j sin 
2
2

 Ex ( z ) = − jG
At t =

2
, we have
4c

 2  
 2
Ex ( z ) = Ge − j 
c  +
 
4c  4c 
 
 Ex ( z ) = Ge

− j ( )

 Ex ( z ) = G ( cos  )
 E x ( z ) = −G
At t =

3
, we have
4c

3 2 3 
 2
Ex ( z ) = Ge − j 
c +
 
4c  4c 
 
 Ex ( z ) = Ge

 3 
− j

 2 

3
3 

 Ex ( z ) = G  cos
− j sin

2
2 

 Ex ( z ) = G ( 0 − j ( −1) )
 Ex ( z ) = jG
(ii) For z = ( 0.2)  :
At t = 0 , we have:

Ex ( z ) = Ge

2 
 2
− j
ct +
z
 
 

 Ex ( z ) = Ge
 Ex ( z ) = Ge

2
 2

− j
c0 + ( 0.2 )  

 

 2
− j
( 0.2 ) 
 


 Ex ( z ) = G ( 0.3 − j 0.95 )
At t =


4c

, we have

Ex ( z ) = Ge

 2  2
...