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Prоblem 1 (15 pоints) A unifоrm аrrаy hаs (hypothetical) isotropic elements placed along the z-axis which have constant current magnitudes and a phase progression of β{"version":"1.1","math":"β"} radians, i.e., for neighboring elements, the one at the larger distance along the z-axis is driven at some frequency such that Ii+1=Iiejβ{"version":"1.1","math":"Ii+1=Iiejβ"}. The normalized array factor for such an N-element array is AF=1Nsin(Nψ/2)sin(ψ/2){"version":"1.1","math":"AF=1Nsin(Nψ/2)sin(ψ/2)"}, where ψ=kdcosθ+β{"version":"1.1","math":"ψ=kdcosθ+β"} with d the distance between elements. Design a 5-element array with half wavelength between the elements and a beam maximum at θ=π/3{"version":"1.1","math":"θ=π/3"}, i.e., find β{"version":"1.1","math":"β"}, and sketch the radiation pattern as a function of θ{"version":"1.1","math":"θ"}. Hint: The AF(ψ){"version":"1.1","math":"AF(ψ)"} has N - 1 complete lobes for ψ∈0, 2π{"version":"1.1","math":"ψ∈0, 2π"}. Problem 2 (15 points) a. (10 points) An infinite current sheet on the z = 0 plane in unbounded free space has an electric current density J=(x^+jy^)δ(z){"version":"1.1","math":"J=(x^+jy^)δ(z)"}, assuming exp(jwt){"version":"1.1","math":"(jwt)"} time dependence and with δ(·){"version":"1.1","math":"δ(·)"} the Dirac delta. Find expressions for the electric and magnetic field everywhere. b. (5 points) Describe completely the polarization for the wave propagating in the half-space z>0{"version":"1.1","math":"z>0"}. Problem 3 (15 points) A plane wave is normally incident on an aperture in an infinite perfect electric conductor (PEC) screen. The aperture is circular with radius five wavelengths (a=5λ){"version":"1.1","math":"(a=5λ)"}. Consider that the screen is infinitesimally thin and that the field in the aperture is equal to the incident field (which is incorrect but becomes more reasonable as the aperture size increases, with a given wavelength). a. (10 points) Show the use of the principle of (Huygens') equivalent currents in the representation of this problem, describing each step clearly. The goal here is to represent the radiation through the aperture into a semi-infinite free space region using the free space Green's function. Please explain your reasoning. b. (5 points) What is the approximate directivity of this aperture antenna? Hint: You may find the relation D0=4πAe/λ2{"version":"1.1","math":"D0=4πAe/λ2"} useful in this regard. Make note of any approximations and their significance. Problem 4 (15 points) An infinitesimal z-directed dipole located at the origin has far-fields, assuming exp(jwt){"version":"1.1","math":"(jwt)"} given by E=jwμ0(I∆z)e-jkr4πrsinθ θ^{"version":"1.1","math":"E=jwμ0(I∆z)e-jkr4πrsinθ θ^"} H=jk0(I∆z)e-jkr4πrsinθ ϕ^{"version":"1.1","math":"H=jk0(I∆z)e-jkr4πrsinθ ϕ^"}, where k0{"version":"1.1","math":"k0"} is the free space wave number, I{"version":"1.1","math":"I"} is the uniform current, and ∆z{"version":"1.1","math":"∆z"} is an infinitesimal length. Consider an antenna of this type located vertically directly (at z=0+{"version":"1.1","math":"z=0+"}) above an infinite planar PEC in the z = 0 plane. a. (5 points) Find an expression for the time-average radiated power. Note: ∫0πsin3θdθ=4/3{"version":"1.1","math":"∫0πsin3θdθ=4/3"}. b. (5 points) Find the maximum gain of this antenna. c. (5 points) Find an expression for the radiation resistance. Problem 5 (15 points) This question relates to Lorentz reciprocity in free space and as it pertains to antenna concepts, given by a. (7 points) In class, we derived the following reaction integral. Explain the key steps and assumptions, and the meaning of the quantities in this equation. b. (4 points) Under what assumptions is ∮(E1×H2-E2×H1)·ds=0{"version":"1.1","math":"∮(E1×H2-E2×H1)·ds=0"}, i.e., prove this result and indicate the assumptions. c. (4 points) On what physical basis are antenna radiation characteristics reciprocal and how is this used in the analysis of antenna systems? Problem 6 (10 points) Consider the numerical solution of the thin wire antenna in free space with a current density of the form J=z^I(z')2πaδ(p'-a)P(z', l/2){"version":"1.1","math":"J=z^I(z')2πaδ(p'-a)P(z', l/2)"}, where I(z'){"version":"1.1","math":"I(z')"}is unknown, a is the wire radius, the wire length is l, and P is a unit amplitude pulse function centered at the origin and of half-length l/2. Assuming that we have exp(jwt){"version":"1.1","math":"(jwt)"} time dependence and define B=∇×A{"version":"1.1","math":"B=∇×A"}, E=-jwA-jwμ0ε0∇(∇·A){"version":"1.1","math":"E=-jwA-jwμ0ε0∇(∇·A)"}, where A satisfies (∇2+k02)A=-μ0J{"version":"1.1","math":"(∇2+k02)A=-μ0J"}, with k0=wμ0ε0{"version":"1.1","math":"k0=wμ0ε0"} the free space wave number, leading to a solution in unbounded free space given by A(r)=μ0∫J(r')e-jk0r-r'4πr-r'dv'{"version":"1.1","math":"A(r)=μ0∫J(r')e-jk0r-r'4πr-r'dv'"} Develop an integral equation that can be used to solve for I(z'){"version":"1.1","math":"I(z')"} without going through the details of evaluating the Green's function on the surface of the cylinder. In this process, assume that there is an incident field (due to some gap voltage) given by Ei{"version":"1.1","math":"Ei"} that is known. Hint: You can assume a PEC cylinder to form an equation in the unknown current. The current on the antenna would give rise to the scattered field, Es{"version":"1.1","math":"Es"}. The total electric field is then E=Ei+Es{"version":"1.1","math":"E=Ei+Es"}. The resulting equation could then be solved on a computer using a basis function expansion where the coefficients are extracted using testing functions. Problem 7 (15 points) Penzias and Wilson estimated the cosmic microwave background as having a temperature of 3.5 K. Assume that they had a 6-meter diameter square horn antenna and made a measurement at 4 GHz looking at a dark region of the sky. Estimate the antenna temperate, TA{"version":"1.1","math":"TA"}, assuming no other sources of noise, and then draw the equivalent circuit for the lossless receiving antenna with some load impedance. Note that with the Nyquist picture, the noise power available from a resistor is kTB, assuming a bandwidth B and k is Boltzmann's constant, making the noise voltage source vn=4kTBR{"version":"1.1","math":"vn=4kTBR"}. Potentially relevant information: kBTA=CpAe∫ΩB(θ,ϕ)Un(θ, ϕ)dΩUn(θ, ϕ)=U(θ, ϕ)U(θ,ϕ)maxΩA=∫Un(θ, ϕ)dΩD0=4πΩAAe=λ2D04πB(θ,ϕ)=2kBTS(θ, ϕ)λ2{"version":"1.1","math":"kBTA=CpAe∫ΩB(θ,ϕ)Un(θ, ϕ)dΩUn(θ, ϕ)=U(θ, ϕ)U(θ,ϕ)maxΩA=∫Un(θ, ϕ)dΩD0=4πΩAAe=λ2D04πB(θ,ϕ)=2kBTS(θ, ϕ)λ2"} Congratulations, you are almost done with this exam. DO NOT end the Honorlock session until you have submitted your work. When you have answered all questions: Use your smartphone to scan your answer sheet and save the scan as a PDF. Make sure your scan is clear and legible. Submit your PDF as follows: Email your PDF to yourself or save it to the cloud (Google Drive, etc.). Click this link to submit your work: Final Exam Return to this window and click the button below to agree to the honor statement. Click Submit Quiz to end the exam. End the Honorlock session.
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