1School of Electrical and Computer Engineering, Cornell University ECE 303: Electromagnetic Fields and Waves Fall 2007 Homework 12 Due on Nov. 28, 2007 by 5:00 PM Reading Assignments: i) Review the lecture notes. ii) Review sections 9.1-9.5, 9.7, 9.8 of the paperback book Electromagnetic Waves. Problem 12.1: (A wireless link of two half-wave dipoles) Consider the following setup for a wireless link (base station and a mobile station) consisting of two half-wave dipoles. The powers shown are the NET powers delivered to or received from the antennas. Assume that: GHz 0.1cm 15km 1150302121====== fddLooθθ The antennas are always assumed to be matched to their respective driving/receiving circuits. The total impedance of each antenna was measured and was found to equal: Ω+=43100 jZA (the reactive part of the impedance of an ideal half-wave dipole is always Ω43j ). a) What is the dissipative part dissR of the antenna impedance? b) What is the radiative efficiency radη of each antenna? Pin-1 d1 Pout-2 d2 1θ 2θ L Base station Mobile station Pout-1 Pin-22 c) What is the gain ()φθ,1G of the base station antenna in the direction of the mobile station? Give the expression and the numerical value as an answer. Make sure you are using the IEEE definition of the antenna gain. d) What is the effective area ()φθ,2A of the mobile station antenna looking in the direction of the base station? Give the expression and the numerical value as an answer. e) If the base station transmits and the mobile station receives, find the ratio 12 −− inoutPP . Give the expression and the numerical value as an answer. f) If the mobile station transmits and the base station receives, find the ratio 21 −− inoutPP . Give the expression and the numerical value as an answer. g) Now suppose some engineer made a mistake in designing the mobile station so that the impedance looking into the transmission line from the antenna end is not the matched value: Ω−= 43100*jZA but is Ω100 . Find the ratio 12 −− inoutPP in this case when the mobile station antenna is not matched to its circuit. Problem 12.2: (N-turn small wire loop antenna) Consider the N-turn small wire loop antenna shown below. In this problem you will verify the antenna theorem: () ()φθπλφθ,4,2GA = for the small wire loop antenna in the same way as was done in the lecture notes for a short-dipole antenna. Since the expression for gain has already been derived in the lecture notes, you will need to calculate the antenna effective area directly and then verify the above relation. Consider plane wave radiation with Poynting vector magnitude P incident from the θ direction. You will need to find the power received Prec by a matched circuit connected to the antenna. a) Find the magnitude |H | of the H-field for the incident plane wave given its Poynting vector magnitude P. xzN turns Radius a θ+ - Incident radiation Vth H3b) Using Faraday’s law, find the voltage phasor Vth generated across the open-circuited ends of the loop when a plane wave with magnetic field phasor H passes through the loop. Assume that the plane wave is incident from the θ direction and the H-field is polarized in the θˆ-direction (as shown in the figure above). Hint: don’t forget the angular dependence here. Note that when angle θ is 0-degrees no magnetic flux passes through the loop and when the angle θ is 90-degrees maximum magnetic flux passes through the loop. c) From you result in part (b), find the maximum power Prec delivered to a matched load by the antenna. Assume that the impedance of the antenna equals its radiation resistance that was calculated in the lecture notes. d) Cast your answer in part (c) in terms of the incident power per unit area (i.e. the Poynting vector magnitude P) and find the effective area ()φθ,A of the antenna. Verify that your answer satisfies the antenna theorem. Problem 12.3: (Electromagnetic scattering from a conducting loop) Consider a conducting loop of a wire with radius a (a << λ) and with its axis oriented along the z-axis. The total resistance of the wire loop is dissR . A plane wave is incident on the loop as shown in the figure below. The H-field phasor of the incident plane wave at the location of the loop is given as, ()() ()()zxHrHiiˆsinˆcos0θθ−==rr a) Find the phasor I for the current that is induced in the loop due to the incident plane wave. b) Find the far-field expression ()rEffsrr− for the scattered E-field? Think of the scattered field as the field radiated by the induced current in the antenna. c) Find the total scattered power Ps. d) Find the scattering cross-section σs of the conducting loop. Problem 12.4: (Electromagnetic scattering from a perfect metal sphere) Consider a perfect metal sphere with radius a (a << λ). A plane wave is incident on the sphere as shown in the figure below. The E-field of the incident plane wave at the location of the sphere is given as, a Incident plane wave E z Incident radiation H θ4()iiEzrEˆ0==rr a) Find the dipole moment phasor pr (“small p”) that is induced in the sphere due to the incident plane wave. b) Find the far-field expression ()rEffsrr− for the scattered E-field? c) Find the total scattered power Ps. d) Find the scattering cross-section σs of the metal sphere and compare it to its actual full cross-sectional area which equals 2aπ. Problem 12.5: (Hertzian dipole linear array) Consider an array of N Hertzian dipoles separated by distance d and pointing towards the +z-direction. The wavelength of the radiation is λ. All the dipoles have the same currents (magnitudes and phases). The angle φ’ is defined from the +ve y-axis (as shown above). The results from this problem will be used in the next problem. A very general plot of the radiation pattern is shown below. The specifics of the plot are not related to this problem. The plot is only for illustrative purposes. a Incident plane wave Eo x y 'φd5 a) Show (based on physical reasoning) that there is a maximum in the radiation pattern for φ’ = 0, irrespective of the wavelength λ and the distance d. This is the 0-th order maximum. b) Find a condition (without calculating the array factor) in terms of the wavelength λ and the distance d that gives the angles 1'φ for the 1-st order maxima (see the figure above). c) Find a condition (without calculating the array factor) in terms of the