School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007

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1 School of Electical and Compute Engineeing, Conell Univesity ECE 303: Electomagnetic Fields and Waves Fall 007 Homewok 1 Due on Nov. 8, 007 by 5:00 PM Reading Assignments: i) Review the lectue notes. ii) Review sections , 9.7, 9.8 of the papeback book Electomagnetic Waves. Poblem 1.1: (A wieless link of two half-wave dipoles) Conside the following setup fo a wieless link (base station and a mobile station) consisting of two halfwave dipoles. The powes shown ae the NET powes deliveed to o eceived fom the antennas. Base station θ P out- P in-1 θ 1 L d P in- Mobile station P out-1 d 1 Assume that: o o θ 1 = 30 θ = 150 L = 1km d1 = d = 15 cm f = 1.0 GHz The antennas ae always assumed to be matched to thei espective diving/eceiving cicuits. The total impedance of each antenna was measued and was found to equal: Z A = j 43 Ω (the eactive pat of the impedance of an ideal half-wave dipole is always j 43 Ω ). a) What is the dissipative pat R diss of the antenna impedance? b) What is the adiative efficiency η ad of each antenna? 1

2 G 1 of the base station antenna in the diection of the mobile station? Give the epession and the numeical value as an answe. Make sue you ae using the IEEE definition of the antenna gain. c) What is the gain ( θ,φ) A of the mobile station antenna looking in the diection of the base station? Give the epession and the numeical value as an answe. d) What is the effective aea ( θ,φ) e) If the base station tansmits and the mobile station eceives, find the atio P out Pin 1. Give the epession and the numeical value as an answe. f) If the mobile station tansmits and the base station eceives, find the atio P out 1 Pin. Give the epession and the numeical value as an answe. g) Now suppose some enginee made a mistake in designing the mobile station so that the impedance * looking into the tansmission line fom the antenna end is not the matched value: Z A = 100 j 43 Ω but is 100 Ω. Find the atio P out Pin 1 in this case when the mobile station antenna is not matched to its cicuit. Poblem 1.: (N-tun small wie loop antenna) Conside the N-tun small wie loop antenna shown below. In this poblem you will veify the antenna theoem: λ A ( θ, φ) = G( θ, φ) 4π fo the small wie loop antenna in the same way as was done in the lectue notes fo a shot-dipole antenna. Since the epession fo gain has aleady been deived in the lectue notes, you will need to calculate the antenna effective aea diectly and then veify the above elation. Conside plane wave adiation with Poynting vecto magnitude P incident fom the θ diection. You will need to find the powe eceived P ec by a matched cicuit connected to the antenna. z θ Incident adiation H + V th - N tuns Radius a a) Find the magnitude H of the H-field fo the incident plane wave given its Poynting vecto magnitude P.

3 b) Using Faaday s law, find the voltage phaso V th geneated acoss the open-cicuited ends of the loop when a plane wave with magnetic field phaso H passes though the loop. Assume that the plane wave is incident fom the θ diection and the H-field is polaized in the θˆ -diection (as shown in the figue above). Hint: don t foget the angula dependence hee. Note that when angle θ is 0-degees no magnetic flu passes though the loop and when the angle θ is 90-degees maimum magnetic flu passes though the loop. c) Fom you esult in pat (b), find the maimum powe P ec deliveed to a matched load by the antenna. Assume that the impedance of the antenna equals its adiation esistance that was calculated in the lectue notes. d) Cast you answe in pat (c) in tems of the incident powe pe unit aea (i.e. the Poynting vecto magnitude P) and find the effective aea A ( θ,φ) of the antenna. Veify that you answe satisfies the antenna theoem. Poblem 1.3: (Electomagnetic scatteing fom a conducting loop) Conside a conducting loop of a wie with adius a (a << λ) and with its ais oiented along the z-ais. The total esistance of the wie loop is R diss. A plane wave is incident on the loop as shown in the figue below. The H-field phaso of the incident plane wave at the location of the loop is given as, Hi ( = 0) = Hi ( cos( θ ) ˆ sin( θ ) zˆ ) z θ Incident adiation E H Incident plane wave a a) Find the phaso I fo the cuent that is induced in the loop due to the incident plane wave. b) Find the fa-field epession E s ff ( ) fo the scatteed E-field? Think of the scatteed field as the field adiated by the induced cuent in the antenna. c) Find the total scatteed powe P s. d) Find the scatteing coss-section σ s of the conducting loop. Poblem 1.4: (Electomagnetic scatteing fom a pefect metal sphee) Conside a pefect metal sphee with adius a (a << λ). A plane wave is incident on the sphee as shown in the figue below. The E-field of the incident plane wave at the location of the sphee is given as, 3

4 Ei ( = 0 ) = zˆ Ei E o Incident plane wave a a) Find the dipole moment phaso p ( small p ) that is induced in the sphee due to the incident plane wave. b) Find the fa-field epession E ( ) c) Find the total scatteed powe P s. s ff fo the scatteed E-field? d) Find the scatteing coss-section σ s of the metal sphee and compae it to its actual full coss-sectional aea which equals π a. Poblem 1.5: (Hetzian dipole linea aay) Conside an aay of N Hetzian dipoles sepaated by distance d and pointing towads the +z-diection. The wavelength of the adiation is λ. All the dipoles have the same cuents (magnitudes and phases). The angle φ is defined fom the +ve y-ais (as shown above). The esults fom this poblem will be used in the net poblem. y φ' d A vey geneal plot of the adiation patten is shown below. The specifics of the plot ae not elated to this poblem. The plot is only fo illustative puposes. 4

5 1-st y 0-th -nd p(θ =π /, φ ) a) Show (based on physical easoning) that thee is a maimum in the adiation patten fo φ = 0, iespective of the wavelength λ and the distance d. This is the 0-th ode maimum. b) Find a condition (without calculating the aay facto) in tems of the wavelength λ and the distance d that gives the angles φ ' 1 fo the 1-st ode maima (see the figue above). c) Find a condition (without calculating the aay facto) in tems of the wavelength λ and the distance d that gives the angles φ ' fo the -nd ode maima (see the figue above). F fo the dipole aay and, noting that φ' = π φ, veify that the angles in pats (b) and (c) do indeed yield maima in the adiation patten. d) Calculate the aay facto ( θ = π, φ) e) Conside the fist ode maima. You need to find the angula full-width φ' of the lobes coesponding to the fist ode maima. If you plot the aay facto on a linea plot you will get something shown below. Find an epession fo the value φ' in tems of the paametes N, d, φ ' 1 and λ. -nd 1-st 0-th 1-st -nd F ( θ = π, φ ) Nomalized to unity Plot fo: N = 10 9 kd = π φ φ 5

6 Poblem 1.6: (Diffaction gating spectomete) In this poblem you will look at the opeation of a diffaction gating spectomete. A diffaction gating consists of N openings (o lines) uled in a metal coating on a glass substate (as shown in the figue below). Each opening (o apetue) has width L in the -diection and L z in the z-diection. The length L z is vey lage and could be consideed infinite fo all pactical puposes. So the openings ae long and naow slits. Each opening can be consideed an apetue antenna, and so the diffaction gating can be consideed a linea aay of apetue antennas. CCD camea y L φ' L d Incident adiation of unknown wavelength H E a The element facto fo this antenna aay is just the fa-field calculated in the lectue notes fo a single ectangula apetue: ( ) ( ) ( ˆ j k j sin ) ( ) ( ) ( ) sin( ),, sin k k L kzlz Ε θ φ = Eff = θ Ea θ e L Lz π k L kzlz Note that in the above epession the θ, φ dependence is embedded in the k and k z tems: = k. ˆ = kˆ. ˆ k sin( θ ) cos( φ) = k. zˆ = kˆ. zˆ k cos( θ ) k = k z = The slit width in the z-diection is vey lage. Fo L z, sin( kzlz ) 1 fo θ = π kzlz 0 othewise which means that the output beam does not diffact significantly in the θ -diection (only in the φ- diection). A CCD camea is placed in font of the diffaction gating at a distance L fom it. The CCD camea can ecod the image of the adiation patten on a flat plane pependicula to the y-ais. 6

7 The spectomete (as the name suggests) is used to find the spectal content of a light souce (i.e. which wavelengths ae pesent in the light fom a souce). It functions as follows. Plane wave of unknown wavelength is incident upon the gating (as shown in the figue). The adiation coming out fom each slit is in phase (since it comes fom the same incident plane wave). Radiation coming out fom diffeent slits intefees to give a adiation patten with maima and minima. The locations of the maima in the hoizontal diection ae ecoded by the CCD camea. The locations of the maima in the hoizontal diection can be used to figue out the wavelength of the incident adiation since diffeent wavelengths will give maima fo diffeent angles. Usually only the 1-st ode maima ae impotant fo this application. Assume a diffaction gating with the following specifications: L = 0.1µ m d = 1.5 µ m N = 300 L = 300mm Suppose light of wavelength 0.5 µm (geen light) is incident upon the spectomete. You need only concen youself with the maima and minima in the (y > 0, > 0) egion. Satisfy youself that the angula locations of the maima and minima will be detemined completely by the aay facto and not the element facto because of the small slit size assumed. a) Find the angula location of the fist ode maimum, and find the hoizontal displacement (fom the y- ais) of the spot (in mm) ecoded on the CCD camea due to the 1-st ode maimum. b) Find the hoizontal size of the spot (in mm) on the CCD camea due to the 1-st ode maimum. Recall that the lobe coesponding to the 1-st ode maimum has a finite angula width and this angula width tanslates into a finite spot size of the image on the CCD camea. Now conside two diffeent wavelengths, 0.5 µm and µm, that ae 1 nm apat in wavelength and both ae incident at the same time on the spectomete. c) Find the hoizontal sepaation (in mm) between the spots on the CCD camea coesponding to the 1-st ode maima fo the two wavelengths. If the spectomete is to have a wavelength esolution of 1 nm (i.e. it can esolve wavelengths as closely spaced as 1 nm) the spot size calculated in pat (b) must be smalle than the sepaation between the spots of the two wavelengths calculated in pat(c). If the spots ovelap then the CCD will ecod one bight spot (as opposed to two distinct spots) and the spectomete will not be able to esolve the two wavelengths that ae 1 nm apat. d) Based on you calculation in pats (b) and (c) does the spectomete have a wavelength esolution of 1 nm? e) What can you do to incease the esolution of the spectomete? (e.g. change the distance L, change the distance d, change the numbe of apetues N, etc.). 7