1- Uzunluğu b olan (b > λ) tek kutuplu bir tel anten xy düzlemindeki iletken plakaya dik olan z ekseninde bulunmakta olup I

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ÏἈχαϊκός Λειβαδάς
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1 1- Uzunluğu b olan (b > λ) tek kutuplu bi tel anten xy düzlemindeki iletken plakaya dik olan z ekseninde bulunmakta olup I sabit akımı taşımaktadı. a) ( θφ,, ) noktasındaki elektik alanı hesaplayın. 1- A monopole wie antenna has a length b (b > λ). It lies along the z-axis pependicula to pec gound plane (z=). It is excited with a unifom cuent I. a) Find the adiated electic field at point ( θφ,, ) by using the image method. b b b jkz cosθ A e jkz cos e jkz cos e e θ θ ( ) = µ ( z ) e dz ˆµ I e dz ˆµ I 4π I = z b 4π = z b 4π jkcosθ b e jsin( kbcos θ) e sin( kbcos θ) = ˆ zµ I Eθ = jω( sin θ) Az = jωµ I sinθ 4π jkcosθ 4π kcosθ b) Show that the effective length is h = bθˆ when θ= π. c) Given that its adiation esistance is R = η( bλ), calculate its gain when θ= π.

2 1- Bi antendeki aşağıdaki akımdan kaynaklanan uzak elektik alanı E ( θφ, ) küesel koodinatlada hesaplayın. 1- The cuent on a taveling wave antenna is given by the following equation. Detemine the fa electic field E ( θφ, ) in spheical coodinates adiated by this cuent. jβy yˆ Iδ( x) δ( ze ) y L J ( ) = elsewhee ˆ = xˆ sinθcosφ+ yˆ sinθsinφ+ ˆz cosθ xˆ = ˆ sinθ cosφ+ θˆ cosθ cosφ φˆ sinφ yˆ = ˆ sinθ sinφ+ θˆ cosθ sinφ+ φˆ cosφ ˆz = ˆcos θ θˆ sinφ L L jkˆ jβy jky sinθsin φ jy ( ksinθsin φ β) e = yˆ y A µ ( ) e dv = ˆIe e dy = ˆI e dy 4 π J V y y e e 1 e = yˆ µ I = µ ( ˆ sinθsinφ+ θˆ cosθsinφ+ φˆ cos φ) I 4 π jk ( sinθsin φ β) 4π jlk ( sinθsin φ β) jlk ( sinθsin φ β) jlk ( sin sin ) 1 ( ˆ ˆ e θ φ β e E= jωa A ) j I ˆ ˆ θθ+ φφ = ωµ cosθsinφ cosφ 4 π θ + φ jk ( sinθsin φ β) e 1 jk ( sinθsin φ β)

3 1- A unifom line souce at the oigin along the z-axis is given as Iz ( ) = I fo z L. a) Evaluate the adiation integal fo the fa field using e = 4π jβ jβz cosθ A ˆ z Iz ( ) e dz. A z jβ L/ jβ e jβz cosθ ILe sin [( βl/)cosθ] = Ie dz 4π = L/ 4 π ( βl/)cosθ b) Find fa electic field due to this souce. jβ ˆ jωµ ILe sin [( βl/)cosθ] E= jωµ sinθa sin ˆ z θ= θ θ. 4 π ( βl/)cosθ - A linea antenna in fee space occupies L z L and caies the taveling wave cuent j z I ( z) = I e β, whee β is constant. a) Find the fa-zone electic field poduced by the antenna L ˆjηke sin cos ˆ [( cos j z jkz j ki k ) L] e sin Ie e dz sin θ β β η E= θ θ θ θ 4π = θ π kcosθ β L b) Let β = k, f = 5 MHz, I = ma and L = cm. Compute the maximum electic field magnitude when = 1 km. β ˆjηkI sin[ kl( cos 1) ] e θ = k E= θ Lsinθ π kl(cosθ 1) kl = ωl c =.1 << 1 sin kl ( cosθ 1) kl( cosθ 1) jη ki e π jk ˆ E θ L sinθ is maximum when ηki L θ = π. E max = 5.1 μv/m π

4 1- Conside a staight wie antenna of length L oiented along the z-axis fom z = L to z = L. The antenna caies a cuent densityj ( ) = ˆz I cos( πzl) δ( x) δ( y). a) Detemine the fa-zone electic and magnetic field adiated by this antenna. L e cosα e jkz cosθ ( ) µ ( ) e dv µ ˆ cos( π ) 4π = V 4π L A J z I z Le dz kl e πlcos( cosθ) A ( ) ˆz Iµ, ˆz = ˆcos θ θˆ sinθ 4 π ( kl) cos θ π kl ˆωµ ILe cos ( cosθ) E ( ) = θ sinθ ( kl) cos θ π kl ˆ E ˆωµ ILe cos ( cosθ) H ( ) = = φ sinθ η η ( kl) cos θ π b) Detemine the powe adiated by the antenna if kl=π. π π π kl 1 1 ωµ IL cos ( cosθ) 3 P = Eθ sinθdd θ φ π sin θdθ η = η ( kl) cos θ π IL 1 π π π ωµ cos ( cosθ) 1. ωµ IL P = dθ 36.6I 4 3 η = π sinθ ηπ fo kl=π c) Detemine the adiation esistance of the antenna if kl=π. R P = = 73.Ω I d) Detemine the diectivity of the antenna if kl=π. π 4 πu ( θφ, ) 1 4πE cos( ) θ cosθ D( θφ, ) = = = 1.64 P η 36.6I sinθ D= [ Dθφ (, )] max = 1.64=.15 db

5 - Deive the fa field appoximation. Hint: use the definitions of the α vectos, and then use the fist two tems of the Taylo expansion (1 + x) 1+αx seveal times. 1- An infinitesimal magnetic dipole of constant cuent M is placed symmetically about the oigin and diected along the y-axis. a) Deive the fa-zone electic field b) Deive the fa-zone magnetic field

6 1- Conside the adiation fom a long wie antenna along the z-axis with cuent: jkz I e, L z L I ( z ) =, z > L jβ e jβ z cosθ a) Evaluate the adiation integal fo the fa field using A = z ˆ I ( z ) e dz 4π. b) Find the adiated fa electic field using E = jωµ sinθ A θ ˆ z c) Appoximate you esults fo small angles θ << 1 adian ( sinθ θ, cosθ 1 θ ). Show that the magnitude of electic field is appoximately E ωµ I L klθ 4π klθ 4 = sin( 4) θ θ.

7 5- Conside the cuent density J= ( αax+ iβa y) δ( x) δ( y) δ( ze ) jωt whee α and β ae eal constants. a) Find the fa-field pointing vecto. b) Find the value of β/α that makes the adiation patten as nealy isotopic as possible. Calculate the esulting diectivity (should be as close to 1 as possible). 6- Thee is a z-diected cuent element of length h extending fom ( x, y, h ) to ( x, y, h ). The element cuent is chaacteized by I(z). Detemine the vecto potential A (, θ, φ) fo. Since I is not given explicitly, you should not, of couse, attempt to pefom the integation. Howeve, you should incopoate whateve simplifications ae possible due to the fa field condition.

9 - Obtain the nomalized adiation patten fo a staight wie antenna along the z axis centeed about the oigin with cuent assumed to have a quadatic distibution given by I ( z ) = I m 1 ( z L). 1- If = ˆ and = ˆ ae vectos of lengths and, locating diffeent points in space, show that, if >>, then ˆ ˆ. Povide sufficient comments to convince the instucto that you undestand the steps.

10 3- If the magnetic vecto potential is given as below detemine E and H. A ˆjkµ ai 1 e ( ) = φ 1+ sinθ 4 jk.

11 9- The E-field of a spheical wave in fee space is given as ˆ E sin θ E= θ cos( ωt kr R ). a) Detemine k (in tems of fequency ω ). b) Detemine the coesponding magnetic field intensity H (, θ, φ).

12 1- dl uzunluğunda bi dipole antenin elektik alan ifadesi aşağıda veilmişti. a) Bu antenin yakın alan ifadesini bulun. a) Bu antenin uzak alan ifadesini bulun. c) Bu antenin kutuplaşma tipini bulun. d) Bu antenin uzak alanındaki güç yoğunluğunu bulun. e) Bu antenin uzak alana yaydigi toplam gücü bulun. f) Bu anten için Uθφ (, ), Dθφ (, ) ve Ω paameteleini hesaplayın. 1- Conside the electic field expession of an antenna of length dl. e 1 kdl E (, θφ, ) = jkηidl { θˆ jφˆ } + { φˆ + jθˆ } + ˆ sinθ 4 π jk ( k) a) Detemine which tems contibute to the nea field. nea field k 1 A (,, ) e E θφ jkηidl ˆsin θ 4 π ( k) jk kdl b) Detemine which tems contibute to the fa field. fa field k 1 e (,, ) θ φ sinθ 4π E θφ = jkηidl { ˆ jˆ} c) Detemine the polaization of this wave in the fa field. d) Detemine the powe density of this wave in the fa field. ˆ E e H = = jkidl ( ˆ jˆ) sin η 4π φ + θ θ 1 1 dl 1, S= Re ( E H ) = ( ηi ) sin 4 λ e) Detemine the total powe deliveed by this wave to the fa field. θˆ P = ˆ sinθdd θ φ ηi sin θdd θ φ ηi S = 4 = λ 3 λ π π 1 dl π π 3 π dl f) Detemine the adiation intensity, the diectivity and the beam solid angle. ηi dl U( θφ, ) = sin 4 λ θ, 4 πu ( θφ, ) 3 D( θφ, ) sin P = = θ, 4π Ω A = = 8π 3 D

13 1- A fictitious magnetic souce can be defined by adding a souce tem to Faaday s law, so that E= jωb M. a) Assume that thee ae no electic souces (J =, ρ = ). Since D= in this case thee must exist a vecto field F such that D= F. Using this elationship, togethe with Maxwell s equations and the gauge pinciple, deive a wave equation fo F.

14 b) Using the Geen s function fo this type of wave equation, give a fomula fo F. c) Finally, deive expessions fo E and H in tems of the souce M. How do these compae to pevious esults?

15 1- Given δa ( ) = µδ I L 4 π µδ I L R evaluate δb ( ) = δa ( ) to show δb ( ) =. 3 4πR

17 1- Show that twice diffeentiable functions given below ae the solutions of the homogeneous g( x, y, z, t) wave equation g( x, y, z, t) µε =. t f1( t R µε ) f( t + R µε ) g1( R, t) =, g( R, t) = whee R = x + y + z. R R

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