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# c3 - Antennas and Radiation 2 Arbitrary Hertzian Dipoles(r...

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Antennas and Radiation 2 φ ( r, t ) - 1 c 2 2 φ ( r, t ) ∂t 2 = - ρ ( r, t ) ε 0 φ ( r, t ) = ρ ( r , t - | r - r | /c ) 4 πε 0 | r - r | dv 2 A ( r, t ) - 1 c 2 2 A ( r, t ) ∂t 2 = - μ 0 J ( r, t ) A ( r, t ) = μ 0 J ( r , t - | r - r | /c ) 4 πε 0 | r - r | dv k = ω/c φ ( r ) = ρ ( r ) 4 πε 0 | r - r | e - jk | r - r | dv A ( r ) = μ 0 J ( r ) 4 πε 0 | r - r | e - jk | r - r | dv Hertzian Dipole I ( t ) = dq ( t ) dt , J ( r ) = ˆ zIdδ 3 ( r ) A ( r ) = ˆ r cos( θ ) - ˆ θ sin( θ ) μ 0 Id 4 πr e - jkr μ 0 H ( r ) = ∇ × A ( r ) H ( r ) = ˆ φ jkId 4 πr e - jkr 1 + 1 jkr sin( θ ) E ( r ) = 1 jωε 0 ∇ × H ( r ) = η 0 jkid 4 πr e - jkr ˆ r 1 jkr + 1 jkr 2 2 cos( θ ) + ˆ θ 1 + 1 jkr + 1 jkr 2 sin( θ ) E nf ( r ) = qd 4 πε 0 r 3 ˆ r 2 cos( θ ) + ˆ θ sin( θ ) H nf ( r ) = ˆ φ Id 4 πr 2 sin( θ ) E ff ( r ) = ˆ θ 0 kId 4 πr e - jkr sin( θ ) H ff ( r ) = ˆ φ jkId 4 πr e - jkr sin( θ ) S ( r, t ) = ˆ r η 0 2 kId 4 πr 2 sin 2 ( θ ) P = η 0 12 π | kId | 2 G ( θ, φ ) = S ( r, t ) · ˆ r P/ 4 πr 2 3 2 sin 2 ( θ ) p ( θ, φ ) = G ( θ, φ ) G | max R rad = P | I | 2 / 2 = η 0 6 π ( kd ) 2 Hertzian Dipole at Arbitrary Point ( h ) E ff ( r ) = ˆ θ 0 kId 4 πr e - jk | r - ˆ r · h | sin( θ ) H ff ( r ) = ˆ φ jkId 4 πr e - jk | r - ˆ r · h | sin( θ ) Arbitrary Hertzian Dipoles E ff ( r ) = ˆ θ 0 k 4 πr sin( θ ) e - jkr I 1 d 1 e jk ˆ r · ˆ h 1 + I 2 d 2 e jk ˆ r · ˆ h 2 H ff ( r ) = ˆ φ jk 4 πr sin( θ ) e - jkr I 1 d 1 e jk ˆ r · ˆ h 1 + I 2 d 2 e jk ˆ r · ˆ h 2 ˆ r = ˆ x sin( θ ) cos( φ ) + ˆ y sin( θ ) sin( φ ) + ˆ z cos( θ ) Two Hertzian Dipoles on X-axis I 1 = I, I 2 = IAe , h 1 = ˆ xl/ 2 , h 2 = - ˆ xl/ 2 E ff ( r ) θ = π/ 2 = ˆ θ 0 kId 4 πr e - jkr e jk l 2 cos( φ ) + Ae e - jk l 2 cos( φ ) H ff ( r ) θ = π/ 2 = ˆ φ jkId 4 πr e - jkr e jk l 2 cos( φ ) + Ae e - jk l 2 cos( φ ) S ff ( r ) θ = π/ 2 = ˆ 0 jkId 4 πr 2 e jk l 2 cos( φ ) + Ae e - jk l 2 cos( φ ) 2 G ( θ = π/ 2 , φ ) = 3 2 1 + A 2 + 2 A cos[ kl cos( φ ) - α )] 1 + A 2 p ( θ = π/ 2 , φ ) = 1 + A 2 + 2 A cos[ kl cos( φ ) - α )] (1 + A ) 2

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c3 - Antennas and Radiation 2 Arbitrary Hertzian Dipoles(r...

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