Topic 4. Linear Wire and Small Circular Loop Antennas. Tamer Abuelfadl

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1 Topic 4 Linear Wire and Small Circular Loop Antennas Tamer Abuelfadl Electronics and Electrical Communications Department Faculty of Engineering Cairo University Tamer Abuelfadl (EEC, Cairo University) Topic 4 ELC45A, ELCN45 1 / 35

2 Linear Wire Antennas and Small Circular Loop Antenna 1 Innitesimal Dipole Small Dipole 3 Finite Length Dipole 4 Conductor Losses and Loss Resistance 5 Linear Elements Near or on Innite Conductor 6 Loop Antennas Tamer Abuelfadl (EEC, Cairo University) Topic 4 ELC45A, ELCN45 / 35

3 Outline 1 Innitesimal Dipole Small Dipole 3 Finite Length Dipole 4 Conductor Losses and Loss Resistance 5 Linear Elements Near or on Innite Conductor 6 Loop Antennas Tamer Abuelfadl (EEC, Cairo University) Topic 4 ELC45A, ELCN45 3 / 35

4 Innitesimal Dipole l λ (l λ/5) End plates are to maintain uniform current, however they are very small to aect radiation! Not very practical, however they are considered as a basic building blocks for complex structures. I e (z ) = ^a z I A(r,θ,φ) = µ 4π ˆ C e jkr I e R dl µi l = ^a z 4πr e jkr Tamer Abuelfadl (EEC, Cairo University) Topic 4 ELC45A, ELCN45 4 / 35

5 Innitesimal Dipole Transformation from rectangular spherical components, A r sinθ cosφ sinθ sinφ cosθ A x A θ = cosθ cosφ cosθ sinφ sinθ A y A φ sinφ cosφ A z Magnetic eld H = 1 µ A = ^a φ A r = µi le jkr 4πr cosθ, A θ = µi le jkr 4πr sinθ [ 1 µr r (ra θ ) A ] r = j ki le jkr [ sinθ ] ^a φ θ 4πr jkr Electric eld E = 1 jωε H Tamer Abuelfadl (EEC, Cairo University) Topic 4 ELC45A, ELCN45 5 / 35

6 Innitesimal Dipole E = jη ki [ ( l 1 4πr e jkr cosθ jkr 1 ) ( â k r r + sinθ H = j ki le jkr 4πr sinθ [ ] â φ jkr Radiation Power Density, W = 1 (E H ) = 1 (âr E θ Hφ â θ E r H ) φ Radiated Power, { P rad = R = ηπ 4 S I l λ } W ds = η k I l 3π [ cosθ + cos3 θ 3 ˆ π ] π dφ = ηπ 3 ˆ π jkr 1 ] )â k r θ sin θ sinθdθ I l λ Radiation Resistance P rad = ηπ 3 I l λ = 1 I R r = ( l R r = 8π λ )

7 Radiation Regions Near eld region kr 1, E = jη I l 4πkr 3 e jkr [cosθâ r + sinθâ θ ] H = I le jkr 4πr sinθâ φ Intermediate-eld (Fresnel) region kr > 1, E = jη ki l 4πr e jkr [ jkr cosθâ r + sinθâ θ H = j ki le jkr sinθâ φ 4πr Far-eld (Fraunhover) region kr 1, E = jη ki l sinθ e jkrâ θ, 4πr H = 1 ηâ r E H = j ki le jkr 4πr ] sinθâ φ Tamer Abuelfadl (EEC, Cairo University) Topic 4 ELC45A, ELCN45 7 / 35

8 Directivity W rad = E θ η U = r W rad = r E θ η = η k I l 3π sin θ = η 8 I l λ sin θ P rad = ηπ 3 I l λ D = 4πU P rad = 1.5sin θ Tamer Abuelfadl (EEC, Cairo University) Topic 4 ELC45A, ELCN45 8 / 35

9 Outline 1 Innitesimal Dipole Small Dipole 3 Finite Length Dipole 4 Conductor Losses and Loss Resistance 5 Linear Elements Near or on Innite Conductor 6 Loop Antennas Tamer Abuelfadl (EEC, Cairo University) Topic 4 ELC45A, ELCN45 9 / 35

10 Small Dipole ( ) ( I e z ) â z I 1 z l = ( ) â z I 1 + z l [ 1 µi le jkr ] A = â z 4πr E = jη ki (l/)e kr 4πr H = j ki (l/)e kr 4πr z l/ l/ z sinθâ θ sinθâ φ ( ) l/ Radiation resistance R r = 8π λ Small dipole of length l is equivalent to innitesimal dipole of length l/.

11 Outline 1 Innitesimal Dipole Small Dipole 3 Finite Length Dipole 4 Conductor Losses and Loss Resistance 5 Linear Elements Near or on Innite Conductor 6 Loop Antennas Tamer Abuelfadl (EEC, Cairo University) Topic 4 ELC45A, ELCN45 11 / 35

12 Finite Length Dipole I e ( z ) = {^az I sin [ k ( l z )], z l ^a z I sin [ k ( l + z )], l z

13 Finite Length Dipole E jω [ A θâ θ + A φâ φ ], A(r,θ,φ) µ 4π A = µ 4π E = jη ke jkr 4πr e jkr r e jkr ˆ l/ r sinθ ˆ C l/ ˆ l/ I e ( r,θ,φ ) e j k r dl I e ( z ) e jkz cosθ dz â z l/ I e ( z ) e jkz cosθ dz â θ

14 ( I e z ) {âz I sin [ k ( l = z )], z l â z I sin [ k ( l + z )], l z E = jη ke jkr 4πr ˆ l/ ( sinθ I e z ) e jkz cosθ dz â θ l/ E = jη ki e jkr 4πr + ˆ l/ sin {ˆ [ ( l sinθ sin k l/ [ ( l k z )] + z e jkz cosθ dz } )]e jkz cosθ dz The integrals can be evaluated using ˆ e αx sin(βx + γ)dx = eαx [α sin(βx + γ) β cos(βx + γ)] α + β ( ) ( ) kl kl E = jη I e jkr cos cosθ cos πr sinθ â θ â θ

15 Radiation Intensity, and Radiation Resistance P rad = U = r η E θ = η I 8π ˆ π ˆ π cos U sinθdθdφ = η I 4π ( ) kl cosθ cos ˆ π sinθ ( kl ) [ ( ) ( )] kl kl cos cosθ cos dθ sinθ P rad = 1 I in R r = 1 I sin (kl/)r r 1 η R r = sin (kl/) π ˆ π [ cos ( ) ( )] kl kl cosθ cos dθ sinθ

16 Outline 1 Innitesimal Dipole Small Dipole 3 Finite Length Dipole 4 Conductor Losses and Loss Resistance 5 Linear Elements Near or on Innite Conductor 6 Loop Antennas Tamer Abuelfadl (EEC, Cairo University) Topic 4 ELC45A, ELCN45 16 / 35

17 Conductor Losses Skin depth: δ s = 1/ πf µσ Resistance of the segment with dimensions ( x, y, δ s ): R = y σδ s x Current owing through length x: I s = J s x

18 Conductor Losses R = y σδ s x Current owing through length x: I s = J s x Power loss in the area x y, P loss = 1 I s R = 1 J s 1 σδ s x y Total Power loss in a conductor area S, P loss = 1 ˆ S J s R s ds, where R s = 1 πf µ = σδ s σ

19 Conductor Losses P loss = 1 ˆ S J s R s ds, where R s = 1 πf µ = σδ s σ R s is called the surface resistance of the conductor. The surface current can be obtained using the approximation of a perfect conductor boundary conditions, J s = n H = J s = H where n is the normal unit vector on the conductor. Total Power loss in a conductor area S, P loss = 1 ˆ S H R s ds, where R s = 1 πf µ = σδ s σ

20 Loss resistance P loss = 1 P loss = 1 = 1 ˆ l/ ˆ l/ S ˆ l/ J s R s ds l/ I e (z ) (πb) R s (πb)dz I e (z ) πb R sdz = 1 I e () R L R L = R ˆ l/ s πb l/ I e (z ) I e () dz Tamer Abuelfadl (EEC, Cairo University) Topic 4 ELC45A, ELCN45 / 35

21 Half-wavelength (λ/) dipole P rad = ( π ) U = r η E θ = η I cos cosθ 8π sinθ ˆ π ˆ π D = 4πU P rad = R r = η π η πr r U sinθdθdφ = η I ˆ π 4π P rad = 1 I R r ( π ) cos cosθ sinθ ( π ) cos cosθ sinθ ˆ π cos ( π cosθ ) dθ = 73 Ω sinθ dθ ( π ) = 1.64 cos cosθ sinθ

22 Outline 1 Innitesimal Dipole Small Dipole 3 Finite Length Dipole 4 Conductor Losses and Loss Resistance 5 Linear Elements Near or on Innite Conductor 6 Loop Antennas Tamer Abuelfadl (EEC, Cairo University) Topic 4 ELC45A, ELCN45 / 35

23 Linear Elements Near or on Innite Conductor Image Theory E = jηki le jkr 4πr +sinθ H = j ki le jkr 4πr [ ( cosθ ( jkr 1 k r jkr 1 k r ] )â θ [ sinθ ] â φ jkr ) â r

24 Linear Elements Near or on Innite Conductor Image Theory

$Vertical Innitesimal Dipole Above Ground Plane A(r,θ,φ) µ 4π e jkr r ˆ C I e e j k r dl H â r E η, E jω [ ] A θâ θ + A φâ φ {jη ki le kr E = 4πr sinθ$

25 Vertical Innitesimal Dipole Above Ground Plane A(r,θ,φ) µ 4π e jkr r ˆ C I e e j k r dl H â r E η, E jω [ ] A θâ θ + A φâ φ {jη ki le kr E = 4πr sinθ [cos(kh cosθ)]â θ z z < η U = I l λ sin θ cos (kh cosθ) < θ π/ π/ < θ π ˆ ˆ P rad = UdΩ = πη I l π/ λ sin 3 θ cos (kh cosθ)dθ ˆ = πη I l 1 ( ) λ 1 x cos (khx)dx

26 Vertical Innitesimal Dipole Above Ground Plane U max = η I l λ P rad = πη I l λ [ 1 3 cos(kh) (kh) = D = 4πU max P rad = ] + sin(kh) (kh) 3 ( ) [ ] l 1 R r = πη λ 3 cos(kh) + sin(kh) (kh) (kh) 3 [ ] 1 cos(kh) + sin(kh) 3 (kh) (kh) 3 kh, R r = 8π (l/λ), R r is identical to isolated Innitesimal dipole. kh, D = 3, R r = 16π (l/λ), D and R r are twice the isolated Innitesimal dipole. Tamer Abuelfadl (EEC, Cairo University) Topic 4 ELC45A, ELCN45 6 / 35

27 λ/4 Monopole on Innite Electric Conductor The radiation elds are identical in the upper half plane (z > ). However the total radiated for the monopole is half its value for the dipole. P rad (m) = 1 P rad (d) D m = D d, R r (m) = 1 R r (d)

28 Outline 1 Innitesimal Dipole Small Dipole 3 Finite Length Dipole 4 Conductor Losses and Loss Resistance 5 Linear Elements Near or on Innite Conductor 6 Loop Antennas Tamer Abuelfadl (EEC, Cairo University) Topic 4 ELC45A, ELCN45 8 / 35

29 Angle Between Two Directions in Space The unit vector describing the the direction (θ, φ), â r = sinθ cosφâ x + sinθ sinφâ y + cosθâ z The unit vector describing the the direction (θ, φ ), â r = sinθ cosφ â x + sinθ sinφ â y + cosθ â z The angle ψ between the two directions, cosψ = â r â r cosψ = sinθ sinθ cos ( φ φ ) + cosθ cosθ Tamer Abuelfadl (EEC, Cairo University) Topic 4 ELC45A, ELCN45 9 / 35

30 Small Circular Loop Circumference C < λ 1 A(r,θ,φ) = µ ˆ e jkr I e 4π R dl C The source current is along â φ : coordinates, I e = I ^a φ. Transforming to Cartesian I x = I sinφ, I y = I cosφ Transforming to the spherical coordinates at the observation point, I r = I x cosφ sinθ + I y sinφ sinθ I θ = I x cosφ cosθ + I y sinφ cosθ I φ = I x sinφ + I y cosφ

31 Small Circular Loop I x = I sinφ, I y = I cosφ Transforming to the spherical coordinates at the observation point, I r = I x cosφ sinθ + I y sinφ sinθ I θ = I x cosφ cosθ + I y sinφ cosθ I φ = I x sinφ + I y cosφ I r = I sinθ sin ( φ φ ) I θ = I cosθ sin ( φ φ ) I φ = I cos ( φ φ ) I e = â r I sinθ sin ( φ φ ) +â θ I cosθ sin ( φ φ ) +â φ I cos ( φ φ )

32 Small Circular Loop A r = µi a 4π sinθ A θ = µi a 4π cosθ R = r + a ar cosψ, cosψ = sinθ cos ( φ φ ) A φ = µi a 4π ˆ π ˆ π sin ( φ φ ) e jk r +a ar sinθ cos(φ φ ) r + a ar sinθ cos(φ φ ) dφ = sin ( φ φ ) e jk r +a ar sinθ cos(φ φ ) r + a ar sinθ cos(φ φ ) dφ = ˆ π cos ( φ φ ) e jk r +a ar sinθ cos(φ φ ) dφ

33 Small Circular Loop A = A φ^a φ, A φ = µi a 4π ˆ π cos ( φ φ ) e jk r +a ar sinθ cos(φ φ ) r + a ar sinθ cos(φ φ ) dφ f () = e jkr r A φ = µi a e jkr 4π r f = e jk r +a ar sinθ cos(φ φ ) r + a ar sinθ cos(φ φ ) f (a) f () + f ()a, f () = e jkr (jkr + 1)sinθ cos ( φ φ ) r [ ( f = e jkr a )sinθ r r + jk cos ( φ φ )] ˆ π cos ( φ φ )[ ( a r + jk A φ = a jkµi e jkr ( ) 1 4 r jkr + 1 sinθ ) sinθ cos ( φ φ )] dφ

34 Small Circular Loop A = A φâ φ, A φ = a jkµi e jkr ( ) 1 4 r jkr + 1 sinθ H = 1 µ A, E = 1 jωε H = η jk H H = (ka) I e jkr 4r [ ( jkr ) ( ] cosθâ (kr) r + )sinθâ 1 (kr) + 1 jkr + 1 θ E = η (ka) I e jkr 4r ( ) 1 jkr + 1 sinθâ φ Tamer Abuelfadl (EEC, Cairo University) Topic 4 ELC45A, ELCN45 34 / 35

35 Small Circular Loop Far Fields, H = (ka) I e jkr 4r sinθ^a θ, E = η (ka) I e jkr 4r sinθ^a φ Radiated Power and Radiation Resistance U = η (ka)4 I ˆ sin θ, P rad = UdΩ = η π (ka)4 I 3 1 ( π ) R r = η (ka) 4 6 For N-turns loop, Loss resistance R L, ( π ) R r = R r = η (ka) 4 N 6 R L = N a b R s, where b is the wire radius, and R s is the conductor surface impedance. Increasing the number of turns, increases the antenna radiation eciency.

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