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									6.013 - Electromagnetics and Applications

Fall 2005

Lecture 20 - Dipole Arrays
Prof. Markus Zahn I. Two Element Array in θ =
π 2

December 1, 2005 plane (x-y plane)

y

r2

sφ aco
φ φ

r a a

r1 x I1

I2

Far field (kr � 1, r � a) ˆ ˆ π E1 −jkr1 E2 −jkr2 π ˆ ˆ Eθ (r, θ = , φ) = e + e = ηHφ (r, θ = , φ) 2 jkr1 jkr2 2 2 η
 ˆ I1 dlk ˆ E1 = − 4π
 ˆ2 dlk 2 η
 I ˆ E2 = − 4π r2 ≈ r + a cos(φ), r1 ≈ r − a cos(φ) 2 π π k�ηdl −jkr � ˆ +jka cos(φ) ˆ −jka cos(φ) � ˆ ˆ Eθ (r, θ = , φ) = ηHφ (r, θ = , φ) ≈ − e I1 e +I e � �� 2 � 2 2 k 4πj�r � �� � array factor
element factor

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Assume: I1 = I, I2 = Iejχ ⇒ E1 = E0 , E2 = E0 ejχ � ˆ π E0 −jkr � +jka cos(φ) π ˆ Eθ (r, θ = , φ) = ηHφ (r, θ = , φ) = e e + ejχ e−jka cos(φ) 2 2 jkr � ˆ χ E0 −jkr jχ/2 � −j( χ −ka cos(φ)) = e e e 2 + ej( 2 −ka cos(φ)) � �� � jkr 2 cos(− χ +ka cos(φ)) 2 � χ � ˆ0 2E −jkr jχ/2 = e e cos − + ka cos(φ) jkr 2 � � 1 |E |2 � 2 ˆθ ˆ π 2|E0 | χ� Sr (t, θ = , φ) = cos2 ka cos(φ) − = 2 2 η η(kr)2 2 1

Broadside: λ π , χ = 0, ka = 2 2 �π � ˆ0 |2 2|E �Sr � = cos2 cos(φ) η(kr)2 2 2a = Endfire: λ π , χ = π, ka = 2 2 �π � ˆ0 |2 2|E �Sr � = cos2 (cos(φ) − 1) η(kr)2 2 2a =

Maxima: ka cos(φ) − Minima: ka cos(φ) − Case Studies:	

χ 2 = ±mπ, m = 0, 1, 2, . . . χ π 2 = ±(2m + 1) 2 , m = 0, 1, 2, . . .

2a =	

λ 2πa 2πa π ⇒ ka = = = 2 4a 2 λ 2a = λ ⇒ ka = π

2a =

λ π χ	 χ ⇒	 cos(φ) − = ±mπ (maxima) ⇒ cos(φ) = ± 2m 2 2 π 2 π χ π χ cos(φ) − = ±(2m + 1) (minima) ⇒ cos(φ) = ± (2m + 1) 2 2 2 π

2


λ = 4a χ 0
π 4 π 2 3π 4

cos(φmax ) cos(φmin ) 0
1 4 1 2 3 4

φmax ±
π 2 ±60◦

φmin 0, π ±120◦ ±90.◦ Broadside


1 −3 4 −1 2 −1 4 0
χ 2π

±75.5◦ ±138.6◦ ±41.4◦ ±104.5◦ 0, π Endfire

π λ = 2a ⇒
cos(φmax ) = χ 0
π 4 π 2 3π 4 χ 2π

1

± m, cos(φmin ) =

± 1 (2m + 1)
 2 φmax 0, ±90◦ , 180◦ 82.8◦ , 151◦ 75.5◦ , 138.6◦ 68.0◦ , 128.7◦ 60◦ , 120◦ φmin ±60◦ 51◦ , 112◦ 41.4◦ , 104.5◦ 29.0◦ , 97.2◦ 90◦ , 0◦

cos(φmax ) cos(φmin ) 0, 1
1 7 8, −8 1 3 4, −4 3 5 8, −8 1 1 2, −2 1 1 2 , −
2 −
3 ,
5 8 8 −
1 ,
3 4 4 1 7 −
8 ,
8

π


0, 1

3


4


II. An N Dipole Array (θ = π ) 2

r�na

lim rn ≈ r − na cos(φ) −N ≤ n ≤ N

� � π � π � ˆ ˆ Eθ r, θ = , φ = ηHφ r, θ = , φ 2 2 +N �� � kηdl ˆ =− In ejkna cos(φ) e−jkr 4πjr −N � �� �
Array factor = AF

Example: ˆ In = I0 e−jnχ0 AF = I0 Let β ≡ e
+N �

−N ≤n≤N

ejn(ka cos(φ)−χ0 )

−N
 j(ka cos(φ)−χ0 )


AF � n S= = β = β −N + β −N +1 + . . . + β −2 + β −1 + 1 + β + β 2 + . . . + β N −1 + β N I0
−N

+N

S(1 − β) = β

−N

−β

N +1

β −N − β N +1 β −N − 2 − β N + 2 ⇒S= = 1−β β −1/2 − β 1/2 � �� �
multiply by
β −1/2 β −1/2

1

1

� � 1 sin (N + 2 )(ka cos(φ) − χ0 ) � � S= sin 1 (ka cos(φ) − χ0 ) 2 Maxima: ka cos(φ) − χ0 = 2nπ, n = 0, 1, 2, . . . Principal maximum at n = 0 ⇒ cos(φ) =
χ0 ka 1 Minima: (N + 2 )(ka cos(φ) − χ0 ) = nπ, n = 1, 2, 3, . . .

5

Demonstration, N = 2 (2 dipole array) 3 2a = λ, χ0 = 0 2 � � � � � 2 2 2π 3 2 3π I ∝ cos (ka cos(φ)) = cos λ cos(φ) � cos(φ) = cos 2 λ � � 42 3� π π 1 � Minima: cos(φ) = ⇒ cos(φ) = ⇒ φ = 70.5◦ 2 2 3 � � 3� 3� �π �π cos(φ) = ⇒ cos(φ) = 1 ⇒ φ = 0◦ 2 2 � � 3π Maxima: cos(φ) = 0 ⇒ φ = 90◦ 2 π 2 3� cos(φ) = � ⇒ cos(φ) = ⇒ φ = 48.2◦ π 2 3

ф

Intensity pattern


6


7



								
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