lecture30 - Lecture 30 An Array of Two Hertzian Dipole...

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1 ECE 303 – Fall 2005 – Farhan Rana – Cornell University Lecture 30 An Array of Two Hertzian Dipole Antennas In this lecture you will learn: • Hertzian dipole antenna arrays • Interference and far-field radiation patterns ECE 303 – Fall 2005 – Farhan Rana – Cornell University Characteristics of a Single Hertzian Dipole Antenna Antenna Gain: For a Hertzian dipole the gain is: ( ) ( ) ( ) θ π φ θ 2 2 sin 2 3 4 ˆ . , , = = r P r t r S G rad r r Antenna Radiation Pattern: For a Hertzian dipole the radiation pattern is: ( ) ( ) ( ) θ φ θ φ θ 2 max sin , , = = G G p ( ) 0 , = φ θ p θ (degrees) 0 180 90 30 60 120 150 ( ) 0 , = φ θ p θ
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2 ECE 303 – Fall 2005 – Farhan Rana – Cornell University A Single Hertzian Dipole Antenna Not at Origin - I y z ( ) ( ) h r d I z r J r r r r = 3 ˆ δ What if one has a Hertzian dipole sitting at some arbitrary point? If one is interested in radiation far-fields only, then assume: r h d r r << << π λ 2 , θ h r r r h r r h r r h r r r h r h h r r h r r r r r r r r r r r r r r r r r r . ˆ . 2 1 . 2 . 2 . . 2 . . 2 2 = = + = ( ) h r r k j o e r Id z r . ˆ 4 ˆ π µ ( ) ( ) [ ] h r k j r k j ff e e r Id k j r H r r r . ˆ sin 4 ˆ = θ π φ ( ) ( ) [ ] h r k j r k j o ff e e r Id k j r E r r r . ˆ sin 4 ˆ = θ π η θ So we get: Additional phase factor x r r h r φ ( ) ( ) ( ) h r k j o r r k j o e h r Id z r A dv e r r r J r A r r r r r r r r r r r r r r = ∫∫∫ = π µ π µ 4 ˆ ' ' 4 ' ' ( ) h r k j o e h r Id z r A r r r r r r = π µ 4 ˆ ECE 303 – Fall 2005 – Farhan Rana – Cornell University A Single Hertzian Dipole Antenna Not at Origin - II y z ( ) ( ) h r d I z r J r r r r = 3 ˆ δ θ x r r a φ x a h ˆ = r Example: ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) z y x r e e r Id k j r E h r k j r k j o ff ˆ cos ˆ sin sin ˆ cos sin ˆ sin 4 ˆ . ˆ θ φ θ φ θ θ π η θ + + = = r r r Suppose: Note that: ( ) ( ) ( ) ( ) [ ] φ θ θ π η θ cos sin sin 4 ˆ a k j r k j o ff e e r Id k j r E = r r Therefore: ( ) ( ) ( ) ( ) [ ] φ θ θ π φ cos sin sin 4 ˆ a k j r k j ff e e r Id k j r H = r r ( ) ( ) [ ] h r k j r k j ff e e r Id k j r H r r r . ˆ sin 4 ˆ = θ π φ
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3 ECE 303 – Fall 2005 – Farhan Rana – Cornell University Two Hertzian Dipoles – General Case y z ( ) ( ) 1 3 1 1 ˆ h r d I z r J r r r r = δ x 1 h r 2 h r 1 J r 2 J r ( ) ( ) 2 3 2 2 ˆ h r d I z r J r r r r = δ ( ) ( ) ( ) ( ) ( ) ( )
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