1ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityLecture 30An Array of Two Hertzian Dipole AntennasIn this lecture you will learn:• Hertzian dipole antenna arrays• Interference and far-field radiation patternsECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityCharacteristics of a Single Hertzian Dipole AntennaAntenna Gain:For a Hertzian dipole the gain is:()()()θπφθ22sin234ˆ.,, ==rPrtrSGradrrAntenna Radiation Pattern:For a Hertzian dipole the radiation pattern is:()()()θφθφθ2maxsin,, ==GGp()0, =φθpθ (degrees)0180903060120150()0,=φθpθ2ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityA Single Hertzian Dipole Antenna Not at Origin - Iyz()()hrdIzrJrrrr−=3ˆδWhat if one has a Hertzian dipole sitting at some arbitrary point?If one is interested in radiation far-fields only, then assume:rhdrr<<⎭⎬⎫⎩⎨⎧<<πλ2,θhrrrhrrhrrhrrrhrhhrrhrrrrrrrrrrrrrrrrrr.ˆ.21.2.2..2..22−≈−=−=−≈−+=−()hrrkjoerIdzr.ˆ4ˆ−−≈πµ()()[]hrkjrkjffeerIdkjrHrrr.ˆsin4ˆ−=θπφ()()[]hrkjrkjoffeerIdkjrErrr.ˆsin4ˆ−=θπηθSo we get:Additional phase factorxrrhrφ()()()hrkjorrkjoehrIdzrAdverrrJrArrrrrrrrrrrrrr−−−−−=⇒∫∫∫−=πµπµ4ˆ ''4' '()hrkjoehrIdzrArrrrrr−−−=πµ4ˆ ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityA Single Hertzian Dipole Antenna Not at Origin - IIyz()()hrdIzrJrrrr−=3ˆδθxrraφxahˆ=rExample: ()()[]() () () () ()zyxreerIdkjrEhrkjrkjoffˆcosˆsinsinˆcossinˆsin4ˆ.ˆθφθφθθπηθ++==−rrrSuppose: Note that:()()() ()[]φθθπηθcossinsin4ˆakjrkjoffeerIdkjrE−=⇒rrTherefore:()()() ()[]φθθπφcossinsin4ˆakjrkjffeerIdkjrH−=rr()()[]hrkjrkjffeerIdkjrHrrr.ˆsin4ˆ−=θπφ3ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityTwo Hertzian Dipoles – General Caseyz()()1311ˆhrdIzrJrrrr−=δx1hr2hr1Jr2Jr()()2322ˆhrdIzrJrrrr−=δ()()()()()()⎥⎦⎤⎢⎣⎡+=+=−−−−−2121.ˆ12.ˆ1.ˆ2.ˆ1sin4ˆ sin4ˆsin4ˆhrkjhrkjrkjohrrkjohrrkjoffeIIeerkdIjerdIkjerdIkjrErrrrrrθπηθθπηθθπηθCan write the E-field and the H-field in the far-field directly:()()⎥⎦⎤⎢⎣⎡+=−21.ˆ12.ˆ1sin4ˆ hrkjhrkjrkjffeIIeerdkIjrHrrrrθπφ() ()()()()θφθφθcosˆsinsinˆcossinˆˆzyxr++=Remember that:ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityTwo Hertzian Dipoles – General Caseyz()()1311ˆhrdIzrJrrrr−=δx1hr2hr1Jr2Jr()()2322ˆhrdIzrJrrrr−=δ()()⎥⎦⎤⎢⎣⎡+=−21.ˆ12.ˆ1sin4ˆ hrkjhrkjrkjoffeIIeerdkIjrErrrrθπηθCan write the E-field in the far-field as:Depends only on the radiating properties of the individual antennas(ELEMENT FACTOR)Depends on the relative positions as well as the relative current amplitudes of the two antennas (ARRAY FACTOR) Describes INTERFERENCE in the far-field between the radiation emitted by the two dipoles4ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityTwo Hertzian Dipoles on X-Axis – Gain and Radiation Pattern - Iyzx2ˆ1lrxh =2ˆ2lrxh −=1Jr2JrlαjeAII =2yxlThe amplitudes and phases of the currents in the two dipoles are not the same:Question: What is the radiation pattern in the x-y plane?()φπθ,2=p()() ()⎥⎥⎦⎤⎢⎢⎣⎡+=−−=φαφπθπηθcos2cos224ˆ llrrkjjkjrkjoffeeAeerdIkjrEII=1()() ()⎥⎥⎦⎤⎢⎢⎣⎡+=−−=φαφπθπφcos2cos224ˆ llrrkjjkjrkjffeeAeerdIkjrH()() ()2cos2cos2224ˆ φαφπθπηllrrkjjkjoffeeAerdIkrrS−=+=⇒φECE 303 – Fall 2005 – Farhan Rana – Cornell University1Jr2JryxlII=1αjeAII =2()() ()2cos2cos2224ˆ φαφπθπηllrrkjjkjoffeeAerdIkrrS−=+=()()()[]⎟⎟⎠⎞⎜⎜⎝⎛+−++===2221coscos21234ˆ.,,2AkAArPrtrSGradαφπφπθlrrTotal Power Radiated:()22112AIdkPorad+=πηsum of the power radiated by individual dipoles()()()[]()⎟⎟⎠⎞⎜⎜⎝⎛+−++====22max1coscos21,2,2AkAAGGpαφφπθφπθlGain:Pattern:Poyntingvector:φTwo Hertzian Dipoles on X-Axis – Gain and Radiation Pattern - II5ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityCase I: Two Hertzian Dipoles on X-Axis1Jr2JryxlII=1αjeAII =2Case I: A = 1ℓ = λ / 4α= -π/ 2() ()⎟⎠⎞⎜⎝⎛⎥⎦⎤⎢⎣⎡++==2cos2cos121,2πφπφπθpPattern:yx4λ22πjeII−=II=1φφ()φπθ,2=pECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityCase I: Two Hertzian Dipoles on X-Axis1Jr2JryxlII=1αjeAII =2Case I: A = 1ℓ = λ / 4α= -π/ 2φφ()φπθ,2=p()φθ,pIn the x-yplaneIn 3D6ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityTwo Hertzian Dipoles on X-Axis – Some Physical Reasoning1Jr2Jryx()φcoslφII=1αjeAII =2When traveling in the direction φ:• The waves from dipole 2 have a phase lead of α compared to those from dipole 1 • But the waves from dipole 2 travel a distance ℓ cos(φ) more than from dipole 1. This means they would lag by a phase of kℓ cos(φ) Consequently:• The net phase difference between waves from dipole 2 and dipole 1 in the direction φis:• One could therefore expect a maximum in the radiation pattern in the direction φif the net phase difference is a integral multiple of 2π:()φαcoslk−(){KKl 1,2,3,n 2cos=±=− nkπφαlECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityCase II: Two Hertzian Dipoles on X-Axis1Jr2JryxlII=1αjeAII =2Case II: A = 1ℓ = λ / 2α= 0φφ() ()[]()φπφπθcoscos121,2 +==pPattern:()φπθ,2=pyx2λII=1II=27ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityCase III: Two Hertzian Dipoles on X-Axis1Jr2JryxlII=1αjeAII =2Case III: A = 1ℓ = λ / 2α= -π/ 2φφ() ()⎟⎠⎞⎜⎝⎛⎥⎦⎤⎢⎣⎡++==2coscos121,2πφπφπθpPattern:()φπθ,2=pyx2λII=122πjeII−=ECE 303 – Fall 2005 – Farhan Rana – Cornell UniversityTwo Hertzian Dipoles on Z-Axis – Gain and Radiation Pattern - Izx2ˆ1lrzh =2ˆ2lrzh −=1Jr2JrlαjeAII =2The amplitudes and phases of the currents in the two dipoles are not the same:Question: What is the radiation pattern ?()φθ,pII=1()()() ()⎥⎥⎦⎤⎢⎢⎣⎡+=−−θαθθπηθcos2cos2sin4ˆ llrrkjjkjrkjoffeeAeerdIkjrE()()() ()⎥⎥⎦⎤⎢⎢⎣⎡+=−−θαθθπφcos2cos2sin4ˆ llrrkjjkjrkjffeeAeerdIkjrH()()() ()2cos2cos222sin4ˆ θαθθπηllrrkjjkjoffeeAerdIkrrS−+=⇒8ECE 303 – Fall 2005 – Farhan Rana – Cornell Universityzx2ˆ1lrzh =2ˆ2lrzh −=1Jr2JrlTotal Power Radiated:()22112AIdkPorad+=πηsum of the power radiated by individual dipoles()()() ()2cos2cos222sin4ˆ