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ece4990notes3 - Determination of Antenna Radiation Fields...

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Determination of Antenna Radiation Fields Using Potential Functions 6 J - vector electric current density (A/m 2 ) M - vector magnetic current density (V/m 2 ) Some problems involving electric currents can be cast in equivalent forms involving magnetic currents (the use of magnetic currents is simply a mathematical tool, they have never been proven to exist). A - magnetic vector potential (due to J ) F - electric vector potential (due to M ) In order to account for both electric current and/or magnetic current sources, the symmetric form of Maxwell’s equations must be utilized to determine the resulting radiation fields. The symmetric form of Maxwell’s equations include additional radiation sources (electric charge density - D and magnetic charge density D m ). However, these charges can always be related directly to the current via conservation of charge equations. Sources of Antenna Radiation Fields
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Maxwell’s equations (symmetric, time-harmonic form) The use of potentials in the solution of radiation fields employs the concept of superposition of fields. Electric current Y Magnetic vector Y Radiation fields source ( J , D ) potential ( A ) ( E A , H A ) Magnetic current Y Electric vector Y Radiation fields source ( M , D m ) potential ( F ) ( E F , H F ) The total radiation fields ( E , H ) are the sum of the fields due to electric currents ( E A , H A ) and the fields due to the magnetic currents ( E F , H F ). Maxwell’s Equations (electric sources only Y F = 0 )
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Maxwell’s Equations (magnetic sources only Y A = 0 ) Based on the vector identity, any vector with zero divergence (rotational or solenoidal field) can be expressed as the curl of some other vector. From Maxwell’s equations with electric or magnetic sources only [Equations (1d) and (2c)], we find so that we may define these vectors as where A and F are the magnetic and electric vector potentials, respectively. The flux density definitions in Equations (3a) and (3b) lead to the following field definitions: Inserting (3a) into (1a) and (3b) into (2b) yields
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Equations (5a) and (5b) can be rewritten as Based on the vector identity the bracketed terms in (6a) and (6b) represent non-solenoidal (lamellar or irrotational fields) and may each be written as the gradient of some scalar where N e is the electric scalar potential and N m is the magnetic scalar potential. Solving equations (7a) and (7b) for the electric and magnetic fields yields Equations (4a) and (8a) give the fields ( E A , H A ) due to electric sources while Equations (4b) and (8b) give the fields ( E F , H F ) due to magnetic sources. Note that these radiated fields are obtained by differentiating the respective vector and scalar potentials. The integrals which define the vector and scalar potential can be
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This document was uploaded on 04/11/2010.

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ece4990notes3 - Determination of Antenna Radiation Fields...

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