15 references 16 problems 26 chapter 2 theory of

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1.5 References 1.6 Problems 26
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Chapter 2 Theory of simple wire antennas This chapter discusses simple wire antennas based on electrical dipoles. Starting from Maxwell’s equations, the electromagnetic fields due to a short current element are calculated. The elemental current then serves as a building block for more complicated antennas. A variety of practical wire antennas will be analyzed and evaluated. None of these will turn out by themselves to have very high gain, however. Methods of achieving the high gain required by radars are then taken up in chapters 3 and 4. While much of the analysis concerns using antennas for transmission, the reciprocity theorem will provide a recipe for evaluating their reception properties. 2.1 Hertzian dipole All time-varying currents radiate as antennas, and the simplest radiating system is an elemental current, sometimes called an ideal or Hertzian dipole. This is a uniform current density J flowing is an element with cross-section A and differential length dl . The current is assumed to have a sinusoidal time variation given by the implicit exp( jωt ) factor, and so phasor notation applies. The differential length of the element is regarded as being very small compared to a wavelength. We also regard the current density as being distributed uniformly in space on the wire. This is an idealization; one does not set out to construct an ideal dipole, although larger antennas can be regarded as being assembled from them. It is expedient to use a combination of spherical and rectangular coordinates and to alternate between them as needed. Be aware that the unit vectors associated with spherical coordinates vary with position. Calculating the fields arising from an elemental dipole is a fundamental physics problem addressed by most text- books on antennas and electricity and magnetism. An abbreviated, intuitive derivation is provided below. Readers interested in more details should consult the appendix or one of the references given. 2.1.1 Vector potential: phasor form Maxwell’s equations in their native form are not amenable to driven wave problems, and derivations usually begin instead with vector and scalar potentials and the gauge or condition that completes their definitions. The magnetic vector potential A , scalar potential φ , and the Lorentz condition relating them are specified by (see appendix): B = ∇ × A E = −∇ φ A ∂t ∇ · A = 1 c 2 ∂φ ∂t 27
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dl J A θ φ r e θ e φ e r x y z Figure 2.1: Diagram of an elemental current element antenna. Working with potentials rather than electric and magnetic fields allows Maxwell’s equations in a vacuum to be recast into the following second-order partial differential equations (in phasor notation): 2 A + k 2 A = μ J 2 φ + k 2 φ = ρ/ǫ where the time dependence is bound up in k , the free space wavenumber: k = ω/c = ω μ ǫ = 2 π/λ The potential equations are coupled only by the fact that the current density J and charge density ρ obey the continuity equation ∇· J +
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