1-2 Vector and scalar potentials

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Rodney Johnson
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1 1-2 Vector and scalar potentials z a f a r q a q J y x f Maxwell's equations μ (1) ε (2) ρ (3) (4) - Derivation of E and H field by source current J Approach 1 : take (1) and plug into (2) μ ε μ μ (5) Electric field E due to current J source problem However, (5) cannot be solved easily(analytically).

2 Approach 2 : Since, define a magnetic vector potential as (6) Substitute B in (1) by (6) ω (7) (7) can be written as ω Φ (8) Φ : scalar potential take the curl of (6) in both sides : μ ωμ ε μ ωμ ε ω Φ μ (9) Simplify (9) : μ ωμ ε Φ (10) In order to define a unique : Both and must be defined. is defined by (6), is defined from (10) as ωμ ε Φ (11) Lorentz condition Using (11), (10) can be written as

3 μ (12) convenient form to be solved The vector potential A is derived from (12) ; Take the divergence of (8) and use (11) and (3) : ω Φ Φ Φ ρε (13) Once and Φ are obtained, then is given as ω Φ ω ω μ ε (14) Now H is derived either from (1) or from (6) μ (1) (6)

4 For a point source(infinitesimal current element) z a f a r q a q Idl y f x spherical coordinate system is used for convenience : coordinate ; (r, q, f) differential length; (dr, rdq, rdf) where differential volume ; r 2 dr dq df differential area on the sphere ; r 2 dq df use solid angle ; area on the sphere/r 2 (c.f. radian ; arc length/r ) differential area in solid angle ; dq df

5 1-3. Radiation from a short current element(short dipole) z a f a r q a q Idl y f x Fig 1.5 The short current filament Current element (short dipole) : component only μ (15) μ π (16) retarded potential : Use Spherical coordinate : θ θ θ (17) μ π θ θ θ (18) μ

6 θ π φ (19) and from equation or (14) μ ε π θ π θ θ θ θ (20) 1) In the region where r>> λ : radiation field in the far zone. θ π θ (21a) θ π φ (21b) (22a) (22b) where, free space intrinsic admittance. Note that and are function of r and θ and independent of φ. Symmetry Complex Poynting vector for the radiation field θ π (23)

7 2) near-zone field and For << 1 θ φ π θ θ θ (25) Since << 1, ε with ω Then πε θ θ θ (26) 1-4. Some basic antenna parameters 1) Radiation Pattern The relative distribution of radiated power as a function of direction ( φθ ) in space. E-plane pattern : as a function of θ with φ = constant H-plane pattern : as a function of φ with θ = constant

8 Half power beam width(hpbw) for E- and H-plane patterns θ θ π (27) for current element z z (a) y x (b) (c) Fig. 2.6 radiation patterns : E-plane and H-plane 2) Directivity and Gain The variation of the intensity with direction in space : The intensity of radiation = power radiated per unit solid angle;

9 Ω for current element θ π (28) Definition of Directivity function θφ θφ Ω π π Ω (29) for current element : total radiated power π θ θ θ φ π (30) θφ θ (31) at θ π Gain of the antenna G is defined as Ω π π Ω (32) η θφ η : efficiency of the antenna η where, : radiation resistance : antenna input resistance

3) Radiation Resistance Equivalent resistance which would dissipate the same amount of power as the antenna radiates when the current in that resistance equals the input current at the antenna

10 3) Radiation Resistance Equivalent resistance which would dissipate the same amount of power as the antenna radiates when the current in that resistance equals the input current at the antenna terminals Radiation resistance of shor dipole π π λ (35) Review i) Radiation pattern three dimensional quantities involving the variation of field(power) as a function of spherical coordinate q and f.

11 * two-dimensional plots a) and b) : plotted as a function of q in cylindrical coordinate system. c) : in the rectangular coordinate system.

12 - Half Power Beam Width(HPBW) and First Null Beamwidth(FNBW) ii) Beam Area(Beam Solid Angle)

13 iii) Antenna Gain d directional P o (W/W ) omidirectional directivity = P d / P o gain = P d / P in where P in : input power iv) Radiation resistance antenna as a circuit element I P d (W/W ) dissipating power P d = I 2 R rad R rad : radiation resistance

14 1-5. Radiation from a small current loop(magnetic dipole) z y x Fig. 2.6 small current loop of radius r 0, area π with current I : Magnetic dipole moment π (36) Current element at φ is given as φ and φ φ φ Vector potential for the differential element ; where Therefore μ φ π φ φ φ φ

15 μ π π φ φφ (37) Approximations of R under λ ( ) and : θ φφ Drop which is small relative to and for leads to θ φφ and θ φφ By using these approximations, μ π π φ φ θ φφφ μ π π θ φ (38) μ μ φ φ θ π θ (39) and θ π φ (40) From (39) and (40)

16 small magnetic dipole is dual of the short dipole : the radiation pattern and directivity are not changed. Total radiated power : π π φ θ θθφ π π π θθθφ π (41) π λ (42) Ex. = 10cm at 1 MHz Ω

ANTENNAS. Vector and Scalar Potentials. Maxwell's Equations. E = jωb. H = J + jωd. D = ρ (M3) B = 0 (M4) D = εe

ANTENNAS. Vector and Scalar Potentials. Maxwell's Equations. E = jωb. H = J + jωd. D = ρ (M3) B = 0 (M4) D = εe ANTENNAS Vector and Scalar Potentials Maxwell's Equations E = jωb H = J + jωd D = ρ B = (M) (M) (M3) (M4) D = εe B= µh For a linear, homogeneous, isotropic medium µ and ε are contant. Since B =, there