Antennas & Propagation
Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
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Overview (entire lecture) -
Mathematical & Physical Fundamentals
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Fundamentals of Antennas
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Practical Antennas
Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
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Part I Mathematical & Physical Foundations
Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
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Physical Basics
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Physical Experience
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Maxwell’s Equations
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Wave equation
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Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
Physical Basics: a) Coulomb’s Law
F
Q ⋅Q F∝k 1 2 2 r r 1 Q1 ⋅ Q2 = r 4πε0 ε r 2
Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
F Q1
Q2
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Physical Basics: b) Electric Field (Intensity) E Force ß à Field
Q1
F 1 Q1 E= = r Q2 4πε0ε r 2
Q2
Test charge Q2
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Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
Physical Basics: c) Electric Flux (Density) D
Å=
1 Q1 r 4πε0 ε r 2
D = ε0 ε ⋅ E
D=
Q1 r 4πr 2 Area
Q1
Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
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Physical Basics: d) Charge Density ?
ρ=
Q V
∆Q dQ = ∆V →0 ∆V dV
ρ = lim
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Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
Physical Basics: e) Current I
I=
Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
∆Q dQ = ∆t dt
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Physical Basics: f) Current Density J
A I
J=
I dI = A dA 11
Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
Physical Basics: g) Magnetic Flux (Density) B In 1820 Biot and Savart found out: I
?L
θ
∆B = k
dB =
Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
I ⋅ ∆L µ0 µ I ⋅ ∆L = r2 4π r 2
µ 0 µ I ⋅ r × dl = µ 0 µ ⋅ f (I ) 4π r 2 12
Physical Basics: h) Magnetic Field H
B = µ0 µ ⋅ f ( I )
H=
B µ0 µ 13
Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
Maxwell’s Equations In differential form: Gauss’ Law
Gauss’ Law
div D = ρ
div B = 0
Faraday’s Law
rot E = −
∂B ∂t
Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
Ampere’s Law
rot H = J +
∂D ∂t
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r E r H
r B
r D
r J
ρ
: : : : : :
electric field (volt/meter; V/m) magnetic field (ampere/meter, A/m) magnetic flux density (tesla; T) electric displacement, electric flux density (coulomb/meter 2; C/m2) electric current density (ampere/meter2; A/m2) electric charge density (coulomb/meter3; C/m3)
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Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
Constitutive relations •
• u
The flux densities, D and B, are related to the field amplitudes E and H by the constitutive relations. The nature of the medium defines the functional form of the relationship. For linear, isotropic media, the relations are simply given by r r
D = εE
In vacuum:
r B
r = µ H
εo = 8.854 x 10-12 Farads/m. µo = 4 π x 10-7 Henrys/m. Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
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Stokes’s theorem
∫
S ( area )
r r (∇ × A) ⋅ ds = ∫ A ⋅ dl C ( loop )
where dS and dl are unit vectors oriented normal to the surface, or tangential to the loop, respectively.
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Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
Gauss's divergence theorem
∫
closed surface
r F ⋅ ds =
∫
volume enclosed
r ∇ ⋅ F dv
∂B ∂B ∫S (∇ × E ) ⋅ ds = − ∫S ∂t ⋅ ds ⇒ ∫C E ⋅ dl = − ∫S ∂t ⋅ ds ∂D ∂D ∫S (∇ × H ) ⋅ ds = ∫S ( J + ∂t ) ⋅ ds⇒∫C H ⋅ dl = ∫S ( J + ∂t ) ⋅ ds
Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
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r r ( ∇ ⋅ D ) dv = ρ dv ⇒ D ∫V ∫V ∫S ⋅ ds = ∫V ρdv
r r ∫V (∇ ⋅ B)dv = 0⇒∫S B ⋅ ds = 0
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Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
Maxwell’s Equations In Integral form:
r ∫ S D ⋅ ds = ∫V ρdv
(Gauss’s Law for electric field)
S : is the unit vector, normal to a surface, and the Area integrals cover only the area enclosed Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
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r ∫S B ⋅ ds = 0
(Gauss’s Law for magnetic field)
r r r d ∫C H ⋅ dl = ∫S D ⋅ ds + ∫ S J ⋅ ds dt
(Ampere’s Law)
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Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
r r d ∫C E ⋅ dl = − ∫S B ⋅ ds dt
(Faraday’s Law)
Physical Experience
Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
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Maxwell’s Equations div D = ρ
div B = 0 D = ε 0ε ⋅ E B = µ0 µ ⋅ H
∂B rot E = − ∂t
rot H = J +
∂D ∂t
They seem coupled.
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Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
Maxwell’s Equations rot E = −
∂B ∂t
rot H = J +
∂D ∂t
THE KEY TO ANY OPERATING ANTENNA 1. You create a time variant current density J 2. This causes a varying magnetic field H 3. This causes a varying electric field E 4. This causes varying magnetic field H Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
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The Wave equation The electromagnetic wave equation comes directly from Maxwell's equations.
r ∇⋅D = 0
r r ∂ B ∇× E = − ∂t
r ∇⋅B = 0
r ∂Dr ∇× H = ∂t
Under these conditions: l l l
Source free (ρ ρ = 0 , J =0) Linear medium ( ε and µ independent of E and H) Isotropic medium
Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
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In an isotropic medium, the permittivity is independent of orientation and is described accurately by the scalar relation D =ε ε E.
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The permittivity can be a function of position, ε(r).
u
In an inhomogeneous medium, the electric field will encounter a different permittivity, ε, depending upon spatial location in the material.
Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
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Wave equations The wave equation for electric field:
2r r ∂ E 2 ∇ E − µε =0 2 ∂ t The wave equation for magnetic field:
2r r ∂ H 2 ∇ H − µε =0 2 ∂t The simplest wave equation solutions of Maxwell are uniform plane waves. 27
Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
Wave equations For a uniform plane wave, only one component is present and rest two are zero: r ∂E =0 ∂x
and
r ∂E =0 ∂y
and
E =0 z
The wave equation for electric field: ∂ 2E y ∂2E x ∂2 ax +ay = µε (a x E x + a y E y ) ∂z 2 ∂z 2 ∂t 2 The simplest wave equation solutions of Maxwell are uniform plane waves.
Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
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Wave equations This gives two second-order partial equation: ∂2E x ∂2E x = µε ∂z 2 ∂ t2
and
∂2E y ∂2E y = µε ∂z2 ∂t 2
With similar analysis for the magnetic field H , the wave equations
∂2 Hx ∂2 Hx = µε ∂z2 ∂t 2
and
∂ 2H y ∂z 2
= µε
∂ 2H y ∂t 2
Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
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Solution of the wave equation for electric field E ( r , t ) = E o f ( ωt − β ⋅ r ) Where Eo is a constant and f is any function of argument (ωt -β⋅r)
Example: E(z,t)=Eo cos(ωt -βz)
Prepared by Dr. Abbou Fouad Mohammed, Multimedia University
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