Wire Antennas: Maxwell’s equations
Maxwell’s equations govern radiation variables:
(
E electric field v m- 1
(
)
H magnetic field a m- 1
)
(
D = εE electric displaceme nt Coulombs m- 2
)
B = µH magnetic flux density (Teslas)
(ε is permittivity; εo = 8.8542 × 10-12 farads/m for vacuum)
(µ is permeability; µo = 4π • 10-7 henries/m)
(1 Tesla = 1 Weber m-2 = 104 gauss)
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1/11/01
E1
Maxwell’s Equations: Dynamics and Statics
Maxwell’s equations
Statics
∂B
∇ ×E = −
∂t
→
=0
∂D
∇ ×H = J +
∂t
→
= J a m- 2
∇ •D = ρ
→
= ρ C m-3
∇ •B = 0
→
=0
(
(
)
)
∂ ˆ1 ∂ ˆ 1 ∂ ⎞
⎛ ∆
∆
+φ
⎟
⎜ ∇ x̂∂ ∂x + ŷ∂ ∂y + ẑ∂ ∂z ; ∇ r̂ + θ
r ∂θ
r sin θ ∂φ ⎠
∂r
⎝
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1/11/01
E2
Static Solutions to Maxwell’s Equations
Maxwell’s equations govern variables:
(
E electric field v m- 1
(
)
H magnetic field a m- 1
)
E = −∇φ since ∇ × E = 0
p
rpq
vq volume of source
B = ∇ × A since ∇ • B = 0
ρq
1
φp =
dv q volts is electrostatic potential at point p
∫
v
4πε q rpq
Jq
µ
Ap =
dv q is the vector potential at p
∫
v
4π q rpq
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1/11/01
E3
Dynamic Solutions to Maxwell’s Equations
B = ∇ × A since ∇ • B = 0
(
)
Jq t − rpq c
µ
Ap ( t ) =
dv q (static solution, delayed)
∫
v
4π q
rpq
c = 1 µoεo ≅ 3 × 108 ms −1 (velocity of light)
− jkrpq
Jqe
µ
Sinusoidal steady state: A p =
4π ∫v q
rpq
dv q
where propagation constant k = ω µoεo = ω c = 2π λ
Solution method : Jq (r ) → A (r ) → B(r ) → E(r )
where E = − ( ∇ × H) jωε from Faraday’s law
Lec 09.6- 4
1/11/01
E4
Elementary Dipole Antenna (Hertzian Dipole)
z
θ
Q(t)
d << λ/2π
Io
d
S=E×H
φˆ , H
ˆ
r θ, E
λ
,
r >> d
2π
and the quasistatic fields are strongest
(store active power, (We ) >> (Wm ) here )
In " near field" r <<
y
φ
x
-Q(t)
Far field: r >> λ 2π ,
Lec 09.6- 5
1/11/01
kI dsin θ − jkr
E ≅ θˆ jηo o
e
4πr
where k = 2π λ , ηo ≡ µo εo = 377Ω
characteristic impedance of free space
kIodsin θ − jkr
ˆ
H≅φ
e
4πjr
E5
Elementary Dipole Antenna (Continued)
E
2D2
r≥
" far field of aperture D"
λ
r≅
r <<
λ
2π
r >>
λ
2π
λ
" near field"
2π
∗
∆
S E × H W m- 2 " Poynting vector"
(
1
where average power density = S(t ) = Re{ S} W m- 2
2
(
and S(t ) = E(t ) × H(t ) W m- 2
Lec 09.6- 6
1/11/01
)
)
E6
Elementary Dipole Antenna (Continued)
∗
∆
S E × H W m- 2 " Poynting vector"
(
1
where average power density = S(t ) = Re{ S} W m- 2
2
(
and S(t ) = E(t ) × H(t ) W m- 2
)
)
z
S
θ
2
ηo Iod sin θ
S(t ) = r̂
for a short dipole
2 2λr
{
y
}
2
1
π Iod
Total power
ˆ
Pt = R e ∫ S • r dΩ = η
Watts
4
π
transmitted
2
3 λ
Lec 09.6- 7
1/11/01
E7
Equivalent Circuit for Short Dipole Antenna
jX
+
Io →
V
– Reactance
Rr = “Radiation
resistance”
–
Reactance is capacitive
for a short dipole antenna
and inductive for a small
loop antenna
z
Io → +
–
d
V
–
deff
I(z)
0
Iod → Iodeff =
Io
1 2
πη Iodeff
Pt = Io Rr =
2
3
λ
Rr =
∞
∫ I(z )dz
−∞
Here deff = d 2
Since d << λ 2π
2
2π
ηo (deff λ )2 ohms for a short dipole antenna
3
Example: λ = 300m(1 MHz), deff = 1m ⇒ Rr ≅ 0.01Ω
such a mismatch requires transformers or hi-Q
resonators for a good coupling to receivers or transmitters
Lec 09.6- 8
1/11/01
E8
Wire Antennas of Arbitrary Shape
Finding self-consistent solution is difficult
(matches all boundary conditions)
Approximate solutions are often adequate
Io
r( )
+
V –
I( )
θ( )
p
Approach
Lec 09.6- 9
1/11/01
1)
Use TEM - line reasoning to guess I(r )
2)
I(r ) → A farfield → Hff → E ff
3)
jkη ˆ
− jkr ( A )
θ
(
A
)
I
(
A
)
e
sin θ(A ) dA
E ff ≅
∫
4πr L
G1
Estimating Current Distribution I( )
on Wire Antennas
E
I(A )
~
→
coax
H( )
H
I
E( )
σ=∞
[
]
1
1
1
2
2
−3
We = εo E , Wm = µo H Jm , We,m ∝
4
4
r2
Most energy stored within 1–2 wire radii.
Therefore V,I on thin wires is TEM - like
Lec 09.6- 10
1/11/01
G2
Examples: Current Distributions and Patterns
I(z)
“Half-wave dipole”
λ/2
z
Rr ≅ 73Ω, jX= 0
Io ↑↓
“Full-wave antenna”
jηIo e
(
− jkr
)
⎡cos kA cos θ − cos kA ⎤
E ff ≅ θˆ
2πr sin θ ⎢⎣
2
2 ⎥⎦
λ
⎛π
⎞
= cos⎜ cos θ ⎟ if A =
2
⎝2
⎠
z
θ
d
for center-fed
wire of length A
⏐E(θ)⏐
D = λ/2
Io
Short dipole, d << λ/2
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1/11/01
G3
Examples: Current Distributions and Patterns
I(z)
z
Io ↑↓
2Io
Current transformer, jX ≠ 0
Long-wire antenna
Radiation decay
+
↑↓
Io
Traveling
wave
Standing
wave
Vee (broadband)
=
+
Io ↑ ↓
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1/11/01
G4
Mirrors, Image Charges and Currents
We can replace planar mirrors (σ = ∞) with image charges and currents;
E, H solution unchanged.
+
J
–
(σ = ∞)
H
Sources
–
Image charges
and currents
J
Image current
+
Note: Anti-symmetry of image charges and currents guarantees E⊥(σ = ∞),
H//(σ = ∞), matching boundary conditions. Also, the uniqueness theorem
says any solution is the valid one.
Io
≅ Ground plane
Say 1 MHz (λ = 300m)
D ≅ 1m (short dipole); D << λ
Deff ≅ 2D/2 = D; X is capacitive
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1/11/01
D
D
Image
current
Antenna
pattern
G5
Antenna Arrays
p E
3
1
↑
2
A1Io
↑
θi
z
y
x
A2Io
4
ri
e-jϕi
B
ϕi = 2πri/λ
↑
AiIo
N identically oriented antenna elements
Ep ≅ E(θ, φ)e
N
jϕo
from Io at
− jϕi ( θ,φ )
A
e
∑ i
i =1
x=y=z=0
2
G(θ, φ) ∝ E = Eo {θ, φ} • ∑ A ie − jϕi ( θ,φ)
i
element
2
2
factor
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1/11/01
array factor
J1
Examples of Antenna Arrays
Full-wavelength antenna:
i(z)
+ ←λ/2→ –
Array factor
Element factor
Pattern
Array
⇒
t=0
×
=
↑↓
i(t)
Linear array:
e− jφi
zi cos θ
1
2
3
0
z2
z3
θ
…
N
z
zi
2π
zi cos θ = 0 at z = 0;
λ
αi is phase of ith element current
Let ϕi = αi +
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1/11/01
J2
Linear Array Example:
Half-Wave Dipole Plus Reflector
y
Image
current
z ⇒
x
x
λ/4
λ/2
Lec 09.6- 16
1/11/01
Reflector >> λ/2
y
Io
y
null
z ×
Array
factor
x
null
Element
factor
z =
Antenna
pattern
J3
Linear Array Example: Jansky Antenna
Used in 1927 to discover galactic radiation at 27 MHz while seeking
radio interference on AT&T transatlantic radio telephone circuits.
x
λ/2
Io
λ/2
x
z
y
z
∝ 72 = 49(x – pol.)
y
∝ 6 = 36(z – pol.)
∝ 12 = 1(x – pol.)
2
x
To cancel y-direction lobes, drive two duplicate
antennas in phase, λ/2 apart in y-direction ⇒
Lec 09.6- 17
1/11/01
z
y
J4
Genetic Algorithms for Designing Wire Antennas
1. Need performance metric, e.g. target gain or pattern plus cost function.
2. Need software tool to compute that metric for wire antennas.
3. Need vector description to represent each possible antenna.
4. Need genetric algorithm to randomly vary vector so its metric can be
computed. The algorithm hill-climbs efficiently toward optimum design,
especially if the antenna is not too reactive.
Yields rats-nest configurations that perform well.
e.g.
Ground plane
5. Metrics and vector descriptions favoring simplicity bias the
design accordingly.
Lec 09.6- 18
1/11/01
J5