Maxwell's Equations

Wire Antennas: Maxwell’s equations

Maxwell’s equations govern radiation variables:

D = εE electric displaceme nt (Coulombs m 2 )

B = µH magnetic flux density (Teslas)

-12 (ε is permittivity; εo = 8.8542 × 10 farads/m for vacuum) -7 (µ is permeability; µo = 4π• 10 henries/m) (1 Tesla = 1 Weber m-2 = 104 gauss)

Lec 09.6- 1 1/11/01 E1 Maxwell’s Equations: Dynamics and Statics

Maxwell’s equations Statics

∂B ∇ × E = − → = 0 ∂t ∂D ∇ × JH += → = J (a m 2 ) ∂t -3 ∇ • D = ρ → =ρ (C m )

∇ • = 0B → = 0

⎛ ∂ 1 ∂ 1 ∂ ⎞ ⎜ ∇ ∆ ˆ ∂ ∂xx ˆ∂ ∂yy ˆ∂ ∂ ;zz ∇++ ∆ rˆ + θˆ + φˆ ⎟ ⎝ ∂r r θ∂ r sin θ φ∂ ⎠

Lec 09.6- 2 1/11/01 E2 Static Solutions to Maxwell’s Equations Maxwell’s equations govern variables: p 1 electric E electric field (v m ) rpq H magnetic field (a m 1 )

vq volume of source E = φ−∇ ∇ × = 0Esince

B = ∇ × A ∇ • = 0Bsince 1 ρ =φ q dv volts is electrosta tic potential at point p p ∫v q r4 πε q r4 pq

µ Jq A p = dv is the vector potential at p ∫v q 4π q rpq

Lec 09.6- 3 1/11/01 E3 Dynamic Solutions to Maxwell’s Equations B = ∇ × A ∇ • = 0Bsince

µ Jtq ( − r pq c) A p ()t = ∫ dvq ( static so l ution, delayed) 4 π v q rpq 18 −18 1c = 1c µ ε oo ×≅ ms103 (velocity of light) − jkrpq µ Jeq Sinusoidal steady state: A p = ∫ dv q 4 π v q rpq

where propagation constant k = ω µ εoo = ω = 2c π 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) S = E × H z φˆ , H θ θˆ , E λ Q(t) r In " near field" r << >> dr, 2 π d << λ/2π and the quasistati c fields are strongest Io d y (store active power, (W e ) >> ( m )hereW ) x φ -Q(t)

kI dsinθ Far field: r >> λ 2π , E ≅ θˆ j η o e−jkr o 4rπ

k where k = π λ,2 η ≡ µ εooo = 377Ω characteristic impedance of free space kI dsinθ H ≅φˆ o e −jkr Lec 09.6- 5 4jrπ 1/11/01 E5 Elementary Dipole Antenna (Continued)

E D2 2 "r ≥ "r far field of aperture D" λ λ λ r >> r ≅ 2π 2π λ r << " near field" 2π

∗ HES ∆ ×HES W m 2 " Poynting vector" 1 where average power density S ()t == Re{} S (W m 2 ) 2 m W t H t E t S and S() t E ()t ×= H () t (W m 2 )

Lec 09.6- 6 1/11/01 E6 Elementary Dipole Antenna (Continued)

∗ HES ∆ ×HES W m 2 " Poynting vector" 1 where average power density S ()t == Re{} S (W m 2 ) 2 m W t H t E t S and S() t E ()t ×= H () t (W m 2 )

z S 2 θ ηo Io d sin θ t S ()t = rˆ for a short dipole y λ r22

Total power 1 π I d 2 P = { SR rˆ d }=Ω• η o Watts transmitted 2 et ∫4 π 3 λ

Lec 09.6- 7 1/11/01 E7 Equivalent Circuit for Short Dipole Antenna

+ jX Reactance is capacitive for a short dipole antenna I → V R = “Radiation o – Reactance r and inductive for a small resistance” – loop antenna z

∞ 2 d d Here deff = d 2 + deff I d I d =→ I() z dz Io → d I(z) effoo ∫ Since d << λ 2 π – 0 − ∞ V – Io 1 πη I d 2 P = 2 RI = o eff t 2 o r 3 λ

2π R = (d λη )2 ohms for a short dipole antenna effor 3 effor

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( ) – p V θ( ) I( ) Approach 1) Use TEM - line reasoning to guess I( r )

2) I( r )→ A farfield →→ EH ffff

jkη ˆ − A )(jkr 3) E ff ≅ ∫ θ AA θ d)(sine)(I)( AA Lec 09.6- 9 πr4 1/11/01 L G1 Estimating Current Distribution I( ) on Wire Antennas

→~ E coax H( ) H I() A I E( ) σ = ∞

1 2 1 2 − 3 1 W ε= oe E , W µ= om [JmH ], W me, ∝ 4 4 r 2 Most energy stored within 1–2 wire radii.

Therefore I,V on thin wires is TEM like-

Lec 09.6- 10 1/11/01 G2 Examples: Current Distributions and Patterns

“Half-wave dipole” I(z) λ/2 z Rr ≅ 73Ω, jX= 0

Io ↑↓ “Full-wave antenna”

jIη e − jkr for center-fed ˆ o ⎡ k A kA ⎤ E ff ≅θ cos cos θ − cos wire of length A 2rπ s in θ ⎣⎢ ( 2 ) 2 ⎦⎥

⎛ π ⎞ λ = cos⎜ θ⎟ ifcos A = ⎝ 2 ⎠ 2 z ⏐E(θ)⏐ θ d D = λ/2 Io

Short dipole, d << λ/2

Lec 09.6- 11 1/11/01 G3 Examples: Current Distributions and Patterns

I(z) z

2Io Io↑↓ Current transformer, jX ≠ 0 Long-wire antenna Radiation decay

+ ↑↓ Io Traveling Standing wave wave

Vee (broadband)

= +

Io ↑↓

Lec 09.6- 12 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 + – H (σ = ∞) Sources

Image charges Image current – J and currents + 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 D

Image Antenna Say 1 MHz (λ = 300m) D D ≅ 1m (short dipole); D << λ current pattern Deff ≅ 2D/2 = D; X is capacitive

Lec 09.6- 13 1/11/01 G5 Antenna Arrays

p E 1 3 ri θi z -jϕ ↑ 2 e i A1Io y 4 ϕ = 2π r /λ ↑ x B i i A I 2 o ↑ AiIo N identically oriented antenna elements N j ϕo −ϕj i (, θφ ) Ep ≅ E( θ , φ )e ∑ Ai e I from Io at i1= 0zyx === 0zyx

2 2 2 − ϕi θ φ ),(j },{EE),(G θ φ =∝ o θ φ},{EE),(G • ∑ A i e i element factor array factor

Lec 09.6- 14 1/11/01 J1 Examples of Antenna Arrays

Full-wavelength antenna: Array factor Element factor i(z) + ←λ /2→ – Array Pattern ⇒ × = t = 0 ↑↓ i(t)

Linear array: e − jφi zi cos θ θ 1 2 3 … N z 0 z2 z3 zi 2π Let +α=ϕ z cos θ = 0 at z = 0; iii λ iii α i is phase of ith element current Lec 09.6- 15 1/11/01 J2 Linear Array Example: Half-Wave Dipole Plus Reflector

Reflector >> λ/2 y y y null

z ⇒ z × z = Image x x current x null λ/4 I o Array Element Antenna λ/2 factor factor pattern

Lec 09.6- 16 1/11/01 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 z Io y λ/2 x z ∝ 72 = 49(x – pol.)

y ∝ 62 = 36(z – pol.) ∝ 12 = 1(x – pol.) x z To cancel y-direction lobes, drive two duplicate antennas in phase, λ/2 apart in y-direction ⇒ y

Lec 09.6- 17 1/11/01 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