Wire Antennas: Maxwell’s equations Maxwell’s equations govern radiation variables: 1• E electricfield ( v ) m 1• H magneticfield ( a ) m • 2 E D = ε electricdisplaceme nt (Coulombs m ) 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 • 2 ∇ ×H = J + → J = a( m) ∂t -3 ∇ • D = ρ → =ρm ( C ) ∇ •B = 0 → = 0 ⎛ ∂ 1 ∂ 1 ∂ ⎞ ⎜ ∇ ∆xˆ ∂ +∂ xyˆ∂ ∂ y +zˆ∂ ∂ z ; ∆ ∇rˆ + θˆ + φˆ ⎟ ⎝ ∂r r∂ θr sinθ∂ φ⎠ -09.6c Le 2 /01 1/11 E2 Static Solutions to Maxwell’s Equations Maxwell’s equations govern variables: p • 1 r E electricfield ( v ) m pq • 1 H magneticfield ( a ) m vq volume of source E =−∇ since φ ∇ × = E 0 B = ∇ × sinceA ∇ • = B 0 1 ρ φ = q dv volts electrostais tic potentialat point p p ∫v q 4 πε q pqr µ Jq A p = is dv the vectorpotential at p ∫v q 4π q rpq -09.6c Le 3 /01 1/11 E3 Dynamic Solutions to Maxwell’s Equations B = ∇ × sinceA ∇ • = B 0 µ Jtq ( − r pq c) A p ()t = ∫ dvq ( static so l ution, delayed) 4 π v q rpq 8 − 1 c = 1 µo 3ε ≅o 10 × ms(velocityof light) − jkrpq µ Jeq Sinusoidal steady state: A p = ∫ dv q 4 π v q rpq where propagation constant k = ω µoε o= ωc= 2π λ methodSolution : J →q (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) Inr " near <<field" , rr>> d 2 π d << λ/2π and cquasistatithe fields strongestare Io d y (store activepower, ( e )W >> W( m ) here) x φ -Q(t) kI dsinθ Far field: r >> λ 2π , E ≅ θˆ j η o e−jkr o 4rπ where =2 π kλ oη, ≡ µ oε = o377Ω 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 2 2D r ≥far " field apertureof D" λ λ λ r >> r ≅ 2π 2π λ " r << near field" 2π ∗ S W ∆ Em× H "• 2 Poyntingvector" 1 • 2 where average power density= S Re() t = S{} ( W ) m 2 • 2 and S t ()=E ()t × () H( t ) W m -09.6c Le 6 /01 1/11 E6 Elementary Dipole Antenna (Continued) ∗ S W ∆ Em× H "• 2 Poyntingvector" 1 • 2 where average power density= S Re() t = S{} ( W ) m 2 • 2 and S t ()=E ()t × () H( t ) W m z S 2 θ I ηo do θsin S ()t = rˆ for a short dipole y 2 2λ r 2 Total power 1 πI o d P = RS{ •rˆ d Ω } =η Watts transmitted t 2 e ∫4 π 3 λ -09.6c Le 7 /01 1/11 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 ∞ Here effd = d 2 + deff I d I → d = ()I z dz Io → d I(z) o o eff ∫ Since << λ d2 π – 0 − ∞ V – Io 1 πηI d 2 P = IR2 = o eff t 2 o r 3 λ 2π 2 R = η(d λ) ohms for a short dipole antenna r 3 o eff 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 -09.6c Le 8 /01 1/11 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 Use 1) TEM- reasoningto line guess ( ) I r 2) I ( )r → A farfield→ HEff → ff jkη ˆ −jkr A ( ) 3) E( ff ≅ ) I∫ (θAA ) e sinθAA ( ) d -09.6c Le 9 4 π r /01 1/11 L G1 Estimating Current Distribution I( ) on Wire Antennas →~ E coax H( ) H E( ) I() A I σ = ∞ 1 2 1 2 − 3 1 We = ε,E o mW = H µ o [ Jm, ] e, W m∝ 4 4 r 2 Most energy stored within 1–2 wire radii. V , ITherefore on thin wires - likeis TEM -09.6c Le 10 /01 1/11 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 ≅ ( Eθ , φee ) ∑ i A from o at I i1= x = y = z = 0 2 2 2 j− ϕ (i θ ,φ ) G ( ,θ )φ∝ E = o θ E φ {• ∑A , i } e i element factor array factor -09.6c Le 14 /01 1/11 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 Letϕ z = αcos + θ = 0 =at 0; z i iλ i is αphase i elementof currentith -09.6c Le 15 /01 1/11 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 .