Constantes e rela¸
c˜
oes fundamentais
ε0≃ 10 −9 36π F/m µ0= 4π × 10 −7H/m 1 Np/m = 8.69 dB/mOndas harm´
onicas
vf = ω β vg= dω dβ = vf+ β dvf dβ = vf(λ) − λ dvf dλ
Linhas de transmiss˜
ao
−dV (z)dz = (R + jωL)I(z) −dI(z)dz = (G + jωC)V (z) d2V (z) dz2 − γ 2V (z) = 0 d2I(z) dz2 − γ 2I(z) = 0 γ = α + jβ =p(R + jωL)(G + jωC) Z0= s R + jωL G + jωC V (z′) = V0+eγz ′ + V0−e−γz ′ I(z′) = 1 Z0 V0+eγz ′ − V0−e−γz ′ Z(z′) = Z0 ZL+ Z0tanh γz′ Z0+ ZLtanh γz′ Pmed(z′) = 1 2Re {V (z ′ )I ∗ (z′)} ΓL= ZL− Z0 ZL+ Z0 = −ΓI Γ(z′) = ΓLe−2γz ′Linha sem perdas
γ = jβ = jω√LC Z0= r L C vf = 1 √ LC = 1 √µε Z(z′) = Z 0 ZL+ jZ0tan βz′ Z0+ jZLtan βz′ S = 1 + |ΓL| 1 − |ΓL|
z|V′ max|;|Imin|=
1
2β(θΓ+ 2nπ) z
′
|Vmin|;|Imax|=
1
2β(θΓ+ (2n + 1)π)
Linha sem distor¸c˜ao (R/L = G/C)
γ = α + jβ =r C L(R + jωL) vf = 1 √ LC Z0= r L C Linhas em circuitos Γg= Zg− Z0 Zg+ Z0 V (z′) = Z0Vge −γ(l−z′) Z0+ Zg · 1 + ΓLe−2γz ′ 1 − ΓgΓLe−2γl I(z′) =Vge −γ(l−z′) Z0+ Zg · 1 − ΓLe−2γz ′ 1 − ΓgΓLe−2γl Linhas de transmiss˜ao 1
Parˆametros de linhas de transmiss˜ao δ = r 1 πf µcσc Rs= 1 δσc
Coaxial Bifilar Condutor sobre terra Tiras
R(Ω/m) Rs 2π 1 a+ 1 b R s πa D/(2a) p(D/(2a))2− 1 Rs 2πa s h/a + 1 h/a − 1 2Rs W L(H/m) µ 2πln b a µ πarcch D 2a µ 2πarcch h a µ2h W G(S/m) 2πσ ln(b/a) πσ arcch(D/2a) 2πσ arcch(h/a) σ W 2h C(F/m) 2πε ln(b/a) πε arcch(D/2a) 2πε arcch(h/a) ε W 2h
Ondas electromagn´
eticas planas
Equa¸c˜oes de Maxwell
∇ × E = −jωµH ∇ × H = (jωε + σ)E ∇ · E = ρε ∇ · H = 0 Ondas planas E(r) = E0e−γˆ a n·r H(r) = H0e−γˆan·r H = 1 η ˆan× E E = −η ˆan× H
Meios sem perdas
γ = jβ = jω√µε η =r µ
ε vf =
1 √µε Meios com perdas
γ = α + jβ =pjωµσ − ω2µε η = s jωµ σ + jωε tan δc= σ ωε Meios bons condutores (σ ≫ ωε)
α = β =r ωµσ 2 η = r ωµ σ ∠45 ◦ δ = 1 α= 1 √ f πµσ Potˆencia e energia
S(t) = E(t) × H(t) I A S(t) · da = −∂t∂ Z V w dv − Z V pσdv w = we+ wm= 1 2ε |E(t)| 2 +1 2µ |H(t)| 2 pσ= σ |E(t)| 2 Smed= 1 2Re{E × H ∗} = 1 2 |E| 2 Re 1 η ˆ an Pmed= Z A Smed· da
Leis de Snell
θi= θr n1sin θi= n2sin θt n =
c vf
=√µrεr
Incidˆencia normal
Γ = Er0 Ei0 = η2− η1 η2+ η1 τ = Et0 Ei0 = 2η2 η2+ η1 = 1 + Γ
Incidˆencia obl´ıqua
Γ⊥=η2cos θi− η1cos θt η2cos θi+ η1cos θt = cos θi− q (ε2/ε1) − sin2θi cos θi+ q (ε2/ε1) − sin2θi τ⊥= 1 + Γ⊥ Γk= η2cos θt− η1cos θi η2cos θt+ η1cos θi =−(ε2/ε1) cos θi+ q (ε2/ε1) − sin2θi (ε2/ε1) cos θi+ q (ε2/ε1) − sin2θi 1 + Γk= cos θt cos θi τk
Condi¸
c˜
oes fronteira
ˆ
an× (E1− E2) = 0 aˆn· (D1− D2) = ρS D = εE
ˆ
an× (H1− H2) = JS aˆn· (B1− B2) = 0 B = µH
ˆ
an aponta do meio 2 para o meio 1
Guias de onda
E(x, y, z) = E0(x, y)e−γz H(x, y, z) = H0(x, y)e−γz ∇2xyE 0(x, y) + h2E0(x, y) = 0 ∇2xyH 0(x, y) + h2H0(y, z) = 0 h2= γ2+ ω2µε ∇2xy = ∂ 2/∂x2+ ∂2/∂y2 E0x= − 1 h2 γ∂E 0 z ∂x + jωµ ∂H0 z ∂y Hx0= − 1 h2 γ∂H 0 z ∂x − jωε ∂E0 z ∂y E0y= − 1 h2 γ∂E 0 z ∂y − jωµ ∂H0 z ∂x Hy0= − 1 h2 γ∂H 0 z ∂y + jωε ∂E0 z ∂x Z = E 0 x H0 y = −E 0 y H0 x Pmed= 1 2 Z A Re 1 Z Ex0 2 + Ey0 2 da H = 1
Zˆz × E (modos TEM ou TM) E = −Z ˆz × H (modos TEM ou TE)
Guias met´alicos fc= h 2π√µε γ =ph 2− ω2µε = h s 1 − ff c 2 vf = 1/√µε q 1 − (fc/f )2 vg= 1 √µε s 1 − ffc 2 ZTEM= r µ ε ZTM= −j r µ ε s fc f 2 − 1 ZTE= jpµ/ε q (fc/f )2− 1
Guias de placas paralelas h = nπ b Modo TMn Modo TEn Ez0(y) = Ansin nπy b Hz0(y) = Bncos nπy b H0 x(y) = jωε h Ancos nπy b E0 x(y) = jωµ h Bnsin nπy b Ey0(y) = − γ hAncos nπy b Hy0(y) = γ hBnsin nπy b Guias rectangulares (a > b) h2=mπ a 2 +nπ b 2 Modo TMmn Modo TEmn Ez0(x, y) = E0sin mπx a sinnπy b Hz0(x, y) = H0cos mπx a cosnπy b Ex0(x, y) = −hγ2mπa E0cos mπx a sinnπy b Hx0(x, y) = γ h2 mπ a H0sin mπx a cosnπy b Ey0(x, y) = − γ h2 nπ b E0sin mπx a cosnπy b Hy0(x, y) = γ h2 nπ b H0cos mπx a sinnπy b Hx0(x, y) = jωε h2 nπ b E0sin mπx a cosnπy b Ex0(x, y) = jωµ h2 nπ b H0cos mπx a sinnπy b Hy0(x, y) = − jωε h2 mπ a E0cos mπx a sinnπy b Ey0(x, y) = − jωµ h2 mπ a H0sin mπx a cosnπy b Guias circulares Modo TMnp Modo TEnp (h)TM np = p − ´esimo zero de Jn a (h)TEnp = p − ´esimo zero de J′ n a Ez0(r, φ) = E0Jn(hr) cos(nφ) Hz0(r, φ) = H0Jn(hr) cos(nφ) Er0(r, φ) = −γhE0Jn′(hr) cos(nφ) H 0 r(r, φ) = − γ hH0J ′ n(hr) cos(nφ) Eφ0(r, φ) = γn h2rE0Jn(hr) sin(nφ) H 0 φ(r, φ) = γn h2rH0Jn(hr) sin(nφ) Hr0(r, φ) = −jωεnh2r E0Jn(hr) sin(nφ) E 0 r(r, φ) = jωµn h2r H0Jn(hr) sin(nφ) H0 φ(r, φ) = − jωε h E0J ′ n(hr) cos(nφ) Eφ0(r, φ) = jωµ h H0J ′ n(hr) cos(nφ) zero J0(x) J1(x) J2(x) J3(x) 1 2.4048 3.8317 5.1336 6.3802 2 5.5201 7.0156 8.4172 9.7610 3 8.6537 10.1735 11.6198 13.0152 4 11.7915 13.3237 14.7960 16.2235 5 14.9309 16.4706 17.9598 19.4094 zero J′ 0(x) J1′(x) J2′(x) J3′(x) 1 3.8317 1.8412 3.0542 4.2012 2 7.0156 5.3314 6.7061 8.0152 3 10.1735 8.5363 9.9695 11.3459 4 13.3237 11.7060 13.1704 14.5858 5 16.4706 14.8636 16.3475 17.7887
Guias diel´ectricos planares
Modos Rela¸c˜ao caracter´ıstica Frequˆencia de corte
Pares TM ν = − nn2 1 2 h1cot h1b 2 fc = (n − 1 2)c bpn2 1− n22 TE ν = −h1cot h1b 2 ´Impares TM ν = n2 n1 2 h1tan h1b 2 fc = (n − 1)c bpn2 1− n22 TE ν = h1tan h1b 2 ν = r ω c 2 (n2 1− n22) − h21 n = 1, 2, 3 . . . Fibras ´opticas NA = sin θA= q n2 1− n22 V = 2πa λ0 q n2 1− n22 Monomodo: V < 2.4048 Cavidades rectangulares h2=mπ a 2 +nπ b 2 fmnp= 1 2√µε r m a 2 +n b 2 +p d 2 Modo TMmnp Modo TEmnp Ez= E0sin mπx a sinnπy b cospπz d Hz= H0cos mπx a cosnπy b sinpπz d Ex= − 1 h2 mπ a pπ d E0cos mπx a sinnπy b sinpπz d Hx= − 1 h2 mπ a pπ d H0sin mπx a cosnπy b cospπz d Ey = − 1 h2 nπ b pπ d E0sin mπx a cosnπy b sinpπz d Hy= − 1 h2 nπ b pπ dH0cos mπx a sinnπy b cospπz d Hx= jωε h2 nπ b E0sin mπx a cosnπy b cospπz d Ex= jωµ h2 nπ b H0cos mπx a sinnπy b sinpπz d Hy= − jωε h2 mπ a E0cos mπx a sinnπy b cospπz d Ey= − jωµ h2 mπ a H0sin mπx a cosnπy b sinpπz d Cavidades circulares ωnpq = 1 √µε r h2 np+ qπ d 2 Modo TMnpq Modo TEnpq (h)TMnp = p − ´esimo zero de Jn a (h)TEnp= p − ´esimo zero de Jn′ a Ez= E0Jn(hr) cos(nφ) cos qπz d Hz= H0Jn(hr) cos(nφ) sin qπz d Er= − 1 h qπ d E0J ′ n(hr) cos(nφ) sin qπz d Hr= 1 h qπ d H0J ′ n(hr) cos(nφ) cos qπz d Eφ= n h2r qπ d E0Jn(hr) sin(nφ) sin qπz d Hφ= − n h2r qπ d H0Jn(hr) sin(nφ) cos qπz d Hr= − jωεn h2r E0Jn(hr) sin(nφ) cos qπz d Er= jωµn h2r H0Jn(hr) sin(nφ) sin qπz d Hφ= − jωε h E0J ′ n(hr) cos(nφ) cos qπz d Eφ= jωµ h H0J ′ n(hr) cos(nφ) sin qπz d
Antenas e radia¸
c˜
ao
A(r) = µ 4π Z V′ J(r′)e−jβR R dv ′, R = |r − r′| H = 1 µ∇ × ACoordenadas esf´ericas
∇ × X = r2sin θ1 ˆr r ˆθ r sin θ ˆφ ∂ ∂r ∂ ∂θ ∂ ∂φ Xr rXθ r sin θ Xφ
Dipolo el´ectrico elementar E = −I dl β 2η 0 4π 2 cos θ 1 (jβr)2 + 1 (jβr)3 ˆ r + sin θ 1 jβr + 1 (jβr)2 + 1 (jβr)3 ˆ θ e−jβr H = −I dl β 2sin θ 4π 1 jβr + 1 (jβr)2 e−jβrφˆ
Dipolo magn´etico elementar E = jωµ0Ib 2β2sin θ 4 1 jβr + 1 (jβr)2 e−jβrφˆ H = −jωµ0Ib 2β2 4η0 2 cos θ 1 (jβr)2 + 1 (jβr)3 ˆ r + sin θ 1 jβr+ 1 (jβr)2 + 1 (jβr)3 ˆ θ e−jβr
Parˆametros de antenas Pr=
Z 2π
0
Z π
0
Smedr2sin θdθdφ U = r2Smed
Rr= 2Pr I2 GD= 4πU Pr D = (GD)max
Antenas finas lineares
I(z′) = Imsin [β (h − |z′|)] F (θ) =cos (βh cos θ) − cos(βh)
sin θ E = jη0Im 2π e−jβr r F (θ)ˆθ H = jIm 2π e−jβr r F (θ) ˆφ Agrupamentos de antenas |E| = 2 |Em|
r |F (θ, φ)| |A(ψ)| ψ = βd sin θ cos φ + ξ
Agrupamentos lineares uniformes
|A(ψ)| = N1 sinN ψ2 sinψ2
