WELLENAUSBREITUNG Formelsammlung INSTITUT FÜR NACHRICHTENTECHNIK UND HOCHFREQUENZTECHNIK 5. Auflage, Sept. 14 Gregor Lasser
Maxwellsche Theorie S 1 = lim V V Σ ( H S d F = t ρ H = S + t D S = t ( D = t ρ S + t ρ = D=ρ B= E= t B H = S + t D D df = ρdv Σ τ B df = Σ E d l= B df t Σ H d l= S df + Σ t ( F = q E + v B Σ D d F S = σ E S = ρ v D = ε E = ε r ε E B = µ H = µ r µ H t ρ( r,t+ σ ε ρ( r,t = ρ( r,t = ρ ( re σ ε t τ D = ε σ 1
E = E(x,y,z,t = E( r,t = Re{ E( re jωt } = 1 H = S +jω D = σ E +jωε E = jωδ E δ = ε+ σ jω = ε jσ ω tanθ = ε ε = σ ωε E = H = E = jωµ H H = jωδ E ( E( re jωt + E ( re jωt P(t = E(t H(t P(t = σe(t } {{ E(t ε E(t } t( E(t + µ H(t }{{} H(t }{{} E (t E (t H (t t V V w e (t = ε E (t w m (t = µ H (t p v (t = σe (t P(tdV = P(t d F ( w e (t+w m (t dv = Σ Σ P(t d F + V p v (tdv E( r,t= 1 ( E( re jωt + E ( re jωt E(t E(t= 1 ( E( r E( re jωt +E( r 4 E ( r+ E ( r E ( re jωt E(t E(t= 1 E(t w e (t = w e = ε 4 E(t w m (t = w m = µ 4 H(t p v (t = p v = σ E(t T = 1 E H = T w +j T b
E d l = E t1 l +E n1 x+e n x E t l E n x E n1 x = (E t1 E t l E n1 +E n = B df t = F E t1 = E t H t1 = H t D d F = (D n1 D n F = ρ S F D n1 D n = ρ S B n1 = B n ε 1 E n1 ε E n = ρ S µ 1 H n1 = µ H n n D 1 = ρ S n B 1 = n E 1 = n H 1 = K n ( D 1 D = ρ S n ( B 1 B = n ( E 1 E = n ( H 1 H = K E µε t E µσ t E = E +(ω µε jωµσ E = E +ω µδ E = H +ω µδ H = = x + y + z k = ω µδ Ψ(x,y,z = X(xY(yZ(z 1 1 1 X(x x X(x+ Y(y y Y(y+ k Z(z zz(z+ }{{} = 1 X(x x X(x = k x k x +k y +k z = k = ω µδ x X(x+k xx(x = const. 3
y H z z H y=jωδe x z H x x H z=jωδe y x H y y H x=jωδe z y E z z E y= jωµh x z E x x E z= jωµh y x E y y E x= jωµh z E x = j ( k κ z x E z +ωµ y H z E y = j ( k κ z y E z ωµ x H z H x = j ( k κ z x H z ωδ y E z H y = j ( k κ z y H z +ωδ x E z 4
Die homogene ebene Welle (HEW + z e y=µ t h x z e x=µ t h y =µ t h z z h y=ε t e x + z h x=ε t e y =ε t e z z e x µε t e x = e x (z,t = c 1 f 1 (z vt +c }{{} f (z +vt }{{} =e + x(z,t =e x (z,t η = η = v = 1 εµ = ω k µ η = ε µ 1πΩ 377Ω ε h + x = e+ y η e + x h + y = e+ y h + x = η E H i z e x h y = e y h x = η w e (t= ε (e x +e y w m (t= µ (h x +h y 5
w m (t = µ 1 η (e x +e y = w e(t p + x p + y p + z =e+ x h+ y e+ y h+ x p + z = 1 η (e+ x +e + y P(t = E H = 1 ( e η + x +e + y i z = 1 ( Ex +E η y i z e x (z,t = Re{E x (ze jωt } = E cos(k(vt z = E cos(ωt kz k = ω v = ω µε λ = π k = π ω εµ E 1 = E x =(E 1 i x + i y e jkz E = E y =( i x +E i y e jkz e x (z,t=e 1 cos(ωt kz e y (z,t=e cos(ωt kz +ψ e x (,t=e 1 cos(ωt e y (,t=e cos(ωt+ψ e x (,t=ecos(ωt, e y (,t=ecos(ωt π = ±Esin(ωt e x (z, = Ecos( kz = Ecos(kz e y (z, = Ecos( kz ± π = ±Esin(kz P = = 1 Re{ E=E y e jkz i y η H= E y e jkz i x P d F = 1 Re{ ( E H d F} (E x H y E y H xdf} = wd Re{ E ye jkz ( E y η e+jkz } = E y η wd P = U Z PV 6
U = E y dy = E y d d P = E y d Z PV E= jωµ H H=σ E +jωε E δ = ε j σ ω = ε(1 js s = 1 Q = tanθ = σ εω η = µ δ = µ 1 ε 1 js η = R+jX = η E 1 1 js jk z = jω µδ = jω µε 1 js = γ = α+jβ jk z = jk E 1 js 1+s +1 R = η E (1+s 1+s 1 X = η E (1+s α = k E 1+s 1 β = k E 1+s +1 η η E (1+j s η η E 1+j s jk z k E ( s +j jk z k E s (1+j d = 1 α 1 k E s = ωµσ 7
Reflexion an glatten Grenzflächen, die Parallelplattenleitung sinθ 1 ε = = n sinθ ε 1 n 1 Γ TM = T TM = ε cosθ 1 ε 1 cosθ ε cosθ 1 + = n cosθ 1 n sin Θ 1 ε 1 cosθ n cosθ 1 + n sin Θ 1 ε 1 cosθ 1 ε cosθ 1 + ncosθ 1 = ε 1 cosθ n cosθ 1 + n sin Θ 1 Γ TE = T TE = ε1 cosθ 1 ε cosθ ε1 cosθ 1 + = cosθ 1 n sin Θ 1 ε cosθ cosθ 1 + n sin Θ 1 ε 1 cosθ 1 ε1 cosθ 1 + cosθ 1 = ε cosθ cosθ 1 + n sin Θ 1 ε Γ TM = tanθ B = = n = n ε 1 n 1 π z m cosθ m = mπ d = z m = λ 1m m = λ 1 cosθ m f µ 1 ε 1 cosθ m λ x = λ 1 sinθ m λ H,m = λ sinθ m d = m λ f G,m = mc d λ G,m = c = d f G,m m λ H,m = λ 1 ( mλ d λ H,m = sinθ m = λ λ H,m = λ 1 ( λ λ G,m 1 ( mλ d 8
Die Oberflächenwelle s 1 = σ 1 ωε 1 1 s = σ ωε 1 k E = ω ε µ ( k z k E 1 j 1 ε s 1 ε 1 1 ε ( R1 β 1 ε α k E = k E = s 1 ε 1 R s 1 ε 1 β k E = ω ε µ ωµ σ 1 k x1 ( 1+j k x ωε ωµ σ 1 (1 j = ωε η 1 k x k x1 ωε σ 1 E x1 = k E d 1 1 jε s 1 ε 1 1+j A 1e jk x1x e jkzz ωε E x1 = E z1 (1+j σ 1 E x = 1 ( 1 j E σ 1 ωε z j ωε σ 1 Z W = E x H y Z W = k k E (1 j 1 ε z s 1 = ωδ ωε (1 js ε 1 ( η E 1+j s Z W1 = k z ωδ 1 9
I z = S 1 df = σ 1 b E z1 dxdy Σ =σ 1 A 1 be jkzz x=y= e jk x1x dx x= =j σ 1A 1 b k x1 e jkzz du z = I z dz dp = 1 I z dz T = 1 E H = 1 E z Hy E x Hy T x = 1 ( E z Hy = 1 ωδ1 A kx1 1 dp = T x bdz T x bdz = 1 I z dz dz = ωδ 1k x1 σ 1 dz b dz η 1 dz b dz = dr+jdx, dr = dz b R 1, dx = dz b X 1 dp W = T W bdz = 1 I z dr I z = bh y1 ( dp W = 1 H y1( br 1 dz bzw. p = 1 b R 1 = 1 σ 1 d 1 = R R = dr = dp W dz = 1 H tang( R 1 l R = l σa dp W dz = 1 H y1( br 1 (lies: R square R b dz = 1 σ 1 d 1 l b ω R = l ωµ, X = l ωµ πa σ 1 πa σ 1 R = X = a ωµσ1 = a ω 1 R X d 1 1
Resonatoren λ G,m,n = (m a 1 + ( n b κ = λ λ H = 1 ( λ λ G c v P = 1 ( λ λ G v G = c 1 ( λ λ G ( mπ ( nπ + ω εµ kz a b P = Re{ T} d F = = ωk zµ 4 ab ( aa π T z dxdy = 1 a b E y Hxdxdy = ωk ( zµ aa π a b sin ( π a xdx dp = 1 H tang R M df z P(z = 1 R M ( a = R M A ( ( k za π [ H x + H z ] a y= sin ( π a a xdx+ b dx+ [ H y ( a ( = A a R M 1+( +b λ H α = π ωµ R λ H a ( a M 1+( a b( +b 3 λ H c = p λ H, bzw. k z = π = pπ λ H c ( nπ ( pπ + + b c ( mπ a 11 }{{} + H z ] cos ( π b a xdx+ = ω mnp εµ dy x= dy
(m ( n ( p ω mnp = πv + + a b c Q,mnp = ω mnpw P Q = πw mit T = 1 PT f mnp W = 1 (εe 4 E +µ H H dτ τ P = 1 R M H tang H tang df Σ λ 11 = ac a +c ω 11 = π εµ a +c ac E y = ωm a π Asin(π a x sin(π c z H x =jk z a π Asin(π a x cos(π c z H z = jacos( π a x sin(π c z W e = W m =A µ a +c 4c abc P =A R M ac(a +c +b(a 3 +c 3 Q = πη b (a +c 3 R M ac(a +c +b(a 3 +c 3 Q = πη 6R M c 1
Koaxialleitungen E = E r e r H = H ϕ e ϕ z U(z+Z I(z =, z I(z+Y U(z = mit Z = R +jωl, Y = G +jωc z U(z Y Z U(z =, U(z = U v e jkzz +U r e +jkzz, L = 1 I z I(z Y Z I(z = I(z = I v e jkzz +I r e +jkzz U v = Z L I v, U r = Z L I r Z R Z L = bzw. Z Y L = +jωl G +jωc jk z = Y Z = (G +jωc (R +jωl ra r i R = G L C v P = ω k = ω Re{k z } 1 L C L = 1 B n A da I B na dr = 1 I C τ = E e C r dr = A ra r i B ϕ dr = µ I H ϕ = I πr L = µ ra I I r i πr dr L = µ π ln r a r i Q C = E e C r dr E = τ πε ra r i τ e r πε 13 ra r i H ϕ dr e r r τ r e r dr = πε ra 1 dr r i r
C = πε ln ra r i Z L,verlustlos = η π ln r a mit η = r i ωµl R = σ R = R innen +R aussen l R ωµl = σ R l πr i + R l πr a l 1 π (1 r i + 1 r a G = ωc tanδ ε = ω πε ln ra r i µ ε = R π (1 r i + 1 r a tanδ ε jk z = γ = α+jβ = (G +jωc (R +jωl R α = α R +α G = ( L C }{{} (1 + G L C }{{ } ( 1 cosh δ R δ G }{{} (3 mit sinhδ R = R ωl, sinhδ G = G ωc α R R = L C R 1+ ra ri ηr a ln ra Z L,min. Dämpfung = η π ln r a = 77 Ω ε r r i εr U max = E max r i ln r a ln ra = E max r ri a r i r i Z L,max.Spannungsfest = 6 Ω εr P max = U max = πe max r i Z L η r a ri ln r a = πe max r a r i η Z L,max. Leistung = 3 Ω εr ln ra r i ( ra r i p v (z = dp dz = d dz P e αz = αp e αz = αp(z mit α = α R +α G 14
Dielektrische Wellenleiter ξ=k x1 d η=k x d ξcotξ = η ξ +η = ω µ d (ε 1 ε = V V = πd λ n 1 n k x1,m = (m 1π, m = 1,,... d (m 1π ω c,m = d ε µ (ε r1 ε r 15
Streifenleitungen Z L µ 1 ( 8h ε π ln w + w 4h Z W = Z L εeff λ = λ εeff ε eff = 1+q(ε r 1 c f c,tem = 4h ε r 1 λ h max = 4 ε r 1 c f c,qtem = (w+,8h ε r α = α L +α D z P(z = H x Rw, P = R = α L = P df = ωµ σ, 1 ωµ P(z σ w H x = T w,z dxdy = 1 ωµ σ 1 ηh E y H x dxdy = 1 η E y hw α L = α L α L = Z W = η h w. ωµ σ 1 Z W w. ( 1+ π arctan(1,4 d 1 α D = k E s, ε ε = tanθ = s α D = π λ tanθ α D = π ( λ tanθ εr ε eff 1 ε eff ε r 1 16
Wellen und Hindernisse Γ rauh = Γ glatt exp [ (kσcosθ e ] E/E = 1/ exp( jπ/4[c(v+js(v]/ C(v = v cos ( πt / dt S(v = v v = h /λ(1/d s +1/d e sin ( πt / dt 17
Antennen A( r = µ e jkr 4πr A( r = µ V Se ( r e jk r r 4π r r dv V Se ( r e +jkr cosϑ dv = µ e jkr 4πr N (ϑ r r = r +r rr cosϑ = (r r cosϑ +r sin ϑ [ = (r r cosϑ 1+ 1 r sin ] ϑ (r r cosϑ +... α = k r = π λ E ϑ (ϑ,ϕ E ϑ (ϑ max,ϕ max = α max = π λ π = π λ r sin ϑ (r r cosϑ D r R r r r R = D λ (+λ H ϕ (ϑ,ϕ H ϕ (ϑ max,ϕ max = f (ϑ,ϕ ϑ max = π/ und ϕ max = beliebig H ϕ E ϑ E ϑ (π/ = = f (ϑ,ϕ = sinϑ H ϕ (π/ { } φ = r Re T e r 1 f T = 1 E H ( E H e r df 18
P r = 4π P r = 1 Re f ( E H e r df df = r sinϑdϑdϕ = r dω { } Re T r e r dω = φ dω = φ max f (ϑ,ϕ = 4π E(ϑ,ϕ E(ϑ max,ϕ max φ = f (ϑ,ϕ φ max P r = φ max f (ϑ,ϕ dω = φ max Ωä 4π φ φ max dω 4π π π Ωä = f (ϑ,ϕ sinϑdϑdϕ D = 4π Ωä = π 4π dϕ π f(ϑ,ϕ sinϑdϑ P LHD P LDUT = P r HD P rdut e DUT = E ϑhd = η I s sinϑ. λr P rhd = πη ( s I 3 λ 3 η sinϑ E ϑhd = Pr,HD 4 π r P rhd = 4πr 3η E ϑhd 1 sin ϑ E ϑhd E ϑhd max = f HD (ϑ = sin ϑ P rdut = 1 Re f P rhd = 4πr 3η E ϑ,hd max E ϑ = jηi πr e jkr F(ϑ,ϕ H I ϕ = j πr e+jkr F (ϑ,ϕ ( E H e r df = 1 π π η I 4π r F(ϑ,ϕ r sinϑdϑdϕ = 19
π π = η I 8π F(ϑ,ϕ sinϑdϑdϕ E ϑ max = η I πr F(ϑ max,ϕ max E ϑ E ϑ max = F(ϑ,ϕ = f (ϑ,ϕ F(ϑ max,ϕ max F(ϑ,ϕ = f (ϑ,ϕ F(ϑ max,ϕ max = f (ϑ,ϕ πr η I P rdut = r η E ϑ max P rhd P rdut = 4πr 3η r η E ϑ max π π π f (ϑ,ϕ sinϑdϑdϕ E ϑhd max π f (ϑ,ϕ sinϑdϑdϕ G REF = P L REF E ϑ DUT max P LDUT E ϑref max 8π/3 G HD = e DUT π π f (ϑ,ϕ sinϑdϑdϕ G HD = P L HD P LDUT E ϑ DUT max E ϑhd max G HD = π = e DUT P rhd P rdut E ϑ DUT max E ϑhd max 8π/3 π f(ϑ,ϕ sinϑdϑdϕ G HD = 8π/3 = 3 G ISO Ωä T E (r = G P S 4πr E E = A λr E E ϑ max T E (r = E E η = ( A E λr η P S = E η A G = 4πr T E (r P S = 4π A λ G ISO = 4π λ Aw P E = AT E
G DUT/ISO = EIRP P LDUT E ϑ DUT max E ϑiso max EIRP = P L G ISO L = ns l = πd cosψ s = lsinψ = πdtanψ k wendel l k s = πν ν = 1,,3,... k wendel = ω v ( l ω v s = πν ν = 1,,3,... c ω = π c λ und l πd λ l = (λ +s v c 3 4 λ < λ < 4 3 λ P = 1 I Z A T = 1 E H P r = 1 T r = 1 E ϑh ϕ T ϑ = 1 E rh ϕ π ϕ= T ϕ H ϕ = E ϑ /η π ϑ= E ϑ = jη Is λ E ϑ η P r = η I s 8λ π e jkr r π λ r sinϑdϑdϕ sinϑ sin 3 ϑdϑ P r = 1 ( s 3 πη I R A = ( s 3 πη λ m = 1+ ρ 1 ρ = U max U min 1
Q = ω R A Q = ω G A Z(z = U(z I(z ( XA mit Z A = R A +jx A ω ω=ω ω = ω Q ( BA mit Y A = G A +jb A ω ω=ω r = r exp(aψ P S1 P E = P S P E1 G(ϑ,ϕ= P L REF P LDUT E ϑ DUT (ϑ,ϕ E ϑref (ϑ,ϕ = = P L REF E ϑdut max f DUT (ϑ,ϕ P LDUT E ϑref max f REF (ϑ,ϕ = =G REF f DUT(ϑ,ϕ f REF (ϑ,ϕ MEG = G(ϑ,ϕ P (ϑ,ϕdω 4π
Wellen im freien Raum r. = dλ 4 T e,iso = P s 4πd T e = P sg s 4πd P e = T e A e P e = T e A e = P sg s 4πd A e A = λ 4π G iso P e = P sg s 4πd λ 4π G e = P s P e = P s ( 1 λd ( λ 4πd A s A e L db = 1 log P s P e G s G e P dbw e = P dbw s +G db s L db ISO +G db e ( λ L ISO = log 4πd L s = 1 log P s P n = 1 log P s P e,min P e,min P n T i = P sg s 4πd P e = T e A e = T iσ 4πd A e = P sg s σ λ (4πd 4π G e P e P s = σg s ( λ 1 4π 4πd 4 σ = AG = A 4π λ A = 4πA λ = L db +SNR min db 3
Mehrwegeausbreitung H(jω = H(jω = τ 1 = d 1 /c, und τ = d /c h(τ = A 1 δ(τ τ 1 +A δ(τ τ h(τe jωτ dτ = A 1 e jωτ 1 +A e jωτ A 1 +A +A 1A cos(ω τ mit τ = τ τ 1 f Notch = 1 τ H(jω = H(jω e jφ H (jω τ Gr = dφ H dω E( r = E 1 e j k 1 r + E e j k r E(t = E cos(ωt kd E (t = E cos(ωt k[d +vt] = E cos(t[ω kv] kd ( = E cos t [πf πλ ] kd v = E ( [ cos πt f v ] kd λ f D = v λ = f v c f D = v v cos(γ = f λ c cos(γ p(e = 1 σ π e E σ Varianz: = E ( E Varianz = E = E p(ede = σ σ = Re(E = P m p(a = a [ ] σ exp a σ 4
Mittelwert a = σ π quadrat. Mittelwert a = σ Varianz a (a = σ σ π =.49σ Medianwert a 5 = σ ln = 1.18σ p(a = a σ exp [ a +A σ ] I ( aa σ quadrat. Mittelwert a = σ +A ( ( n P e λ d = G s G e P r 4πd d ] 1 (F M p(f = exp [ σ F π σf 5