272963120 Problemas de Antenas y Propagacion PDF
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Universidad de Granada
Curso 2010-2011
Antenas y Propagación
Problemas de Antenas y Propagación
Profesor: Ignacio Sánchez García 4º Curso de Ing. de Telecomunicación Curso 2010-2011
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Escuela Técnica Superior de Telecomunicacione Telecomunicacioness Antenas y Propagación Propagación Curso 2010-2011
RELACIÓN DE PROBLEMAS Tema 1.- Fundamentos Físicos de la Radiación 1. Un campo magnético en el espacio libre, en ausencia de densidades de carga y de corriente, viene dado por (con a, n y ω constantes)
B a sen(t nx) i
ˆ
a n y cos( t nx) j
ˆ
Usar las ecuaciones de Maxwell para obtener la expresión, en el dominio del tiempo, del campo eléctrico en la región. Solución: E ( x, y, z, t )
an2 y 0 0
cos t nx k ˆ
2. Un medio se dice dispersivo cuando la relación entre la frecuencia y el número de ondas no es lineal, generalmente porque su permitividad y permeabilidad son funciones de la frecuencia. Suponga que la permitividad de un medio puede escribirse como
0 1 2 1 2 0
Determínese la relación de dispersión que liga ω y k para una onda plana que se propaga en este medio. Supóngase que la permeabilidad del medio es la del vacío para todas las frecuencias y que la conductividad es nula. Haga un análisis del comportamiento en frecuencia de este medio
Solución:
k 0 0 1 2 1 2 0 0 k 0 0 1 medio con r (1 (1 ) análisis asintótico k 0 0 onda propagándose en el vacío
3. En la región 1 de la figura, se tiene un campo eléctrico en la interface dado por
E 2 j 3k V/m. Obtener el campo eléctrico en la interface de la región 2 ˆ
ˆ
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Solución: E
1
i
ˆ
2
3
j 3k [V/m]
ˆ
2
ˆ
4. Considere los campos:
E a sen ky t k ˆ
B b sen( ky t )i
ˆ
a) Encontrar las condiciones que deben cumplir a, b, k y ω para que estos campos se correspondan con verdaderos campos electromagnéticos en el vacío (ausencia de cargas y corrientes) b) Calcular los potenciales V y A para los campos anteriores tales que cumplan el Gauge de Lorenz (invariante de Lorenz) Solución:
a)
b ; =ck ; a=bc 0 ; A a cos ky t k
ak
b) V
ˆ
5. El potencial vector magnético de una onda viajera es A A x sen t k 0 z i donde Ax es ˆ
una constante. Encontrar el campo electromagnético asociado
E ( z, t ) A x cos t k0 z i [V/m] k 0 H ( z , t ) A x cos t k 0 z j [A/m] 0 ˆ
Solución:
ˆ
6. En una región del vacío libre de cargas y de corrientes, el campo eléctrico es:
E E0 cos( t ) cos(kz)i
ˆ
A partir de dicho campo, calcular: a) El campo magnético b) La relación de dispersión que liga ω con k
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Escuela Técnica Superior de Telecomunicacione Telecomunicacioness Antenas y Propagación Propagación Curso 2010-2011 g) El vector de Poynting h) ¿A qué corresponde este campo eléctrico? Solución:
a) B( z , t )
E 0 c
sen( t )sen(kz ) j
ˆ
ck 1 2 2 2 c) ue 0 E0 cos ( t ) cos (kz kz ) 2 1 2 2 2 d) um 0 E0 sen ( t )sen (kz ) 2 1 2 e) u ue um 0 E0 1 cos(2 t ) cos(2kz ) 4 1 1 f) ue 0 E02 cos 2 (kz ) ; um 0 E02 sen 2 (kz ) ; u 4 4 b)
g)
S
EH j *
1 E 02 0 c
1 4
0E02
sen(kz ) cos(kz)k S 0 ˆ
h) Corresponde a una onda estacionaria 7. El campo electromagnético debido a una antena corta tiene por potencial
0 I 0 L e j ( t kr )k 4 r
A Re
ˆ
Siendo I0 la amplitud de la intensidad oscilante y L la longitud de la antena a) Hallar el campo magnético y, a partir de este, el campo eléctrico en todos los puntos del espacio b) ¿A qué se reducen los campos E y B cuando r es muy grande? c)
Hallar el valor del vector de Poynting para r muy grande, su promedio sobre un periodo y su flujo a través de una superficie esférica.
Solución:
1 2 e jkr s en 4 r r 1 1 I L 1 1 E 0 2 c o s r r r2 4 c jkr 3 r 2 B
a)
0 I 0 L jk
ˆ
ˆ
sen jkr 3 1
e jkr ˆ
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0 I 0 L jk jkr B e s en 4 r kr 1 E I 0 L jk e jkr s en 4 0c r ˆ
b)
ˆ
c)
S
I 02 L2k 2
sen r S 2
32 2 r 2 0c
ˆ
I 02 L2k 2 32 2 r 2 0c
sen r 2
ˆ
S
dS
I 02 L2k 2
12 0c
8. El fasor campo magnético de una onda plana propagándose por un medio dado es 4
H ( x, t ) 0.02e10 x sen(2 10 109 t 104 x)k A/m ˆ
Se pide: a) Decir que tipo de medio es. Justificar la respuesta. b) Calcular la conductividad y la impedancia intrínseca del medio. Suponer μ=μ0. c) Escribir los fasores correspondientes a los campos magnético y eléctrico. d) Calcular los vectores densidades de potencia instantánea y media. Solución: a) El medio es un buen conductor.
b) c)
25.33 103 ( S / m) ; =0.55845º () 104 x
H ( x) j 0.02e
104 x
k ( A / m) ; E( x) j0.0111e ˆ
e
j 104 x
s en t 10 4 x sen t 10 4 x i W 2.232 104 e 210 xse 4 S 8.2024 105 e 210 xi W / m 2
S d)
e
j104 x
4
ˆ
e
j
4
j (V / m)
ˆ
/ m2 ;
4
ˆ
9. Obtener la expresión de los campos electromagnéticos producidos por distribuciones de equivalentes de carga y corriente magnéticas. Solución: Teoría
10. Haciendo uso de la aproximación de campo lejano, obtener las expresiones de los campos de radiación producidos por distribucio nes equivalentes de carga y corriente magnéticas. Solución: Teoría
11. En un medio no magnético, el campo eléctrico viene dado por
E ( x, t ) 4sen(2 10 107 t 0.8x )k V/m ˆ
Se pide:
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Escuela Técnica Superior de Telecomunicacione Telecomunicacioness Antenas y Propagación Propagación Curso 2010-2011
Solución:
a)
r 14.59 ; = 98.7
b)
81.05 mW/m2 P 724.48 W
c)
12. Dos ondas planas se propagan en la misma dirección. Ambas tienen la misma frecuencia y tienen igual amplitud. La onda A está polarizada linealmente en la dirección del eje x y la onda B está polarizada en la dirección i
ˆ
j . Suponiendo que la onda B tiene un pequeño ˆ
retraso, θ, respecto a la onda A, ¿cuál es la polarización de la suma de ambas ondas? Solución: La polarización es RHEP
13. La intensidad del campo magnético de una onda viene dada por
H ( x, t ) j100 cos(t x ) k 200 cos(t ˆ
ˆ
x
2
)
¿Cuál es la polarización de esta onda? Solución: La polarización es RHEP
14. Una antena produce una intensidad de campo eléctrico en zona de campo lejano de
E j
V 0 R
e
jk0 R
sen donde k 0 es la constante de fase en el espacio libre, R es la ˆ
distancia punto fuente al punto campo y θ es el ángulo con respecto a la vertical (eje z). Calcular: a) La densidad de potencia media b) La potencia total radiada por la antena Solución:
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Tema 2.- Parámetros Básicos de Antenas
()
1. La intensidad de radiación de una antena viene dada por
a) Dibujar el diagrama de radiación en representación polar b) A partir de la intensidad de radiación, obtener la directividad Solución:
a) b) D = 5.23 (D=7.18 dB)
̂
2. Obtener la directividad de una antena que radia un campo cuya expresión es
2
G(θ,φ)=1.5 sin θ , D = 1.5 (D=1.76 dB) Solución: G(θ
3. Suponga una antena antena que tiene una potencia de entrada de 40π 40π *W+ y una eficiencia del 98%. Si la intensidad de radiación máxima es de 200 [W/str], calcular la directividad y la ganancia de la antena en dB G(θ,φ)=20 (G(θ (G(θ,φ)=13.01 dB) Solución: D = 20.4 (D=13.098 dB), G(θ
( ) ( ) () ̂
4. Una antena radia un campo magnético que viene dado por la expresión
a) Representa el diagrama de radiación para el campo eléctrico y para la
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Escuela Técnica Superior de Telecomunicacione Telecomunicacioness Antenas y Propagación Propagación Curso 2010-2011 a) La directividad de la antena b) El área eficaz c)
La resistencia de radiación
Solución:
a) D=21.85 (D=13.39) b) Aef 1.74
2
c)
Rrad 279.25
6. El diagrama de radiación normalizado de una antena es de la forma: n t(θ,φ)=cos θ para θ < π/2 a) Calcular la directividad de la antena. b) Representar gráficamente el diagrama de radiación para n = 1, 2, 3. c) Para los valores anteriores de n calcular el ancho de haz. Solución:
a) D=2(n+1) b)
n 1 A c)
180 º n 2 A 120º n 3 A 90º
7. La intensidad de radiación emitida por cierta antena es de la forma: 2 3 K(θ K(θ,φ) = 8 (sen θ)(sen φ) cuando 0 < θ < π y 0 < φ < π Calcular la resistencia de radiación de la antena para una corriente eficaz a la entrada de 3A. Solución: Rrad 1.58
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10. Un campo eléctrico normalizado de una antena viene dado por la expresión
sen sen . La amplitud de campo en la componente cenital toma un valor máximo en el intervalo 0 y 0 y cero fuera de este intervalo. E
a) Encontrar el valor exacto de la directividad b) Encontrar el valor aproximado de la directividad c)
Obtener la diferencia en dB
Solución:
a) D=6 (D=7.78 dB) b) D = 5.09 (D= 7.067 dB) c)
DdB 0.7dB
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Escuela Técnica Superior de Telecomunicacione Telecomunicacioness Antenas y Propagación Propagación Curso 2010-2011
Tema 3.- Propagación de Ondas en Medio Natural 1. Determinar la relación entre la potencia que recibe una antena receptora separada de la antena transmisora por una montaña a la frecuencia de 1 MHz. MHz. La altura máxima de la montaña es de 800 m. La antena receptora está separada de la montaña por una distancia de 2 km y la antena transmisora por una distancia de 1 km. Solución: La pérdida de transmisión es de -14 dB
2. Empleando las cartas de la UIT-R, obtenga la distancia a la que se alcanza un campo de 100 μV/m eficaces transmitiendo con un mástil de 75 m que a 1 MHz radia una potencia de 91.5 kW. La directividad de la antena es de 3.28. a) Sobre la superficie del mar b) Sobre tierra seca Solución: a) b)
1100 km d 100 km d
3. Considere un radioenlace a 5.625 GHz de corto alcance (d = 10 Km) sobre una llanura que utiliza como antenas pequeños reflectores, de unas 20λ 20 λ de diámetro, situados sobre sendas torres de 20 metros de alto. Considerando que el coeficiente de reflexión es igual a 0.5 , calcule en cuanto cambia la amplitud del campo incidente sobre la antena receptora respecto al caso en que la propagación fuera en espacio libre.
Solución:
E E directo
1.5
E 3.52dB E directo
4. Suponga un radioenlace como el mostrado en la figura. Las dimensiones están dadas sobre el nivel del mar y las características del mismo son:
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Solución:
37.045 dB P rec 75.12 dB
a) P rec b)
5. Un radioenlace de microondas tiene una longitud de trayecto de 30 km y una frecuencia de trabajo de 10 GHz. El radioenlace está diseñado para soportar unas pérdidas máximas por propagación de 167 dB. La antena transmisora está montada en una torreta de 20 m sobre el nivel de tierra. A 10 km de la antena transmisora existe una colina de 41 metros de altura. Calcular: a) La pérdida total en trayecto si la altura de la antena receptora es de 20 metros. Sobre el nivel de tierra b) La altura que ha de tener la antena receptora para obtener unas pérdidas totales por propagación iguales a las dadas en las especificaciones. Solución: a) Ltotal b)
161.97 dB
h 40 m
6. Un radar biestático trabaja a una frecuencia de 8 GHz usando antenas transmisora y
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Escuela Técnica Superior de Telecomunicacione Telecomunicacioness Antenas y Propagación Propagación Curso 2010-2011
c) Calcule las pérdidas del radioenlace del caso b) en condiciones de lluvia intensa (100 litros/hora) en dB. d) ¿Qué intensidad de lluvia puede soportar el radioenlace para que el nivel de señal no baje en 4.8 dB con respecto al nivel sin lluvia? Solución: a) L 92.4 dB b) L 88.5 dB c) L 103.5 dB d) 50 mm / h
8. Se dispone de un radioenlace con dos bocinas sectoriales plano E de 1λ x 3λ de apertura, separadas 1 km, con eficiencia de radiación igual a 0.9 y eficiencia de apertura igual a 0.6. El radioenlace funciona a 10 GHz en espacio libre. La antena receptora se encuentra girada tal como se presenta en la figura 1, en el plano del papel. El diagrama de radiación en dicho plano es el de de la figura 2. Calcule: Calcule: a) Calcule las pérdidas de inserción del radioenlace en dB. b) Si la antena transmisora transmite con una PIRE de 30 dBW, calcule la potencia (en dBm) que la antena receptora es capaz de entregar al receptor.
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a) ¿Qué intensidad de campo se mediría, respecto al valor medido por el inquilino, si la pared no estuviera? b) ¿Qué altura ha de tener la pared para que la intensidad de campo eléctrico medido en el balcón del edificio cumpla la reglamentac reglamentación ión sobre salud pública? c) Mientras los albañiles modifican la pared, se ha producido un incremento máximo en la intensidad del campo eléctrico medido en el balcón. ¿Cómo justificaría este aumento? Respecto a la altura original de la pared, ¿cuánto habían construido o destruido los albañiles cuando se produjo éste aumento? Solución:
a) El campo sin la pared es el mismo que el medido sin la pared b) La altura de la pared ha de ser de 3.2309 m, esto es, hemos de subir la pared una altura de 1.331 m. c)
Los albañiles han construido una altura de 0.6338m por encima de la pared original.
10. Calcule el alcance visible sobre el mar para una antena situada en el borde del mismo, sobre una torre de 100 m de altura, para una atmósfera cuyo coíndice de refracción vale N(h)= 289-78 h(Km). Dato: radio de la Tierra 6370 km. Solución: d=50.32 km
11. Una antena linealmente polarizada que funciona a 3 GHz posee un rendimiento de radiación de un 75% y presenta un diagrama de radiación con simetría de revolución respecto a θ = 0º a) Estime la ganancia de potencia para una dirección situada a 5º respecto de la de máxima radiación. b) Calcule la potencia disponible en bornes de entrada cuando incide sobre ella en
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Escuela Técnica Superior de Telecomunicacione Telecomunicacioness Antenas y Propagación Propagación Curso 2010-2011
13. Un radio enlace terrestre de corto alcance (d=5 Km) en banda X (10 GHz), utiliza dos antenas parabólicas de 50 cm de diámetro que poseen eficiencias de 0.7. Si la temperatura de ruido de antena es de 250 K, calcular la potencia necesaria del transmisor que asegure a la salida del receptor, y en un ancho de banda de 1MHz, una relación señal-ruido de 30 dB cuando sobre el trayecto cae una lluvia torrencial de 150 litros/hora. Dato: La constante de Bolztmann vale 1.38x10-23 Julios/K Solución: P trans
1.3 mW
14. Dos antenas de bocina idénticas están separadas una distancia 100 m. Ambas antenas poseen una ganancia directiva de 15 dB en la dirección de la máxima transmisión y sus dimensiones físicas son 12 cm y 6 cm, respectivamente. La antena transmitiendo (Tx) envía una señal con una potencia de 5 W a 3 GHz. a) Determinar la potencia recibida suponiendo que ambas antenas están adaptadas a sus respectivos circuitos de transmisión y recepción. b) ¿Es aplicable la fórmula de Friis en las l as condiciones del problema? c) Calcular la intensidad del campo eléctrico en la antena receptora
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16. Una comunicación en onda media (4 Mhz) utiliza como antena transmisora un monopolo vertical de 25 m y un receptor que requiere una señal de 0.1 mV/m. Calcule la potencia que debe radiar el monopolo transmisor si el receptor está situado a 100 km de distancia. Datos: el tipo de terreno es agua dulce con una constante dieléctrica relativa de 3 y una conductividad de 3x10-5 S/m . Solución: P trans
11.95 kW
17. Para un radioenlace de 35 km de longitud, a una frecuencia de 15 GHz, calcular la atenuación adicional que produce la presencia de componentes gaseosos (básicamente vapor de agua y oxigeno). Solución: 1.155 dB
18. Considerar un radioenlace radioenlace de 30 km de longitud que funciona a una frecuencia frecuencia de 13 GHz con polarización horizontal. Calcular la atenuación por lluvia cuando ésta se considera moderada (25 mm/h) Solución: 31.73555 dB
19. Un radioenlace de 1.5 Km a 60 GHz utiliza sendas antenas parabólicas de 1 metro de diámetro, que poseen eficiencias eficiencias de 0.7. La antena transmisora radia una potencia de 100 W hacia el horizonte. ¿Cuánto valen las pérdidas adicionales respecto del mismo radioenlace en condiciones de vacío (espacio libre) cuando el trayecto de propagación está bajo una lluvia de 25 mm/h de intensidad? Solución: total
37.725 dB
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Escuela Técnica Superior de Telecomunicacione Telecomunicacioness Antenas y Propagación Propagación Curso 2010-2011
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Tema 4.- Antenas Lineales y Antenas de Lazo 1. Obtener el vector de radiación, los campos radiados y el diagrama de radiación de dos dipolos elementales de corriente I y longitud h, situados simétricamente a una distancia d=λ\2 d=λ\2 del origen, en los siguientes casos:
a) Dos dipolos situados en el eje z, con corrientes en el mismo sentido. b) Dos dipolos, paralelos al eje y, y con corrientes en sentidos opuestos
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Escuela Técnica Superior de Telecomunicacione Telecomunicacioness Antenas y Propagación Propagación Curso 2010-2011
a) El diagrama de radiación de la antena en los planos z=0, y=0 y el plano φ=135. b) La polarización de las antenas en la dirección de los ejes coordenados. c)
La directividad de la antena
d)
La longitud efectiva
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a) La resistencia de radiación del dipolo b) La potencia media total radiada por el dipolo c)
La eficiencia de la antena
d) La ganancia máxima de la antena e) La longitud y apertura efectiva f)
El ancho del Haz
Solución:
0.3158 Ω P rad 0.1579 W e f 75.95 %
a) Rrad b) c)
d) D = 1.1388 (D=0.564 dB) 5. Un dipolo Hertziano tiene una longitud de 0.02λ y lleva una corriente de 2 A a 150 MHz. Calcular: a) Los campos cercanos
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Escuela Técnica Superior de Telecomunicacione Telecomunicacioness Antenas y Propagación Propagación Curso 2010-2011 7. Una antena de lazo está compuesta de 10 espiras pequeñas recorridas pos una corriente I(t)=I0 cos (ω t). a) Encontrar la potencia media radiada por la antena b) Encontrar la resistencia de radiación de la antena c)
Las expresiones de los apartados anteriores al caso de una antena con n espiras
Solución: 2
a) Prad
0.07 kI0 a 4
b) Rrad
0.14 k
n rad
P
c) n rad
R
n kI 0
2
2
a4
a 4
12
n k
2
a 4
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Solución:
a)
e f 22.23%
b)
e N f
c)
Q 525.49
Ne f e f N 1 1
; N=6
d) Ambos centrados en el origen con p y m paralelos. L N
2
a
2
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a) La resistencia de radiación b) La expresión de la ganancia directiva en función de las coordenadas esféricas (θ, ( θ, φ). φ). c) Represente la ganancia directiva y calcule el valor de la directividad
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Solución: a) 2
b) Prad
I
0 Prad 73.2 I 2
cos cos 2 d cos 2 d
cos 2
sen
Rrad 146 c)
2 c o s c o s
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Escuela Técnica Superior de Telecomunicacione Telecomunicacioness Antenas y Propagación Propagación Curso 2010-2011
c)
1 I
8 2 r
2 r sen4 2
sen2 2 sen2
2 0 2
sen2
ˆ
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16. Suponga un dipolo simétrico de longitud L. Los brazos del dipolo están cargados con sendos inductores en serie (de longitud despreciable) tal y como se muestra en la figura. La estructura está funcionando con una distribución de corriente que varía sinusoidalmente y cuya distribución espacial es la mostrada en la figura.
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e) ¿Cómo modificaría las fases de las corrientes de los dipolos para girar 90º el
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