# Artículo de Friss/Índice/Derivación de la fórmula de transmisión

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# Derivación de la fórmula de transmisión

Explicación figura 1

Al representar un circuito de radioenlace con una antena transmisora isotrópica, y una antena receptora, se pueden visualizar los patrones de radiación de potencia. Con esta figura podemos determinar que:
Pt= Potencia de alimentación de la antena transmisora
Pr= Potencia disponible en los terminales de salida de la antena receptora
d = Distancia entre las antenas transmisora y receptora

Demostración de las ecuaciones 8 y 9

${\displaystyle Ar=A_{efect}\rightarrow }$ igualamos la variable ${\displaystyle A_{efect}}$ como Ar
${\displaystyle A_{efect}={\frac {P_{r}}{P_{o}}}\rightarrow }$ (3) ${\displaystyle p_{o}={\frac {P_{r}}{Aefect}}\rightarrow }$ despejamos la ecuacion de ${\displaystyle A_{efect}}$ para obtener ${\displaystyle P_{o}}$
(4) ${\displaystyle P_{o}={\frac {Pt}{4\pi d^{2}}}\rightarrow }$ flujo de potenci por unidad de area a una distancia d
(5) ${\displaystyle {\frac {P_{r}}{A_{r}}}={\frac {P_{t}}{4\pi d^{2}}}\rightarrow }$ utilizamos la ecuación 4 y reemplazamos ${\displaystyle P_{o}}$
(6) ${\displaystyle P_{r}=({\frac {P_{t}}{4\pi d^{2}}})*(A_{r})\rightarrow }$ Aquí vemos cómo se puede ir pasando variables para balancear la ecuación y despejar ${\displaystyle {\frac {P_{r}}{P_{t}}}}$
(7) ${\displaystyle {\frac {P_{r}}{P_{t}}}={\frac {A_{r}}{4\pi d^{2}}}\rightarrow }$ Finalmente tenemos la ecuación que define cuando se trata de un frente de onda plano.
Demostración de la ecuación 10
(8) ${\displaystyle {\frac {A_{t}}{A_{isot}}}\rightarrow }$ se remplaza una antena transmisora isotrópica por una antena transmisora con una relación (8)
(9) ${\displaystyle A_{isot}={\frac {\lambda ^{2}}{4\pi }}\rightarrow }$ se define (9) como el área efectiva de una antena isotrópica.
(10) ${\displaystyle {\frac {P_{r}}{P_{t}}}={\frac {A_{r}}{4\pi d^{2}}}\rightarrow }$ tomamos la ecuación anteriormente despejada.
(11) ${\displaystyle {\frac {P_{r}}{P_{t}}}={\frac {A_{r}}{4\pi d[2}}*{\frac {A_{t}}{A_{isot}}}\rightarrow }$ se multiplica la ecuación (7) con la relación de antena transmisora (8)
${\displaystyle {\frac {P_{r}}{P_{t}}}={\frac {A_{r}}{4\pi d^{2}}}*{\frac {A_{t}}{\frac {\lambda ^{2}}{4\pi }}}\rightarrow }$ Así mismo se reemplaza la ecuación de ${\displaystyle A_{isot}}$(9)
${\displaystyle {\frac {P_{r}}{P_{t}}}={\frac {A_{r}A_{t}}{\lambda ^{2}d^{2}}}\rightarrow }$ al reemplazar, se cancelarán algunas variables y esta es la función final, la cual es igual a la fórmula de transmisión simple.

# Derivation of transmission formula

Having defined the effective area of an antenna, it is a simple matter to derive (1). As shown in Fig. 1, consider a radio circuit made up of an isotropic transmitting

antenna and a receiving antenna with effective area Ar. The power flow per unit area at the distance d from the transmitter is
${\displaystyle P_{0}={\frac {P_{t}}{4\pi d^{2}}}.}$
Assuming a plane wave front at the distance d, definition (2) for the effective area and formula (8) give
${\displaystyle {\frac {P_{r}}{P_{t}}}={\frac {A_{r}}{4\pi d^{2}}}}$
Replacing the isotropic transmitting antenna in the illustration with a transmitting antenna with effective area At will increase the received power by the ratio ${\displaystyle At/A_{i}sotr}$, and we obtain
${\displaystyle {\frac {P_{r}}{P_{t}}}={\frac {A_{r}}{4\pi d^{2}A_{isotr}}}}$
Introducing the effective area (6) for the isotropic antenna, we have (1).

# Limitations of transmission formula

In deriving (1), a plane wave front was assumed at the distance d. Formula (1), therefore, should not be used when d is small. W. D. Lewis, of these Laboratories, has made a theoretical study of transmission between large antennas of equal areas with plane phase fronts at their apertures and he finds that (1) is correct to within a few per cent when
${\displaystyle d\geq {\frac {2a^{2}}{\lambda }}}$
where a is the largest linear dimension of either of the antennas.
Formula (1) applies to free space only, a condition which designers of microwave circuits seek to approximate. Application of the formula to other conditions may require corrections for the effect of the “ground,” and for absorption in the transmission medium, which are beyond the scope of this note.
he advantage of (1) over other formulations is that, fortunately, it has no numerical coefficients. It is so simple that it may be memorized easily. Almost 7 years of intensive use has proved its utility in transmission calculations involving wavelengths up to several meters, and it may become useful also at longer wavelengths. It is suggested that radio engineers hereafter give the radiation from a transmitting antenna in terms of the power flow per unit area which is equal to ${\displaystyle P_{t}A_{t}/\lambda ^{2}d^{2}}$, instead of giving the field strength in volts per meter. It is also suggested that an antenna be characterized by its effective area, instead of by its power gain or radiation resistance. The ratio of the effective area to the actual area of the aperture of an antenna is also of importance in antenna design, since it gives an indication of how efficiently the antenna is utilizing the physical space it occupies. *The directional pattern, which has not been discussed in this note,is, of course, always an important characteristic of an antenna.