19_Friis

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Prof.Dr. Rodrigo P. Lemos
Ondas
Fórmula de Friis
1
Prof.Dr. Rodrigo P. Lemos
Antenas Comunicantes
• A comunicação entre uma antena transmissora e uma antena receptora 
pode ser compreendida a partir dos conceitos de ganho e de área 
efetiva.
• Considere-se duas antenas orientadas na direção de máximo ganho uma 
da outra e distantes de r entre si.
• Sejam Ptrans, Gtrans, e Atrans a potência, o ganho e a área efetiva da 
antena de transmissão, enquanto que Prec, Grec, e Arec são a potência, o 
ganho e a área efetiva da antena de recepção.
15.5. Effective Length 611
The definition of h is motivated by the case of a z-directed Hertzian dipole antenna,
which can be shown to have h = l sinθθˆθθ. More generally, for a z-directed linear antenna
with current I(z), it follows from Eq. (16.1.5) that:
h(θ)= h(θ) θˆθθ , h(θ)= sinθ 1
Iin
∫ l/2
−l/2
I(z′)ejkz′ cosθdz′ (15.5.3)
As a consequence of the reciprocity principle, it can be shown [996] that the open-
circuit voltage V at the terminals of a receiving antenna is given in terms of the effective
length and the incident field E i by:
V = E i · h (15.5.4)
The normal definition of the effective area of an antenna and the resultG = 4piA/λ2
depend on the assumptions that the receiving antenna is conjugate-matched to its load
and that the polarization of the incident wave matches that of the antenna.
The effective length helps to characterize the degree of polarization mismatch that
may exist between the incident field and the antenna. To see how the gain-area relation-
ship must be modified, we start with the definition (15.3.1) and use (15.4.5):
A(θ,φ)= PRPinc =
1
2
RL|IL|2
1
2η
|Ei|2
= ηRL|V|
2
|ZL + ZA|2|Ei|2 =
ηRL|Ei · h|2
|ZL + ZA|2|Ei|2
Next, we define the polarization and load mismatch factors by:
epol = |E i · h|
2
|E i|2 |h|2
eload = 4RLRA|ZL + ZA|2 = 1− |Γload|
2 , where Γload = ZL − Z
∗
A
ZL + ZA
(15.5.5)
The effective area can be written then in the form:
A(θ,φ)= η|h|
2
4RA
eload epol (15.5.6)
On the other hand, using (15.1.4) and (15.4.1), the power gain may be written as:
G(θ,φ)= 4piU(θ,φ)
PT
=
4piηk
2|F⊥|2
32pi2
1
2
RA|Iin|2
= piη|h|
2
λ2RA
⇒ η|h|
2
4RA
= λ
2
4pi
G(θ,φ)
Inserting this in Eq. (15.5.6), we obtain the modified area-gain relationship [997]:
A(θ,φ)= eload epol λ
2
4pi
G(θ,φ) (15.5.7)
Assuming that the incident field originates at some antenna with its own effective
length hi, then E i will be proportional to hi and we may write the polarization mismatch
factor in the following form:
epol = |hi · h|
2
|hi|2 |h|2 = |hˆi · hˆ|
2 , where hˆi = hi|hi| , hˆ =
h
|h|
612 15. Transmitting and Receiving Antennas
When the load is conjugate-matched, we have eload = 1, and when the incident field
has matching polarization with the antenna, that is, hˆi = hˆ∗, then, epol = 1.
15.6 Communicating Antennas
The communication between a transmitting and a receiving antenna can be analyzed
with the help of the concept of gain and effective area. Consider two antennas oriented
towards the maximal gain of each other and separated by a distance r, as shown in
Fig. 15.6.1.
Fig. 15.6.1 Transmitting and receiving antennas.
Let {PT,GT,AT} be the power, gain, and effective area of the transmitting antenna,
and {PR,GR,AR} be the same quantities for the receiving antenna. In the direction of
the receiving antenna, the transmitting antenna has PEIRP = PTGT and establishes a
power density at distance r:
PT = dPTdS =
PEIRP
4pir2
= PTGT
4pir2
(15.6.1)
From the incident power density PT, the receiving antenna extracts power PR given
in terms of the effective area AR as follows:
PR = ARPT = PTGTAR4pir2 (Friis formula) (15.6.2)
This is known as the Friis formula for communicating antennas and can be written in
several different equivalent forms. Replacing GT in terms of the transmitting antenna’s
effective area AT, that is, GT = 4piAT/λ2, Eq. (15.6.2) becomes:
PR = PTATARλ2r2 (15.6.3)
A better way of rewriting Eq. (15.6.2) is as a product of gain factors. Replacing
AR = λ2GR/4pi, we obtain:
PR = PTGTGRλ
2
(4pir)2
(15.6.4)
Ptrans, Gtrans, Atrans Prec, Grec, Arec
Antena receptoraAntena transmissora
Prof.Dr. Rodrigo P. Lemos
Antenas Comunicantes
• À distância r, a densidade de potência transmitida é dada por:
• Strans corresponde à densidade de potência incidente sobre a antena 
receptora. Assim, a potência recebida é proporcional à área efetiva 
da antena de recepção:
que, substituindo Strans, resulta na Fórmula de Friis para propagação:
Strans =
dPtrans
dA =
PEIRP
4!r2 =
PtransGtrans
4!r2
Prec = Sinc ! Arec = Strans ! Arec
15.5. Effective Length 611
The definition of h is motivated by the case of a z-directed Hertzian dipole antenna,
which can be shown to have h = l sinθθˆθθ. More generally, for a z-directed linear antenna
with current I(z), it follows from Eq. (16.1.5) that:
h(θ)= h(θ) θˆθθ , h(θ)= sinθ 1
Iin
∫ l/2
−l/2
I(z′)ejkz′ cosθdz′ (15.5.3)
As a consequence of the reciprocity principle, it can be shown [996] that the open-
circuit voltage V at the terminals of a receiving antenna is given in terms of the effective
length and the incident field E i by:
V = E i · h (15.5.4)
The normal definition of the effective area of an antenna and the resultG = 4piA/λ2
depend on the assumptions that the receiving antenna is conjugate-matched to its load
and that the polarization of the incident wave matches that of the antenna.
The effective length helps to characterize the degree of polarization mismatch that
may exist between the incident field and the antenna. To see how the gain-area relation-
ship must be modified, we start with the definition (15.3.1) and use (15.4.5):
A(θ,φ)= PRPinc =
1
2
RL|IL|2
1
2η
|Ei|2
= ηRL|V|
2
|ZL + ZA|2|Ei|2 =
ηRL|Ei · h|2
|ZL + ZA|2|Ei|2
Next, we define the polarization and load mismatch factors by:
epol = |E i · h|
2
|E i|2 |h|2
eload = 4RLRA|ZL + ZA|2 = 1− |Γload|
2 , where Γload = ZL − Z
∗
A
ZL + ZA
(15.5.5)
The effective area can be written then in the form:
A(θ,φ)= η|h|
2
4RA
eload epol (15.5.6)
On the other hand, using (15.1.4) and (15.4.1), the power gain may be written as:
G(θ,φ)= 4piU(θ,φ)
PT
=
4piηk
2|F⊥|2
32pi2
1
2
RA|Iin|2
= piη|h|
2
λ2RA
⇒ η|h|
2
4RA
= λ
2
4pi
G(θ,φ)
Inserting this in Eq. (15.5.6), we obtain the modified area-gain relationship [997]:
A(θ,φ)= eload epol λ
2
4pi
G(θ,φ) (15.5.7)
Assuming that the incident field originates at some antenna with its own effective
length hi, then E i will be proportional to hi and we may write the polarization mismatch
factor in the following form:
epol = |hi · h|
2
|hi|2 |h|2 = |hˆi · hˆ|
2 , where hˆi = hi|hi| , hˆ =
h
|h|
612 15. Transmitting and Receiving Antennas
When the load is conjugate-matched, we have eload = 1, and when the incident field
has matching polarization with the antenna, that is, hˆi = hˆ∗, then, epol = 1.
15.6 Communicating Antennas
The communication between a transmitting and a receiving antenna can be analyzed
with the help of the concept of gain and effective area. Consider two antennas oriented
towards the maximal gain of each other and separated by a distance r, as shown in
Fig. 15.6.1.
Fig. 15.6.1 Transmitting and receiving antennas.
Let {PT,GT,AT} be the power, gain, and effective area of the transmitting antenna,
and {PR,GR,AR} be the same quantities for the receiving antenna. In the direction of
the receiving antenna, the transmitting antenna has PEIRP = PTGT and establishes a
power density at distance r:
PT = dPTdS =
PEIRP
4pir2
= PTGT
4pir2
(15.6.1)
From the incident power density PT, the receiving antenna extracts power PR given
in
terms of the effective area AR as follows:
PR = ARPT = PTGTAR4pir2 (Friis formula) (15.6.2)
This is known as the Friis formula for communicating antennas and can be written in
several different equivalent forms. Replacing GT in terms of the transmitting antenna’s
effective area AT, that is, GT = 4piAT/λ2, Eq. (15.6.2) becomes:
PR = PTATARλ2r2 (15.6.3)
A better way of rewriting Eq. (15.6.2) is as a product of gain factors. Replacing
AR = λ2GR/4pi, we obtain:
PR = PTGTGRλ
2
(4pir)2
(15.6.4)
Prec =
PtransGtransArec
4!r2
Ptrans, Gtrans, Atrans Prec, Grec, Arec
Antena receptoraAntena transmissora
Prof.Dr. Rodrigo P. Lemos
Antenas Comunicantes
• Substituindo-se Gtrans em termos da área efetiva da antena transmissora, 
tem-se:
• Alternativamente, pode-se escrever a área efetiva da antena receptora 
em termos do seu ganho:
Prec =
Ptrans
4!Atrans
"2
#
$%
&
'( Arec
4!r2 =
PtransAtransArec
"2r2
Prec =
PtransGtrans
!2Grec
4"
#
$%
&
'(
4"r2 =
PtransGtransGrec!2
4"r( )2
Prof.Dr. Rodrigo P. Lemos
Antenas Comunicantes
• Analisando a fórmula de Friis, pode-se identificar o termo 
associado à atenuação da potência recebida em função do 
quadrado da distância r percorrida no espaço livre:
• Alternativamente, pode-se escrever esse termo como Ganho:
Assim, a Fórmula de Friis pode ser escrita como:
ou, em decibéis:
Gespaço! livre =
1
Lespaço! livre
=
"
4#r
$
%&
'
()
2
Lespaço! livre =
4"r
#
$
%&
'
()
2
Prec = PtransGtrans
1
Lespaço! livre
Grec
Prec.dB = Ptrans.dB +Gtrans.dB ! Lespaço! livre.dB +Grec.dB dB
Prof.Dr. Rodrigo P. Lemos
Exemplo
• Um satélite em órbita geoestacionária transmite sinais de TV para uma estação terrena distante 
40.000 km. Considere que as antenas parabólicas do satélite e da estação têm diâmetros de 0,5 m e 
de 5 m, respectivamente. A eficiência de apertura de ambas é de 60%. O satélite transmite com 
potência de 6 W na frequência de descida de 4GHz. Calcule os ganhos das antenas em dB e a 
potência total recebida.
O comprimento de onda em 4 GHz é λ = 7,5 cm. Os ganhos das antenas são dados por:
A potência de transmissão do satélite é dada em dB por:
A perda no espaço livre é:
 
Assim, pela Fórmula de Friis, a potência recebida será:
6
Gmáx.rec = ea
!d
"
#
$%
&
'(
2
)
Gsat = 263,2 = 24dB
Gterra = 26320 = 44dB
Ptrans.dB = 10 log 6( ) = 8 dB
Lespaço! livre =
4"r
#
$
%&
'
()
2
= 4 *1019 + Lespaço! livre.dB = 196dB
Prec.dB = Ptrans.dB +Gtrans.dB ! Lespaço! livre.dB +Grec.dB
= 8 + 24 !196 + 44 = !120dB " Prec = 10!12W
!
Obrigado!
7