|Antenna Effective Aperture Friis Equation|
Antenna Effective Aperture Friis Equation.
Up to this point in our discussion about antennas we have only discussed their transmitting characteristics. In a communication system, we need both transmitting and receiving antennas. There are many, many similarities between the transmitting and receiving properties of antennas, but there is one difference, which we’ll discuss in this lecture.
Antenna Effective Aperture
In a receiving mode, antennas function as transducers to convert some of the energy in a passing electromagnetic wave into the motion of charges (i.e., a current) in an electrical circuit connected to the antenna.
We define the maximum effective aperture of the antenna, Aem, as the ratio of the maximum time average power (PR) delivered to a conjugated matched load ( ZL = ZA* ) connected to a lossless receiving antenna to the time average power density of an incident EM wave (linearly polarized UPW) illuminating the antenna, SiAV :
(The adjective “maximum” in maximum effective aperture indicates the assumption there are no losses in the antenna.) Alternatively, from (1)
P R = Aem S i AV
which states that the time average power delivered to a conjugate matched load connected to a lossless receiving antenna equals the power flow through an area equal to Aem of the incident UPW. It can be shown that for any antenna, the maximum directivity D is related to the maximum effective aperture as
In the case that there are losses in the antenna, the power delivered to the matched load is further reduced to the fraction er (i.e., the radiation efficiency) of what would have been received in the lossless situation according to
Ae = er Aem
where Ae is the effective aperture of the receiving antenna. Using (3) in (2) and the definition of antenna gain, we arrive at
In words, this extremely important equation (4) states that the ratio of an antenna gain as a transmitter and its effective aperture area as a receiver is a constant (dependant on wavelength) for any antenna. Amazing!
Example N34.1. Verify equation (2) for a Hertzian dipole antenna. Imagine a uniform plane wave (UPW) is incident on a Hertzian dipole antenna as shown in Fig. 1. From the definition of Aem in (1), we first calculate IAV S for the UPW
To calculate PR, we assume the antenna terminals are connected to a matched load:
Assuming the Hertzian dipole antenna is aligned with Ei , then it can be shown that
Voc = - Ei . T L [v]
For a conjugate matched load ( ZL = ZA* ), maximum time average power will be delivered to the load circuit attached to the antenna. Using voltage division in this circumstance
If ZA = R A+ jXA , then
The time average power delivered to this load is then
where RA = Rr for a lossless antenna. Substituting (5) and (8) into the definition of Aem in (1):
We solved for Rr in (20) of Lecture 32 to be
Substituting (10) into (9) we find
As we saw in Example N33.1, D = 3 2 for the Hertzian dipole antenna. Using this in (11), we find that
which verifies (2) for the Hertzian dipole antenna.
The Friis Equation
Antennas are often used in a wireless communication system, or often called a communication “link,” as illustrated in the figure below. There is an important equation that is used to design such communication links, called the Friis equation. We imagine that two antennas are oriented towards each other for maximum transmitted and received power:
The Friis equation is simple to derive. First, if PT = Pin is the time average power accepted by (or delivered to) a transmitting antenna, we saw from equation (6) in Lecture 33 that the gain of the transmitting antenna GT is defined as
We can rearrange this equation (12) for an expression of the radiated time average power density, SAV, at the position of a receiving antenna as
where d is the separation distance between the transmitting and receiving antennas. The receiving antenna will capture some of this incident power density in (13) converting it to time average power delivered to a matched load according to the definition of antenna aperture in
where AeR is the antenna aperture of the receiving antenna. Using (14) in (13) gives
As we saw in (4), however, effective aperture and antenna gain are related to each other by the same factor for all antennas. Using (4) here in (15) gives
Equation (16) – and to a lesser extent (15) – is called the Friis equation. It can be used to design communication links, which we’ll illustrate in the next example.
Example N34.2. Consider a direct broadcast satellite (DBS) television system typical of Dish Network or DirecTV. These typically use transmitting satellites with 120 W of radiated power, frequency between 12.2 GHz and 12.7 GHz, and an EIRP of ~55 dBW in each 24-MHz transponder that handles several compressed digital video channels. The receiving system uses an 18” (0.46 m) diameter offset-fed reflector antenna. What is the approximate receiver power delivered to a matched load? We’ll use (16) for this calculate at a mid-band frequency of f = 12.45 GHz. The Effective IsotropicRadiated Power (EIRP) is defined as
EIRP = PT GT [W]
It is the same time average power in a certain direction that would be radiated by an isotropic radiator ( Gi = 1) it if had an input power of PT GT . For this example, EIRP = 55 dBW. We can write the Friis equation in terms of dBW. Taking 10log10 of (16) we obtain
PR (dB) = PT (dB) + GT (dB) + GR (dB) + 20log10(C0)
PR (dB) = PT (dB) + GT (dB) + GR (dB) - 20log10(f) -20log10 (d) +147.6
In this example
PT(dBW) = 10log10(120) = 20.8 dBW
and using the definition of EIRP in (17)
GT (dB) = EIRP ? PT(dBW) = (dBW) ?55 20.8 =34.2 dB
For geosynchronous orbit, a typical “slant path” distance is d = 38,000 km.
For this parabolic dish receiver antenna and its very large electrical size (~20 wavelengths in diameter), AeR is approximately 70% of the physical aperture area of the dish. Therefore,.
Using these values in (18):
PR(dB) = - 116.9 dBW = PVR =2.04*10-12 W
This is a very small power! Without the combined gains of the two antennas (~68.2 dB = 34.2 dB + 34.0 dB) this received signal would be hopelessly lost in noise. This example illustrates one use of the Friis equation: What antenna gains are required for a communication link? There are other applications, of course.