Lecture 6: Friis Transmission Equation and Radar Range Equation (Friis equation. EIRP. Maximum range of a wireless link. Radar cross section. Radar equation. Maximum range of a radar.)

1. Friis Transmission Equation Friis transmission equation is essential in the analysis and design of wireless communication systems. It relates the power fed to the transmitting antenna and the power received by the receiving antenna when the two antennas are 2 separated by a sufficiently large distance ( RD 2/max  ), i.e., they are in each other’s far zones. We derive the Friis equation next. A transmitting antenna produces power density Wt t( t , ) in the direction ( , )tt. This power density depends on the transmitting antenna gain in the given direction G( , )tt, on the power of the transmitter Pt fed to it, and on the distance R between the antenna and the observation point as PPtt WGetttttttt== D (,)(,) . (6.1) 44RR22 Here, et denotes the radiation efficiency of the transmitting antenna and Dt is its directivity. The power Pr at the terminals of the receiving antenna can be expressed via its effective area Ar and Wt : PAWrrt= . (6.2)

(,) tt (,) R rr

To include polarization and dissipation in the receiving antenna, we add the radiation efficiency of the receiving antenna er and the PLF:  2 PeArrr== WA trPLF|| t W rtr e ρρˆˆ , (6.3) 2   2 Pr = D r( r , r )   W t e r |ρρˆˆ t  r | . (6.4) 4

Ar

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Here, Dr is the directivity or the receiving antenna. The polarization vectors of the transmitting and receiving antennas, ρˆ t and ρˆ r , are evaluated in their respective coordinate systems; this is why, one of them has to be conjugated when calculating the PLF. The signal is incident upon the receiving antenna from a direction ( , )rr, which is defined in the coordinate system of the receiving antenna: 2  Pt  2 =PDerrrrttttrtr De(,)(,)|| ρρˆˆ . (6.5) 44R2

Wt The ratio of the received to the transmitted power is obtained as 2 Pr  2  =etrtrtttrrr eDD||(,)(,)ρρˆˆ  . (6.6) PRt 4 If the impedance-mismatch loss factor is included in both the receiving and the transmitting antenna systems, the above ratio becomes 2 Pr 222   =(1 − || − )(1trt  rtrt || )||( ttr , rr e )( eDD ,)ρρˆˆ    . (6.7) PRt 4 The above equations are variations of Friis’ transmission equation, which is widely used in the design of wireless systems as well as the estimation of antenna radiation efficiency (when the antenna directivity is known). For the case of impedance-matched and polarization-matched transmitting and receiving antennas, Friis equation reduces to 2 Pr    = GGtttrrr(,)(,) . (6.8) PRt  4  The factor ( / 4R )2 is called the free-space loss factor. It reflects two effects: (1) the decrease in the power density due to the spherical spread of the wave through the term 1/ (4 R2 ), and (2) the effective aperture dependence on the wavelength as 2 / (4 ) .

2. Effective Isotropically Radiated Power (EIRP) From the Friis equation (6.8), it is seen that to estimate the received power Pr we need the product of the transmitter power Pt and the transmitting antenna gain Gt. If the transmission line introduces losses in addition to those of the antenna system, these need to be accounted for. This is why often a

Nikolova 2020 2 transmission system is characterized by its effective isotropically radiated power (EIRP): E IR P PG= ettTL , W (6.9) where eTL is the loss efficiency of the transmission line connecting the transmitter to the antenna. Usually, the EIRP is given in dB, in which case (6.9) becomes EIRPdB= Ptt dB + G dBi + e TL,dB . (6.10)

Bearing in mind that Pintt e,TL P= and GUPttint= 4/ max,, , the EIRP from (6.9) can also be written as EIRP U= 4 max,t , W (6.11) where Um ax,t is the maximum transmitted radiation intensity. It is now clear that the EIRP is a fictitious amount of power that an isotropic radiator would have to emit in order to produce the peak power density observed in the direction of the maximum radiation. As such, and as evident from (6.9), the EIRP is greater than the actual power an antenna needs in order to achieve a given amount of radiation intensity in its direction of maximum radiation. The EIRP is often used to specify the radiation limits on wireless communication or radar devices in relation to EM interference regulations.

3. Maximum Range of a Wireless Link Friis’ transmission equation is frequently used to calculate the maximum range at which a wireless link can operate. For that, we need to know the nominal power of the transmitter Pt , all the parameters of the transmitting and receiving antenna systems (polarization, gain, losses, impedance mismatch), and the minimum power at which the receiver can operate reliably Pr min . Then, 2 2 2 2 2  Pt Rmax =(1||)(1||) − t −  rtrtr e e |ρρˆˆ  | D tttrrr (,)(,)  D   . (6.12) 4 Pr min The minimum power at which the receiver can operate reliably is dependent on numerous factors, of which very important is the signal-to-noise ratio (SNR). There are different sources of noise but we are mostly concerned with the noise of the antenna itself. This topic is considered in the next lecture.

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4. Radar Cross-section (RCS) or Echo Area The RCS is a far-field characteristic of a radar target, which creates an echo by scattering (reflecting) the radar EM wave.

The RCS of a target σ is the equivalent area capturing that amount of power, which, when scattered isotropically, produces at the receiver power-flux density, which is equal to that scattered by the actual target:

W ||E 2 ==lim4lim4RR22ss, m2. (6.13) 2 RR→→Wii||E Here, R is the distance from the target to the receiving antenna, m; 2 Wi is the incident power density, W/m ; 2 Ws is the scattered power density at the receiver, W/m .

To understand better the above definition, we can re-write (6.13) in an equivalent form: Wi 1 lim() = WRs . (6.14) R→ 4 RR22 The product Wi represents the equivalent intercepted power, which is assumed to be scattered (re-radiated) isotropically to create a fictitious spherical wave, the power density Ws of which decreases with distance from target as 1/ R2 in the far zone. The quantity is independent of R because  describes the target itself whereas is a number describing the incident-wave power flux density at the target, which depends only on the distance to the transmitting antenna, not the receiving one. Ws must be equal to the true scattered power density produced by the real scatterer (the radar target). We note that in general the RCS has little in common with any of the cross- sections of the actual scatterer. However, it is representative of the reflection properties of the target. It depends very much on the angle of incidence, on the angle of observation, on the shape and size of the scatterer, on the EM properties of the materials that it is built of, and on the wavelength. The RCS of targets is similar to the concept of effective aperture of antennas. Large RCSs result from large metal content in the structure of the object (e.g., trucks and jumbo jet airliners have large RCS,  100 m2). The RCS

Nikolova 2020 4 increases also due to sharp metallic or dielectric edges and corners. The reduction of the RCS is desired for stealth military aircraft meant to be invisible to radars. This is achieved by careful shaping and coating (with special materials) of the outer surface of the airplane. The materials are mostly designed to absorb EM waves at the radar frequencies (usually S and X bands). Layered structures can also cancel the backscatter in a particular bandwidth. Shaping aims mostly at directing the backscattered wave at a direction different from the direction of incidence. Thus, in the case of a monostatic radar system, the scattered wave is directed away from the receiver. The Stealth aircraft has RCS smaller than 10−4 m2, which makes it comparable or smaller than the RCS of a penny.

5. Radar Range Equation The radar range equation (RRE) gives the ratio of the transmitted power (fed to the transmitting antenna) to the received power, after it has been scattered (re-radiated) by a target of cross-section  . In the general radar scattering problem, there is a transmitting and a receiving antenna, and they may be located at different positions as shown in the figure below. This is called bistatic scattering. Often, one antenna is used to transmit an EM pulse and to receive the echo from the target. This case is referred to as monostatic scattering or backscattering. Bear in mind that the RCS of a target may vary considerably as the location of the transmitting and receiving antennas change. Assume the power density of the transmitted wave at the target location is PG(,)(,)  Pe D   W ==t t t t t t t t t , W/m2. (6.15) t 22 44RRtt

Rt Rr

(,)tt (,)rr

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The target is represented by its RCS  , which is used to calculate the captured power PWct=  (W), which, when scattered isotropically, gives the power density at the receiving antenna that is due to the target. The density of the re- radiated (scattered) power at the receiving antenna is PWPD (,) We===cttttt  . (6.16) rt222 44(4)RRRrrtr R The power delivered to the receiver is then 2 PD (,) PeAWeDe== (,) tttt . (6.17) rrrrrrrrt  2 4(4)RRtr Re-arranging and including impedance mismatch losses as well as polarization losses, yields the complete radar range equation: 2 PDDrtttrrr 222   (,)(,) =−eet − rtrtr(1||  )(1|| ) || ρρˆˆ   . (6.18) PRttr R 44 For polarization matched loss-free antennas aligned for maximum directional radiation and reception, 2 PDDrtr  00 =   . (6.19) PRttr R 44 The radar range equation is used to calculate the maximum range of a radar system. As in the case of Friis’ transmission equation, we need to know all parameters of both the transmitting and the receiving antennas, as well as the minimum received power at which the receiver operates reliably. Then, 2222  ()(1Rt Re rt || emax rtrtr)(1=− || )|| −  ρρˆˆ 2 (6.20) PDDttttrrr  (  ,)(,) .   Pr min 44 Finally, we note that the above RCS and radar-range calculations are only basic. The subjects of radar system design and EM scattering are huge research areas themselves and are not going to be considered in detail in this course.

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