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Equation Section 6

Lecture 6: Friis Transmission Equation and Radar Range Equation

(Friis equation. Maximum range of a wireless link. Radar cross section. Radarequation. Maximum range of a radar.)

1. Friis transmission equationFriis transmission equation is essential in the analysis and design of wireless

communication systems. It relates the power fed to the transmitting antenna andthe power received by the receiving antenna when the two antennas areseparated by a sufficiently large distance ( 2max2 /R D ), i.e., they are in eachothers far zones. We derive Friis equation next.

A transmitting antenna produces power density ( , )t t tW in the direction( , )t t . This power density depends on the transmitting antenna gain in the

given direction ( , )t tG , on the power of the transmitter tP fed to it, and on thedistanceRbetween the antenna and the observation point as

2 2( , ) ( , )

4 4t t

t t t t t t t t

P PW G e D

R R

. (6.1)

Here, te denotes the radiation efficiency of the transmitting antenna and tD isits directivity. The power Pr at the terminals of the receiving antenna can beexpressed via its effective area rA and tW:

r r tP A W . (6.2)

R

( , )t t ( , )r r

To include polarization and heat losses in the receiving antenna, we add theradiation efficiency of the receiving antenna re and the PLF:

2 PLF | |r r r t r t r t r P e A W A W e , (6.3)2

2 ( , ) | |4

r

r r r r t r t r

A

P D W e

. (6.4)

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Here, rD is the directivity or the receiving antenna. The polarization vectors ofthe transmitting and receiving antennas, t and r , are evaluated in theirrespective coordinate systems; this is why, one of them has to be conjugatedwhen calculating the PLF.

The signal is incident upon the receiving antenna from a direction ( , )r r ,which is defined in the coordinate system of the receiving antenna:

22

2 ( , ) ( , ) | |

4 4t

tr r r r t t t t r t r

W

PP D e D e

R

. (6.5)

The ratio of the received to the transmitted power is obtained as2

2 | | ( , ) ( , )4

rt r t r t t t r r r

t

Pe e D D

P R

. (6.6)

If the impedance-mismatch loss factor is included in both the receiving and thetransmitting antenna systems, the above ratio becomes

22 2 2 (1 | | )(1 | | ) | | ( , ) ( , )

4r

t r t r t r t t t r r r

t

Pe e D D

P R

. (6.7)

The above equations are variations of Friis transmission equation, which iswell known in the theory of EM wave propagation and is widely used in thedesign of wireless systems as well as the estimation of antenna radiation

efficiency (when the antenna gain is known).For the case of impedance-matched and polarization-matched transmittingand receiving antennas, Friis equation reduces to

2

( , ) ( , )4

rt t t r r r

t

PD D

P R

. (6.8)

The factor 2( / 4 )R is called thefree-space loss factor. It reflects two effects:(1) the decrease in the power density due to the spherical spread of the wavethrough the term 21/ (4 )R , and (2) the effective aperture dependence on the

wavelength as 2 / (4 ) .

2. Maximum range of a wireless link

Friis transmission equation is frequently used to calculate the maximumrange at which a wireless link can operate. For that, we need to know thenominal power of the transmitter tP , all the parameters of the transmitting and

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receiving antenna systems (such as polarization, gain, losses, impedancemismatch), and the minimum power at which the receiver can operate reliably

minrP . Then,2

2 2 2 2max

min

(1 | | )(1 | | ) | | ( , ) ( , )

4

tt r t r t r t t t r r r

r

PR e e D D

P

. (6.9)

The minimum power at which the receiver can operate reliably is dependent onnumerous 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 noiseof the antenna itself. This topic is considered in the next lecture.

3. Radar cross-section (RCS) or echo area

The RCS is a far-field characteristic of radar targets which create an echo

far field by scattering (reflecting) the radar EM wave. The RCS of a target isthe equivalent area capturing that amount of power, which, when scatteredisotropically, produces at the receiver power density equal to that scattered bythe target itself:

22 2

2

| |lim 4 lim 4

| |s s

R Ri i

WR R

W

E

E, m2. (6.10)

Here,Ris the distance from the target, m;

iWis the incident power density, W/m2;

sW is the scattered power density at the receiver, W/m2.

To understand better the above definition, we can re-write (6.10) in anequivalent form:

2lim ( )

4i

sR

WW R

R

. (6.11)

The product iW

represents some equivalent intercepted power, which isassumed to be scattered (re-radiated) isotropically to create a fictitious sphericalwave, whose power density in the far zone Wsdecreases with distance as

21/R .It is then expected that iW is a quantity independent of distance. sW must beequal to the true scattered power density sW produced by the real scatterer (theradar target).

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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 theangle 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 oftargets 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 RCSincreases also due to sharp metallic or dielectric edges and corners. Thereduction of RCS is desired for stealth military aircraft meant to be invisible toradars. This is achieved by careful shaping and coating (with special materials)of the outer surface of the airplane. The materials are mostly designed to absorbelectromagnetic 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 differentfrom 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 hasRCS smaller than 410 m2, which makes it comparable or smaller that the RCSof a penny.

4. Radar range equation

The radar range equation (RRE) gives the ratio of the transmitted power (fedto 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 areceiving antenna, and they may be located at different positions as it is shownin the figure below. This is calledbistatic scattering.

Often, one antenna is used to transmit an EM pulse and to receive the echofrom the target. This case is referred to as monostatic scattering or

backscattering. Bear in mind that the RCS of a target may considerably differas the location of the transmitting and receiving antennas change.

Assume the power density of the transmitted wave at the target location is

2 2

( , ) ( , )

4 4t t t t t t t t t

t

t t

PG Pe DW

R R

, W/m2. (6.12)

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( , )t t

tR

( , )r r

rR

The target is represented by its RCS , which is used to calculate the capturedpower c tP W (W), which when scattered isotropically gives the powerdensity at the receiving antenna that is actually due to the target. The density ofthe re-radiated (scattered) power at the receiving antenna is

2 2 2

( , )

4 4 (4 )c t t t t t

r t

r r t r

P W PDW e

R R R R

. (6.13)

The power transferred to the receiver is2

2

( , )( , )

4 (4 )t t t t

r r r r r r r r t

t r

PDP e A W e D e

R R

. (6.14)

Re-arranging and including impedance mismatch losses as well as polarizationlosses, yields the complete radar range equation:

2

2 2 2 ( , ) ( , ) (1 | | )(1 | | ) | |4 4

r t t t r r r t r t r t r

t t r

P D De e

P R R

. (6.15)

For polarization matched loss-free antennas aligned for maximum directionalradiation and reception:

2( , ) ( , )

4 4r t t t r r r

t t r

P D D

P R R

. (6.16)

The radar range equation is often used to calculate themaximum range of aradar system. As in the case of Friis transmission equation, we need to know

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all parameters of both the transmitting and the receiving antennas, as well as theminimum received power at which the receiver operates reliably. Then,

2 2 2 2max

2

min

( ) (1 | | )(1 | | )| |

( , ) ( , ) .

4 4

t r t r t r t r

t t t t r r r

r

R R e e

P D D

P

(6.17)

Finally, we note that the above RCS and radar-range calculations are onlybasic. The subjects of radar system design and electromagnetic scattering arehuge research areas themselves and are not going to be considered in detail inthis course.