1

Friis Transmission FormulaFriis Transmission Formula

Pt Pr

The Friis Transmission Formula enables us to calculate thepower received (Pr) given that a known power (Pt) isradiated. Friis assumes that both antennas are isotropicand that they are in free space.

FreeFree--Space LossSpace Loss

• “Free space” implies that there are no objects presentto affect propagation

• To satisfy the Law of Conservation of Energy, theenergy leaving any closed surface around a radiatormust be independent of the distance (R) from thatradiator

R

SphericalSurface

2

FreeFree--Space Loss (2)Space Loss (2)

The spherical surface has an area of 4R2,the power and the power density will beuniform over that surface since radiation isisotropic. Hence, the power density will be:

2

2/

4tP

S Watts mR

For a non-isotropic source, the above equationcan be modified by the antenna pattern toaccount for variation with angle

Friis Transmission FormulaFriis Transmission Formula

This gives us radiated power density as a function of radiatedpower and distance.

2

2W atts

(4 )t

r

PP

R

where G is the gain of the antenna, and is the wavelength.For an isotropic antenna, G=1, and the power received is:

2

4e

GA

For any antenna:

The maximum power that can be delivered to a matched loadby a receiving antenna, Pr, is given by Pr=SAe, where Ae is theEffective Aperture of the receiving antenna.

Friis Transmission Formula: assumes both transmitting andreceiving antennas are isotropic

3

Expressing Friis in DecibelsExpressing Friis in Decibels

since =c/f, where c is the speed oflight (3 x 105 km/s)

2

2(4 )t

r

PcP

Rf

10 4

2 2 6 2 2 2

(9 10 ) 5.6993 10

(4 ) ( 10 )t r

km MHz t MHz km

P P

R f P f R

Taking 10 log of both sides we get the ratio of Pr /Pt in dB:

410log 10log(5.6993 10 ) 20log( ) 20log( )

32.44 20log( ) 20log( )

rMHz km

t

MHz km

Pf R

P

f R

Free-space gain in dB

Including the Effects of AntennaIncluding the Effects of AntennaGainsGains

Free-Space Channel Model

SignalfromXMTR(dBm)

XMTRAntennagain (dBi)

Free-SpaceGain(dB)

RCVRAntennagain (dBi)

ReceivedPower(dBm)

4

Example ProblemExample Problem

A ground-based communication system transmits to a geo-synchronous satellite located 41935 km from the transmitterat a frequency of 1 GHz The gain of the ground-basedantenna is 25 dBi, and the satellite antenna has a gain of15 dBi.

Assuming free-space propagation path loss, what must bethe transmitter power in Watts to produce 5VRMS at theoutput of the satellite antenna? Assume that the satelliteantenna is matched to 50.

Example Problem SolutionExample Problem Solution

? +25 dBi +15 dBi-185 dB

32.44 20log(1000) 20log(41935) 185

-93 dBm

262

13

13

5 105 10

50

5 1010log 93

0.001

RECEIVED

dBm

VP

R

P dBm

93 ? 25 185 15 ? 52 or 158.5 WattsdBm