IC, 



FUNDAMENTAL RELATIONS 



The general relation between the input voltage at 

 the receiver and the received power is Vi = VPo^;. 

 where Ri is the resistance of the receiver load circuit 

 (which is equal to the radiation resistance for maxi- 

 mvun power transfer) and Vt is the input voltage. 

 Hence, using eciuation (14), 



Vi = O.OnSEoK^Ri volts (23) 



1 ,- E„L 



E„\^R, = ^r 



8W5 



2.2.2 



Antenna Gain. Polarization 



The equations of Section 2.1 may be further 

 generalized to apply to any type of antenna through 

 the introduction of a cjuantity called the antenna 

 gain. The term gain, as applied to an antenna, is a 

 measure of the efficiency of the antenna as a radiator 

 or receiver as compared with that of a doublet 

 antenna, \\'ith all antennas located in free space. 



Quantitatively, the gain, ft, of a directive trans- 

 mitting antenna is the ratio of the power P/ radiated 

 by a doublet antenna to the power Pi radiated by the 

 antenna in question to give the same response in a 

 distant receiver, with both transmitting antennas 

 adjusted for maximum transfer of power. Hence 



P/ 

 Pi' 



^rhe gain C^ of a directive receiving antenna is the 

 ratio of the power Pi" radiated by a transmitting 

 antenna, which produces a certain response in the 

 matched load circuit of a distant doublet receiving 

 antenna, to the power Pi radiated by the same 

 transmitting antenna to produce the same response 

 in the matched load circuit of the receiving antenna 

 in question, with both receiving antennas adjusted 

 for maximum transfer of power. Hence 



(n 



(24) 



Gi 



Pi 



(25) 



From the definitions given above it follows that 

 for a transmitting and receiving antenna combina- 

 tion in free space, with gains (n and T/a and adjusted 

 for maximum power transfer, the power ratio is 

 equal to 



— = G1G2 (—\ = 0/;,Ao', (2(5) 



Pi VSttc// 



where Pi, (h are the power output and gain of the 

 transmitter and P2 is the power delivered to the 

 matched load of a receiving antenna of gain G'2. 



If the antennas are not in free space, equation (26) 

 becomes 

 P 



P: 



= CiChA-, 



7 = ^'M^y^'^p" = ^i^'-(AoA,)^ (27) 

 ^ \87rfl/ 



where .1 is the gain factor and Ap is the path-gain 

 factor. Note that for highlj' directive antennas Ap 

 may depend upon the directivity characteristic of 

 the antennas, e.g. when the antenna discriminates 

 Ijetween the direct and reflected waves. 



Since power is proportional to the square of field 

 strength, equation (20), for any transmitting an- 

 tenna, becomes 



E = E.^ThAp. (28) 



In defining gain, the electric doublet is selected 

 here as the comparison antenna in place of the iso- 

 tropic radiator (that is, a hj^pothetical antenna 

 which radiates equally in all directions) which is 

 sometimes used in the literature. Since the gain of 

 an isotropic radiator relative to a doublet is M, the 

 gain of any antenna referred to an isotropic radiator 

 is 3 '2 the value referred to a doublet antenna. 



ulisotropic) = 1. 5 Tf (doublet). 



(29) 



The chief objections to the isotropic radiator are 

 that it does not occur in practice and cannot be 

 produced experimentally, even approximately. 



In experimentally measuring the gain of an an- 

 tenna, a half-wave dipole is often used as a reference 

 antenna. While the gain of a half- wave dipole rela- 

 tive to a doublet is approximately unity, being 1.09 

 for a very thin dipole, it depends somewhat on its 

 actual dimensions so that it is better to express the 

 experimental gain in terms of the doublet antenna 

 even though a longer antenna is used as a reference 

 antenna in making the measurements. 



When antennas are oriented so that the directions 

 of polarization make an angle 7 with each other 



DIRECTION or 

 MAX RADIATION 



TRANS REC 



{;::v;:x;; 



TRANSMITTER 



DIRECTION or 

 MAX RE-RADIATION 



Figure 4. Relation of antenna axes anil wave polariza- 

 tion. 



(while the maxima of their angular patterns still 

 point toward each other), the formulas for power 

 transfer, equations (18), (22), and (27), are multi- 

 plied by a factor cos- 7 (see Figure 4). 



