14 



FUNDAMENTAL RELATIONS 



will be called scattered power. The sum of these is 

 equal to the power abstracted from the wave. 



The calculation of the received and scattered 

 power maj^ be carried out by means of the equivalent 

 circuit of Figure 2. In this figure, Z„ is the impedance 



■i 



OOUBLCr 



LOAD 



VOLTAGE GENERATED IN ANTENNA 



0^ 



Figure 2. Equivalent circuit of antenna and load. 



of the doublet and Z; is the impedance of the load, 

 that is, the impedance connected across the terminals 

 of the antenna when it is acting as a receiver. V is 

 the voltage generated in the antenna. 



The load is supposed to be tuned, which means that 

 the reactance part of Z; is set equal and opposite 

 to the reactance part of Za, so that Z^ + Z; = Ra 

 + Ri, that is, the total impedance is simply the 

 sum of the resistance parts of the impedances of the 

 antenna and the load. Hence 



/ = 



T' 



(10) 



Ra + Rl 



gives the current. But P, = RJ- is the po'ner ab 

 sorbed by the load and hence is equal to 



r-R, 



Pr 



{Ra + Rlf ' 



(11) 



where P, is called the received power. In the same 

 way 



V"-Ra 



P. = 



(12) 



ifia + Rlf 



is the power scattered bj^ the doublet. 



It is easy to show that the maximum power is 

 delivei-ed to the load if Ra = Ri- In this, the matched 

 load case, 



^r — ^s — ^ — , ^ 



iR. 



4P, 



(13) 



Now the resistance of the doublet, neglecting its 

 low ohmic resistance, is only the radiation resistance 

 [equation (9)] and the potential or voltage across 

 the terminals is equal to EoL, where £'o is the field 

 strength of the incident plane wave and L is the 



effective length of the doublet. Inserting these 

 quantities into ecjuation (13), 



Pr=Ps = 



120^ Stt 



(14) 



In these equations it has been assumed that the line 

 of the doublet has been oriented parallel to the 

 electric vector of the incident wave in order to obtain 

 maximum power absorption. 



The factor Ea'/l^Oiv will be recognized from 

 equation (6) as the power per unit area of the inci- 

 dent wave. The formula thus sa3rs that all the power 

 crossing an area 3X"/87r is received, and that all the 

 power crossing an equal area is scattered. The 

 area 3X-/8Tr is therefore called the absorption cross 

 section or scattering cross section of the matched 

 doublet. Since the antenna has been placed parallel 

 to the polarization of the incident wave, this is the 

 maximum absorption cross section. Moreover, this 

 formula holds only when the doublet has been 

 matched to its load, and consequentlj^ 3X-/87r is the 

 maximum absorption cross section. 



It will be noted, however, that 3X-/87r is not the 

 maximum scattering cross section. This maximum 

 is achieved by shorting out the load, that is, setting 

 Rl = 0. In this case, 



Pr =0 



2ir ■ 



P. = 



E,^ 



120 T 



Hence the scattering cross section of the shorted 

 (dummy) doublet is four times the scattering cross 

 section of the matched load doublet. 



It should be noted in passing that the cross sections 

 introduced here should not be confused with the 

 radar cro.ss section which is discussed in Section 2.4. 



2-1-3 Transmission between Doublets 

 in Free Space 



Assume that two doublets, one to function as a 

 transmitter and the other as a receiver, a distance 

 d » X apart, are adjusted for maximum power 

 transfer. This means that the axes of the doublets 

 are parallel and lie in their common equatorial plane 

 and that each is matched to its connected circuit. 

 Then the power radiated by the transmitting doublet, 

 from equation (7), is equal to 



45 



Pi = 



watts, 



(16) 



