THE CAI.Cl 1,-VTION OK VKKTICAI. COVKRA(;K 



171 



Table 13. Lobes for a medium-height radar. (Example 20. ) 



15.6.19 



The General Lobe Formula 



The assumption of a sinusoidal lobe shape and the 

 neglect of the phase of reflection and diffraction in 

 the preceding section may in some cases lead to 

 considerable error, especially when the direct and 

 reflected waves are very different in strength. In 

 general a more accurate method is required for sites 

 over 1,000 ft in height, where vertical polarization 

 is used or where it is desired to know the lobe shape 

 in detail. The method given in this section provides 

 a general solution of the coverage problem in the 

 optical region (except along the bottom of the 

 first lobe). 



The development of the lobe formula will be 

 reviewed, and equation (99) will be given in a some- 

 what different form. The expression for the electric 

 vector due to the direct wave is 



£ d = ^/(7)exp(-i2x 



For the reflected wave 



Br = ^/(7 



26») flexp - j2ir 



(100) 



(101) 



where E d = electric field intensity at the target 

 due to the direct wave, microvolts 

 per meter; 



/(7 



E r = electric field intensity at the target 

 due to the reflected wave, microvolts 

 per meter; 

 Ei = electric field intensity at 1 mile in 

 the equatorial plane of the antenna, 

 microvolts per meter; 

 f(y) = modified antenna factor for the direct 

 wave (Section 15.6.12); 

 26) = modified antenna factor for the 

 reflected wave (Section 15.6.15); 

 R = a complex factor for the reflected 



wave given by 

 R = Dpz |exp[-i(0 + f)]} ; (102) 



D = divergence factor (Section 



15.6.17); 

 p exp ( — j4>) = complex reflection factor (Section 



15.6.16); 

 z exp ( — jf ) = complex diffraction factor (Section 



15.6.11). 



The net field at the target is 



E T = Ed + E T , 

 Ei 



where 



B, 



d 



/(t) +/(7 -26) 



Dpz {exp[ -j(cp+ r + 5)]} 



(103) 



considering only the absolute value of E T and taking 

 r = ra = d except where the path difference is 

 involved. The path difference phase shift is 



2tt, 



U) 



(104) 



Equation (103) may for convenience be written 



l»,l-$4. 



(105) 



The target is assumed to have a complicated form 

 and to be changing its aspect constantly. The 

 reflected energy is considered to be of random phase 

 and magnitude. The magnitude of the reradiated 

 field (microvolts per meter at a distance of 1 mile 

 from the target) is found by using a coefficient of 

 reradiation, p T , which varies with the target and 

 aspect. 



The received field intensity is by the reciprocity 

 theorem : 



Pt I E T | 



E = 



d 



(106) 



Substituting from equation (105) 



