92 



CALCULATION OF RADIO GAIN 



different for the two states of polarization. For 

 vertically polarized radiation, 



, 27r e, - 1 



A \ir. 



where e,. is the complex dielectric constant e^— jGOo-X, 



, 27rV(e, - 1)H-(6(VX)= .,,,. 



or p = — . U-l^J 



X «;- + [GOaxy 



For horizontal polarization, 



, 27r 



1 



277 





1)- + (GOcrX)-. (145) 



Ai depends also on the phase of the complex di- 

 electric constant. The phase is determined by the 

 parameter 



Q = 



60o-X 



(146) 



For iilti-a-short waves, with the exception of 

 vertically polarized waves over sea water at distances 

 less than 50/ p' (see Section 5.7.4 and Figure 45) and 

 a wavelength greater than 1 meter, Ai is inde- 

 pendent of Q and, to a sufficient approximation, is 

 given by 



1 



Ar = 



p'd 



(147) 



Ai as a function of p'd is plotted in Figure 56. 



In the case excepted above, Ai deviates substan- 

 tially from 1/p'd (see Figure 45) for distances less 

 than 50/p'. Table 3 gives 50/ p' as a function of X. 

 It appears that the deviations are immaterial for 

 practical purposes as long as the wavelength is 

 smaller than 3 meters, since we are usually con- 

 cerned with ranges larger than 7 km. It should 

 further be mentioned that, in the above case, Ai 

 depends to a small degree on Q. However, the 

 variations are less than 1 db and may be neglected 

 for wavelengths less than 10 meters. 



Table 3. Sea water (vertical polarization). 



The condition that the earth may be considered 

 plane is that the shadow factor Fs [see equations 

 (149) and (207)] .shall be approximately unity. For 

 Fs = 0.9, which is approximately 1 db below unity, 

 y, = f/(5) = 0.4 from Figure 58; /(5) in Figure 57 



is approximately unity, so that ^ = sd^ 0.4. 

 Since s, from equation (150) for k = 4/3, is given by 

 4.43 X 10"^ X"'■'^ it follows that d, for the plane-earth 

 approximation to hold, must be less than 10'*X^^^, 



d < m\^'^ (plane earth). (147a) 



3. Curved earth. The screening effect of the earth 

 curvature results in a further decrease in gain. Well 

 within the diffraction region, the shadow effect pro- 

 duces an exponential drop in field strength with dis- 

 tance, which is much greater for higher modes than 

 it is for the first mode. 



Denoting the screening or shadow factor by Fs 

 and the distance gain factor by $i, we have 



$1 = 2AoAiFs. (148) 



For the dielectric case (see Sections 5.1.7 and 5.7.3) 

 5 > > 1 and for distances greater than 1.5/s, the 

 shadow factor is 



Fs = 2.507 (sd)^/' e-'-'^' '^'", 



where 



L X (fca)-J ' 



= 4.43 



10 - V 





(149) 



(150) 



Equation (149) gives the value of the shadow 

 factor for the first mode only. In Figure 32, the 

 curve marked "dielectric earth" is a plot of the 

 shadow factor evaluated by using all terms or modes. 

 However for sd > 1.5 only the first mode is impor- 

 tant. Consequently, equation (149) represents this 

 curve accurately for all values of sd larger than 1.5. 



Height-Gain 



For antennas at zero height, the height-gain 

 functions / are equal to unity, so that equation (148) 

 represents the actual value of the first mode for 

 both antennas at zero height. 



\Vlien the antennas are raised above the ground, 

 it is convenient to distinguish between low and high 

 antennas, the division between the two cases being 

 given by the critical height h^ = 30X'''^. 



1. Loiv nntcnna; h < h^ = 30X"''^. For a low 

 antenna, / is a function of Ih and of Q, where I is a 

 quantity that depends on the complex dielectric 

 constant and is given by 



= V?fi': 



(151) 



