156 BELL SYSTEM TECHNICAL JOURNAL 



APPENDIX 



Calculations of Three-Dimensional Directivity 



Antenna: Horizontal Rhombus. 



W'ave Polarization: Horizontal. 



Far End Termination: Characteristic Impedance. 



Ground Constants: General Case. 



Assumptions: Negligible Wire Attenuation and Leakage. 

 Negligible Mutual Coupling. 

 Uniform Characteristic Impedance. 



Nomenclature: (See Fig. 1) 



P = point under consideration. 



/ = element length. 



^ = one-half of side apex angle. 



A = angle made with ground by wave-direction in incident 

 plane. 



j3 = angle made by horizontal component of wave-direction 

 with principal axis drawn through receiver and termi- 

 nation. 



// =: height above ground. 



X = wire distance of point P from R for element (2) as well as 

 for element (1). 



e = r.m.s. space voltage. 



E = r.m.s. space voltage and phase at P due to «, direct 

 propagation. 



E' = r.m.s. space voltage and phase at P due to e, after ground 

 reflection. 



£/'i and Ep2 = wire voltage at Pin element (1) or element (2), 



direct propagation. 

 Epi and Epi' — same as above but for ground reflection. 

 a — amplitude ratio of ground reflection to direct space voltage. 

 a = apparent phase angle change after ground reflection. 

 X = wave-length. 

 Ir = receiver current. 

 Zo = characteristic impedance. 

 R = receiver impedance. 

 T = far end termination impedance. 



