Chapter 2 

 FUNDAMENTAL RELATIONS 



21 THE ELECTRIC DOUBLET 



LN FREE SPACE 



^■' ' Radiation of an Electric Doublet 



IT IS C'OXVEXiEXT to present the basic relation- 

 ships of radiation and reception by antennas in 

 their simplest form, that of the radiation and recep- 

 tion of electric doublets in free space. The resul ting- 

 formulas ^^^ll later be generalized to include other 

 types of antennas and their positions relative to the 

 earth. 



An electric doublet is a rectilinear antenna, \\'hich 

 is symmetrical about the point or points of connec- 

 tion thereto and is so short that its directive proper- 

 ties are independent of its length. The field of such 

 an anteima dries not depend on the distribution of 

 current along the wire, because the wire is so short 



Figure 1. Polar coordinate system. 



that there is no phase difference between waves 

 reaching a point in space from different portions of 

 the wire. In sj^mbols, I « X where I is the length 

 of the antenna and X is the wavelength of the radia- 

 tion. 



To facilitate the analysis of the field of the doublet 

 antenna, the spherical polar coordinate system 

 shown in Figure 1 is introduced. The upper half 

 of the doublet is shown in the figure. The distance 

 from the center of the antenna to a point in space 

 is here denoted by r. Elsewhere in this volume 

 this cjuantity is written d. 



Let dl be an infinitesimal portion of /, the length of 

 the doublet, and let the current in this portion be 

 the real part of Ie'^-'"^^K Let dEr, dEe, dE^ and dHr. 

 dHg, dH,f, be the components of the electric and 

 magnetic field strengths at any point P{r, d, (/>) due 

 to the current in dl. A straightforward solution of 

 the fundamental equations of electromagnetic theory 

 gives the following values for these components, 

 valid at distances large compared with the length of 

 the doublet: 



dEr = 007 



dEe = (IOtt/ 



2irr'J 



dl cos 



+ 



1 



./X 

 47r-/-'. 



^;i2ir Ai(d-r) 



\'olts per meter, 

 rf/sin9f^"'-"^^'='-" 

 volts per meter, (1) 



dE^ = 

 dH,, = 



0, (///, = 0, dHe = 0: 

 I 



.Xr 2Tr'- 



r/Lsin^e^''-'^""^'-'-' 



amperes per meter, 



where 



c = velocity of light = 3 X 10* meters per second 



J = v^i,' 



and all distances are measured in meters. Electric 

 field strengths are in volts per meter and magnetic 

 field strengths are in amperes per meter. Unless 

 otherwise explicitly stated, the mks rationalized 

 units are used throughout this volume. 



Equation (l) can be simplified at once. Since the 

 time ^'ariation of the field is assumed sinusoidal. 



^;{2tc(/a) 



mav be omitted. The term e 



-j(2rrTl\) 



gives the 



phase, and it too can be omitted when only the 

 amplitufle is required. From here on, unless other- 

 wise stated, it is understood that root-mean-square 

 (rms) -^'allies \\i\\ be used for dEr, dEe, clH^, and I. 



The field in the neighborhood of the doublet is 

 called the induction field and is given by the terms 

 in equation (1) which include the highest powers of 

 r in the denominators. This field is important when 

 mutual effects between closely spaced antennas, or 

 antennas and reflectors or directors, are in^■olved. 



The radiation field, of greater interest for most of 

 the purposes of this volume and the only important 

 field at large distances (/•>>X), is given l:)y the 



12 



