26 



ANTENNAS 



2. Gain of half-wave dipole. The gain of the dipole 

 relative to a doublet is the ratio of the power supplied 

 to the doublet to the power supplied to the dipole to 

 produce the same field strength at the same distance 

 in the direction of maximimi radiation (here the 

 equatorial plane, 6 = 90 degrees). 



For ecjual maximum fields, comparing ecjuations 

 (3) in Chapter 2 and (G) in this chapter, 



/ 



Idl = - 7, . 



The power per unit area for the doublet, using 

 equation (3), in Chapter 2, is 



W 



doublet 



E- 

 120ir 



307r sin- 6 

 ircp 



m 



and for the dipole the power per unit area is gi\'en 

 by equation (5). 



The dipole gain is then 



G-- 



Jw 



doublet 



dA 



l^dinnlp dA 



Power radiated by doublet 

 Power radiated bj' dipole 



dipole 



where the integration is carried out over spheres 

 surrounding the antennas. Carrying out this opera- 

 tion, 



rMipoie = l-09 (or0.4db). (9) 



3. Radiation Resistance. The radiation resistance 

 of the half- wave dipole is 



Rr = 





W'dipoief'^ = "3.1 ohms. (10) 



4. Impedance of an Infinitely Thin Dipole. IThe 

 formulas given here are valid only for a half-«"a^-e 



CYLINDRICAL COORDINATES 



Figure 8. Half-wave dipole field components. 



dipole composed of wire of vanishing thickness. 

 For wire of finite dimensions, see Section 3.2.7. 



Here (type A in Figure 6) it is necessary' to calcu- 

 late the voltage F, required at the input to establish 

 a current distribution /,■ cos [(27r,/X)3], as shown in 

 Figure 8. To do this, the total field of the dipole 

 must be known, including the induction field which is 

 significantly large at short distances as well as the 

 radiation field. In cylindrical coordinates, the total 

 field is given bv 



(7) Er= + j30Ii 



X/4 -;(?I2) , + x/4 - 

 e ^ ^ ' -|- — e 



b\-\ 



'(^ 



L ar 



br 



E,= - j307, 

 7, 



K't) 



+ 



77. = 



47r 



e T e 



b' i' 



(11) 



(12) 

 (13) 



By the reciprocity theorem a small current length 

 I^dz = li cos [(27r/X)3] • f/s induces a voltage (— dV,) 

 at the input point which is equal to the \-oltage 

 f/T'j = E^ds induced in ds by a small current length 

 lidz taken at the input point. Hence 



E^dz — dVi 



Ldz 



dz 



dz. 



li cos I — z I ■ 

 and the total input voltage is 



,, = 2/;;'-E.cos(?-'.). 



Carrying out the operation indicated and dividing 

 l)y 7, gives the impedance of the half-wave dipole as 



Z = 73.1 +,;42.5 ohms. (14) 



The dipole thus has an inductive reactance of 42.5 

 ohms if a sine distribution of current amplitudes is 

 assumed. 



The reactance can he altered by changing the 

 length of the wire. Increasing the length increa.ses 

 the inductance; decreasing the length decreases the 

 inductance, first to zero for resonance, and then 

 for still shorter lengths to a capacitive reactance. 

 Changes in length of only 4 to 5 per cent will pro- 

 duce large changes in the reactance. 



*^* Modifications of the 



Half- Wave Dipole 



Two modifications \nll be given. 

 1. Quarter-wave dipole with artificial ground. A 

 convenient device for doubling the effective length 



