FUNDAMENTALS OF ELECTROMAGNETIC RADIATION

2.7 HALF-WAVE DIPOLE ANTENNAS

 One of the simplest practical antennas is the half-wave dipole antenna shown in Fig. 2.10. It is usually fed from a two-wire transmission line. Each arm of the antenna is very nearly one-quarter wavelength long. It has been found both theoretically and experimentally that the current distribution on a thin half-wave dipole antenna is closely approximated by a sinusoidal standing wave of the form

  � ���

 The current is, of necessity, zero at the ends where z= � . We can find the far-zone radiated field from the half-wave dipole antenna by using Eq. (2.50), given in the previous section.

  �

 Figure 2.10 The half-wave dipole antenna.

With reference to Fig. 2.10, it is seen that the unit vector , , and . Hence Eq. (2.50) gives

We now note that , so . Thus we find that

The magnetic field is given by Eq. (2.49b) and is

The power flux per unit area is given by

The total radiated power is obtained by integrating this expression over the surface of a sphere of radius r; thus

This integral can be evaluated in terms of the cosine integral �

The values of this integral are tabulated .� The result of carrying out the integration is

 
When we equate this expression to , we find that the radiation resistance of the half-wave dipole is 2x36.56, or 73.13 . The transmission line that feeds the half-wave dipole should have a characteristic impedance of 73.14 for maximum power transfer.

 The directivity function for the half-wave dipole is obtained by using Eq. (2.55) multiplied by r` and Eq. (2.57) in the defining relation (2.34) and is

 The maximum directivity is 1.64, which is only a modest increase over the value of 1.5 for the short current filament. The radiation pattern in the E plane is shown in Fig. 2.11. The half-power beam width is 78�, which is only a small amount less than the 90� for the short current filament. The most important difference between the short electric dipole and the half-wave dipole antenna is that the latter has a radiation resistance of 73.13 �which is much larger than the ohmic resistance would be for most practical antenna structures. Thus the

  �

Figure 2.11 Principal E-plane radiation pattern for a half-wavelength dipole antenna.

 � See J. A. Stratton, Electromagnetic Theory, McGraw-Hill Book Company New York, 1941. Sec. 8.7.
�� E. Jahnke and F. Emde. Tables of Functions, Dover� PubIications, New York, 1945.

 gain is also very nearly equal to 1.64. It also turns out that when the dipole is a half-wavelength long (actually a few percent less) the input reactance is zero, so the input impedance is essentially equal to the radiation resistance.