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Chapter 5 245

uniform excitation and low SLL in some instances. It can be done, for example, by putting the
more directive elements at the edge position, thereby reducing the radiation in the direction of
sidelobe peaks, or playing with separation between elements, their number and position in array.

5.4.6 Linear Arrays with Progressive Phase Distribution

Previously we restricted our analysis and synthesis assuming that all linear array elements are
excited in phase. Meanwhile, the phase shifting is a powerful tool to steer the direction of an
array main beam in space and time, synthesize beams of a particular shape with pre-fixed
excitation (for example, all elements are fed equally), regulate sidelobe level, etc. The growing
interest in the phase-only analysis [12] and synthesis is mainly dictated by the fact that the phase
is much easier to control than the amplitude. We will concentrate our attention on the beam-
steering / beam-scanning topic only referring the reader for more details to [13, 14] and the
extensive list of references in these publications.

Suppose that the phase distribution in linear array in Figure 5.4.9a is progressive meaning
that = , = . ≠ 0, i.e. the phase shift between two successive radiators is constant

and equal to . We may assume that such progressive phase excitation is supported by some
real or fictional electromagnetic wave traveling along the linear array, i.e. the z-axis, with some
equivalent phase velocity and the wave phase constant = . If so, the phase shift can
⁄

be written as = . The negative corresponds to the wave traveling in the positive

th
direction of z-axis from the zeroth element to the array end while the N element is excited later
in time than the zeroth one. Meanwhile, the positive corresponds to the wave propagating in
opposite direction with the maximum lag-delay of the zeroth element. Depending on excitation,
the equivalent phase velocity may exceed the speed of light, i.e. || < , be equal, i.e. || =
or bellow, i.e. || > . Here = is the wavenumber in free space. An antenna where
⁄
the progressive phase excitation supported by a real EM wave propagating along an antenna’s
geometrical axis is customarily called a traveling wave antenna. Note that the latter can consist
of discrete elements as shown in Figure 5.4.9a or be continuous as well when the infinite sum /
integral of infinitesimal radiators adjoining one another. We will start from the discrete case.
The simplest way to physically develop the wave traveling along the linear array is based on
the geometrical factor as a delay line. Such approach is demonstrated schematically in Figure
5.4.9a on the example of classical relatively high gain and very traditional Yagi or Yagi-Uda
antenna forming the narrow main beam in the positive z-direction. It belongs to a broad class
of linear arrays customary called end-fire antennas. In general, this antenna consists of only
one active dipole #0 driven by RF generator as shown on the bottom of Figure 5.4.9a. The
voltage polarity, initial electric field orientation, and matching electric current direction are
shown at some moment of time. Loosely speaking, the electric current in the driven element
exerts EM waves propagating with the speed of light in two opposite direction: z > 0 and z < 0.
The longer dipole #-1 as a combination of several metal roads parallel to the electric vector E
(see the top picture in Figure 5.4.9a) acts as an equivalent metal mirror reflecting the incident
wave back to the front of the periodic structure thereby reducing antenna back radiation, z < 0.
As a result, most of the EM wave energy is directed along the periodic structure, i.e. in the
direction z > 0. Going this way the EM wave reaches the dipole #1 and exerts an electric current
in it with some delay in time.