Q4Antenna And Wave Propagation

(a) Endfire vs Broadside Arrays

The fundamental distinction between an end-fire array (EFA) and a broadside array (BSA) is the direction of the array's main radiation beam relative to the physical line (axis) along which the array elements are arranged. In a broadside array, the main beam of radiation points in the direction perpendicular (broadside, at 90 degrees) to the array's own axis, achieved by feeding all elements with equal (zero) progressive phase difference, psi=0, so that in the broadside direction, all elements' radiated fields arrive with zero relative phase difference (since the additional path length between successive elements, projected along the broadside direction, is itself exactly zero) and hence add up perfectly in phase. In an end-fire array, the main beam of radiation points directly along the array's own physical axis (either forward, theta=0, or backward, theta=180 degrees), achieved by feeding the elements with a progressive phase difference psi that exactly compensates for the spatial phase delay corresponding to the physical element spacing, i.e., psi=-betad (for a forward-pointing end-fire beam) or psi=+betad (for a backward-pointing end-fire beam), so that the total phase difference between successive elements' contributions (excitation phase difference plus spatial path-length phase difference) becomes exactly zero specifically along the array's own axis.

Physically, a broadside array can be thought of as analogous to a two-slit or multi-slit optical interference pattern viewed face-on (constructive interference straight ahead, perpendicular to the slit plane), while an end-fire array is analogous to a traveling wave feeding a chain of successively-delayed radiators such that the emitted wavefronts reinforce along the direction of wave travel itself — end-fire arrays are commonly used where a compact, elongated antenna structure with a beam pointing along its own physical length is needed (such as Yagi-Uda antennas, log-periodic antennas, and traveling-wave antennas), whereas broadside arrays are used where the desired beam direction is perpendicular to a naturally elongated or planar mounting structure (such as building-mounted panel antennas or stacked dipole curtain arrays for shortwave broadcasting).

Proof: Two Isotropic Sources, Equal Amplitude, Opposite Phase (psi=180 degrees) Acts as End-Fire Array

Setup: consider two isotropic point sources separated by distance d along the z-axis, both fed with equal current amplitude but with a phase difference of psi=180 degrees between them (source 2 leading or lagging source 1 by exactly 180 degrees, i.e., opposite/antiphase excitation). The total far-field, as a function of the observation angle theta measured from the array axis, is the phasor sum of the two individual source contributions, each carrying its own excitation phase plus the spatial phase corresponding to the extra path length betadcos(theta) that source 2's contribution must additionally travel (or save) relative to source 1, depending on the observation direction:

Simplifying using the standard phasor-sum identity (1+e^(jx) = 2cos(x/2)e^(jx/2)), the magnitude of the array factor is:

Substituting the given psi=180 degrees (opposite phase excitation):

Evaluating for d=lambda/2 (so that betad=(2pi/lambda)*(lambda/2)=pi=180 degrees), the array factor becomes:

Checking theta=0 (forward end-fire direction): cos(theta)=1, so the argument becomes 90(1+1)=180 degrees, giving AF=2cos(180 degrees)=2*(-1)=-2, with magnitude |AF|=2, the maximum possible array factor value for two equal-amplitude sources — confirming maximum radiation exactly along theta=0.

Checking theta=180 degrees (backward end-fire direction): cos(theta)=-1, so the argument becomes 90(-1+1)=0 degrees, giving AF=2cos(0)=2, again the maximum possible magnitude |AF|=2 — confirming maximum radiation also exactly along theta=180 degrees.

Checking theta=90 degrees (broadside direction): cos(theta)=0, so the argument becomes 90(0+1)=90 degrees, giving AF=2cos(90 degrees)=0 — confirming a complete null broadside to the array, exactly as expected for a genuine end-fire (as opposed to broadside) radiation pattern.

Conclusion: since the array factor magnitude is maximum (|AF|=2) at both theta=0 and theta=180 degrees (i.e., along the array's own physical axis, in both forward and backward directions) and exhibits a complete null broadside to the array (theta=90 degrees), this array configuration — two equal-amplitude, opposite-phase (psi=180 degrees) isotropic sources spaced at d=lambda/2 — is definitively confirmed to behave as a (bidirectional) end-fire array, radiating its maximum power along its own physical axis rather than perpendicular to it, which completes the required proof.

(b) Radiation Pattern of 4-Isotropic Source Array: Broadside vs End-Fire

For a uniform linear array of N=4 isotropic sources with equal excitation amplitude, the normalized array factor magnitude is given by the standard uniform-array formula:

Broadside configuration (psi=0, equal amplitude and phase): here psi_t=betadcos(theta), which equals zero exactly when theta=90 degrees, giving the maximum array factor value AF=1 broadside to the array — the resulting radiation pattern is a relatively broad main lobe centered at theta=90 degrees (perpendicular to the array axis), flanked by smaller side lobes and nulls at angles where 4*psi_t/2 is a non-zero integer multiple of pi (but psi_t/2 is not itself a multiple of pi), with the pattern symmetric about the broadside direction and also symmetric front-to-back (identical lobes exist on both the theta<90 and theta>90 sides of the array, since a broadside array has no inherent forward/backward directional preference along its own axis).

*End-fire configuration (psi=-betad, equal amplitude, ordinary end-fire condition):* here psi_t=betad*(cos(theta)-1), which equals zero exactly when theta=0 degrees, giving the maximum array factor value AF=1 along the array's own forward axis — the resulting radiation pattern is a single, comparatively narrow main lobe directed along theta=0 degrees (along the array's own physical axis), typically narrower in the immediate vicinity of the main beam than the corresponding broadside pattern's main lobe (for the same N and spacing) but generally exhibiting a less symmetric overall lobe structure and comparatively higher relative side-lobe levels, with essentially no significant radiation in the theta=180 degree (backward) direction for the ordinary end-fire condition (in contrast to the two-source opposite-phase case discussed in part (a), which was inherently bidirectional; for larger N with the ordinary end-fire phasing, the pattern becomes progressively more strongly unidirectional toward theta=0).

Comparison: both configurations use the identical underlying array factor formula AF=sin(4psi_t/2)/(4sin(psi_t/2)), the only difference being the value of psi (and hence psi_t as a function of theta) substituted into this common formula — this again illustrates how a single uniform-array mathematical framework, differing only in the choice of inter-element progressive phase psi, produces the two qualitatively very different broadside and end-fire radiation pattern types, underscoring phase excitation control as the single most important design parameter governing an array's main-beam pointing direction.