Q4Antenna And Wave Propagation

Question

Q.2. Define the principle of pattern multiplication of an antenna array. A uniform linear array consists of 12 isotropic point sources with spacing of lambda/4. Calculate:

(a) Directivity
(b) Effective aperture, if the phase difference psi = -90 degrees

Answer 1

Principle of Pattern Multiplication

The principle of pattern multiplication states that the total far-field radiation pattern of an array composed of N identical, similarly-oriented antenna elements can be obtained as the product of two separate factors: the element factor (the individual radiation pattern of a single isolated array element, considered on its own) and the array factor (the radiation pattern that would result if each individual element were replaced by an isotropic point source, with the array's specific geometric arrangement, element spacing, and excitation amplitude/phase distribution preserved):

E_{t o t a l} (θ, ϕ) = E_{e l e m e n t} (θ, ϕ) \times A F (θ, ϕ)

This principle is of major practical importance in antenna array analysis and design because it allows the two contributing factors to be analyzed independently — the element factor depends only on the specific type of radiating element used (dipole, patch, horn, etc.), while the array factor depends only on the number of elements, their geometric arrangement/spacing, and their relative excitation amplitudes and phases, entirely independent of the specific element type chosen. This separation greatly simplifies both the analysis of existing arrays and the design of new arrays, since the array factor (which determines the overall beam-steering, beamwidth, and side-lobe characteristics resulting from the array geometry and excitation) can be studied and optimized using the simple isotropic-source array factor formula, independently of whatever specific element pattern is eventually chosen to populate the array.

Hansen-Woodyard Condition: Enhanced End-Fire Directivity

The ordinary end-fire condition (psi=-beta*d, used in this problem) achieves maximum radiation exactly along the array axis, but Hansen and Woodyard showed that a further, deliberately increased progressive phase shift can produce a still-larger directivity than the ordinary end-fire array of the same length, at the cost of a small reduction in the main-to-side-lobe ratio and the appearance of a small backward-direction lobe. The Hansen-Woodyard condition modifies the required phase shift to:

ψ = - (β d + \frac{2.94}{N}) (approximately, for large N)

This additional phase increment beyond the ordinary end-fire value narrows the main beam further and can increase the achievable directivity by roughly 1.8 times (about 2.5 to 3 dB) over the ordinary end-fire value for the same array length and element count, making the Hansen-Woodyard design attractive whenever the small increase in side-lobe level and back-lobe radiation is an acceptable trade-off for the higher forward gain, such as in traveling-wave end-fire antennas (e.g., long Yagi-Uda or helical antennas) designed for maximum forward gain in a fixed physical length.

Directivity Comparison: Broadside vs End-Fire Arrays

For the same number of elements N and the same inter-element spacing d, a broadside array and an ordinary end-fire array generally achieve different directivity values because their array factor beamwidths differ in the two principal cuts through the pattern. The approximate directivity of a large uniform broadside array (elements in phase, d of the order lambda/2) is given by D0(BSA)=2N(d/lambda), the same functional form as the ordinary end-fire directivity formula used in part (a) of this problem, but the two arrays realize this directivity differently: the broadside array's directivity arises from a pattern narrow only in the plane containing the array axis while remaining a full ring (omnidirectional in azimuth about the array axis for a linear array), whereas the end-fire array's directivity arises from a narrower, pencil-like beam concentrated along the single end-fire axis direction. For equal N and d/lambda, ordinary end-fire and broadside arrays of this simple uniform type can give comparable directivity values, but the Hansen-Woodyard-optimized end-fire array exceeds the ordinary broadside array's directivity for the same size, which is why end-fire geometries (with Hansen-Woodyard phasing) are generally preferred over broadside geometries whenever a single, highly directive pencil beam along one fixed axis (rather than a fan-shaped broadside pattern) is the desired design objective, such as in a Yagi-Uda antenna optimized purely for maximum forward gain.

(a) Directivity of 12-Element Array, Spacing lambda/4, psi=-90 degrees

Given: N=12 elements, spacing d=lambda/4, progressive phase difference psi=-90 degrees.

First, identifying the type of array this configuration represents: the spatial phase delay corresponding to the element spacing is betad=(2pi/lambda)(lambda/4)=pi/2=90 degrees. Since the given psi=-90 degrees exactly equals -betad, this satisfies the ordinary end-fire array condition (psi=-beta*d, for maximum radiation along theta=0), confirming this is an end-fire array configuration.

For a uniform end-fire array (ordinary end-fire condition), the directivity can be approximated (Kraus' approximate formula, valid for reasonably large N and the ordinary end-fire condition) as:

D_{0} \approx 2 N (\frac{d}{λ})

Substituting N=12 and d/lambda=0.25:

D_{0} \approx 2 \times 12 \times 0.25 = 6

D_{0} (dB) = 10 lo g_{10} (6) \approx 7.78 dB

(b) Effective Aperture

The effective aperture is related to directivity by the universal antenna relation:

A_{e m} = \frac{D _{0} λ ^{2}}{4 π} = \frac{6 λ ^{2}}{4 π} \approx 0.4775 λ^{2}

Result: the directivity of the 12-element end-fire array is D0=6 (approximately 7.78 dB), with a corresponding effective aperture of approximately 0.4775*lambda^2 (expressed in terms of the operating wavelength, since no specific operating frequency was given in the problem statement, only the ratio of element spacing to wavelength). This result illustrates the general array design principle that increasing either the number of elements N or the element spacing (in wavelengths) d/lambda directly and proportionally increases an ordinary end-fire array's directivity, up to the practical limits imposed by grating-lobe formation at excessive spacing and by the array's physical size constraints.

Physical Significance of Effective Aperture in This Result

The effective aperture Aem=0.4775lambda^2 obtained here represents the equivalent capture area that this 12-element end-fire array would present if operated in receive mode, meaning that for a given incident power density (in watts per square metre) arriving from the array's own end-fire boresight direction, the power actually delivered to a matched receiver load connected to the array equals this incident power density multiplied by 0.4775lambda^2. This value can be directly compared against the effective aperture of a single isotropic radiator (Aem=lambda^2/(4pi) approximately 0.0796lambda^2) and of a single half-wave dipole (Aem approximately 0.13lambda^2), showing that this 12-element end-fire array configuration captures roughly 6 times more power from an incident wave along its main-beam direction than an isotropic antenna would, and roughly 3.7 times more than a single half-wave dipole, directly reflecting the directivity gain of D0=6 achieved by combining the 12 elements coherently along the array's end-fire axis. This reciprocity between transmit-mode directivity and receive-mode effective aperture (both scaling by the identical factor D0) is a general and fundamental property of all antennas, following directly from the universal relation Aem=D0lambda^2/(4*pi) used in this calculation, and underlies why any array or reflector design intended to concentrate transmitted power into a narrow beam automatically and equally improves its ability to selectively capture power from a wave arriving along that same preferred direction.

Practical Design Trade-offs for End-Fire Arrays

While the approximate formula D0=2N(d/lambda) used above shows that directivity can, in principle, be increased without limit simply by increasing the number of elements N or the element spacing d, several practical constraints limit how far this scaling can usefully be pursued in an actual end-fire array design. Increasing the element spacing d beyond roughly 0.25-0.4 wavelength in an end-fire configuration risks the earlier onset of grating lobes (as discussed in the corresponding broadside/end-fire array answer elsewhere in this paper), since the required progressive phase shift for end-fire operation brings a repeat (grating) lobe into the physically visible angular region at a smaller spacing than would occur for a simple broadside array. Increasing N instead lengthens the overall physical array (for fixed spacing), which for many practical installations (such as rooftop or tower-mounted HF/VHF end-fire arrays) becomes constrained by available physical space, mechanical wind-loading considerations, and the added feed-network complexity and loss associated with distributing correctly phased excitation to a larger number of elements. In practice, therefore, end-fire array designs settle on a spacing in the neighborhood of lambda/4 (as used in this problem) as a reasonable compromise that avoids grating lobes while still achieving useful directivity gain per additional element, with further directivity improvement, where required beyond what a purely uniform-excitation ordinary end-fire array can provide, more commonly pursued through the Hansen-Woodyard phasing enhancement (discussed in the corresponding pattern-multiplication answer earlier in this response) rather than through indefinitely increasing N or d alone.