Applied Antenna Mathematics

Formula reading: λ is wavelength in meters, c₀ is propagation speed in free space, and f is frequency in hertz. The second form is the quick engineering version when frequency is in GHz: divide 0.3 by f_GHz to get wavelength in meters. Wavelength is the ruler used by the antenna. Element spacing, aperture size, phase shifter design, far-field distance and target scattering all scale from λ. A radar at 10 GHz has a wavelength of about 3 cm; a radar at 3 GHz has a wavelength of about 10 cm.

Boundary formulas

Formula reading: D is the largest physical aperture dimension and λ is wavelength. R_reactive estimates where stored electric and magnetic energy dominates near the antenna. R_far is the Fraunhofer distance; beyond it, the outgoing wavefront is close enough to a plane wave that gain, sidelobe level and beamwidth specifications become meaningful. The larger the antenna or the shorter the wavelength, the farther away the true far field begins.

Pattern quantities

How to read the graphic

The drawings are simplified original teaching diagrams. They show the physical principle: a dipole radiates broadly, a horn launches a controlled waveguide aperture, a parabolic reflector focuses energy from a feed, and a phased array forms/steers beams by controlling element phase.

Dipole and Monopole

Dipoles and monopoles are fundamental resonant antennas. A half-wave dipole is compact, simple and useful as an element in arrays or as a reference antenna. A monopole uses a ground plane and behaves like half a dipole above that plane.

Formula reading: l_dipole is the approximate total physical length of a resonant half-wave dipole. Real antennas are usually slightly shorter than λ/2 because conductor diameter, end effects and dielectric loading change the electrical length. G_dipole is the ideal peak gain relative to an isotropic radiator; it is modest, but the pattern is broad and useful for coverage or as an array element.

Horn Antenna

A horn is a flared waveguide aperture. It is widely used at microwave frequencies because it is predictable, handles power well and can have clean patterns with relatively low sidelobes. Horns are also common as feeds for reflectors and as calibration antennas.

Formula reading: G is linear gain, A is physical or effective aperture area, η is aperture efficiency, and λ is wavelength. A larger horn mouth captures and concentrates more wavefront area; shorter wavelength also increases gain for the same physical aperture. Θ is beamwidth: as aperture dimensions grow in wavelengths, the main lobe narrows.

Parabolic Reflector

A parabolic reflector turns spherical waves from a feed into a narrow beam. It is a high-gain solution for search radar, tracking radar, satellite links and radio astronomy. Mechanical scan is common, though shaped reflectors and multiple feeds can create special beam shapes.

Formula reading: D is dish diameter. The ratio D/λ tells how many wavelengths fit across the aperture; more wavelengths across the dish means higher gain and a narrower beam. η collects real losses: feed taper, spillover, blockage, surface error and ohmic loss. Θ_deg is an approximate half-power beamwidth in degrees; the constant 70 is a practical rule of thumb for common reflector illumination.

Phased Array, PESA and AESA

A phased array forms beams by combining many radiating elements with controlled phase and amplitude. A PESA uses a central transmitter with phase shifting distribution. An AESA uses many transmit/receive modules, improving agility, reliability and multi-function operation.

Formula reading: d is element spacing. Keeping d near or below λ/2 reduces the risk of grating lobes when the beam scans away from boresight. Δφ is the progressive phase shift between adjacent elements. Increasing Δφ tilts the combined wavefront, steering the beam to θ_scan. Large scan angles reduce projected aperture and can raise sidelobes, so array geometry is an EW performance constraint, not only an RF design choice.

Slot and Slotted Waveguide Arrays

A slot antenna radiates from openings cut into a conducting surface or waveguide. Slotted waveguide arrays are common in radar because they are rugged, efficient and can create fan beams for surveillance.

Lens, Luneburg and Retrodirective Concepts

Lens antennas bend wavefronts using material or graded-index structures. Luneburg lenses can focus plane waves onto different points depending on arrival direction. Retrodirective arrays, such as Van Atta arrays, reradiate energy back toward the source and are important in decoy and calibration thinking.

Why feed illumination matters

If the feed under-illuminates the dish, useful aperture is wasted. If it over-illuminates, spillover raises sidelobes and wastes power. Feed blockage also reduces efficiency and can increase sidelobes.

Formula reading: A_physical is the geometric area of the circular dish. The gain equation then converts that physical aperture into electrical directivity using efficiency η. A dish can have large physical area but poor realized gain if feed illumination is wrong, surface accuracy is poor, or spillover sends energy outside the reflector.

Steering equation

Formula reading: θ is the desired steering angle measured from array broadside. The phase shift Δφ creates a controlled path-length difference between neighboring elements. The d ≤ λ/2 rule is a sampling rule in space: if elements are too far apart, the array can create additional main beams called grating lobes. At large scan angles, projected aperture becomes smaller, gain falls and sidelobe behavior worsens. This scan loss is a real coverage-planning constraint in AESA radars.

Technical reading of an antenna pattern

• Main beam: direction of maximum useful gain.
• Half-power beamwidth: angular width between -3 dB points; a practical resolution metric.
• Sidelobe level: unwanted gain away from boresight; critical for jammer entry and clutter pickup.
• Backlobe: radiation or reception behind the antenna; important for platform integration and EM compatibility.
• Polarization: orientation of the electric field; mismatch reduces link or target return.

Specification Checklist for Radar/EW Antennas

Polarization loss

Formula reading: PLF is polarization loss factor. p_TX and p_RX are unit polarization vectors describing transmit and receive electric-field states. The dot product measures how well the two electric-field orientations overlap. For two linear antennas misaligned by angle ψ, the coupling is cos²ψ. If ψ = 0°, loss is 0 dB. If ψ = 90°, ideal coupling is zero, though real antennas have finite cross-polar isolation.

Main polarization types

Why it matters in EW

A jammer with the wrong polarization can waste much of its power before the victim receiver even sees it. An ESM receiver with only one polarization can miss or under-measure emitters with orthogonal or circular polarization. Radar polarimetry uses differences between HH, VV, HV and VH returns to infer target structure, surface roughness, vegetation, rain and chaff behavior.

Transmit Antenna Chain

A transmitting EW antenna must handle RF power, thermal load, voltage breakdown and pattern control. For jammers, the operational metric is often effective radiated power in the victim direction, not transmitter output alone.

Formula reading: EIRP is the equivalent isotropic radiated power: the power an ideal isotropic antenna would need to radiate to produce the same power density in the main-beam direction. P_TX is transmitter output power, G_ant is antenna gain, and L_feed represents feed loss. In dB form, gains add and losses subtract. For a jammer, EIRP in the victim radar direction is often more important than raw transmitter power.

Receive Antenna Chain

A receiving EW antenna must preserve sensitivity, phase stability and amplitude accuracy. Direction finding needs channel matching; ESM intercept needs wide bandwidth; radar receivers need controlled sidelobes and protection from high-power pulses.

Formula reading: S_inc is incident power density at the antenna in W/m². A_e is effective aperture area, the receiving area that converts incident field power into receiver input power. Gain and wavelength determine A_e: a high-gain antenna has a larger effective capture area, but that usually comes with narrower angular coverage.

Loop and Ferrite Loop Antennas

Loop antennas respond strongly to magnetic-field coupling and are useful at lower frequencies and in compact receiving systems. Small loops are often inefficient transmitters but can be useful receive sensors. Ferrite loading increases effective magnetic coupling but narrows bandwidth and introduces loss/Q tradeoffs.

Traveling-Wave, Beverage, Vee and Rhombic Antennas

Traveling-wave antennas use current that progresses along a conductor rather than standing as a simple resonant pattern. They are useful for wideband HF/VHF receiving and long baseline directional behavior. Termination controls reflections and pattern cleanliness.

Yagi-Uda Antennas

A Yagi-Uda antenna uses a driven element with parasitic reflector and director elements. Mutual coupling between elements creates a directional beam without an active feed network for every element.

Frequency-Independent Antennas: LPDA, Spiral and Discone

Frequency-independent antennas maintain similar electrical geometry over scale. Log-periodic dipole arrays use a sequence of scaled elements; spirals and discones support broad bandwidth and are common in intercept and monitoring roles.

Microstrip Patch Antennas

Patch antennas are low-profile printed radiators. They integrate well on aircraft, missiles, vehicles and conformal arrays. Their basic form is narrowband, but stacked patches, slots and arrays can extend bandwidth or shape polarization.

Active Antennas and Integrated Apertures

Active antennas place amplification, phase control or conversion close to the radiating element. In AESA systems, distributed transmit/receive modules improve agility and graceful degradation but require calibration and thermal management.

Reflection Coefficient and VSWR

Formula reading: Z_L is load impedance and Z₀ is transmission-line characteristic impedance, often 50 Ω in RF systems. Γ is reflection coefficient: zero means perfect match, while larger magnitude means more reflected voltage wave. VSWR converts |Γ| into the standing-wave ratio on the line. Mismatch reflects power back toward the transmitter or creates standing waves. In high-power EW transmitters this can cause protection trips or hardware stress.

Feed Loss Directly Reduces System Performance

Formula reading: P_ant,dB is power actually reaching the antenna terminals after feed loss. Every dB lost in the feed is a dB removed from EIRP. G/T is receive figure of merit: antenna gain divided by system noise temperature, expressed in dB/K. Higher G/T means better receive sensitivity. Receive feed loss before the low-noise amplifier worsens noise figure, which is why receiver LNAs are placed as close to the antenna as practical.

Interferometer Direction Finding

Formula reading: d is baseline spacing between antenna elements. A wave arriving at angle θ reaches one element before the other, producing phase difference Δφ. The inverse-sine equation estimates angle from measured phase. Longer baselines improve angular sensitivity but create phase ambiguity when spacing exceeds about λ/2 unless multiple baselines or ambiguity resolution are used.

Amplitude Monopulse Concept

Formula reading: Σ is the sum channel and represents total received energy. Δ is the difference channel and changes sign depending on which side of boresight the target is located. Dividing Δ by Σ normalizes out target strength so the receiver estimates angle error rather than just amplitude. Near boresight, the normalized difference-over-sum signal is approximately proportional to angular error. This is why monopulse can estimate angle from a single pulse, unlike sequential lobing.

Formula reading: A is effective aperture area, η is efficiency and λ is wavelength. The 4π factor connects aperture capture area to isotropic radiation over a sphere. Gain increases with aperture area and with decreasing wavelength. The efficiency term η keeps the formula honest: no real antenna illuminates its aperture perfectly. Feed loss, taper, spillover, blockage and surface error all reduce the useful aperture.

Formula reading: Θ_rad is beamwidth in radians, D is aperture width in the scanned plane, and k is a shape factor set by illumination taper and aperture type. The second formula converts angular beamwidth into cross-range resolution S_a at range R. A narrow beam helps separate two targets at the same range but different bearing.

Formula reading: R_FF is the minimum far-field distance, D is maximum antenna dimension, and λ is wavelength. The D² term is why large radar apertures need long test ranges. Antenna gain and beam pattern are far-field concepts. Too close to the aperture, phase and amplitude vary across space in a way that breaks simple plane-wave assumptions.