Introduction to Antennas | MMUST Electrical Engineering Study Guide

Course Orientation

1. Course Overview

An antenna is the interface between a guided electromagnetic wave on a transmission line and a wave propagating through free space. Every wireless system—from FM broadcasting and mobile communications to radar, satellite links, Wi-Fi, and 5G—depends on antennas to transmit and receive information efficiently.

Why antennas matter

Convert electrical signals into radiated electromagnetic fields.
Collect weak incident fields and convert them into received voltages or currents.
Shape radiation in preferred directions, improving coverage and link reliability.
Enable frequency reuse through spatial selectivity and polarisation discrimination.

Core engineering questions

How much power is radiated rather than lost?
In which directions is radiation strongest?
What frequency range can the antenna support?
How well is it matched to the transmitter or receiver?
What antenna provides the required gain, size, beamwidth, and cost?

Prerequisite knowledge: Vector calculus, complex phasors, Maxwell’s equations, transmission-line theory, impedance, decibels, and Fourier analysis.

Learning Outcomes

2. Learning Objectives

After completing this guide, the learner should be able to:

Explain the physical mechanism by which time-varying currents radiate electromagnetic waves.
Identify the principal regions around an antenna: reactive near field, radiating near field, and far field.
Define and calculate radiation resistance, efficiency, directivity, gain, beamwidth, sidelobe level, front-to-back ratio, bandwidth, and polarisation.
Differentiate between isotropic, omnidirectional, directional, wire, aperture, reflector, lens, loop, and microstrip antennas.
Explain mutual coupling, pattern multiplication, array factor, scanning, beam steering, and grating lobes.
Analyse a wireless link using Friis transmission and radar range equations.
Evaluate impedance mismatch using reflection coefficient, return loss, VSWR, and bandwidth criteria.
Select suitable antenna technologies for broadcasting, mobile communication, satellite, Wi-Fi, IoT, and radar systems.
Outline practical antenna measurement and laboratory safety procedures.
Interpret antenna specification sheets and perform elementary antenna design trade-offs.

Electromagnetic Foundations

3. Fundamental Theory of Radiation

3.1 Maxwell’s equations in brief

Antennas obey Maxwell’s equations. In free space with no charge or current, Faraday’s and Ampère’s laws couple time-varying electric and magnetic fields:

∇ × E = −∂B/∂t
∇ × H = ∂D/∂t
Time-varying electric and magnetic fields sustain one another and propagate as an electromagnetic wave.

At a metallic antenna boundary, tangential electric field must vanish. Surface currents and charges redistribute so that these boundary conditions and Maxwell’s equations are satisfied. Accelerating charges—not steady currents—produce radiation.

3.2 How an antenna radiates

A transmitter produces a time-varying voltage along a conductor. The resulting current creates a magnetic field, while separated charges create an electric field. When the conductor dimensions become appreciable relative to wavelength, opposite charges/currents no longer cancel perfectly in the far field, and energy escapes as radiation.

Transmission line mode

Fields are guided and remain close to the conductors. Input power flows toward the load rather than freely into space.

Transition region

The line opens out or changes geometry. Fields spread, and part of the guided wave transforms into a radiating wave.

Radiation mode

The fields detach from the structure and propagate outward, carrying real power through space.

3.3 Basic electromagnetic wave properties

Quantity	Relationship	Meaning
Wavelength	λ = c / f	Spatial period of the wave; c ≈ 3×10⁸ m/s in free space.
Wave impedance	\|E\| / \|H\| = η₀ ≈ 377 Ω	Ratio of electric to magnetic field magnitudes in free-space far field.
Intrinsic velocity	c = 1/√(μ₀ε₀)	Propagation speed in vacuum.
Refractive index	n = √(εᵣμᵣ)	Wave speed and wavelength decrease in material media.

3.4 Antenna regions

Reactive near field: r < 0.62√(D³/λ)
Radiating near field: 0.62√(D³/λ) ≤ r < 2D²/λ
Far field: r ≥ 2D²/λ
D is the largest antenna dimension. The far-field approximation is also subject to a practical lower distance limit.

Reactive near field: stored electric and magnetic energy dominates; coupling is strongly affected by nearby objects.
Radiating near field: radiation is significant but angular field distribution still changes with distance.
Far field: radial field component becomes negligible, E and H are orthogonal and transverse, and field strength falls approximately as 1/r.

Conceptual illustration of a centre-fed dipole and outward-propagating electromagnetic waves.

Physical insight: An electrically small antenna is not a good radiator because the currents over its small structure nearly cancel. Radiation performance generally improves when the antenna is comparable to a substantial fraction of a wavelength, subject to pattern and matching requirements.

Antenna Characterisation

4. Essential Antenna Parameters

4.1 Radiation pattern

A radiation pattern is a graphical or mathematical description of radiated field strength or power density as a function of direction. Patterns are normally plotted in two principal planes and normalised to their maximum.

Main lobe: direction of maximum radiation.
Sidelobes: smaller lobes outside the main beam; they waste power and can receive interference.
Back lobe: radiation in the direction opposite the main beam.
Nulls: directions where the radiation ideally reaches zero.
Beamwidth: angular width between specified power levels, commonly the half-power or −3 dB points.

4.2 Directivity

Directivity measures how concentrated the radiated power is in a particular direction compared with an isotropic radiator radiating the same total power.

D(θ,φ) = 4π U(θ,φ) / P_rad
D₀ = 4π U_max / P_rad
U is radiation intensity in W/sr. Isotropic directivity = 0 dBi.

An ideal isotropic antenna is a mathematical reference and cannot be physically realised. An omnidirectional antenna radiates uniformly in one plane, such as the azimuth plane, but not in all planes.

4.3 Gain and radiation efficiency

Gain accounts for both directional concentration and ohmic/conduction/dielectric losses. It is referenced to the accepted input power.

G(θ,φ) = e_r D(θ,φ)
e_r = R_r / (R_r + R_loss)
eᵣ is radiation efficiency, Rᵣ radiation resistance, and Rloss equivalent loss resistance.

Gain is often specified in dBi (relative to an isotropic source) or dBd (relative to a lossless half-wave dipole). An ideal half-wave dipole has about 2.15 dBi gain, so:

G(dBi) = G(dBd) + 2.15 dB

4.4 Beamwidth, sidelobes, and front-to-back ratio

Metric	Definition	Engineering significance
Half-power beamwidth (HPBW)	Angle between the two directions where normalised power falls to one-half, or −3 dB.	High gain usually implies narrow beamwidth.
First-null beamwidth (FNBW)	Angle between the first nulls on either side of the main beam.	Useful in array and aperture analysis.
Sidelobe level (SLL)	Level of the largest sidelobe relative to the main lobe, normally in dB.	Lower SLL reduces interference and noise pickup.
Front-to-back ratio	Main-lobe power divided by back-lobe power, in dB.	Important for sector base-station and point-to-point antennas.

For a uniform rectangular aperture, a useful approximation is:

HPBW ≈ k λ / D_ap
Typical values of k are about 51° for uniform illumination and 70° for tapered illumination; Dap is aperture length.

4.5 Polarisation

Polarisation describes the time-varying orientation of the electric field vector in the plane transverse to propagation.

Linear: E-field remains in one plane: vertical, horizontal, or slanted.
Circular: magnitude is constant while orientation rotates; right-hand (RHCP) or left-hand (LHCP).
Elliptical: most general case; circular and linear are special limits.

Polarisation mismatch factor: p = |ê_rx · ê_tx*|²
Perfectly aligned polarisations: p = 1. Orthogonal linear polarisations: p = 0.

Design example: A vertical monopole and a horizontal dipole are cross-polarised for broadside incidence, giving theoretically zero coupled power. Circularly polarised links tolerate rotation but require matching handedness.

4.6 Input impedance and bandwidth

At the feed terminals, the antenna presents an input impedance Z_A = R_A + jX_A. The resistive term includes radiation resistance and equivalent loss resistance, while reactance represents stored near-field energy.

Bandwidth may be specified through acceptable return loss, VSWR, gain variation, axial ratio, or pattern performance. Common criteria are S₁₁ ≤ −10 dB, corresponding approximately to VSWR ≤ 2, although stricter requirements are often used.

4.7 Effective aperture and aperture efficiency

A receiving antenna captures incident power over an equivalent area.

A_e = G λ² / (4π)
A_e = η_apA_physical
Receiving capability is directly linked to gain by reciprocity.

4.8 Reciprocity

For linear passive media, transmitting and receiving antenna characteristics are reciprocal. The impedance, radiation pattern, directivity, and effective aperture remain the same whether the antenna transmits or receives. Reciprocity does not imply that a communication link is symmetric if transmit powers, noise figures, cable losses, or antenna orientations differ.

Antenna Types

5. Major Antenna Families

5.1 Isotropic and elementary dipoles

The Hertzian or infinitesimal dipole is an idealised current element much shorter than wavelength. It is valuable for theory and array analysis. A short dipole has a doughnut-shaped pattern with nulls along its axis and maximum broadside radiation.

Short dipole directivity ≈ 1.5 (1.76 dBi)
Radiation resistance of short dipole: R_r ≈ 80π²(l/λ)²
l is total dipole length, valid for l ≪ λ.

5.2 Half-wave dipole

A centre-fed half-wave dipole has an approximate free-space length of λ/2 and a feed resistance near 73 Ω at resonance. Its doughnut pattern has a directivity of approximately 1.64 or 2.15 dBi. The antenna is resonant, relatively simple, and widely used as a reference and as an element in arrays.

5.3 Monopole

A monopole is one half of a dipole mounted over a conducting ground plane. An ideal quarter-wave monopole radiates above the plane and has half the radiation resistance of a corresponding dipole, approximately 36.5 Ω, with about 2.15 dBi directivity in its upper hemisphere. Practical ground systems strongly affect performance.

5.4 Loop antennas

Loops range from electrically small multi-turn receiving loops to resonant half-wave and full-wave loops.

Small loop: low radiation resistance, strong magnetic-field coupling, used in RFID, AM reception, and direction finding.
Large loop: can provide useful gain and shaped patterns; a half-wave loop offers approximately 1.1–1.2 dBi gain depending on geometry and feed arrangement.

5.5 Aperture antennas

Aperture antennas radiate through an opening in a conducting surface or through a guided-wave transition. Horn antennas are common examples. Aperture dimensions determine gain and beamwidth, while illumination taper affects sidelobes, efficiency, and beamwidth.

5.6 Reflector antennas

Parabolic reflectors convert spherical waves from a feed at the focal point into narrow, approximately planar beams. They provide high gain and are used in satellite earth stations, radio telescopes, radar, and point-to-point microwave links.

G ≈ η_ap(πD/λ)²
D is reflector diameter, not directivity. Aperture efficiency ηap commonly ranges from roughly 50% to 75%.

5.7 Microstrip or patch antennas

A patch antenna consists of a conducting patch printed over a ground plane and dielectric substrate, usually fed by a microstrip line, coaxial probe, aperture, or proximity coupling.

Advantages

Low profile and light weight
Conformal and easy mass manufacture
Suitable for arrays and integration
Moderate fabrication cost

Limitations

Narrow bandwidth unless modified
Lower gain than large apertures
Surface-wave and feed-network losses
Polarisation and cross-polarisation control can be challenging

5.8 Slot, leaky-wave, and travelling-wave antennas

Slots cut into waveguides or ground planes act as magnetic-current radiators. Arrays of slots produce shaped beams with controlled amplitude and phase. Leaky-wave antennas gradually release guided power and can provide frequency-dependent beam scanning.

5.9 Antenna comparison

Antenna	Typical gain	Main features	Typical applications
Short dipole	~1.8 dBi theoretical	Simple, electrically small, low radiation resistance	Low-frequency reception, test references
Half-wave dipole	~2.15 dBi	Resonant, balanced, easy reference	Broadcasting, base antennas, array element
Quarter-wave monopole	~2.15 dBi ideal above ground	Unbalanced feed; depends on ground plane	AM/FM, handhelds, vehicle communications
Loop	Negative to several dBi	Compact, magnetic coupling, directional options	RFID, AM radio, direction finding
Horn	~10–25 dBi	Broadband, well-controlled pattern, moderate gain	Test ranges, feeds, microwave systems
Parabolic reflector	~25–50+ dBi	Very high gain, narrow beam, large structure	Satellite, radar, radio astronomy
Microstrip patch	~5–9 dBi	Low profile, printable, moderate bandwidth	Wi-Fi, GNSS, mobile devices, arrays
Yagi–Uda	~7–15 dBi	Passive directors increase gain and directivity	TV, point-to-point links, radio telemetry
Helix	~10–15 dBi	Circular polarisation, axial-mode operation	Satellite communications, space links

Gain values are representative rather than universal; real values depend on frequency, size, materials, feed loss, and design.

Array Theory

6. Antenna Arrays and Beamforming

6.1 Why arrays?

An antenna array combines the fields of multiple radiating elements. Proper amplitude and phase control allows engineers to increase gain, electronically steer the beam, shape nulls, suppress interference, and form multiple beams.

AF(ψ) = Σ_n=0^N−1 a_ne^{j nψ}
ψ = k d cosθ + β
N = number of elements, an = excitation coefficient, d = spacing, β = progressive phase shift, k = 2π/λ.

Under the pattern multiplication principle, the total array pattern is approximately the product of the single-element pattern and the array factor, provided mutual coupling is neglected.

6.2 Array classifications

Linear array: elements arranged along one axis.
Planar array: elements arranged over a surface; supports two-dimensional beam control.
Conformal array: elements follow a curved platform surface, such as an aircraft fuselage.
Phased array: phase shifters steer the beam without mechanical movement.
Adaptive array: signal-processing weights dynamically place nulls or optimise a desired response.

6.3 Beam steering

For a uniform linear array, the main-beam direction θ₀ satisfies:

β = −k d cosθ₀
β is the phase increment between adjacent elements. The sign convention depends on array orientation and phase definition.

Electronic scanning is fast and reliable but raises complexity, insertion loss, calibration demands, and cost. Analogue phase shifting, digital beamforming, and hybrid architectures are common.

6.4 Element spacing and grating lobes

If spacing is too large, additional maxima called grating lobes appear. For a broadside uniform linear array with progressive phase control, the common no-grating-lobe condition is:

d ≤ λ / (1 + |cosθ₀|)
Broadside case (θ₀ = 90°) therefore gives d ≤ λ/2.

6.5 Mutual coupling

Nearby elements exchange energy through their near fields and through surface or substrate waves. Mutual coupling changes input impedances, active element patterns, scanned impedance, gain, and sidelobes. Accurate array design therefore requires full-wave simulation or measured coupling data, especially for wideband and closely spaced arrays.

6.6 Array design trade-offs

Choice	Benefit	Possible penalty
More elements	Higher gain and finer spatial control	More feed loss, complexity, power consumption, cost
Wider spacing	Lower element count for an aperture	Grating lobes and stronger mutual coupling variations
Amplitude taper	Lower sidelobes	Lower gain and wider main beam
Wide scan range	Flexible coverage	Element design difficulty and scan loss
Dense digital beamforming	Flexible simultaneous beams and nulls	High data rate, hardware cost, calibration burden

Modern relevance: Massive MIMO in 5G/6G, satellite phased arrays, automotive radar, and electronically scanned radar all exploit array principles. The physics remains rooted in phase-coherent superposition of radiated fields.

Wireless Link Analysis

7. Propagation and Link-Budget Analysis

7.1 Free-space path loss

Free-space path loss describes spreading loss between ideal isotropic antennas. It is not an absorption loss; power density simply falls with area as the wavefront expands.

P_r = P_tG_tG_r(λ / 4πR)²
Pt and Pr are transmit and receive powers; Gt and Gr are antenna gains in linear form; R is separation.

FSPL(dB) = 20 log₁₀(d_km) + 20 log₁₀(f_MHz) + 32.44
This compact form uses distance in kilometres and frequency in megahertz.

7.2 Complete link budget

P_r(dBm) = P_t + G_t − L_txfeed − L_path − L_misc + G_r − L_rxfeed
Link margin = P_r − sensitivity
All quantities except linear power ratios must be in dB or dBm consistently.

Additional losses may include atmospheric absorption, rain attenuation, polarisation mismatch, pointing error, body loss, foliage, clutter, shadowing, and interference degradation.

7.3 Noise and received signal-to-noise ratio

N = kTB
k = 1.38×10⁻²³ J/K, T is system noise temperature in kelvin, B is bandwidth in hertz.

SNR(dB) = P_r(dBm) − [10 log₁₀(kTB) + 30]
The +30 converts watts to dBm.

High antenna gain can improve link performance by increasing received signal power, but receiver noise figure, cable placement, front-end losses, and interference also matter. A low-loss feed path is especially important at millimetre-wave frequencies.

7.4 Radar range equation

For a monostatic radar viewing a target with radar cross-section σ:

P_r = P_tG² λ² σ / [(4π)³ R⁴]
G is antenna gain; for separate transmit and receive antennas, use GtGr.

The R⁴ dependence makes radar links extremely sensitive to range. Reflector area, aperture efficiency, transmitter power, integration time, and receiver noise all strongly influence detection.

7.5 Propagation mechanisms

Line-of-sight

Direct path between transmitter and receiver. Dominant in satellite links, microwave backhaul, and many high-frequency urban links.

Reflection

Ground, walls, and buildings generate multipath replicas that can reinforce or cancel the direct signal.

Diffraction

Waves bend around obstacles and around the Earth’s curvature, enabling limited non-line-of-sight coverage.

Scattering

Rough surfaces and small objects redirect energy in many directions, producing multipath fading.

Refraction

Atmospheric gradients change effective propagation paths and may cause fading or ducting.

Absorption

Atmospheric gases, rain, foliage, and materials convert some electromagnetic energy into heat.

Feed Design

8. Impedance Matching and Feeding Networks

8.1 Reflection and mismatch loss

If the antenna impedance Z_L differs from the source or transmission-line impedance Z₀, part of the incident power reflects.

Γ = (Z_L − Z₀) / (Z_L + Z₀)
RL = −20 log₁₀|Γ|
VSWR = (1 + |Γ|) / (1 − |Γ|)
Mismatch loss = −10 log₁₀(1 − |Γ|²)
ZL and Z0 should be complex-consistent in the Γ formula for complex loads.

\|Γ\|	Return loss	VSWR	Reflected power	Transmitted power into load
0	∞ dB	1.00	0%	100%
0.10	20 dB	1.22	1%	99%
0.316	10 dB	1.93	10%	90%
0.50	6.02 dB	3.00	25%	75%

8.2 Matching techniques

Quarter-wave transformer: transforms a real load using a line whose impedance is Z_T = √(Z₀Z_L) at centre frequency.
Stub tuning: adds open- or short-circuited transmission-line sections to cancel reactance.
L-networks: use lumped inductors and capacitors for narrowband matching.
Baluns: provide balanced-to-unbalanced transformation and may also perform impedance conversion.
Bandwidth enhancement: thicker conductors, parasitic elements, slots, stacked patches, coupled resonators, and multi-resonant structures.

8.3 Balanced and unbalanced systems

A dipole is typically balanced, whereas coaxial cable is unbalanced. Connecting them directly can drive current onto the cable outer conductor, distorting the pattern and increasing loss. A balun suppresses this common-mode current and improves measurement repeatability.

8.4 Smith-chart interpretation

The Smith chart represents normalised impedance or admittance on the complex reflection plane. Designers use it to choose matching-network element values and to visualise how impedance changes with frequency. Moving around a lossless transmission line traces a circle of constant |Γ|, while adding series reactance or shunt susceptance follows constant-resistance or constant-conductance contours.

Bandwidth caution: A narrow S11 dip at one frequency does not by itself prove useful antenna bandwidth. Gain, efficiency, polarisation, pattern stability, and front-end performance should all remain acceptable across the operating band.

System Context

9. Antennas in Modern Communication Systems

9.1 Broadcasting

AM broadcast stations often use tall monopoles and ground systems. FM and TV systems use horizontally, vertically, circularly, or elliptically polarised arrays to shape regional coverage and control interference. Pattern nulls can protect co-channel stations.

9.2 Cellular and Wi-Fi

Base-station antennas typically use vertically stacked collinear elements or panel arrays to produce sector beams. User devices integrate multiple compact antennas to support MIMO diversity and spatial multiplexing. Wi-Fi access points may use dipoles, patches, or multi-element arrays with vertical, dual-linear, or slant polarisations.

9.3 Satellite communication and navigation

Earth stations use parabolic reflectors or phased arrays for high gain and narrow beams. Satellite terminals may use patches, helix antennas, or electronically steered arrays. GNSS receivers require circularly polarised antennas and wide-angle coverage, while suppressing multipath where possible.

9.4 Radar and sensing

Radar antennas must provide target sensitivity, angular resolution, scanning speed, and low sidelobes. Reflector, slotted-waveguide, phased-array, and multiple-input multiple-output radar systems each offer distinct cost and performance trade-offs. Automotive radar often uses planar antenna arrays integrated into vehicle panels.

9.5 IoT, RFID, and wearable systems

Device miniaturisation, cost, battery life, and uncertain platform geometry often dominate design. Common choices include monopoles, inverted-F antennas, loops, chip antennas, and small printed antennas. Ground-plane size, human-body proximity, and housing materials can strongly detune performance.

9.6 Millimetre-wave and 5G/6G systems

Millimetre-wave arrays pack many compact elements into a small aperture, enabling beamforming to overcome higher path loss and blockage. Massive MIMO uses many antenna elements to serve users spatially, increase spectral efficiency, and improve reliability. Reconfigurable intelligent surfaces and integrated sensing and communication are active research directions.

Selection framework: Define frequency band, bandwidth, gain, pattern, polarisation, power handling, size, environment, matching, cost, and manufacturing constraints before selecting an antenna topology. A theoretically elegant antenna may fail in a real product because of packaging, feed loss, or platform coupling.

Laboratory Practice

10. Antenna Measurements and Laboratory Practice

10.1 Common measurements

Measurement	Typical method	Purpose
S-parameters	Vector network analyser	Impedance matching, bandwidth, feed loss, mutual coupling
Radiation pattern	Anechoic chamber or outdoor range	Beamwidth, sidelobes, nulls, cross-polarisation, front-to-back ratio
Gain	Gain comparison, absolute or substitution method	Quantify radiated directional performance
Efficiency	Reverberation chamber, Wheeler cap, or pattern integration	Separate radiation from loss
Axial ratio	Rotating linear source or polarisation measurement	Assess circular-polarisation purity
Power handling	Calibrated high-power source and thermal monitoring	Check voltage breakdown and heating limits

10.2 Suggested undergraduate laboratory sequence

Inspect and identify. Record antenna type, dimensions, connectors, labels, and nominal frequency. Measure physical length and estimate operating wavelength.

Calibrate the instrument. Perform a VNA calibration at the measurement plane to remove cable and connector errors. Use torque-appropriate connectors.

Measure S11. Record reflection coefficient, return loss, and VSWR across the band. Identify resonance and −10 dB bandwidth.

Measure pattern. Rotate the antenna under test while keeping the source fixed. Capture co- and cross-polarised cuts through the main beam.

Compare with theory. Plot measured and simulated patterns. Account for cable loss, range geometry, alignment, multipath, and finite chamber performance.

Document uncertainty. State instrument settings, calibration quality, alignment tolerance, frequency, distance, environment, and repeatability.

10.3 Safety and good practice

Confirm that RF power is off before changing connectors or moving an antenna under test.
Respect RF exposure limits and never look directly into an energized high-gain antenna beam.
Use suitable attenuators and power ratings to protect receivers and VNA front ends.
Ground equipment appropriately and follow institutional electrical-safety procedures.
Do not touch powered antenna elements; voltage and current can be high even at moderate power.
Control reflective clutter and maintain the required far-field distance during pattern measurements.
Use an RF-shielded environment when measuring sensitive low-level signals.

10.4 Typical report structure

Title, student details, date, and equipment used.
Objectives and theoretical background.
Antenna geometry, specifications, and measurement setup.
Procedure, calibration method, and instrument settings.
Results, plots, calculations, and comparison with expected values.
Error analysis, uncertainty, limitations, and safety observations.
Conclusions and recommendations for design improvement.

Interactive Learning Tools

11. Interactive Antenna Calculators

Enter values and select “Calculate.” All results are immediate and are intended for preliminary engineering insight.

11.1 Wavelength, dipole, and monopole dimensions

Frequency (MHz) Velocity factor

Results will appear here.

11.2 Reflection coefficient, return loss, VSWR, and mismatch loss

Reference impedance Z₀ (Ω) Antenna resistance R (Ω) Antenna reactance X (Ω)

Results will appear here.

11.3 Aperture gain and approximate beamwidth

Frequency (GHz) Circular aperture diameter (m) Aperture efficiency η_ap (0–1)

Results will appear here.

11.4 Friis link budget

Transmit power (dBm) Transmit antenna gain (dBi) Receive antenna gain (dBi) Distance (km) Frequency (MHz) Additional losses (dB)

Results will appear here.

11.5 Uniform linear-array pattern

Number of elements N Element spacing d/λ Progressive phase β (degrees)

The plot uses the standard uniform linear-array factor |Σ exp(jnψ)|/N and assumes identical isotropic elements.

Revision Sheet

12. Key Formula Summary

Topic	Formula	Notes
Wavelength	λ = c/f	c ≈ 3×10⁸ m/s in free space.
Far-field distance	r_ff ≈ 2D²/λ	D is largest antenna dimension.
Radiation efficiency	e_r = R_r/(R_r+R_loss)	Ratio of radiated to accepted power, neglecting mismatch unless included.
Gain	G = e_rD	Gain includes radiation losses; realised gain may also include mismatch.
Effective aperture	A_e = Gλ²/(4π)	Links gain to receiving capture area.
Reflection coefficient	Γ = (Z_L−Z₀)/(Z_L+Z₀)	Complex quantities are required for complex loads.
VSWR	(1+\|Γ\|)/(1−\|Γ\|)	1 is perfect match; infinity is total reflection.
Friis equation	P_r = P_tG_tG_r(λ/4πR)²	Applies to free-space far-field conditions.
Aperture gain	G = η_ap(πD/λ)²	D is circular aperture diameter.
Noise power	N = kTB	Use consistent linear units before converting to dBm.
Array phase	ψ = kd cosθ + β	Coordinate conventions may change the sign of the cosine term.

Exam strategy: Draw the geometry, convert units before substitution, distinguish power from field quantities, and state assumptions. In dB calculations, add gains and subtract losses; never add dBm power levels directly unless combining powers explicitly.

Self-Assessment

13. Revision Questions

Attempt each question before opening the answer. Use the calculators and formula sheet to check numerical work.

1. Why does a steady direct current not radiate efficiently?

Radiation requires time-varying or accelerated charge. A steady DC produces static fields that store energy near the conductor but do not detach as a propagating wave.

2. An antenna has a maximum radiation intensity of 40 W/sr and radiates a total of 100 W. Find its maximum directivity.

D₀ = 4πUmax/Prad = 4π(8)/100 ≈ 1.005 sr-normalised units, or about 0.02 dBi.

Note: Umax must be supplied in W/sr for this calculation.

3. Explain the difference between directivity and gain.

Directivity compares radiation intensity in a direction with the intensity of an isotropic source radiating the same total power. Gain also includes antenna radiation losses, so G = eᵣD. Gain cannot exceed directivity for a passive lossy antenna.

4. A linear antenna has a return loss of 10 dB. Find |Γ| and VSWR.

|Γ| = 10^(−RL/20) = 10^(−0.5) ≈ 0.316.

VSWR = (1+0.316)/(1−0.316) ≈ 1.93.

5. Why is a half-wave dipole preferred to a much shorter dipole for transmission?

A short dipole has very low radiation resistance and high reactance, making matching and efficient power transfer difficult. A resonant half-wave dipole has a practical feed resistance near 73 Ω and much better radiation efficiency.

6. State two advantages and two disadvantages of microstrip patch antennas.

Advantages: low profile, light weight, conformal, printable, and array-friendly. Disadvantages: narrow bandwidth, moderate gain, surface-wave/feed loss, and sensitivity to dielectric and manufacturing tolerances.

7. A 10 GHz link has a path length of 10 km and no additional losses. The transmitting antenna has 30 dBm power and both antennas have 20 dBi gain. Estimate received power.

FSPL = 32.44 + 20log10(10) + 20log10(10000) = 32.44 + 20 + 80 = 132.44 dB.

Pr = 30 + 20 + 20 − 132.44 = −62.44 dBm.

8. What causes grating lobes in a phased array?

Element spacing that is too large relative to wavelength allows additional phase-coherent maxima to occur in visible angular space. The broadside no-grating-lobe limit for uniform spacing is commonly d ≤ λ/2, modified when scanning.

9. Define co-polarisation and cross-polarisation.

Co-polarisation is the intended receive polarisation aligned with the incident field. Cross-polarisation is the orthogonal component, generally unwanted because it indicates polarisation impurity, mismatch, or multipath.

10. Explain reciprocity and give one practical consequence.

For a passive antenna in a linear medium, transmit and receive impedance and radiation characteristics are the same. Consequently, a directional transmitting antenna is also directional when receiving, enabling satellite dishes to use one reflector for both functions.

11. A circular aperture has 0.5 m diameter at 6 GHz and 60% aperture efficiency. Estimate gain.

λ = c/f = 0.05 m.

G = 0.6[π(0.5)/0.05]² ≈ 2961, or 10log10(2961) ≈ 34.71 dBi.

12. Why is cable loss especially serious before a low-noise receiver?

Loss before the first amplifier attenuates both the wanted signal and incoming noise but contributes additional receiver noise. It therefore reduces G/T and can significantly degrade system sensitivity. The antenna should ideally feed a low-noise amplifier through a short, low-loss path.

Quick multiple-choice check

Q1. The −3 dB beamwidth is measured between points where power falls to:

1/4 of maximum 1/2 of maximum 1/10 of maximum

Q2. Which quantity includes antenna ohmic and dielectric losses?

Directivity Radiation pattern angle only Gain

Q3. For an ideal broadside uniform array, the usual element-spacing guideline to avoid grating lobes is:

d ≤ λ/2 d ≤ 2λ d ≤ λ/20

14. Recommended References

Balanis, C. A. Antenna Theory: Analysis and Design, latest available edition. Wiley.
Stutzman, W. L., and Thiele, G. A. Antenna Theory and Design, latest available edition. Wiley.
Kraus, J. D., Marhefka, R. J., and Khan, A. S. Antennas for All Applications. McGraw-Hill.
Pozar, D. M. Microwave Engineering. Wiley, especially chapters on transmission lines, matching, and network analysis.
Saunders, S. R., and Aragón-Zavala, A. Antennas and Propagation for Wireless Communication Systems. Wiley.
IEEE Antennas and Propagation Society publications and educational resources.
Manufacturer datasheets for antenna gain, pattern, VSWR, polarisation, and power-handling specifications.

Study advice: First master the definitions and units, then reproduce each derivation, and finally solve numerical and design problems. Antenna engineering is most effective when mathematics, electromagnetic intuition, measurement, and practical constraints are studied together.