// RF Theory
Antenna Theory — Radiation Patterns
How antennas radiate energy into space — radiation patterns, gain, directivity, near field vs far field, EIRP, and HPBW. Includes interactive 3D radiation patterns and pattern slices for the half-wave dipole and microstrip patch antenna.
// Foundation
What Is an Antenna?
An antenna is a transducer that converts between guided electromagnetic energy (on a transmission line) and unguided electromagnetic waves radiating through free space. Run it forward and it transmits; run it backwards and it receives. The same physical structure does both — reciprocity guarantees that the radiation pattern, gain and impedance are identical in transmit and receive modes.
The fundamental mechanism: accelerating electric charges radiate electromagnetic energy. A current flowing in a wire accelerates the electrons along the wire. When that current oscillates at radio frequencies, the accelerating charges radiate electromagnetic waves that detach from the wire and propagate outward at the speed of light. The antenna's geometry determines how much energy radiates, in which directions, and at what impedance the antenna presents to the transmission line feeding it.
The antenna as an impedance transformer: A transmission line typically has Z₀ = 50 Ω. Free space has an intrinsic impedance η₀ = 377 Ω. The antenna matches these two impedances — transforming the guided wave's 50 Ω environment to the 377 Ω free-space wave impedance — while simultaneously shaping the radiation pattern.
// Fields Around an Antenna
Near Field, Far Field & How Radiation Happens
Near Field vs Far Field
The space around an antenna is divided into three regions based on distance r from the antenna, where λ is the wavelength and D is the maximum antenna aperture dimension:
Reactive near field: The stored electromagnetic energy here is much larger than the radiated energy. The E and H fields are 90° out of phase (reactive). The fields fall off very rapidly with distance (1/r³ and 1/r²). This is the region where inductive charging, RFID and near-field communication (NFC) operate — energy is exchanged rather than radiated.
Radiating near field (Fresnel): Radiation dominates but the pattern still varies with distance. The relative amplitudes and phases of different parts of the antenna wavefront haven't "sorted themselves out" yet.
Far field (Fraunhofer): The radiation pattern is fixed — it no longer changes shape with distance, only amplitude (which falls as 1/r²). The E and H fields are in phase and mutually perpendicular: E/H = η₀ = 377 Ω. All antenna measurements and specifications are made in the far field. The radiation pattern you see on a datasheet is the far-field pattern.
Radiating near field (Fresnel): Radiation dominates but the pattern still varies with distance. The relative amplitudes and phases of different parts of the antenna wavefront haven't "sorted themselves out" yet.
Far field (Fraunhofer): The radiation pattern is fixed — it no longer changes shape with distance, only amplitude (which falls as 1/r²). The E and H fields are in phase and mutually perpendicular: E/H = η₀ = 377 Ω. All antenna measurements and specifications are made in the far field. The radiation pattern you see on a datasheet is the far-field pattern.
How Radiation Happens — Physical Intuition
Consider a half-wave dipole fed at its centre. The current distribution along the wire is sinusoidal — maximum at the feed point, zero at the tips. Each infinitesimal element of this current radiates like a tiny Hertzian dipole (a point current source). The total radiation pattern is the vector sum of all these tiny contributions, accounting for their position along the wire, their current amplitude, and the phase of the wave they contribute at each observation angle.
Hertzian Dipole — The Building Block
Electric field (far field): E_θ = j(η₀ I Δl / 2λr) · sin(θ) · e^(−jkr)Pattern factor: sin(θ) — maximum at θ=90° (broadside), zero at θ=0° (end-fire)
Power density: S = |E|²/(2η₀) ∝ sin²(θ)/r² (W/m²)
Directivity: D = 1.5 (1.76 dBi)
// Understanding Patterns
Radiation Patterns — Reading the Map of Radiated Power
3D Radiation Pattern
A radiation pattern is a graphical representation of how an antenna distributes radiated power as a function of direction in space. The 3D radiation pattern is a surface where the distance from the antenna to the surface represents the power radiated in that direction. A perfect isotropic radiator (theoretical only) would be a perfect sphere.
E-Plane and H-Plane Pattern Slices
Because a 3D pattern is hard to read from a datasheet, engineers take two orthogonal 2D cross-sections through the 3D pattern:
E-plane: The plane that contains the electric field vector and the direction of maximum radiation. For a vertical dipole this is any vertical plane through the dipole axis. The E-plane pattern of a dipole is a figure-of-eight — two lobes, nulls at the tips.
H-plane: The plane perpendicular to the E-plane that contains the magnetic field vector. For a vertical dipole this is the horizontal plane. The H-plane pattern of a dipole is a perfect circle — omnidirectional in azimuth.
H-plane: The plane perpendicular to the E-plane that contains the magnetic field vector. For a vertical dipole this is the horizontal plane. The H-plane pattern of a dipole is a perfect circle — omnidirectional in azimuth.
Half-Power Beamwidth (HPBW)
Beamwidth Definitions
HPBW (Half-Power Beamwidth): angular width between the −3 dB points of the main lobeFNBW (First Null Beamwidth): angular width between the first nulls on either side of the main lobe
Half-wave dipole E-plane HPBW: 78°
Microstrip patch E-plane HPBW: ≈ 70–80° (depends on substrate)
Microstrip patch H-plane HPBW: ≈ 100–120°
Parabolic dish (D=1m, f=10 GHz): HPBW ≈ 1.75°
// Key Antenna Metrics
Gain, Directivity & EIRP
Directivity — The Geometric Concentration
Directivity D(θ,φ) describes how much more power is radiated in a given direction compared to an isotropic radiator radiating the same total power. It is a purely geometric property — it depends only on the shape of the radiation pattern, not on the antenna's losses.
Directivity Formula
D(θ,φ) = 4π · U(θ,φ) / P_radU(θ,φ) = radiation intensity in direction (θ,φ) in W/sr
P_rad = total radiated power in W
Maximum directivity: D_max = 4π / Ω_A where Ω_A = beam solid angle
Approximate formula: D_max ≈ 4π / (HPBW_E × HPBW_H) in steradians
Isotropic radiator: D = 1 (0 dBi) — reference point
Half-wave dipole: D = 1.64 (2.15 dBi)
Microstrip patch: D ≈ 5–8 (7–9 dBi) depending on substrate
Gain vs Directivity — The Efficiency Factor
Directivity assumes all input power is radiated. In reality, some power is dissipated as heat in the antenna's ohmic resistance. Gain accounts for this efficiency:
Antenna Gain
G = η_rad × Dη_rad = radiation efficiency = P_rad / P_in (0 to 1)
G(dBi) = D(dBi) + 10·log₁₀(η_rad)
Typical efficiencies:
Half-wave dipole (copper): η ≈ 98–99% → G ≈ 2.13 dBi (essentially lossless)
Microstrip patch (FR4): η ≈ 70–85% → G ≈ 5–7 dBi (dielectric and conductor losses)
Electrically small antenna (<<λ/10): η can be <10% — radiation resistance too small
Chip antenna (IoT device): η ≈ 30–60% — limited by small size and nearby PCB
EIRP and ERP
EIRP (Equivalent Isotropically Radiated Power) is the power that a hypothetical isotropic radiator would need to transmit to produce the same power density in the direction of maximum gain as the actual antenna.
EIRP and ERP
EIRP (dBm) = P_TX (dBm) + G_ant (dBi) − L_cable (dB)ERP (dBm) = P_TX (dBm) + G_ant (dBd) − L_cable (dB)
dBi = gain relative to isotropic radiator
dBd = gain relative to half-wave dipole (0 dBd = 2.15 dBi)
EIRP = ERP + 2.15 dB always
Example: 1 W (30 dBm) TX, 3 dB cable loss, 10 dBi antenna:
EIRP = 30 + 10 − 3 = 37 dBm = 5 W
Why EIRP matters: Regulatory bodies (FCC, Ofcom, ETSI) limit EIRP, not transmit power. A 100 mW transmitter with a 20 dBi dish has EIRP = 10 W — this must comply with the regulatory EIRP limit for the band. You can use less transmit power with a high-gain antenna to stay legal, or you can use more transmit power with a low-gain antenna.
Friis Transmission Equation
The received power P_R at distance d with transmit power P_T, transmit gain G_T and receive gain G_R:
P_R = P_T · G_T · G_R · (λ/4πd)²
The term (λ/4πd)² is the free-space path loss factor. In dB:
P_R(dBm) = P_T(dBm) + G_T(dBi) + G_R(dBi) − FSPL(dB)
FSPL(dB) = 20·log₁₀(4πd/λ) = 20·log₁₀(d) + 20·log₁₀(f) + 20·log₁₀(4π/c)
FSPL(dB) ≈ 20·log₁₀(d_km) + 20·log₁₀(f_MHz) + 32.44 dB
Example: 2.4 GHz WiFi, d=100 m, G_T=G_R=3 dBi:
FSPL = 20·log(0.1) + 20·log(2400) + 32.44 = −20 + 67.6 + 32.44 = 80 dB
P_R = 20 dBm + 3 + 3 − 80 = −54 dBm
P_R = P_T · G_T · G_R · (λ/4πd)²
The term (λ/4πd)² is the free-space path loss factor. In dB:
P_R(dBm) = P_T(dBm) + G_T(dBi) + G_R(dBi) − FSPL(dB)
FSPL(dB) = 20·log₁₀(4πd/λ) = 20·log₁₀(d) + 20·log₁₀(f) + 20·log₁₀(4π/c)
FSPL(dB) ≈ 20·log₁₀(d_km) + 20·log₁₀(f_MHz) + 32.44 dB
Example: 2.4 GHz WiFi, d=100 m, G_T=G_R=3 dBi:
FSPL = 20·log(0.1) + 20·log(2400) + 32.44 = −20 + 67.6 + 32.44 = 80 dB
P_R = 20 dBm + 3 + 3 − 80 = −54 dBm
// Antenna 1 of 2
Half-Wave Dipole — Pattern & Physics
The half-wave dipole (λ/2 dipole) is the fundamental reference antenna. It consists of two quarter-wave conductors fed at the centre. Its radiation pattern, gain of 2.15 dBi, and feed impedance of 73 Ω are the reference point from which all other antennas are compared.
Radiation Pattern Equation
Half-Wave Dipole Far-Field Pattern
F(θ) = [cos(π/2 · cosθ) / sinθ]²θ = elevation angle from the dipole axis (0° = end-fire, 90° = broadside)
Maximum radiation: θ = 90° (broadside, perpendicular to the wire)
Nulls: θ = 0° and θ = 180° (along the wire axis)
HPBW (E-plane): 78°
H-plane: Perfect circle (omnidirectional in azimuth)
Directivity: 1.64 = 2.15 dBi
Radiation resistance: R_rad = 73.1 Ω
Feed impedance: Z_in ≈ 73 + j42.5 Ω (slightly inductive; shorten by ~3% for resonance)
Half-Wave Dipole — Interactive 3D Radiation Pattern
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Pattern Slices
3D pattern surface
E-plane cut
H-plane cut
Dipole antenna
Microstrip Patch Antenna — Interactive 3D Radiation Pattern
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Pattern Slices
3D pattern surface
E-plane cut
H-plane cut
Patch element
// Antenna 2 of 2
Microstrip Patch Antenna — Pattern & Physics
The microstrip patch antenna is a conducting rectangular patch on one side of a grounded dielectric substrate. Current flows along the patch and the electric field fringing at the two radiating edges produces radiation. The patch radiates primarily from the two open radiating edges — which act as two in-phase slot radiators separated by the patch length (≈ λ/2 in the substrate).
Pattern Equation and Physics
Microstrip Patch Far-Field Pattern
E-plane (φ=0°): F_E(θ) = cos(θ) · sinc(k₀h sinθ/2)H-plane (φ=90°): F_H(θ) = sinc(k₀W sinθ/2) · cos(k₀L sinθ/2)
h = substrate thickness, W = patch width, L = patch length ≈ λ_eff/2
Directivity: D ≈ 5–8 (7–9 dBi) for typical substrates
Radiation efficiency: 70–95% (FR4 is lower, Rogers is higher)
E-plane HPBW: ≈ 70–80° (determined by substrate height h)
H-plane HPBW: ≈ 100–120° (determined by patch width W)
Backlobe: 15–25 dB below main lobe (ground plane blocks but doesn't eliminate)
Why patches radiate only in the upper hemisphere: The ground plane reflects backward radiation upward, so the patch is a unidirectional antenna (unlike the omnidirectional dipole). There is no radiation directly at the horizon (θ = 90°) because the ground plane causes a null there. The main lobe peaks at boresight (θ = 0°, looking straight up from the patch).
The cos(θ) factor: The E-plane pattern of a patch has a cos(θ) factor that creates nulls at ±90° (the horizon). This is the ground-plane effect — the element pattern of a radiating slot above a ground plane is always cos(θ). It is NOT present in a dipole pattern, which is why a dipole radiates at the horizon and a patch does not. This makes the patch useful for mobile communications (needs to radiate toward the sky and nearby base stations, not into the ground) but poor for broadcasting where horizon coverage is important.
// Side by Side
Dipole vs Patch — Full Comparison
| Property | Half-Wave Dipole | Microstrip Patch |
|---|---|---|
| Directivity | 2.15 dBi | 7–9 dBi |
| E-plane HPBW | 78° | 70–80° |
| H-plane HPBW | 360° (omni) | 100–120° |
| Pattern shape | Toroid (donut) | Hemisphere (broad beam) |
| Radiation hemisphere | Both (360° in elevation) | Upper only (ground plane) |
| Polarisation | Linear (along wire) | Linear (along patch length) |
| Feed impedance | 73 Ω (resistive at resonance) | 100–200 Ω (edge), 50 Ω (inset/coax) |
| Bandwidth | ~10–15% | 1–5% (narrow, limited by Q) |
| Size | λ/2 in length | ≈ λ/2 × λ/2 / √εr (compact at high f) |
| Profile | 3D structure (wire) | Flat, planar (PCB) |
| Typical applications | Broadcasting, base stations, reference | Mobile handsets, GPS, WiFi, RFID, 5G |
| Key advantage | Simple, omnidirectional, reference | Flat, integrable on PCB, directional |
| Key limitation | Low gain, 3D structure | Narrow bandwidth, backlobe |
// Wave Property
Polarisation — The Orientation of the E-Field
Polarisation describes the direction of the electric field vector of the radiated wave. It is one of the most important parameters in link design — a polarisation mismatch between TX and RX antennas causes direct signal loss.
Polarisation Loss Factor (PLF)
PLF = |ê_TX · ê_RX|² = cos²(ψ)ψ = angle between the polarisation directions of TX and RX antennas
Aligned (ψ=0°): PLF = 1 (0 dB loss) — maximum power transfer
Crossed (ψ=90°): PLF = 0 (−∞ dB) — complete polarisation isolation
45° offset: PLF = 0.5 (−3 dB loss)
Linear polarisation: E-field oscillates in one plane. A vertical dipole is vertically polarised. A horizontal patch is horizontally polarised. Polarisation loss between vertical TX and horizontal RX = −∞ dB (complete rejection).
Circular polarisation (CP): E-field rotates continuously — one complete revolution per wavelength of travel. Produced by two orthogonal linearly-polarised fields with 90° phase difference. CP antennas suffer only −3 dB loss when received by any linearly-polarised antenna at any rotation angle — immune to polarisation rotation. Used in GPS (RHCP) and satellite links.
Polarisation rotation in real environments: Reflections from buildings, the ionosphere (Faraday rotation) and multipath propagation all rotate the polarisation of a linearly-polarised wave. This is why satellite and GPS systems prefer circular polarisation — the signal arrives with unknown polarisation rotation but CP antennas handle it gracefully.
Circular polarisation (CP): E-field rotates continuously — one complete revolution per wavelength of travel. Produced by two orthogonal linearly-polarised fields with 90° phase difference. CP antennas suffer only −3 dB loss when received by any linearly-polarised antenna at any rotation angle — immune to polarisation rotation. Used in GPS (RHCP) and satellite links.
Polarisation rotation in real environments: Reflections from buildings, the ionosphere (Faraday rotation) and multipath propagation all rotate the polarisation of a linearly-polarised wave. This is why satellite and GPS systems prefer circular polarisation — the signal arrives with unknown polarisation rotation but CP antennas handle it gracefully.
Dual-polarisation antennas: Modern cellular base station antennas use two feeds at ±45° to the vertical (cross-polarised). This provides spatial diversity — the two polarisations arrive at the handset with different multipath channels, effectively doubling the data rate through polarisation-division multiplexing (part of MIMO).
// Practical Design
Feed Impedance — Connecting to 50 Ω
The antenna feed impedance Z_ant = R_rad + R_loss + jX_ant must be matched to the transmission line (typically 50 Ω) to maximise power transfer. Any mismatch causes reflections (VSWR > 1) and reduces radiated power.
Feed Impedance Components
Z_ant = R_rad + R_loss + jX_antR_rad = radiation resistance — the "useful" resistance representing radiated power
R_loss = loss resistance — conductor and dielectric losses (heat)
X_ant = reactance — capacitive or inductive, depends on whether antenna is shorter or longer than resonant length
Radiation efficiency: η = R_rad / (R_rad + R_loss)
Half-wave dipole at resonance: Z ≈ 73 + j0 Ω (resistive, X=0)
Short dipole (L << λ/2): R_rad is very small, X is large capacitive — hard to match
Patch antenna impedance matching: At the radiating edge, the patch impedance is typically 100–200 Ω. Common matching techniques:
① Inset feed: Move the feed point inward from the edge — impedance decreases from the edge value to near 50 Ω at the inset distance y₀ = (L/π)·arccos(√(Z₀/R_edge))
② Quarter-wave transformer: A λ/4 line between the patch edge and the 50 Ω feed
③ Aperture coupling: A slot in the ground plane couples a microstrip feed line on the back of the substrate to the patch on top — eliminates the physical connection and reduces spurious radiation
① Inset feed: Move the feed point inward from the edge — impedance decreases from the edge value to near 50 Ω at the inset distance y₀ = (L/π)·arccos(√(Z₀/R_edge))
② Quarter-wave transformer: A λ/4 line between the patch edge and the 50 Ω feed
③ Aperture coupling: A slot in the ground plane couples a microstrip feed line on the back of the substrate to the patch on top — eliminates the physical connection and reduces spurious radiation