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Antennas & Radiation
1. Radiation from time-varying current
Radiation ultimately comes from accelerating charge and time-varying current. In the far field, electric and magnetic fields become approximately transverse and locally related by the free-space impedance,
\[\frac{|E|}{|H|}\approx\eta_0\approx377\ \Omega.\]The radiated power density is described by the time-average Poynting vector,
\[\langle\mathbf S\rangle=\frac12\operatorname{Re}\{\mathbf E\times\mathbf H^*\}.\]2. Near field, Fresnel region and far field
For an antenna with largest dimension $D$, a commonly used far-field criterion is
\[R_{FF}\gtrsim\frac{2D^2}{\lambda}.\]The exact boundary is application dependent. In the reactive near field, stored electric/magnetic energy dominates; in the radiating near field (Fresnel region), the angular field distribution still changes with distance; in the far field, angular dependence is effectively independent of range and fields scale approximately as $1/R$.
3. Radiation pattern, directivity and gain
Radiation intensity is
\[U(\theta,\phi)=r^2S_r.\]Directivity is
\[D(\theta,\phi)=\frac{4\pi U(\theta,\phi)}{P_{rad}},\]and gain includes radiation efficiency,
\[G=\eta_{rad}D.\]Important pattern quantities include main-beam direction, half-power beamwidth, sidelobe level, front-to-back ratio and cross-polarization.
4. Effective aperture
A receiving antenna converts incident power density into available power through its effective aperture,
\[A_e=\frac{G\lambda^2}{4\pi}.\]This relation is one of the cleanest bridges between transmit and receive antenna concepts.
5. Polarization
Polarization describes the trajectory of the electric-field vector: linear, circular or elliptical. For two linearly polarized antennas with relative angle $\psi$, ideal polarization mismatch contributes
\[\eta_p=|\hat e_t\cdot\hat e_r|^2=\cos^2\psi.\]Circular polarization adds handedness; axial ratio is a common quality metric.
6. Common antenna families
| Antenna | Strength | Typical use |
|---|---|---|
| Dipole | simple reference radiator | communications, calibration |
| Monopole | compact over ground plane | mobile/vehicle systems |
| Loop | magnetic-field coupling | LF/HF, RFID, sensing |
| Patch | low profile, PCB integration | phones, GNSS, arrays |
| Horn | broadband, predictable pattern | microwave links, ranges, feeds |
| Reflector | very high gain | satellite, radio astronomy, radar |
| Helix | circular polarization | satellite/space links |
| Phased array | electronic steering | radar, 5G/6G, satellite terminals |
7. Arrays and beamforming
For a linear array,
\[AF(\theta)=\sum_{n=0}^{N-1}w_ne^{jn(kd\sin\theta+\beta_s)}.\]Changing progressive phase $\beta_s$ steers the beam. Element spacing, element pattern, mutual coupling and scan angle determine grating lobes and scan loss. See AESA & Phased Arrays.
8. Link equation
For two matched, polarization-aligned antennas in free space,
\[P_r=P_tG_tG_r\left(\frac{\lambda}{4\pi R}\right)^2.\]This is a far-field power-transfer equation, not a complete channel model.
Worked example — aperture of a 20 dBi antenna at 10 GHz
At 10 GHz, $\lambda\approx0.03$ m. A 20 dBi gain corresponds to $G=100$:
\[A_e=\frac{100(0.03)^2}{4\pi}\approx7.2\times10^{-3}\ \text{m}^2.\]That is about $72\ \text{cm}^2$ of effective receiving area.
9. How antennas are measured
- VNA: impedance, return loss and mutual coupling.
- Anechoic range: gain, pattern, polarization and sidelobes.
- Near-field scanner: measure near field and transform to far field.
- Reverberation chamber: efficiency/diversity/statistical OTA quantities.
- OTA chamber: complete device performance including enclosure and beamforming.
Antenna measurement is strongly affected by reference antenna calibration, chamber reflections, cable movement, positioner accuracy and polarization alignment.
10. Engineering reality
11. Design questions worth asking
- What bandwidth and matching criterion are actually required?
- Is realized gain more important than raw directivity?
- What is the polarization environment?
- Does the platform alter the installed pattern?
- Is the antenna operating in someone’s near field?
- How sensitive is performance to manufacturing tolerance?
- What quantity will be measured to validate the design?
References
- C. A. Balanis, Antenna Theory: Analysis and Design.
- W. L. Stutzman and G. A. Thiele, Antenna Theory and Design.
- R. F. Harrington, Time-Harmonic Electromagnetic Fields.
- MIT OpenCourseWare, Electromagnetics and Applications.
Related: RF & Microwave · Wireless & MIMO · AESA · Measurements
Radio-telescope feeds and receiver chains
Radio astronomy connects antenna aperture, feed illumination, spillover and polarization directly to receiver noise temperature and interferometric sensitivity. See Radio Astronomy — RF Circuits & Receiver Systems for feed horns, OMTs, cryogenic receivers, calibration and correlator backends.
Advanced study — embedded patterns, mutual coupling and array calibration
From isolated element patterns to active element patterns
An array is a coupled multiport network. Each element’s radiation depends on its neighbors, terminations and scan state. If the incident waves at the element ports form vector (\mathbf a), the accepted power, for a common real reference impedance and power-wave normalization, is
\[P_{\mathrm{acc}}=\mathbf a^\dagger(\mathbf I-\mathbf S^\dagger\mathbf S)\mathbf a.\]The embedded-element far fields can be assembled into an array response (\mathbf e(\theta,\phi)). Its realized beam pattern depends on feed-network losses, the active reflection coefficients and the excitation vector—not only the textbook array factor. For (\mathbf a_i\ne0), the active reflection coefficient at port (i) is
\[\Gamma_{\mathrm{act},i}=\frac{\sum_j S_{ij}a_j}{a_i}.\]Consequently an array well matched under single-port excitation can be badly matched when steered to a particular scan angle. Investigate scan blindness, grating lobes, surface waves and active impedance together.
Aperture fields and measurement inversion
The far-field pattern is approximately the spatial Fourier transform of tangential aperture fields under suitable planar-aperture assumptions. Planar near-field scans exploit this connection to reconstruct radiated fields. Probe correction, finite scan truncation, sample spacing and multiple reflections all bias the reconstructed result.
Advanced exercise: calculate the active reflection coefficients versus scan angle using a measured or simulated multiport (\mathbf S) matrix. Compare the isolated-element array-factor prediction with an embedded-element-pattern synthesis.
Further reading: IIT Bombay EE6110 Advanced Engineering Electromagnetics; R. F. Harrington, Time-Harmonic Electromagnetic Fields; R. C. Hansen, Phased Array Antennas.
Advanced topic — coupling-aware arrays and active impedance
In a finite phased array, radiating elements are coupled ports, not independent sources. With incident port-wave amplitudes (\mathbf a) and scattering matrix (\mathbf S), the active reflection coefficient of element (i) (for (a_i\ne0)) is
\[\Gamma_{\mathrm{act},i}=\frac{\sum_j S_{ij}a_j}{a_i}.\]It depends on the excitation phases and therefore on the steering angle. An array that is well matched under single-port measurement can exhibit poor scan-angle-dependent matching, even without any change in physical hardware. Embedded element patterns are needed to predict realized radiation patterns accurately; multiplying an isolated-element pattern by an ideal array factor omits coupling.
For equal real port-reference impedances and power-wave normalization, accepted array power is
\[P_{\mathrm{acc}}=\mathbf a^\dagger(\mathbf I-\mathbf S^\dagger\mathbf S)\mathbf a.\]Advanced experiment: extract the full complex multiport (\mathbf S)-matrix from an EM solver, apply progressive scan-phase excitations, and plot active match, accepted power and realized gain versus scan angle. Compare isolated and embedded-element predictions.
Further reading: R. C. Hansen, Phased Array Antennas; IIT Bombay Advanced Engineering Electromagnetics.