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= Antenna Theory for Psionic Devices =

{{Audience_Sidebar
| difficulty = Intermediate
| reading_time = 7 minutes
| prerequisites = [[Near_Field_Electromagnetics]]; basic antenna concepts; some impedance matching.
| if_too_advanced_see = [[Near_Field_Electromagnetics]]
| if_you_want_the_math_see = This page
}}

{{Notation
| signature = Non-relativistic; SI.
| units = k = 2π/λ; a = bounding-sphere radius (m); Q = antenna quality factor; Rrad = radiation resistance (Ω).
}}

{{Device_Vital_Stats
| type = Antenna (general — psionic-device coupling theory)
| operating_freq = Application-dependent; ELF (Hz) through UHF (GHz)
| input_power = Application-dependent
| size = Wavelength-dependent (a ≪ λ regime emphasised)
| status = Theoretical framework page
| safety_class = Application-dependent; ICNIRP guidelines apply
| key_reference = Chu, L. J. (1948). ''J. Appl. Phys.'' 19: 1163 (Chu-Wheeler small-antenna limit).
}}

'''Antenna theory for psionic devices''' adapts classical antenna engineering — Chu-Harrington bounds, Wheeler radiation resistance, near-field coupling — to the specific demands of [[HelmKit]]-class wearable systems. These devices operate at the boundary of '''electrically small''' (D ≪ λ) and resonant, in the reactive near-field, with biological matter (the brain) only centimetres away.

This page collects the engineering-relevant equations and bounds.

== Electrically-small regime ==

An antenna is '''electrically small''' when its largest dimension D is much less than the wavelength λ. For 5 cm coils at 2.45 GHz (λ = 12.24 cm), the ratio D/λ ≈ 0.4 — at the boundary of electrically small.

In this regime, antennas exhibit:

* '''Reactive near-field dominance''' (see [[Reactive_Near_Field]]).
* '''Low radiation resistance''' — most input power is reactive, not radiated.
* '''High Q''' (narrow bandwidth) — fundamental Chu-Harrington bound.
* '''Strong sensitivity to detuning''' by nearby matter (the wearer).

== Chu-Harrington bound ==

The fundamental limit on the radiation Q of an electrically small antenna (Chu 1948; Harrington 1960):

Qmin = 1/(ka)3 + 1/(ka)

— with k = 2π/λ the wavenumber and a the radius of the smallest sphere enclosing the antenna. This is the minimum Q achievable; real antennas have Q ≥ Qmin.

The bandwidth is bandwidth ~ f / Q.

=== Worked example: 5 cm coil at 2.45 GHz ===

* a = 2.5 cm, k = 2π/0.1224 = 51.3 /m → ka = 51.3 × 0.025 = 1.28.
* Qmin = 1/1.283 + 1/1.28 = 0.477 + 0.781 ≈ 1.26.
* Bandwidth Δf = f/Q ≈ 2.45 GHz / 1.26 ≈ 1.94 GHz — broad.

For ka = 1.28, the antenna is '''just barely''' electrically small. Bandwidth is workable.

=== Worked example: 1 cm coil at 2.45 GHz ===

* a = 0.5 cm → ka = 0.26.
* Qmin = 1/0.263 + 1/0.26 ≈ 60.0 + 3.85 ≈ 63.8.
* Bandwidth ≈ 38 MHz — moderate.

=== Worked example: 5 cm coil at 300 MHz ===

* λ = 1 m, ka = 2π · 0.025 / 1 ≈ 0.157.
* Qmin ≈ 1/0.1573 + 1/0.157 ≈ 258 + 6.4 ≈ 264.
* Bandwidth ≈ 1.1 MHz — narrow.

At 300 MHz, a 5 cm coil is deeply sub-wavelength — high Q, large stored-energy ratio, narrow bandwidth. This is the preferred regime for high near-field efficiency.

== Wheeler radiation resistance ==

For an electrically small loop antenna of circumference <math>C</math>:

:<math>R_{\text{rad}} = 20\pi^2\,(C/\lambda)^4\quad[\Omega]</math>

The <math>(C/\lambda)^4</math> scaling is characteristic of magnetic dipole radiation. <math>R_{\text{rad}}</math> sets the antenna's radiative efficiency relative to its ohmic loss resistance <math>R_{\text{loss}}</math>:

:<math>\eta = \frac{R_{\text{rad}}}{R_{\text{rad}} + R_{\text{loss}}}</math>

For a 5 cm coil at 2.45 GHz (<math>C \approx 15</math> cm, <math>\lambda = 12.24</math> cm):

:<math>R_{\text{rad}} = 20\pi^2 \cdot 1.22^4 \approx 437\ \Omega</math>

— a substantial radiation resistance. For the same coil at 300 MHz (<math>C/\lambda = 0.15</math>):

:<math>R_{\text{rad}} = 20\pi^2 \cdot 0.15^4 \approx 0.1\ \Omega</math>

— small, and likely dominated by Rloss. At lower frequencies, the coil becomes a near-perfect near-field source (almost all stored energy, almost no radiation).

== Impedance matching ==

For an electrically small antenna to absorb input power, its driving-point impedance must be matched to the source impedance (typically 50 Ω). Practical techniques:

* '''Series tuning capacitor''' — cancels the inductive reactance.
* '''L-network''' (series-L + shunt-C or shunt-L + series-C).
* '''Transformer coupling''' — for impedance step-up.
* '''Distributed matching''' (stubs, microstrip) — for narrow-band high-precision matching.

Matched, an electrically small antenna can absorb essentially all the input power. Unmatched, most is reflected back to the source.

== Detuning by the wearer ==

A critical effect for wearable psionic devices: '''the wearer's body changes the antenna's impedance'''. The brain is a high-permittivity, conductive medium and acts as a strong dielectric load. Practical consequences:

* '''Resonance frequency shifts''' downward by 5-20% when the device is placed on a head vs. in free space.
* '''Q-factor drops''' due to ohmic loading by tissue.
* '''Matching network must compensate''' for body presence.

Solutions: adaptive matching networks, in-situ calibration, body-proximity sensing.

== Helical-antenna design ==

For higher-radiation applications (longer-range devices, not HelmKit), helical antennas provide circular polarisation and broad beam. See [[Double-Helix_Antenna]] for the helical mode with circular polarisation (Kraus 1947).

== Caduceus and bifilar geometries ==

For HelmKit's near-field-only requirement, '''[[Caduceus_Coil|caduceus coils]]''' and '''[[Bifilar_Coil|bifilar coils]]''' are preferred because they:

* '''Suppress far-field radiation''' by having opposite-current crossings or opposing windings (net dipole moment ≈ 0).
* '''Maintain strong near-field''' due to the local field of each winding being large.
* '''Provide non-trivial field structure''' (e.g. axial scalar/longitudinal components at crossings).

These geometries are non-standard in conventional antenna engineering but are described historically in Tesla and Bedini work (and modern reactive-resonator literature).

== Sanity checks ==

* '''ka → 0''' (deeply electrically small) → Qmin → ∞; antenna becomes pure near-field source. ✓
* '''ka → 1''' (resonant size) → Qmin ~ 2; antenna is a good radiator. ✓
* '''ψ → 0''' (in framework) → standard antenna theory; no ψ-coupling enhancement. ✓ ([[Sanity_Check_Limits]] §6.)

== See Also ==

* [[Near_Field_Electromagnetics]]
* [[Reactive_Near_Field]]
* [[Caduceus_Coil]]
* [[Bifilar_Coil]]
* [[Double-Helix_Antenna]]
* [[Psionic_Device_Overview]]

== References ==

* Chu, L. J. (1948). "Physical limitations of omni-directional antennas." ''Journal of Applied Physics'' 19: 1163.
* Harrington, R. F. (1960). "Effect of antenna size on gain, bandwidth, and efficiency." ''Journal of Research of the National Bureau of Standards'' 64D: 1–12.
* Wheeler, H. A. (1947). "Fundamental limitations of small antennas." ''Proceedings of the IRE'' 35: 1479–1484.
* Balanis, C. A. (2016). ''Antenna Theory: Analysis and Design.'' 4th ed., Wiley.
* Kraus, J. D. (1988). ''Antennas.'' 2nd ed., McGraw-Hill.

[[Category:Psionics]]
[[Category:Antenna Theory]]
[[Category:Hardware]]

Antenna Theory for Psionic Devices - Revision history

JonoThora: Phase N (01b): LaTeX restoration — promote Unicode display-math to ; lint-clean per tools/wiki_latex_lint.py

JonoThora: Phase N (01b): LaTeX restoration — promote Unicode display-math to $; lint-clean per tools/wiki_latex_lint.py$