Evaluating Harmonic Resonance & The Octave Paradox
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Center-Fed Impedance Mechanics (The Octave Paradox): A standard dipole is physically fed exactly at $0.5L$. At the fundamental frequency ($L = \lambda/2$), the center is a current maximum (belly), producing a low, matching impedance of $\approx 73\Omega$. However, when operated at an even harmonic/octave (e.g., $L = 1.0\lambda$), the center naturally becomes a current node (zero current, massive voltage). Ohms law dictates $Z = V/I$; dividing by near-zero current drives the impedance toward infinity, creating the severe SWR mismatch seen in the spectrum analyzer.
$$ L_{feet} = \frac{492}{f_{MHz}} \times VF \times K_{apex} \quad \text{where} \quad K_{apex} \approx 1 - 0.04 \left| \frac{\delta}{90^\circ} \right| $$ $$ Z_{dipole}(f) \approx \left[50 + 23\cos(2\delta)\right] \times \left[ 1 + Q \cdot \sin^2\left(\pi \left( L_{elec} - 0.5 \right)\right) \right] $$Fan Dipole Bypassing: If the Fan Dipole mode is engaged, an additional parallel wire cut exactly to the operating frequency is connected to the balun. RF energy follows the path of least resistance, ignoring the infinite impedance of the harmonic wire and restoring $L_{elec} = 0.5$.
OCFD Asymmetry Mechanics: The Carolina Windom solves the even-harmonic node problem purely through geometry. By shifting the feedpoint to $37.5\%$, the feedpoint mathematically sidesteps the zero-current nodes of the 2nd, 4th, and 8th harmonics. The impedance stabilizes at a higher, but highly consistent, baseline across these octaves. The mandatory 4:1 voltage balun then executes a constant $Z_{in} / 4$ step-down transformation to match the $50\Omega$ radio.
$$ L_{feet} = \frac{492}{f_{MHz}} \times VF \times K_{apex} \quad \text{where} \quad K_{apex} \approx 1 - 0.04 \left| \frac{\delta}{90^\circ} \right| $$ $$ Z_{windom\_base} = 150 + 100\cos(2\delta) $$ $$ Z_{antenna}(f) \approx \frac{Z_{windom\_base}}{\sin^2\left(0.375 \cdot \pi \cdot \frac{f}{f_{design}}\right) + 0.05} \quad \rightarrow \quad Z_{radio} = \frac{Z_{antenna}}{4} $$Far-Field Superposition: The free-space radiation vectors for both architectures are calculated using the scalar superposition of their independent physical legs $(\gamma_1, \gamma_2)$, allowing the 3D mesh to dynamically warp as the apex angle $(\delta)$ is adjusted:
$$ \cos\gamma_{1,2} = \pm\sin\theta\cos\phi\cos\delta + \cos\theta\sin\delta $$ $$ E(\theta, \phi) \approx \left| \frac{\cos\left(\pi L_{elec} \cos\gamma_1\right) - \cos\left(\pi L_{elec}\right)}{\sin\gamma_1} + \frac{\cos\left(\pi L_{elec} \cos\gamma_2\right) - \cos\left(\pi L_{elec}\right)}{\sin\gamma_2} \right| $$