3 · The feed and the answer

Chapter 2 ended owing two explanations: the feed line we typed without comment, and the question of how many segments the toy’s “working” answer actually cost. Start with the feed, because it is the antenna’s connection to the outside world.

What “1 volt in a gap” became

In the toy, the generator turned into exactly one line:

v[N // 2] = 1.0

That is the delta-gap source: pretend the generator’s 1 volt appears entirely across the feed segment, producing an applied field there and nothing anywhere else. It’s the crudest workable model of a feed — no coax, no balun, just a voltage jump — and it is what virtually every wire code uses by default.

momwire’s delta-gap is the same move. Inside compute_impedance, the right-hand side is built as v[fi] += -V_i / geom["seg_h"][fi] — voltage over feed-segment length, with the sign and constants absorbed into the solver’s conventions. Same idea, a better-tested version of it — and, we are about to see, the feed is where the toy’s last ohm hides.

Solve the system, read off the current where the generator sits, and the number every ham actually wants falls out:

Z_in = V / I(feed) = 1 / I[N // 2]

Drive-point impedance. SWR is a one-line formula away from it; “resonance” means its imaginary part crosses zero. When this chapter ends, we’ll sweep it across frequency and watch that happen.

How many segments did that cost?

Chapter 2 quoted the toy at N = 161: 70.9 − 8.7j, against momwire’s settled 69.6 − 18.3j. Close on the resistance, loose on the reactance. The honest question is what happens as we refine — and the answer is the most instructive picture in naive MoM:

It converges — no window, no collapse, just a steady approach to the right answer. But look how unevenly:

• The resistance settles fast. By a couple hundred segments R is within an ohm, and its error is clean first-order (∝ 1/N). Three coarse solves and a Richardson extrapolation land it dead on 69.6.
• The reactance crawls. At N = 481 — half a thousand unknowns — X is still only −14.4, several ohms shy. And it doesn’t ride a clean 1/N: the self-term log(dz/a) from chapter 2 injects a logarithm, so the reactance follows X ≈ X∞ + (b + c·log N)/N. Fit that curve (the dotted line) and you can extrapolate to X∞ ≈ −17.1 from coarse data — a real rescue, pulling a converged number out of solves that individually look nowhere near it.

So: yes, you can get final values out of the pulse method. But count the cost. The resistance needs a handful of solves and a one-line extrapolation. The reactance needs many solves, a fitted model of its trend, and even then lands about an ohm short of a good basis — because the last ohm isn’t a convergence error at all. Notice R extrapolates exactly onto momwire while X stops an ohm away: an offset in the reactance alone, untouched by refining, is the fingerprint of the feed model. Our bare delta-gap and momwire’s tested one simply disagree about the near-field a little, and that disagreement lives entirely in X.

And brute force is no escape hatch. To reach the converged reactance without extrapolating, you’d shrink the segments until dz approached the wire radius — but the thin-wire kernel has its own limit, roughly dz > 8a for 1% accuracy (the red line, thousands of segments out). Push past it and the kernel itself is wrong; you’d be converging a coarse approximation to the wrong number. The pulse method makes you choose between a curve fit and a fortune in unknowns, and hands you an approximate feed either way.

Why so slow — and the thread into Act II

Why does the reactance, of all things, crawl the worst? The clue is where we put the charge. X is near-field, reactive energy — and near-field energy is mostly the scalar potential, the field of those charge piles at the segment endpoints. A pulse current is discontinuous, so its charge lands in concentrated knots at the joints; the potential of a knot is a crude, slowly-improving thing, and it dominates the reactance. Resolving it is exactly what the extra segments are struggling to do.

That single observation is the hinge of the whole primer. Make the current continuous and its charge stops piling into knots and spreads into honest, distributed charge — the scalar potential converges, the reactance snaps into place, and you need a handful of segments instead of hundreds. Every basis momwire ships is continuous, and Act II is the story of what that buys: SinusoidalSolver and BSplineSolver nail this same dipole — resistance and reactance — at twenty-one unknowns, no extrapolation, in milliseconds. You cannot cheaply out-refine a staircase; you can only stop drawing one.

The payoff plot

One number is an appetizer. The answer a ham actually plots is the impedance across the band — and now that solves in one call. The swept solver (compute_impedance_swept) takes an array of wavenumbers:

import numpy as np
from momwire import SinusoidalSolver

wire = np.array([[0.0, -5.291, 0.0], [0.0, 5.291, 0.0]])
solver = SinusoidalSolver(wires=[wire], nsegs=81, wavelength=22.0,
                          wire_radius=0.0005,
                          feed_wire_index=0, feed_arclength=5.291)  # 1 V gap at the center
lams = np.linspace(18.0, 27.0, 46)                    # 46 frequencies
Z = np.asarray(solver.compute_impedance_swept(2 * np.pi / lams)).ravel()

46 frequencies in 65 ms, and the classic textbook curve appears — generated by your specimen, not copied from a book:

Everything a ham knows by feel is in that picture, computed from first principles: the dipole is capacitive when short, inductive when long, and resonates (X = 0) at L ≈ 0.487 λ — a hair under a half wave, exactly the “cut it a few percent short” rule of thumb, which for this wire is the thickness correction made visible. Our specimen sits just below resonance at λ = 22 m, which is why its reactance is −18 Ω.

Act I, closed

You now own the core of the method of moments, honestly:

• an antenna question is an integral equation for the current (ch. 1),
• a basis + testing scheme turns it into Z·I = V, and even the crudest choice — pulses on the 2n+1-point grid — works (ch. 2),
• the feed model supplies V, and Z_in = V/I(feed) is the answer (ch. 3),
• but the naive basis converges slowly and unevenly, buying its final digits only with extrapolation or a flood of unknowns — which is why the craft of Act II, better bases, is not a luxury.

Act II opens inside that craft: what makes NEC’s strange little constant + sine + cosine basis so unreasonably effective that twenty-one unknowns nail a dipole to a tenth of an ohm — reactance and all.

Turn the knob yourself

Drag the length knob of a dipole in the antennaknobs simulator and watch X sweep through zero live — the sweep figure above, played as an instrument.