Metal Detector Coil Calculator and PI Pulse Timing

Coil and transmit pulse

Wheeler's rectangular-section air-core approximation is

with mean radius $r$, axial height $b$ and radial build $c$ in inches. The coefficient is explicitly converted for millimetre inputs. See Wheeler (1928), DOI 10.1109/JRPROC.1928.221309; worked examples are attached in CREI 740 pp.33–34 and Kesler, attached p.26. A thin detector bundle is outside Wheeler's original near-equal-denominator-term 1% accuracy claim; measure the completed coil.

Copper length is $\ell=\pi DN+\ell_{leads}$ and $R_{20}=\rho\ell/A$, with annealed-copper resistivity $1/58\;\Omega\,mm^2/m$ at 20 °C (IEC 60028 convention used by the platform material data). Measured L and R override these estimates for pulse timing. With the damping resistor permanently connected, the supply seen by the coil is $V_{th}=V_bR_d/(R_d+R_s)$ and $R_{th}=R_sR_d/(R_s+R_d)$. Thus $I_p=V_{th}[1-e^{-t_{on}(R_c+R_{th})/L}]/(R_c+R_{th})$ and $W=LI_p^2/2$.

Discrete wire packing

The copper-area inequality is only a necessary screening bound; area alone cannot establish winding feasibility. A separate discrete check constructs aligned and staggered layouts of equal bare round wires in the rectangular bundle, trying both axial and radial row stacking.

Aligned rows hold $\lfloor b/d\rfloor\lfloor c/d\rfloor$ wires. Half-diameter staggered rows have pitch $\sqrt{3}d/2$; alternating row counts are $\lfloor w/d\rfloor$ and $\lfloor w/d-1/2\rfloor$. The first row consumes one diameter of height, so the row count is $1+\lfloor(h/d-1)/(\sqrt{3}/2)\rfloor$. We also test equal-count rows with offset $s=\min[d/2,w-d\lfloor w/d\rfloor]$ and pitch $\sqrt{d^2-s^2}$. These are geometric constructions derived here from circle contact, not an empirical fill factor.

The largest tested layout supplies the discrete capacity. When either bundle dimension is exactly one wire diameter, all wire centres lie on a line and capacity reduces to the floor of the other dimension divided by diameter. Thus a 0.63 mm by 5 mm bundle holds at most seven bare 0.63 mm wires, and nine turns need at least 5.67 mm along the row.

This is a conservative regular-row construction check, not a search for every possible irregular circle packing. All counts use bare copper diameter; passing does not account for enamel, winding tolerances or manufacturability. Allow additional space for those effects.

Resonance, damping and recovery allowance

The turn-off network is a series $R_c,L$ branch in parallel with $R_d$ and total shunt C. Kirchhoff's laws give

The lossless lumped resonance is $1/(2\pi\sqrt{LC})$. Setting $\alpha=\omega_n$ gives

These equations and the following planning allowance are derived here, rather than copied from a detector design.

The assumed clamp releases at $I_e=\min(I_p,V_c/R_d)$. Its constant-voltage ramp estimate is $t_f=(L/R_c)\ln[(V_c+R_cI_p)/(V_c+R_cI_e)]$. The tail starts from the conservative voltage $V_0=\min(V_c,I_pR_d)$. Let $\lambda=\alpha$ for an underdamped network, otherwise $\lambda=\omega_n^2/(\alpha+\sqrt{\alpha^2-\omega_n^2})$. A normalized zero-slope voltage tail is bounded by $(1+\alpha t)e^{-\lambda t}$. For $\epsilon=V_{ready}/V_0$, use the deliberately generous allowance

The inequality $(1+cx)e^{-x}\le(1+c)e^{-x/2}$ for $x\ge0$, $c\ge1$, supplies this bound. Add $1/\lambda$ for peak arrival. Suggested sample start is $\max(t_f+1/\lambda+t_b,t_{rx})$. The plotted coil curve starts its decay after the ramp and peak-arrival allowance and caps the settling envelope at unity. It is an allowance curve, not a simulated clamp waveform.

Target decay and calibrated air range

For a solid non-magnetic sphere, the Smythe/Wait magnetic-diffusion idealization has fundamental time constant $\tau=\mu_0\sigma a^2/\pi^2$; Geonics TN32 Eq.2 states this result. Material resistivity is from the platform's 20 °C metal data. The late-mode fraction is $e^{-t/\tau}$; the gate result is its exact rectangular-gate average. No absolute target amplitude is inferred.

For an on-axis small compact target, a circular coil's coupling is proportional to $Nr^2/(r^2+z^2)^{3/2}$. Reciprocity makes the transmit-receive signal proportional to its square at fixed current. Equating it to an empirical reference gives

A negative right side returns zero range with a message. The default anchor is DesAlex's 18 June 2009 Clone PI-W report: 21 cm coil and about 25 cm coin air range. That coin measurement is separate from the sphere timing surrogate.

• Circular, air-core, rectangular winding bundle; all turns carry current in the same sense. Lead inductance and distributed winding details are not calculated.
• Discrete packing checks use bare-wire aligned or staggered rows in both bundle orientations. The area check alone does not establish feasibility; enamel and winding tolerances need additional space. Irregular optimized packing is outside this construction model.
• Copper resistance estimate at 20 °C. Enamel, joints, heating, skin and proximity effects are excluded; use measured resistance at operating temperature when these matter. The material constants follow the platform's IEC 60028 copper and IEC 60889 conductor-aluminium data; check against the standard in force for procurement.
• Transmit current starts from zero. The RL rise ignores the short capacitive switching-edge transient; the damping resistor's loading through source resistance is included.
• The reverse-clamp voltage is an effective design assumption across the coil. Turn-off is approximated by a clamp ramp and a separately budgeted linear RLC tail. Supply-referenced clamp topology, clamp capacitance, voltage-dependent MOSFET capacitance and amplifier recovery are not simulated.
• The sphere calculation concerns only the fundamental late exponential mode of a homogeneous, non-magnetic solid conductor. It is not a model of a coin, steel target or hollow shell.
• The depth comparison is calibrated to one empirical air test. The calibration test's target, orientation, turns, peak current, sampling and receiver threshold stay fixed. Changing electrical inputs does not automatically rescale this independent diameter comparison.
• Drawings are schematic. Moving markers indicate current circulation; their speed is illustrative and their fading follows the sphere mode. Chart axes explicitly retain their stated mm and µs units; physical input and result cards support the viewer's metric/US conversion.

Use for preliminary sizing and timing comparisons of circular mono-coils, followed by measurement. Wheeler's original approximately 1% claim applies only to suitably proportioned rectangular windings, not every thin detector bundle. The self-resonance result requires a supplied lumped capacitance; it does not predict that capacitance from wire spacing. Discrete packing acceptance only establishes a tested bare-wire row layout; verify insulated wire diameter, winding clearance and the finished bundle.

The recovery budget is a conservative engineering allowance within its assumed linear network, not a guaranteed safe receiver delay or a semiconductor stress calculation. Verify the actual peak voltage, avalanche or clamp energy, resistor pulse rating and receiver overload recovery. The energy-rate result excludes transmit copper losses, damping-resistor on-time loss and the rest of the electronics. No thermal rating is selected.

Sphere late-mode retention is meaningful for comparison after higher modes have decayed. At early times, finite current turn-off, multiple diffusion modes and pulse history affect the total signal. A mode fraction is not a metal-discrimination result or a detection threshold.

Air-range extrapolation assumes a small target on the coil axis in a nearly uniform applied field across that target. Keep coil diameter within 0.5–2 times the measured reference and keep the winding thin. Ground mineralization, target geometry, receiver noise, electromagnetic interference and sweep speed can dominate actual depth. The Clone PI-W anchor is a builder report with unspecified test uncertainty; it is not independent validation of the extrapolation law.

Detecting and recovery of finds can require permits. The NRW portal describes permission under §15 DSchG NRW and separate landowner consent, checked 27 October 2026. Other German states and countries differ. Check the rules currently in force before searching or digging.