Cyclotron Info

Magnet Power Calculator

Almost all the magnetomotive force of an iron-core magnet is spent driving flux across the air gap. This calculator turns a desired gap field into required amp-turns, then into coil power by either of two routes: a known winding (turns and resistance), or a copper budget (mass and mean turn length) — the latter gives the classic result that dissipation depends on how much copper you buy, not how you wind it.

Results

Required amp-turns NI—
Gap-only amp-turns (η = 1)—
Current—
Voltage—
Power dissipation—

Note: the requested field is above ~1.5 T. Ordinary low-carbon steel poles begin saturating there; the iron's MMF share grows rapidly and this linear model increasingly understates the required amp-turns. Above ~1.8 T treat the result as invalid.

The math

Ampère's law around the magnetic circuit gives NI = Hgapg + ΣHironℓiron. In the gap H = B/μ₀; iron below saturation has permeability in the thousands, so its term is small but not zero. Lumping the iron contribution and fringing into an efficiency factor η:

NI = B · g / (μ₀ · η)

η ≈ 0.9–0.95 is typical for a well-proportioned magnet with generous iron cross-section operating below ~1.2 T; the 0.85 default adds margin for fringing and imperfect joints. As the iron approaches saturation (~1.5–1.8 T for low-carbon steel poles — the assumption behind the warning above), η collapses and no single factor rescues the linear model.

Route 1 — copper mass

For total conductor length ℓ = Nℓt, cross-section a, copper volume V = M/δ (δ = 8960 kg/m³), resistance is R = ρℓ/a = ρN²ℓt²/V. With I = NI/N:

P = I²R = ρ · δ · (NI)² · ℓt² / M

The turn count cancels: power depends only on the amp-turn demand, the mean turn length, and the copper mass. Turns merely trade current against voltage. Copper resistivity is taken as ρ = 1.724 × 10⁻⁸ Ω·m at 20 °C scaled by (1 + 0.00393 (T − 20)).

Route 2 — known winding

I = NI / N, P = I²R, V = IR

Assumptions and limits

  • Linear iron (constant η). Invalid approaching saturation — see the warning threshold.
  • DC operation, steady-state temperature; cooling is not modeled. Sustained dissipation above a few hundred watts per coil generally needs forced air or water.
  • Uniform gap; shims and pole-face profiling change the local field but not this bulk estimate.

Worked check

B = 0.582 T across a 40 mm gap at η = 0.85 requires NI = 21,800 A-turns (18,500 for the gap alone). With 40 kg of copper at a 1.0 m mean turn and 60 °C, P ≈ 2.1 kW; the same amp-turns as a 500-turn, 2 Ω winding draw 43.6 A at 87 V ≈ 3.8 kW.

Sources

  • D. B. Montgomery, Solenoid Magnet Design, Wiley-Interscience, 1969 — power vs. copper volume relations.
  • J. J. Livingood, Principles of Cyclic Particle Accelerators, Van Nostrand, 1961 — ch. 9 (cyclotron magnet design, gap MMF).
  • Saturation values: typical B–H data for AISI 1006–1020 low-carbon steel (knee ~1.5–1.6 T, hard saturation ~1.8–2.1 T).