Cylindrical Coaxial vs Planar Disc Magnetic Couplings: Torque Calculation Logic and the Exponential Impact of Air-Gap Thickness

Subtitle: From magnetic circuit to torque — how to calculate each configuration, and how torque decays with air gap

When selecting or designing a magnetic coupling, engineers face three questions that can't be skipped: which configuration to choose, whether the torque is sufficient, and how large the air gap can be. Cylindrical coaxial and planar disc are two completely different magnetic-circuit geometries, each with its own torque formula; and the impact of air gap on torque is not "a little more gap, a little less torque" — it is exponential decay: adding just 1 mm can cut torque in half.

This article breaks the two configurations, the torque formulas, and the exponential air-gap effect into calculable engineering language.

1. Geometric Differences in the Two Configurations

Cylindrical coaxial (barrel / through-shaft). The inner and outer magnet rotors share the same axis, with magnetic poles facing each other radially. Flux crosses the containment shell along the radial direction. The shell is cylindrical, relatively simple to manufacture, and can withstand high external pressure.

Best for: chemical-process pumps, high-flow transfer, reactor agitation, large-torque continuous-duty processes. Representative product: API 685 magnetic pump.

Planar disc (axial-flux / face-to-face). Two disc rotors face each other, with magnetic poles facing each other axially. Flux crosses the containment shell along the axial direction. The shell is a flat disc, must withstand the pressure differential across its faces, and generates axial thrust on the bearings.

Best for: compact structures, axially space-constrained applications, high speed, small-to-medium torque, such as small chemical pumps, magnetic stirrers, and laboratory fluid equipment.

Core difference: the cylindrical's magnetic circuit is "radially closed," the disc's is "axially closed"; the torque magnitude, decay characteristics, and sensitivity to air gap are all different and must be calculated separately.

2. Torque Calculation for Cylindrical Coaxial

The cylindrical magnetic circuit's working air-gap cross-section = 2π × r_m × L, where r_m is the mean radius and L the axial length. From the Maxwell stress tensor, the torque formula is:

T = k_t × (B_g² × π × r_m² × L) / (2μ₀)

Where:

· B_g: air-gap flux density at the working gap (T), determined by the magnetic circuit design and the gap itself;

· r_m: mean radius (m), the midpoint between the outer-rotor inner diameter and inner-rotor outer diameter;

· L: effective axial length (m);

· μ₀: vacuum permeability = 4π × 10⁻⁷ H/m;

· k_t: coupling coefficient (0.85–0.95), related to pole count, magnet arc coverage, and assembly concentricity.

Numerical example: Ø100 mm × 100 mm cylindrical (r_m ≈ 0.05 m, L = 0.1 m), B_g = 0.6 T, k_t = 0.9:

T = 0.9 × (0.6² × π × 0.05² × 0.1) / (2 × 4π × 10⁻⁷)

  ≈ 0.9 × (0.36 × π × 2.5 × 10⁻⁴) / (2.513 × 10⁻⁶)

  ≈ 101 N·m

Order-of-magnitude reference: for a 100 mm OD, 100 mm L cylindrical coupling at B_g = 0.6 T, torque is approximately 100 N·m; increasing L or r_m approximately scales torque linearly.

3. Torque Calculation for Planar Disc

The disc circuit's working air-gap cross-section = π × (r_o² - r_i²), where r_o is the outer radius and r_i the inner radius. Mean lever arm = (r_o + r_i) / 2 = r_m. The torque formula is:

T = k_t × (B_g² × π × (r_o² - r_i²) × r_m) / (2μ₀)

Where:

· B_g: axial working-gap flux density (T);

· r_o, r_i: outer and inner radii of the disc (m);

· r_m: mean radius (m);

· μ₀: vacuum permeability;

· k_t: coupling coefficient (0.75–0.90; disc is slightly lower than cylindrical due to edge flux leakage).

Numerical example: Ø80 mm OD, Ø30 mm ID disc (r_o = 0.04 m, r_i = 0.015 m), B_g = 0.5 T, k_t = 0.85:

T = 0.85 × (0.5² × π × (0.04² - 0.015²) × 0.0275) / (2 × 4π × 10⁻⁷)

  ≈ 0.85 × (0.25 × π × 1.375 × 10⁻³ × 0.0275) / (2.513 × 10⁻⁶)

  ≈ 10.0 N·m

Order-of-magnitude reference: at the same OD, a single disc delivers approximately 10 N·m; stacking multiple discs (double, triple) multiplies torque proportionally — the standard way to scale up a disc coupling.

4. The Exponential Impact of Air Gap on Torque

4.1 The spatial decay law of periodic magnetic fields. The inner and outer rotors of a magnetic coupling are periodic multi-pole structures. The distance between adjacent like poles is the pole pitch τ. For such a periodic structure, the field along the air-gap direction decays approximately as:

B(z) ≈ B_0 × exp(-πz/τ)

Where:

· B_0: flux density at the magnet surface (zero gap);

· z: distance from the magnet surface (i.e., the air-gap thickness);

· τ: pole pitch.

Physical meaning: the smaller the pole pitch τ, the faster the field decays in space; the larger τ, the slower the decay.

4.2 The squared dependence of torque. Because torque T ∝ B_g², the torque at gap z is:

T(z) ≈ T_0 × exp(-2πz/τ)

This is exponential decay: each increase of approximately τ/(2π) ≈ τ/6.28 in gap cuts torque to about 37% (1/e) of the previous value.

4.3 Numerical example: decay at pole pitch 20 mm. Assume pole pitch τ = 20 mm, zero-gap torque T_0:

· g = 0 (ideal): T_0

· g = 1 mm: T_0 × exp(-2π × 1/20) ≈ 0.73 T_0 (73% remaining)

· g = 2 mm: T_0 × exp(-2π × 2/20) ≈ 0.53 T_0 (53% remaining)

· g = 3 mm: T_0 × exp(-2π × 3/20) ≈ 0.39 T_0 (39% remaining)

· g = 5 mm: T_0 × exp(-2π × 5/20) ≈ 0.21 T_0 (21% remaining)

· g = 10 mm: T_0 × exp(-2π × 10/20) ≈ 0.043 T_0 (4.3% remaining)

Key observation: a 10 mm gap (only 0.5× the pole pitch) cuts torque to 4.3% — 95.7% of the torque is swallowed by the air gap. This is why magnetic-coupling engineers are "millimeter-sensitive" to air gap.

4.4 Effect of pole-pitch size on decay rate. At the same absolute gap g, a larger pole pitch τ makes the exponent -2πg/τ smaller in magnitude, so decay is slower:

· τ = 10 mm (many-pole small size), g = 2 mm: T ∝ exp(-2π × 2/10) ≈ 0.21 T_0

· τ = 30 mm (few-pole large size), g = 2 mm: T ∝ exp(-2π × 2/30) ≈ 0.66 T_0

Engineering conclusion: large-pole-pitch / few-pole couplings are more tolerant of air gap; small-pole-pitch / many-pole couplings are more sensitive to air gap. But fewer poles also means lower magnetic-circuit efficiency — pole count is a balance between "gap sensitivity" and "magnetic efficiency."

5. The Engineering Composition of Air Gap

The theoretical air gap is not just the containment-shell thickness — it is the sum of multiple error sources:

· Containment-shell wall thickness: metal (0.5–1.5 mm), PEEK (1.5–3.0 mm), ceramic (2.0–4.0 mm);

· Single-side assembly clearance: 0.1–0.3 mm;

· Rotor radial runout: 0.05–0.2 mm;

· Shell thermal expansion and manufacturing tolerance: 0.1–0.3 mm;

· Axial pressure-differential shell deflection (disc type): 0.1–0.5 mm.

Typical engineering air gaps:

· Cylindrical: 1.5–3.0 mm (metal shell + assembly clearance);

· Disc: 1.0–2.5 mm (face-to-face is easier to set small, but axial assembly is demanding).

Engineering design rule: working air gap should be controlled within 0.3–0.7× the pole pitch. Above 0.7× the pole pitch, torque enters the "steep decline zone" — gains are quickly eaten by the gap.

6. Configuration Selection Decision

Choose cylindrical when:

· Torque demand is large (> 50 N·m);

· Continuous duty is long (> 8,000 h/year);

· The shell withstands external pressure; manufacturing and leak-testing are mature;

· Axial space is generous.

Choose disc when:

· Axial space is constrained (compactness mandatory);

· Speed is high (lower disc inertia gives faster acceleration/deceleration);

· Torque demand is medium (< 50 N·m);

· Small chemical pump, magnetic stirrer, or laboratory equipment.

Choose stacked multi-disc when:

· Both compactness and high torque are required: double- or triple-disc stacking is the standard scaling path for disc couplings;

· Note: each additional disc adds one more air gap pair; gap design must be proportionally tighter.

7. Several Conclusions for Design and Selection

· Calculate torque after calculating air gap: B_g is determined by magnetic-circuit design and gap; once gap is set, torque decays as exp(-2πg/τ) — gap is a design input, not a design output.

· Few poles with large pitch tolerate gaps; many poles with small pitch fear gaps: pole-count selection must account for gap sensitivity.

· Gap engineering margin: typical 1.5–3 mm; always express as a fraction of pole pitch (safe below 0.5× τ).

· Configuration follows duty: large torque / continuous → cylindrical; compact / high-speed → disc; both → stacked multi-disc.

· Formulas are estimates; simulation and measurement are truth: all formulas above are engineering estimates; flux leakage, magnet arc coverage, and assembly errors introduce ±15–25% deviation; final torque should be verified by FEA simulation and prototype testing.

8. Conclusion

Cylindrical coaxial and planar disc are two parallel magnetic-coupling configurations, with different magnetic-circuit geometries, different torque formulas, and different sensitivities to air gap. The impact of air gap on torque is exponential — this means "air-gap optimization" is the highest-priority, highest-payoff engineering action in magnetic-coupling design, more direct than simply using larger magnets or switching grades.

For cylindrical / disc magnetic-coupling torque-air-gap-pole-pitch matching design, Halbach array optimization, or prototype torque measurement evaluation, please feel free to reach out — FAIZEAL can provide technical support.