Passive magnetic attitude control for a 1P: sizing the magnet and damper rods

Why 50 mm geometry couples the magnet and the dampers#

Passive magnetic attitude control (PMAC) aligns one axis of a spacecraft with the local geomagnetic field ${\mathbf{B}}_{\oplus }$ via a permanent magnet of dipole moment $\mathbf{m}$ and dissipates rotational kinetic energy in orthogonal soft-magnetic rods or strips[1]. In a $1\text{U}$ or $3\text{U}$ CubeSat the damper rods can be placed $60\text{-}150\text{mm}$ away from the permanent magnet. Inside a $50\text{mm}$ $1\text{P}$ cube, the separation $d$ rarely exceeds $20\text{-}35\text{mm}$[2]. Because the dipole stray field grows as $d^{-3}$, an oversized magnet biases the hysteresis rods into saturation and stops rotational damping[3].

Step 1: Sizing the permanent dipole#

For a maximum pointing offset ${\theta }_{\max }$ under worst-case environmental torque $T_{\text{dist}}$ at minimum orbital field $B_{\min }$ (about $23\mu \text{T}$ at the magnetic equator at $525\text{km}$) with a torque margin $k_m\approx 10$, the required dipole moment is[1]:

$$m=\frac{k_m{T}_{\text{dist}}}{B_{\min }\sin {\theta }_{\max }}$$

Disturbance	Torque (nN·m)	Required moment (A·m²)
Aerodynamic, ρ = 1 × 10⁻¹² kg/m³	1.6	0.005
Aerodynamic, ρ = 5 × 10⁻¹² kg/m³	7.9	0.023
Residual dipole 1 × 10⁻³ A·m² in 30 µT	30.0	0.086
CSSWE 3U CubeSat, for comparison	100.0	0.288

1P at 500 km: 25 cm² projected area, drag coefficient 2.2, 7.6 km/s, 1 cm centre-of-mass offset. The residual-dipole torque does not shrink with the spacecraft, so neither does the magnet.

For a $1\text{P}$ with a deployed UHF tape antenna, aerodynamic torque at moderate solar activity requires only $m\approx 0.015\text{-}0.030\text{A}\cdot {\text{m}}^2$—a tiny AlNiCo-5 or SmCo cylinder of a few millimetres[4].

Step 2: Stray-field coupling and minimum separation#

Along a damper rod placed perpendicular to the magnet axis at lateral distance $d$ and axial offset $\delta$ from the dipole mid-plane, the peak axial stray field is[2]:

$$H_{\parallel ,\max }=\frac{0.859m\delta }{4\pi (d^2+{\delta }^2{)}^2}$$

Moment (A·m²)	Equatorial, 20 mm	Axial, 20 mm	Equatorial, 45 mm	Rod offset allowed at 20 mm (mm)	Rod offset allowed at 45 mm (mm)
0.02	199	398	17	2.2	Any
0.10	995	1,989	87	0.44	13.2
0.31	3,084	6,167	271	0.14	3.7
0.91	9,052	18,104	795	0.05	1.2

Allowed offset: the largest distance of a rod from the magnet mid-plane for which its peak axial stray field (review, Eq. 3) stays below the equatorial geomagnetic field at 525 km, 18.6 A/m.

To keep the hysteresis loop open as the spacecraft tumbles in ${\mathbf{B}}_{\oplus }$, the axial component of the magnet's stray field along the rod must stay well below the rod's apparent saturation field $H_{\text{sat,app}}=N_d{J}_s/{\mu }_0$.

Step 3: Shape demagnetization and despin time#

Short rods suffer strong internal demagnetizing fields. For a cylinder of length-to-diameter ratio $p=L/D$, the Sato–Ishii magnetometric factor[5] and the exact prism formula of Aharoni[6] show that $N_d$ shears the intrinsic $B\text{-}H$ loop:

$$N_d\approx \frac{1}{1+\frac{4}{\sqrt{\pi }}p}$$

Element	N	Apparent saturation N·Mₛ (A/m)	Peak polarization (mT)	Loss per cycle (nJ)	Mass (g)	Saturated within (mm)
HyMu-80 rod, Ø 1 × 45 mm (1P limit)	9.75 × 10⁻³	5,742	3.7	0.51	0.31	11
HyMu-80 rod, Ø 1 × 95 mm (1U limit)	4.64 × 10⁻³	2,734	7.9	2.25	0.65	14
HyMu-80 rod, 0.2 × 0.2 × 33.5 mm (Grizu-263A)	2.82 × 10⁻³	1,660	12.9	0.067	0.012	17
Fe–B–Si strip, 25 µm × 5 × 45 mm	1.14 × 10⁻³	1,413	29.8	2.01	0.040	18
HyMu-80 strip, 25 µm × 5 × 45 mm	1.14 × 10⁻³	670	32.1	0.69	0.049	23

Applied field amplitude 30 A/m, idealized rectangular loop (materials review, Eqs. 5–10). Last column: distance from a 0.1 A·m² magnet, in its equatorial plane, inside which the element is driven to saturation.

Configuration	Inertia (kg·m²)	Damper mass (g)	Loss per cycle (nJ)	Despin time (h)
1P, 250 g, four Ø 1 × 45 mm HyMu-80 rods	1.04 × 10⁻⁴	1.23	2.03	26
1U, 1.33 kg, four Ø 1 × 95 mm HyMu-80 rods	2.22 × 10⁻³	2.60	9.01	125
1P, 250 g, four Fe–B–Si strips	1.04 × 10⁻⁴	0.16	8.05	6.6
Grizu-263A, 237 g, twelve 0.2 mm HyMu-80 rods, no magnet bias	9.88 × 10⁻⁵	0.14	0.80	63

Orientation factor 0.6, homogeneous cube inertia, applied field 30 A/m. These figures compare designs; magnet bias, non-uniform magnetization and minor loops reduce the real damping.

Step 4: Asymptotic dead band#

Once the angular rate drops near orbital rate, the Earth field swing $H_{\perp }$ must exceed the material coercivity $H_c$ to traverse the hysteresis loop. Below ${\theta }_{\text{db}}\approx \arcsin (H_c/H_{\perp })$, the rod behaves elastically and libration damping stops[1]:

Damper alloy	Coercivity (A/m)	H⊥ = 16 A/m	H⊥ = 30 A/m	H⊥ = 40 A/m
80% Ni–Fe (HyMu-80)	0.96	3.4°	1.8°	1.4°
Fe-based amorphous, annealed	3	10.8°	5.7°	4.3°
Grain-oriented Fe–3%Si, low	5	18.2°	9.6°	7.2°
Grain-oriented Fe–3%Si, high	10	38.7°	19.5°	14.5°

H⊥ is the geomagnetic field perpendicular to the magnet axis: about 16 A/m at the weak end of LEO and 30 A/m as a typical value.