Passive magnetic attitude control (PMAC)

Definition

Passive magnetic attitude control (PMAC) combines two magnetic parts. A permanent magnet aligns one body axis with the local geomagnetic field, the way a compass needle does. Hysteresis dampers, rods or strips of a soft-magnetic alloy, turn rotational energy into heat as the field component along them cycles, so the spin and the libration around the field line die out[1]. The system has no power supply, software, sensors or moving parts. That is why it is the most common attitude control of 1P and 1.5P PocketQubes, and why it flew on Grizu-263A.

How it works

As the spacecraft moves along its orbit the field direction turns, roughly twice per orbit relative to an inertial frame, and the magnet drags the stabilized axis along. In a dipole field the inclination $I$ of the field line follows $\tan I=2\tan {\lambda }_m$, where ${\lambda }_m$ is the magnetic latitude: the axis lies almost horizontal over the magnetic equator and almost vertical near the poles[1]. The magnet’s restoring torque $mB\sin \theta$ makes the axis oscillate around the field line with an undamped libration period $2\pi \sqrt{I/(mB)}$; the dampers remove the energy of that oscillation and of the initial spin.

flowchart LR
  M["Permanent magnet"] -->|"restoring torque m × B"| BODY["Spacecraft body"]
  BODY -->|"rotation"| ROD["Hysteresis dampers"]
  ROD -->|"loss per cycle, heat"| BODY
  EARTH["Geomagnetic field"] --> M
  EARTH --> ROD

Why a 1P is different

Methods developed for CubeSats and microsatellites still apply[2][3], but three effects change at 50 mm[1][4]:

1. The magnet does not shrink with the spacecraft. Disturbance torques of a 1P are tiny, but the torque of a residual dipole does not depend on size, so the moment the rules call for stays near a few hundredths to a tenth of an A·m².
2. The magnet sits close to everything. The field of a dipole decays as $r^{-3}$, so at the walls of a 1P it is eight times stronger than at the walls of a 1U. Dampers biased by that field stop working, and magnetometers saturate.
3. The dampers are short. Only about 45 mm fit, against 95 mm in a 1U, which lowers dissipation per element, although the much lower inertia more than compensates.

Sizing the magnet

A common rule, adapted by Gerhardt and Palo from Santoni and Zelli, asks for a restoring torque ten times the disturbance torque at the weakest field and the largest acceptable error[3]:

For a 1P at 500 km[4]:

Drag alone would call for a few hundredths of an A·m²; a residual dipole of 10⁻³ A·m², rarely measured on PocketQubes, pushes the requirement towards 0.1 A·m². The 3U CSSWE value is shown for scale.

Magnet materials

A 100 mm³ magnet delivers about 0.1 A·m² at full remanence with either AlNiCo 5 or sintered NdFeB, but the two behave very differently[4]:

AlNiCo has a low coercivity, so a short magnet self-demagnetizes; it should be elongated. NdFeB keeps its magnetization but drifts about 12% over a 100 K swing and is limited to about 80 °C in standard grades. For the eclipse cycling of a PocketQube, elongated AlNiCo 5 or SmCo are the safer choices.

The Grizu-263A estimate

Grizu-263A used four 8 mm AlNiCo-5 cubes and twelve HyMu-80 rods of 0.2 × 0.2 × 33.5 mm. Intersecting a hyperbolic fit of the AlNiCo-5 demagnetization curve with the load line of a cube ($N=1/3$) gives an operating point near 0.12 T: each cube keeps only about 15% of its remanence, and the four together carry about 0.31 A·m². Stacked into one 8 × 8 × 32 mm column ($N=0.108$) they would carry about 0.91 A·m²[1]. The resulting torque budget at 525 km:

The restoring torque exceeds every disturbance by two to three orders of magnitude, and the libration period of 10–23 s is far shorter than the time over which the field direction changes, so the axis tracks the field closely. A magnet of 0.02 A·m² would still give a tenfold margin.

Dampers and demagnetization

A rod magnetized along its axis feels an internal field $H_i=H_a-NM$. With permeabilities of 10⁴–10⁵, $N{\mu }_r\gg 1$ for every practical element, so the polarization reached is set by geometry, not by the material[4]:

The demagnetizing factor of a cylinder follows the Sato–Ishii approximation and that of a rectangular strip Aharoni’s exact solution[5][6]; the prolate-ellipsoid formula of Osborn is a convenient bound[7].

Thin strips are far more efficient per gram than rods: a 0.04 g Fe–B–Si strip dissipates four times as much per cycle as a 0.31 g rod. They also saturate more easily, as the last column shows.

Despin time

With the estimate of Farrahi and Sanz-Andrés, $t_{\mathrm{d}}=2\pi I{\omega }_0/(k_w\sum W)$[8]:

The length limit of a 1P does not by itself lengthen despin: losing dissipation as $L^{1+n}$ is more than compensated by losing inertia as $L^5$[4].

Coercivity and the dead band

Once the spin is gone, a damper perpendicular to the magnet sees a field of amplitude $H_{\perp }\sin \theta$. Below the coercivity only reversible losses remain, so damping stops at a dead-band half-angle ${\theta }_{\mathrm{db}}\approx \arcsin (H_{\mathrm{c}}/H_{\perp })$[4]:

High-coercivity dampers despin faster but leave a larger residual libration; low-coercivity ones damp down to small angles but dissipate less per cycle. Simulations by Fiorillo et al. found Fe-based amorphous strips the most effective in both regimes[9], which suggests a hybrid set for a 1P: amorphous strips for despin and bias tolerance, plus a few low-coercivity Ni–Fe elements for small angles.

Magnet-to-rod coupling

Hysteresis dampers dissipate only if the geomagnetic field drives them around a loop. For a rod perpendicular to a dipole, at lateral distance $d$ and offset $\delta$ along the dipole axis from its mid-plane, the peak axial field along the rod is[1]

which scales as $m\delta /d^4$. Shrinking the magnet-to-rod distance from 45 mm to 20 mm multiplies the coupling by about 25:

With 0.31 A·m² at 20 mm, an offset of 1 mm already produces about 130 A/m, several times the geomagnetic field; keeping the stray field below it would need a placement tolerance of 0.14 mm, against 2.2 mm for a right-sized 0.02 A·m² magnet.

Saturated dampers

A damper works only while the total axial field stays below its apparent saturation field $N{M}_s$. The magnet does not shrink, so the dampers of a 1P sit in a strong bias field: a 0.1 A·m² dipole saturates the Grizu-263A rod geometry within about 17 mm in its equatorial plane[4]. Flight experience confirms the hazard: on Delfi-C3, despin took about three months instead of the predicted few hours, largely because rods near the magnet were saturated[8].

Magnetometer bias

A magnetometer a few centimetres from the magnet measures the geomagnetic field plus the magnet’s own. At 20 mm from 0.31 A·m² the equatorial field is 3.8 mT and the axial field 7.6 mT, comparable with or above the full-scale ranges of common MEMS magnetometers (±0.4–1.6 mT, ±4.9 mT)[1]. The reading is then saturated or carries a large constant bias that must be calibrated in orbit, for example with attitude-independent methods[10].

Layout rules for a 1P

The papers condense the analysis into four rules[4]:

• Size the moment no larger than the sizing rule requires.
• Place dampers in or close to the equatorial plane of the magnet, where the stray field is parallel to the magnet axis and has no component along a perpendicular damper.
• Choose demagnetizing factors so that $N{M}_s$ exceeds the local axial field by a margin, for example a factor of two.
• Map the stray field of the integrated unit and measure its residual dipole.

Processing the dampers

80% Ni–Fe alloys get their very low coercivity only from a final hydrogen anneal; any cold work afterwards, from bending or rough handling, can multiply it[11]. Fe-based amorphous ribbons are stress-sensitive through their magnetostriction: bonding a ribbon rigidly to FR-4 or aluminium generates thermal stresses of 50–105 MPa over a 60 K swing, enough to disturb the weak anisotropies set by annealing[12][4]. Flight elements should be annealed in their final shape and mounted free to expand.

What telemetry can show

Gyroscope rates after deployment reveal the effective damping, and magnetometer channels show whether the sensor saturated or recorded a constant bias on top of the orbital variation of the field. Because the stabilized axis follows the field inclination, the orientation of a monopole pattern relative to a ground station, and the depth of its nulls, varies predictably with latitude, which signal strengths recorded by SatNOGS stations can test[1].