  CubeSats are miniature satellites used to carry experimental payloads into
orbit, where it is often critical to precisely control their attitude. One way
to do this is through the use of magnetorquers, which can be integrated into
PCBs. This technique saves considerable space and capital when compared with
more common torque-rod magnetorquer systems. Here we derive a method of
analyzing different PCB-integrated magnetorquer geometries, parametrizing them
such that the magnetic moment and efficiency are optimized. Furthermore, by
modulating the trace width, the trace number, and other electrical
characteristics of the magnetorquer coil, this paper optimizes the generated
magnetic moment. Both constant voltage and constant current sources are
analyzed as inputs. These optimizations are then simulated in COMSOL for
multiple geometries, and it is found that there exists an optimal geometry,
given a specified power dissipation. Simulations verify the general trend and
maxima of these derivations, barring small, consistent re-scaling in the
magnitude of the coil resistance. It is also found that these PCB-integrated
magnetorquers provide a sufficient alternative to commercial coil magnetorquers
- particularly in volume-restricted configurations. This study extends such
analysis to larger CubeSat configurations, and finds that these larger
implementations increase magnetorquer efficiency. Optimizations for common
PCB-implementable geometries on small satellites are tabulated in the Appendix.
