“M” Theory

• Each field mill output can be considered as a linear sum of the external electric field, field due to charge on the aircraft, and “other” terms:

  • mi = Mxi*Ex + Myi*Ey + Mzi*Ez + Mqi*Eq + (?) (a)

•  can depend on anything other than Ex, Ey, Ez, or Eq

• The set of equations (a) for all mills on an aircraft can be represented as a matrix equation:

  • m = M*E (b)

• where m (mill outputs) & E (vector electric field and field due to charge on the aircraft) are vectors, M is a 6x4 matrix, and we are “neglecting” the “other” () terms for now

• However, we need the 4x6 matrix C which satisfies the equations:

  • E = C*m (c)
  • C*M = I (d)

• (where I is the 4x4 identity matrix) to calculate the electric fields from the mill outputs

• So we need a way to get from M to C in a logical way

• To calculate C from M, we use the Moore-Penrose pseudoinverse [Penrose, R., A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51, 406-413, 1955] (MP)

  • One major advantage of using the MP is that when properly invoked, it is not sensitive to DC offsets in the data

Notes: