In Article “Mutual Coupling between Antennas” we discussed how the coupling between two antennas can be found from reactance. Given a transmitting antenna that generates a field at the receiving antenna, the reactance is described by an integral equation:

  (1)

The volume integral is over the receiving antenna currents, but it is often reduced to a surface or line integral. A second form of Eq. (1) uses the fields radiated by both antennas. Given a surface that surrounds the receiving antenna, the integral for reactance is taken over this surface:

  (2)

The differential normal ds is pointed away from the receiving antenna. When we represent the two antennas and the transmission between them as an impedance matrix, it implies that we know the input currents to both antennas. By expressing the coupling as an impedance matrix, we compute mutual impedance from the reactance integral:

  (3)

Antennas that we describe by input currents only have electric current densities excited on their surfaces. The mutual impedance formula using reactance reduces to

  (4)

The volume integral reduces to a line integral in most cases. Antennas with given input voltages such as slots can be described using magnetic currents and we use a mutual admittance matrix for the antenna pair:

  (5)

By using reciprocity antennas made of linear, isotropic materials, we have equal crossmatrix terms:

  (6)

We calculate self-impedance terms by integrating over the surface of the antenna: for example, the radius of a dipole with the source of the field located at the center of wires or slots.