The simplest approach for coupling between antennas is to start with a far-field approximation. We can modify Eq. (1)

 (1)

for path loss and add the phase term for the finite distance to determine the S-parameter coupling:

  (2)

Equation (2) includes the polarization efficiency when the transmitted polarization does not match the receiving antenna polarization. We have an additional phase term because the signal travels from the radiation phase center along equivalent transmission lines to the terminals of each antenna. Equations (3) and (2) have the same accuracy except that Eq. (2) eliminates the need to solve the two-port circuit matrix equation for transmission loss. These formulas assume that antenna size is insignificant compared to the distance between the antennas, and each produces approximately uniform amplitude and phase fields over the second element.

 (3)

We can improve on Eq. (2) when we use the current distribution on one of the two antennas and calculate the near-field fields radiated by the second antenna at the location of these currents. Since currents vary across the receiving antenna, we use vector current densities to include direction:  electric and  magnetic. Although magnetic current densities are fictitious, they simplify the representation of some antennas. We compute coupling from reactance, an integral across these currents [see Eq. (5)]:

  (4)

 (5)

The input power to the transmitting antenna  produces fields  and  . The power  into the receiving antenna excites the currents. The scalar product between the incident fields and the currents includes polarization efficiency. If we know the currents on the transmitting antennas, we calculate the near-field pattern response from them at the location of the receiving antenna. Similar to many integrals, Eq. (4) is notional because we perform the integral operations only where currents exist. The currents could be on wire segments or surfaces. A practical implementation of Eq. (4) divides the currents into patches or line segments and performs the scalar products between the currents and fields on each patch and sums the result. A second form of the reactance [see Eq. (6)] involves an integral over a surface surrounding the receiving antenna.

 (6)

In this case each antenna radiates its field to this surface, which requires near-field pattern calculations for both. Equation (4) requires adding the phase length between the input ports and the currents, similar to using Eq. (2). When we use Eq. (4), we assume that radiation between the two antennas excites insignificant additional currents on each other. We improve the answer by using a few iterations of physical optics, which finds induced currents from incident fields.

We improve on Eq. (4) by performing a moment method calculation between the two antennas. This involves subdividing each antenna into small elements excited with simple assumed current densities. Notice the similarity between Eqs. (3) and (2) and realize that Eq. (4) is a near-field version of Eq. (2). We use reactance to compute the mutual impedance  between the small elements as well as their self-impedance. For the moment method we calculate a mutual impedance matrix with a row and column for each small current element. We formulate a matrix equation using the mutual impedance matrix and an excitation vector to reduce coupling to a circuit problem. This method includes the additional currents excited on each antenna due to the radiation of the other.