The vector effective height relates the open-circuit voltage response of an antenna to the incident electric field. Although we normally think of applying effective height to a line antenna, such as a transmitting tower, the concept can be applied to any antenna. For a transmitting tower, effective height is the physical height multiplied by the ratio of the average current to the peak current:
 (1-48)
The vector includes the polarization properties of the antenna. Remember from our discussion of antenna impedance mismatch that the open-circuit voltage is twice that across a matched load for a given received power: . The received power is the product of the incident power density S and the effective area of the antenna, . Gathering terms, we determine the open-circuit voltage from the incident field strength E and a polarization efficiency Γ:
We calculate polarization efficiency by using the scalar product between the normalized incident electric field and the normalized vector effective height:
 (1)Equation (1) is equivalent to Eq. (2):
 (2)because both involve the scalar product between the incident wave and the receiving polarization, but the expressions have different normalizations. You can substitute vector effective height of the transmitting antenna for the incident wave in Eq. (1) and calculate polarization efficiency between two antennas. When an antenna rotates, we rotate h. We could describe polarization calculations in terms of vector effective height. We relate the magnitude of the effective height h to the effective area and the load impedance :
 (3)The mutual impedance in the far field between two antennas can be found from the vector effective heights of both antennas. Given the input current to the first antenna, we find the open-circuit voltage of the second antenna:
 (4)When we substitute Eq. (3) into Eq. (4) and gather terms, we obtain a general expression for the normalized mutual impedance of an arbitrary pair of antennas given the gain of each in the direction of the other antenna as a function of spacing r:
 (5)The magnitude of mutual impedance increases when the gain increases or the distance decreases. Of course, Eq. (5) is based on a far-field equation and gives only an approximate answer, but it produces good results for dipoles spaced as close as 1λ. Figure 1 gives a plot of Eq. (5) for isotropic gain antennas with matched polarizations which shows the 1/R amplitude decrease with distance and that resistance and reactance curves are shifted out of phase. The cosine and sine factors of the complex exponential produce this effect. We multiply these curves by the product of the antenna gains, but the increased gain from larger antennas means that it is a greater distance to the far field. When we bring two antennas close together, the currents on each antenna radiate and excite additional currents on the other that modify the result given by Eq. (5). But as we increase the distance, these induced current effects fade.
 FIGURE 1 Normalized mutual impedance (admittance) from the vector effective length for two antennas with 0 dB gain along the line between them.
Equivalent height analysis can be repeated using magnetic currents (e.g., used with microstrip patches), and Eqs. (4) and (5) become mutual admittance. Figure 1 is also valid for these antennas when we substitute normalized mutual admittance for normalized mutual impedance. For antennas with pattern nulls directed toward each other, the mutual impedance decreases at the rate , due to the polarization of current direction h. |
