The path loss formulas assume that the two antennas have matched polarizations. Polarization mismatch adds an extra loss. We determine polarization efficiency by applying the scalar (dot) product between normalized polarization vectors. An antenna transmitting in the z-direction has the linear components
The incident wave on the antenna is given by
where the wave is expressed in the coordinates of the source antenna. The z-axis of the source is in the direction opposite that of the antenna. It is necessary to rotate the coordinates of the receiving antenna wave. Rotating about the x-axis is equivalent to changing the sign of the tilt angle or taking the complex conjugate of Ea.
The measurement antenna projects the incident wave polarization onto the antenna polarization. The antenna measures the incident field, but we need to normalize the antenna polarization to a unit vector to calculate polarization efficiency:
We normalize both the incident wave and antenna responses to determine loss due to polarization mismatch:
The normalized voltage response is
 (1-40)When we express it as a power response, we obtain the polarization efficiency Γ:
 (1)This is the loss due to polarization mismatch. Given that δ1 and δ2 are the phases of the polarization ratios of the antenna and the incident wave. As expressed in terms of linear polarization ratios, the formula is awkward because when the antenna is rotated to determine the peak response, both the amplitudes and phases change. A formula using circular polarization ratios would be more useful, because only phase changes under rotation.
Two arbitrary polarizations are orthogonal ( Γ = 0) only if
 (2)
This can be expressed as vectors by using unit vectors: a1 · a2∗= 0; a1 and a2 are the orthonormal generalized basis vectors for polarization. We can define polarization in terms of this basis with a polarization ratio ρ. By paralleling the analysis above for linear polarizations, we obtain the polarization efficiency for an arbitrary orthonormal polarization basis:
 (3)It has the same form as Eq. (1) derived for linear polarizations.
We can use Eq. (3) with circular polarizations whose polarization ratio ρc magnitudes are constant with rotations of the antenna. The maximum and minimum polarization efficiencies occur when δ1 − δ2 equals 0â—¦ and 180â—¦, respectively. The polarization efficiency becomes
 (4)In all other vector pair bases for polarization, the magnitude of the polarization ratio ρ changes under rotations.
Figure 1 expresses Eq. (4) as a nomograph. If we have fixed installations, we can rotate one antenna until the maximum response is obtained and realize minimum polarization loss. In transmission between mobile antennas such as those mounted on missiles or satellites, the orientation cannot be controlled and the maximum polarization loss must be used in the link analysis. Circularly polarized antennas are used in these cases.
FIGURE 1 Maximum and minimum polarization loss. (After A. C. Ludwig, A simple graph for determining polarization loss, Microwave Journal, vol. 19, no. 9, September 1976, p. 63.)
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