where 𝑉𝑖 is the corresponding i-th (i = YU1, YU2, YD1, YD2, XR1, XR2, XL1, and XL2) photodiode voltage, 𝑉𝑇 is the thermal voltage given by 𝑉𝑇 = 𝑘B𝑇⁄𝑞 where 𝑘B = 1.38×10-23 J/K is the Boltzmann constant, 𝑇 is the absolute temperature, 𝑞 = 1.602×10-19 C is the elementary charge, ℜ is the photodiode responsivity, 𝑃𝑖 is the i-th photodiode captured optical power, and 𝐼S is the photodiode saturation current.

To keep photodiodes in the photovoltaic mode they must be connected to the high impedance nodes, thus requiring high values of the resistance RL. The corresponding photodiode-captured optical power depends upon the shadow position within the enclosure, i.e. it depends on the shadow distribution over the active photodiode surface. This is presented in Figure 1. The actively illuminated area on the photodiodes’ surface depends upon the pitch and roll angles of the sun with respect to the photodiodes, as presented in Figure 2. This is, naturally, valid only for the photodiodes that are shadowed by the enclosure. For example, if the Sun illuminates the sensor from the first quadrant, shown in Figure 2, only photodiodes PDYU2 and PDXR2 will be in the shadow and their corresponding illuminated area will be:

where small pitch ξ and roll ψ angles were assumed (ξ, ψ ≪ 1) thus giving, in the first approximation, the linear dependence of the illuminated photodiode area with respect to the corresponding angles, where A is the area of the photodiode active surface, K is the positive proportionality constant that depends on the sensor geometry, and where A ≫ Kξ, Kψ is also valid.

In the first approximation the output voltage signals VX and VY are directly proportional to the pitch and roll angles with sensor sensitivity S. As the output signals are independent of the solar irradiance the circuitry has an inherent automatic gain control.