# Orientation and Exposure Efficiency of a Solar Tracking Surface in Clear Sky
## 4 Discussion
Figure 12: Orientation (**a**) and exposure (**b**) efficiency of vertical rotation surface in the periods (own research): summer solstice at the angle of inclination \(26.55^{\circ}\) and \(50^{\circ}\) respectively; 2—spring equinox \(\beta=50^{\circ}\); 3—winter solstice \(\beta=73.45^{\circ}\). A polar-rotating tracker is a device, in which the rotating axis is parallel to the earth axis and orients the tracking surface by the simplest synchronic algorithm with a permanent angle speed \(\omega=15\) grad/hour. In the literature, scientists only consider the variant of construction, in which the tracking surface is combined with the rotating axis \(\beta=\varphi\). Figure 13 presents the scheme of tracking, which corresponds to the period of equinox, when the end of the normal of a tracking surface traces a common path with the Sun. In its turn, it coincides with the equinoctial line. In the case of coinciding of the normal with the direction to the Sun, their azimuth angles are equal \(\gamma(\tau)=\gamma_{s}(\tau)\), and orientation efficiency of the surface during a day is permanent and equal to one. The current angle of inclination of the receiving surface is measured by evident equations:  
\[\mathbf{AB\perp OB\perp DB} \tag{36}\]  
\[OAB=\gamma \tag{37}\]  
\[tg\beta=\frac{AD}{AB} \tag{38}\]  
\[AD=AO\cdot tg\varphi \tag{39}\]  
\[AB=AO\cdot cos\gamma_{s} \tag{40}\]  
hence:  
\[tg\beta\ (\tau)=\frac{tg\varphi}{cos\gamma_{s}\ (\tau)} \tag{41}\]  
In other days of a year, pathways of the Sun and the normal of the tracking surface, combined with the rotating axis, do not coincide and conditions of irradiation are described by the general scheme, presented in Figure 14. The normal \(n\) of the tracking surface is demonstrated, not the very tracking surface. Daily path of the Sun along the skyline is marked by a continuous line of a standard width and the ends of the normal by a dash line, which coincides with the equinoctial line. Both trajectories stay in parallel planes, because their rotating centers are placed on a common polar axis \(OP\). Figure 13: Scheme of irradiation of the tracking surface of a polar-rotating tracker in the period of equinox—designed based on [21; 23; 35; 36]. In the equatorial system of sky coordinates, the synchronic tracking corresponds to the hourly angle \(\omega\tau\), which is the same for the Sun and for the normal and is marked on the scheme by heavy parcels of both trajectories, limited by the arc of an equatorial vertical \(KC_{n}CP\), supported on the diameter \(KL\). In the horizontal system of coordinates, different values of azimuth deflections of the normal \(\gamma\) and the Sun \(\gamma_{s}\) correspond to the same hour angles. The scheme demonstrates that they are limited by the points \(A\) and \(B\) of the intersection of the surface of the horizon with the surfaces of two horizontal verticals \(AC_{n}Z\) and \(BCZ\). Calculation of \(\gamma\) can also be performed for Equation (42) in case the zenith angle of the end of the normal is marked as \(\theta_{zn}\) in a period of a year with a conventional deflection \(\delta_{n}\). Thus, under equality of hour angles of the Sun and the normal, the following equation is performed:  
Figure 12:0  
hence,  
Figure 12:1  
Current values of the azimuths of the Sun and tracking surface of orientation and energy efficiency are presented for the period of summer solstice in Figure 12:2  
In case of combination of the tracking surface with the rotation axis, its normal \(\overrightarrow{n}\) crosses the line of the horizon at the point \(E\) six hours after the full sun, while the Sun stays at the point \(D\) at the crossing of the sun's trajectory with the arc of the equatorial vertical \(EPW\).