# Cubic Composite Sensor with Photodiodes for Tracking Solar Orientation

Authors: Yong-Nong Chang
Journal: Journal of Nanomaterials
 (2013)

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## Abstract

A cubic composite solar sensor with photo diode is proposed for tracking the relative solar orientation. The proposed solar sensor composes of five photodiode detectors which are placed on the front, rear, left, right, and horizontal facets in a cubic body, respectively. The solar detectors placed on five facets can detect solar power of different facets. Based on the geometric coordinate transformation principle, the relationship equations of solar light orientation between measured powers with respect to various facets can be conducted. As a result, the solar orientation can be precisely achieved without needing any assistance of electronic compass and extra orientation angle corrector. Eventually, the relative solar light orientation, the elevation angle, and azimuth angle of the solar light can be measured precisely.


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## Body

## 1. Introduction

Recently, the renewable energy has drawn great attention due to energy shortage. Many renewable energy researches are focused on solar power system [1–4]. For the solar system, a precise sensor is indispensable to tracking the sunlight orientation. Many studies have been focusing on solar tracking technique. However, there still exist some drawbacks on solar light tracking apparatus. Solar light tracker can, in general, be classified into mechanical and electronic category [5, 6]. Mechanical solar tracker experiences much fault probability and maintenance cost, electronic solar tracker depends highly on electronic compass accompanied by perpetual calendar, and precise perpetual calendar heavily relies on complicated computation through the usage of internet, thus leading to inconvenience in application [7, 8]. Therefore, to develop an electronic solar light tracker, without needing electronic compass and perpetual calendar, is essential to researchers and solar power applications.An electronic solar light tracker can only be realized through the application of photo detecting devices. There are many photo devices such as photodiode, photo transistor, and resistive CdS components. The CdS components are passive components and usually cannot play the key role in a precise measurement. In this paper, the active photodiodes with wide measuring range and fast response and low dark current are applied as a solar sensor to measure the solar orientation.The solar light orientation information should consist of the relative elevation angle and azimuth angle between solar source and photo detector. For collecting lights from all possible orientations, this paper employed five photodiodes to compose a cubic structure photo detector in the absence of electronic compass and perpetual calendar to accomplish an electronic solar light orientation tracking system. The composite sensor is composed of five photo diode detectors placed on five different facets: left, right, front, rear, and horizontal facets, to constitute a cubic structure solar light detector. By placing five detectors on five different facets, the sunlight orientation can be derived by geometric coordinate transformation technique.There exists discrepancy in collecting power amounts for photo detectors on different facets. Moreover, the incident light on the photo diode comprises not only direct solar light but also diffuse light and scattered light [8]. The diffuse and scattered light power is also generally called undisciplined light power, denoted by PU. Therefore, the received solar power should be categorized precisely into power of direct incident solar light PD and power of undisciplined incident solar light PU. Through the geometric transformation analysis and accurate evaluation formula, each collecting solar power on different facets can be divided into direct incident solar power PD and undisciplined incident solar power PU, to accomplish the calculation of solar light orientation and solar polar powers on all facets.In this paper, a novel light orientation sensor is proposed to track the solar orientation effectively. No expensive electronic components and no perpetual calendar are required. By appropriate computation based on different solar power quantities collected from five sensors, the sunlight direction can be determined. Also, the solar light orientation and power can easily be attained.


## 2. Optimal Shunt Resistor Matching and Selection

The ideal photo diode should generate an active current source which is proportional to the incident solar light. The total current through the photodiode is the sum of the dark current and the photo active current, so the dark current must be minimized to maximize the sensitivity of the device. In order to obtain the correct photo voltage, an appropriate shunt resistorRP should be shunted across the output terminals. The application circuit is shown in Figure 1 with external shunt resistor RP; the photo voltage is generated at the two ends of the photo diode. The dark current has essential effects on the output voltage. Therefore, a calibration process for the shunt resistor RP is required.Figure 1
Photo diode applied circuit for detecting solar power.The values of the output load resistorRP. The adopted photo detector in this paper is produced by Hamamatsu Inc. with model code S1087-01. The parameters of the shunt resistor are assorted at 10 Ω, 100 Ω, 220 Ω, 510 Ω, 1 kΩ, and 2 kΩ.The solar power is defined asPsolar, the output voltage is defined as Vout, and S is defined as the sensitivity of the detector. In regard to an ideal photo power detector, S should be kept with a constant quantity to maintain the linear relationship between sensing power and sensor output voltage as follows. 						(1)PSolar=VoutS.By shunting appropriate resistorRP in the output terminal, the solar power can be measured as shown in Figure 2, in case that ill condition occurs. The resistor value cannot be used to convert the current into voltage correctly. As shown in Figure 2, during the lower solar power period in the morning or evening periods, the solar power is roughly proportional to the photo current with resistor ranging from 100 Ω to 220 Ω. However, the output power is slightly lower than the standard value during midday with direct sunlight. When the value is above 2 kΩ, the solar power curve lies under the standard curve. When the resistor value is between 510 Ω and 1 kΩ, the output curve is inspected to be better fitting the theoretical solar power.Figure 2
Optimal resistor matching and selection.Since the photo diode has nonlinear equivalent characteristics as shown in Figure2, an appropriate output resistor is required to be selected. When the sunlight is in either weak condition or strong condition, the solar power curve displays nonlinear phenomenon. When the value falls between 510 Ω~1 kΩ, the curve has the best coincident condition with the standard curve. Considering that the identical solar power would generate identical output voltage, larger resistance will require less output current. Therefore, 1 kΩ resistor is selected to shunt with photo detector to enhance the linearity and precision of the proposed circuit.In order to obtain the precise voltage conversion, an OPAMP circuit is designed to ensure the precise voltage gain for the five detectors. The calibration circuit of the detector is shown in Figure3. The 0.01 μF capacitor is connected across shunt resistor RP to eliminate noise.Figure 3
Detector calibration circuit.


## 3.  Geometric Coordinate Transformation and Formulation

As designated in Figure4, photo-detecting device is located at origin of the horizontal plane of 3D coordinate. The polar coordinate representation of position of light source is designated as (r,ϕ,α) which results in the Cartesian coordinate and polar coordinate transformation relationship expressed as follows:
(2)x=rsinϕcosα,y=rsinϕsinα,z=rcosϕ,
where ϕ is the light incident angle, α is the azimuth angle, and r is the distance between light source and photo diode detector. Then, the elevation angle ϕh can be described as
(3)ϕh=90°-ϕ.
The orientation of sunlight is defined as elevation angle ϕh and azimuth angle α that means the orientation of the sunlight can be specified by formulation using the two angles: ϕh and α.Figure 4
Geometric coordinate system.For a 3D coordinate system, the rear direction is defined as theX direction, the right direction is defined as the Y direction, and the normal vector direction of X-Y horizontal plane is defined as the Z direction, as indicated in Figure 4. The solar light incident angle ϕ can also be defined as the angle between the sunlight and the normal vector direction of horizontal plane. Thus, the horizontal detector collects direct light power PHD which can be expressed as
(4)PHD=PDcosϕ.
The collected direct light power PHD of horizontal detector is the multiplication of solar direct radiation power and cosine of incident angle ϕ. Incident angle ϕ is defined as the angle between the normal vector of each facet with its corresponding incident light. Thus, when the horizontal detector rotates an angle θ along X, Y, and Z axis, the associated incident angles of all facets will change accordingly. Figure 5 illustrates the rotational coordinate transformation by rotating along Z-axis with an angle θZ or along X-axis with θx or along Y-axis with θy. By doing so, the original coordinate (x, y, z) is transformed to a new coordinate (x1, y1, z1), and the expression can be written as follows.(1)
By rotatingθz along z-axis, x1, y1, and z1 can be obtained as follows
(5)[x1y1z1]=[cosθz-sinθz0sinθzcosθz0001][xyz]=r[sinϕcosαcosθz-sinϕsinαsinθzsinϕcosαsinθz+sinϕsinαcosθzcosϕ].In (5), new coordinate z1 is identical with the original coordinate z. The angle between incident solar light and normal vector of detector plane of new coordinate still remains ϕ; the projection on normal vector direction is rcosϕ. Therefore, regardless of the rotating angle θz, the incident of solar light keeps unchanged, that is, ϕ.(2)
By rotatingθx along x-axis, x1, y1, and z1 can be obtained as
(6)[x1y1z1]=[1000cosθx-sinθx0sinθxcosθx][xyz]=r[sinϕcosαsinϕsinαcosθx-cosϕsinθxsinϕsinαsinθx+cosϕsinθx].The projection component of solar light on the normal vector of new coordinate planez1 deviates a significant quantity from the original projection value z, so does the incident angle.(3)
By rotatingθy along y-axis, x1, y1, and z1 can be obtained as
(7)[x1y1z1]=[cosθy0sinθy010-sinθy0cosθy][xyz]=r[sinϕcosαcosθy+cosϕsinθysinϕsinα-sinϕcosαsinθy+cosϕcosθy].Like that of rotation alongX-axis, the projection of incident light on normal vector of new coordinate plane z1 deviates noticeably from original one, z. It implies that incident angle changes with rotation angle θy.Figure 5
θ rotation along the X-, Y-, and Z-axis.After installing five detectors on five different facets, the cubic composite sensor is shown in Figure6, Aside from the horizontal facet detector, the other four detectors can be looked upon as those coordinates transformed from the horizontal one by rotating either along X-axis or Y-axis with ±90°. The coordinate transformation can be derived in the following section.Figure 6
Cubic sensors model in a 3D coordinate system.


## 4. Cubic Composite Sensor

The relationship between direct incident solar powerPD, undisciplined power PU, the light incident angle ϕ, and the azimuth angle α can be discussed in this section. Since the five detectors are located on five different facets of a cubic box, the coordinate transformation relationships between five facets will be discussed below.(1)
The horizontal (H) detector is located on the facet which θx=0° along z-axis. Therefore, the transformed orientation coordinate of the sunlight and corresponding collecting power can be derived as
(8)[x1y1z1]=r[sinϕcosαcos0°-sinϕsinαsin0°sinϕcosαsin0°+sinϕsinαcos0°cosϕ]=r[sinϕcosαsinϕsinαcosϕ]=[xyz].The normal vector of horizontal planez1 is the same with the original normal vector z; the incident angle is kept unchanged, that is, ϕH=ϕ,
(9)ϕH=ϕ,(10)PH=PDcosϕ+PUH=PDcosϕH+PUH.The collected powerPH on the horizontal detector can be expressed as (10). Wherein, PD is the maximum direct incident solar power and PUH is the power caused by light diffusion and scattering.(2)
The right sensor is located on facet whichθx=90° along x-axis. Therefore, the orientation of the sunlight can be derived as
(11)[x1y1z1]=r[sinϕcosαsinϕsinαcos90°-cosϕsin90°sinϕsinαsin90°+cosϕcos90°]=r[sinϕcosα-cosϕsinϕsinα]=[x-zy].As shown in (11), right facet detector is equivalent to rotating horizontal facet by θx=90°. The normal vector of right plane is directed toward Z-axis which is the positive Y-axis direction of horizontal facet. The light incident angle ϕR is related to original incident angle ϕ by (12). Next, the collected power on the right facet detector PR is written as
(12)ϕR=cos-1(sinϕsinα),(13)PR=PDsinϕsinα+PUR=PDcosϕR+PUR,wherePUR is the undisciplined light power collected by right detector.(3)
The left sensor is located on facet whichθx=-90° along x-axis. Therefore, the orientation of the sunlight can be derived as
(14)[x1y1z1]=r[sinϕcosαsinϕsinαcos(-90°)-cosϕsin(-90°)sinϕsinαsin(-90°)+cosϕcos(-90°)]=r[sinϕcosαcosϕ-sinϕsinα]=[xz-y].As shown in (14), left facet detector is equivalent to rotating horizontal facet by θx=-90°. The normal vector of left plane is directed toward the negative Y-axis direction of horizontal facet. The light incident angle ϕL is related to original incident angle ϕ by (15). More, the collected power on the right facet detector PL is written as
(15)ϕL=cos-1(-sinϕsinα),(16)PL=-PDsinϕcosα+PUL=PDcosϕL+PUL,wherePUL is the undisciplined light power collected by left detector.(4)
The front sensor is located on facet whichθy=90° along y-axis. Therefore, the orientation of the sunlight can be derived as
(17)[x1y1z1]=r[sinϕcosαcos90°+cosϕsin90°sinϕsinα-sinϕcosαsin90°+cosϕcos90°]=r[cosϕsinϕsinα-sinϕcosα]=[zy-x].As expressed in (16), front facet detector is equivalent to rotating horizontal facet by θy=90°. The normal vector of front plane is directed toward the negative X-axis direction of horizontal facet. The light incident angle ϕF is related to original incident angle ϕ by (18). Next, the collected power on the front facet detector PF is written as
(18)ϕF=cos-1(-sinϕcosα),(19)PF=-PDsinϕcosα+PUF=PDcosϕF+PUF,wherePUF is the undisciplined light power collected by front detector.(5
) The rear sensor is located on facet whichθy=-90° along y-axis. Therefore, the orientation of the sunlight can be derived as
(20)[x1y1z1]=r[sinϕcosαcos(-90°)+cosϕsin(-90°)sinϕsinα-sinϕcosαsin(-90°)+cosϕcos(-90°)]=r[-cosϕsinϕsinαsinϕcosα]=[-zyx].As shown in (20), rear facet detector is equivalent to rotating horizontal facet by θy=-90°. The normal vector of rear plane is directed toward the positive X-axis direction of horizontal facet. The light incident angle ϕRe is related to original incident angle ϕ by (21). Next, the collected power on the rear facet detector PRe is written as
(21)ϕRe=cos-1(sinϕcosα),(22)PRe=PDsinϕcosα+PURe=PDcosϕRe+PURe,wherePURe is the undisciplined light power collected by rear detector.Figure6 defines incident quadrant of the solar light. The assessing criteria of incident quadrant are based on the collecting light power of five facets PH, PR,  PL, PF, and PRe, respectively. By comparing the quantities of the collecting light power, the criteria are described as follows.(1)
When thePR>PL and PF>PRe, the sunlight comes from the first quadrant.(2)
When thePL>PR and PF>PRe, the sunlight comes from the second quadrant.(3)
When thePL>PR and PRe>PF, the sunlight comes from the third quadrant.(4)
When thePR>PL and PRe>PF, the sunlight comes from the fourth quadrant.Once the solar light incident quadrant is determined, the solar power orientation and power information can be obtained by substituting (8) in to (22). When the sunlight is incident to the fourth quadrant, the relationships of ϕF, ϕR, α, and PD can be conducted from (12) to (15) to arrive at (23) as follows:
(23)PDcosϕRe=PDsinϕcosα,PDcosϕR=PDsinϕsinα.
Since the incident light falls on the fourth quadrant, the direct light cannot reach front and left facets. The collecting solar power can merely come from the undisciplined light power. Consequently, the undisciplined light power of (23) can be roughly substituted by collected power quantity of adjacent facets. In other words, the direct solar light falling on the fourth quadrant comes from right and rear directions; the undisciplined light power on the rear facet PURe is approximated by left facet light power PL; the undisciplined light power on the right facet PUR is approximated by front facet light power PF.From (23), the α angle can be derived as follows
(24)PRDPReD=PDsinϕsinαPDsinϕcosα=sinαcosα=tanα⟹α=tan-1PRDPReD.
From (10) and (22), the ϕ angle can be derived as follows
(25)PReDPHD=sinϕcosαcosϕ=tanϕcosα⟹ϕ=tan-1PReDPHDcosα.
After the ϕ angle is decided, the PD can be derived as
(26)PD=PHDcosϕ.
The incident quadrant of incident light is changing with time. Similar derivation can be used to obtain the solar light orientation and power information.


## 5. Experimental Results

In order to verify the formulation, an experimental measurement was performed on June 02, 2013. Table1 lists the collecting solar power of all facet detectors on the hour.Table 1
Solar power with respect to five detectors (unit: (W/m2)).
Time
Facet
Light incident quadrant
P
R
P
Re
P
L
P
F
P
H
08:00
454.7
105.1
92.4
164.1
220.2
I
09:00
526.7
135.3
113.3
170.9
390.2
I
10:00
467.0
128.9
120.4
172.5
552.8
I
11:00
439.9
136.4
127.4
149.0
703.0
I
12:00
267.5
130.0
121.1
118.6
815.9
IV
13:00
91.1
111.9
103.7
92.6
953.9
III
14:00
141.4
147.8
317.3
112.7
860.4
III
15:00
143.3
136.7
452.9
148.8
691.6
II
16:00
168.6
158.4
511.5
198.2
582.9
IIIn Table1, to take data at 12:00 A.M. as an example, the cubic sensor receives the incident light from the fourth quadrant, the detectors on the right, rear, and horizontal facets can absorb power of direct incident solar light PD. However, the left and front facets can only absorb power of undisciplined incident solar light PU. Therefore, the collecting solar power on the left detector PL is near the undisciplined power PUL. Also, the collecting solar power on the front detector PF can be approximated by the undisciplined power PUF. The collected undisciplined power on the horizontal detector equals around the average of these two quantities. Besides, any detector, with direct incident solar power, possesses collected undisciplined solar power which is similar with the power collecting environment of adjacent facet detector. Thus, the collected undisciplined solar power can be substituted by collecting solar power on adjacent facet detector. It means that PURe equals PL and PUR equals PUF. Table 2 lists the undisciplined powers of each facet on the hour. Table 3 lists the direct incident solar power of each detector which is obtained by deducting undisciplined power from measured power. Therefore, the listed quantities in Table 1 are equal to the summation of those corresponding entities in Tables 2 and 3.Table 2
Undisciplined incident solar light power with respect to five detectors (unit: (W/m2)).
Time
Facet
Light incident quadrant
P
UR
P
URe
P
UL
P
UF
P
UH
08:00
105.1
105.1
92.4
92.4
98.7
I
09:00
135.3
135.3
113.3
113.3
124.3
I
10:00
128.9
128.9
120.4
120.4
125.7
I
11:00
136.4
136.4
127.4
127.4
131.8
I
12:00
118.6
121.1
121.1
118.6
119.9
IV
13:00
91.1
91.1
92.6
92.6
91.8
III
14:00
141.4
141.4
112.7
112.7
127.1
III
15:00
143.3
136.7
136.7
143.3
140.0
II
16:00
168.6
158.4
158.4
168.6
163.5
IITable 3
Direct incident solar light power with respect to five detectors (unit: (W/m2)).
Time
Facet
Light incident quadrant
P
RD
P
ReD
P
LD
P
FD
P
HD
08:00
349.6
0
0
71.7
121.5
I
09:00
391.4
0
0
57.6
265.9
I
10:00
338.1
0
0
52.1
427.1
I
11:00
303.5
0
0
21.6
571.2
I
12:00
148.9
8.9
0
0
696.0
IV
13:00
0
20.8
11.1
0
862.1
III
14:00
0
6.4
204.6
0
733.3
III
15:00
0
0
316.2
5.5
551.6
II
16:00
0
0
353.1
29.6
419.4
IIBased on the direct incident solar light power on five facets listed in Table3, the orientation angles at different times can be evaluated. Taking the measured power at 12:00 A.M. on the selected day as an example, the azimuth angle and elevation angle are calculated and obtained as follows.According to (24), it arrives at α:
(27)α=tan-1PRDPReD=tan-1148.98.9=86.58°.
Using (25), incident angle ϕ can be obtained as
(28)ϕ=tan-1PRDPHDsinα=12.1°.
Elevation angle is the complimentary angle of incident angle; thus,
(29)ϕh=90-ϕ=77.9°.
The theoretical azimuth angle and the computed azimuth angle are shown in Table 4, and the theoretical elevation angle and the computed elevation angle are shown in Table 5, respectively. Theoretical azimuth angles and elevation angles are obtained from perpetual calendar.Table 4
Azimuth angles comparison between theoretical and computed values.
Time
Theoretical azimuth anglesα
Computed azimuth anglesα
Error
08:00
105.14°
101.59°
3.55°
09:00
100.82°
98.37°
2.45°
10:00
96.72°
98.76°
−2.04°
11:00
92.30°
94.07°
−1.77°
12:00
85.55°
86.58°
−1.03°
13:00
331.06°
331.91°
−0.85°
14:00
273.15°
271.79°
1.36°
15:00
267.02°
269.00°
−1.98°
16:00
262.70°
265.21°
−2.51°Table 5
Elevation angles comparison between theoretical and computed values.
Time
Theoretical elevation anglesϕh
Computed elevation anglesϕh
Error
08:00
22.61°
18.80°
3.81°
09:00
36.00°
33.91°
2.09°
10:00
49.57°
51.31°
−1.74°
11:00
63.25°
61.96°
1.29°
12:00
76.98°
77.90°
−0.92°
13:00
88.14°
88.43°
−0.29°
14:00
75.20°
74.40°
0.80°
15:00
61.48°
60.17°
1.31°
16:00
47.81°
49.81°
−2.00°In this paper, the light diffusion factor is neglected and not taken into the formulation. Only simple constant offset value is used to present the diffusion power. In Tables4 and 5, the elevation angles and azimuth angles are quoted and calculated. In Table 4, by comparing the theoretical azimuth angles and computed azimuth angles, the errors are kept below 3.55°. In Table 5, by comparing the theoretical elevation angles and the computed elevation angles, the errors are also kept below 3.81° which is within acceptable range.


## 6. Conclusion

This paper has successfully proposed a cubic composite sensor with five sensors placing on five different facets. By coordinate transformation and detailed formulation, the azimuth angles and elevation angles for different sunlight orientations can be derived. Accordingly the sunlight orientation can be determined. No electronic compass and CdS resistors are required in the mechanism. Results show that the proposed method has precise solar tracking performance. This proves the validity of the proposed formulation to estimate the sunlight orientation and solar power. Besides, the solar light power can also be measured and provide valuable information for the solar tracking system.



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