Power
Available Constraints
List of Symbols
E Energy (joules)
r Packing Factor (cell
area/entire side area)
h Solar Cell Efficiency (per cell)
S Solar Constant 1367 W/m2 or 0.8819 W/in2
A Effective Area (area normal to the sun)
P Period of Orbit (seconds)
f Fraction of orbital period in sun
At Total area for a side
q Angle of incidence (angle between sun vector and vector normal to area surface)
w Angular velocity of satellite (rad/s)
t Period of one revolution of satellite (s)
Orbit
The satellite will be delivered to a 492 km. altitude orbit. From the calculations in Appendix A (1.1):
Event Time Percent of Orbit
Length of orbit: 5670 s 100%
Eclipse: 2150 s 38%
Sun: 3520 s 62%
Table 1. Orbit Times
(These are based on worst case eclipse occurring twice yearly)
The energy equation for a solar cell is:
The packing factor and the cell efficiency are based on the type of cell used. The effective area is based on the number of sides facing the sun and their orientation.
The effective area is based on the entire cross section of the satellite facing the sun in one of two possible cases.
Orientation
Case 1 provides the maximum effective area at 666.3 in2. This is the most optimistic scenario, but unlikely in that the satellite would have to be in a torque-free configuration and have no rotation about its major axis so that it would have to remain at precisely the attitude indicated in Figure 1.
The worst-case scenario would be for the satellite to orbit with its end always facing the sun. No cells would be facing the sun to generate power. This event is unlikely for the same reasons as the satellite remaining in a Case 1 attitude.
The satellite will most likely tumble in orbit, and not be stuck in any one orientation. The effective area will vary as the satellite rotates. Case 2 provides an effective area of 615.6 in2. Rotation about the major axis, with that axis perpendicular to the sun vector yields the minimum average power.
Figure 1. Effective Area
The effective area will oscillate between zero for the end-on orientation, and either Case 2 for the side-on orientation. The energy for one rotation about the major axis was calculated to be:
When computed over the length of an entire orbit, the angular velocity of rotation cancels out and:
Case 1 Orientation Factor: 2.6
Case 2 Orientation Factor: 2.4
Rotation Factor: (2/pi)
The following tables compare the trade-offs associated with packing factor and power/energy produced for several common efficiencies of solar cells.
|
Average
Power (W) |
|
|
|
Efficiency |
|
|
|
|
|||
|
|
|
Silicon |
|
|
GaAs |
|
|
|
|
|
|
|
Packing
Factor |
|
0.12 |
0.14 |
0.16 |
0.19 |
0.21 |
0.23 |
0.25 |
0.26 |
0.27 |
0.28 |
|
0.20 |
|
5.1 |
5.9 |
6.8 |
8.1 |
8.9 |
9.8 |
10.6 |
11.0 |
11.5 |
11.9 |
|
0.25 |
|
6.4 |
7.4 |
8.5 |
10.1 |
11.2 |
12.2 |
13.3 |
13.8 |
14.3 |
14.9 |
|
0.30 |
|
7.7 |
8.9 |
10.2 |
12.1 |
13.4 |
14.7 |
16.0 |
16.6 |
17.2 |
17.9 |
|
0.35 |
|
8.9 |
10.4 |
11.9 |
14.1 |
15.6 |
17.1 |
18.6 |
19.3 |
20.1 |
20.8 |
|
0.40 |
|
10.2 |
11.9 |
13.6 |
16.1 |
17.8 |
19.5 |
21.2 |
22.1 |
22.9 |
23.8 |
|
0.45 |
|
11.5 |
13.4 |
15.3 |
18.2 |
20.1 |
22.0 |
23.9 |
24.9 |
25.8 |
26.8 |
|
0.50 |
|
12.7 |
14.9 |
17.0 |
20.2 |
22.3 |
24.4 |
26.6 |
27.6 |
28.7 |
29.7 |
|
0.55 |
|
14.0 |
16.4 |
18.7 |
22.2 |
24.5 |
26.9 |
29.2 |
30.4 |
31.6 |
32.7 |
|
0.60 |
|
15.3 |
17.8 |
20.4 |
24.2 |
26.8 |
29.3 |
31.9 |
33.1 |
34.4 |
35.7 |
|
0.65 |
|
16.6 |
19.3 |
22.1 |
26.2 |
29.0 |
31.8 |
34.5 |
35.9 |
37.3 |
38.7 |
|
0.70 |
|
17.8 |
20.8 |
23.8 |
28.3 |
31.2 |
34.2 |
37.2 |
38.7 |
40.2 |
41.6 |
|
0.75 |
|
19.1 |
22.3 |
25.5 |
30.3 |
33.5 |
36.7 |
39.8 |
41.4 |
43.0 |
44.6 |
|
0.80 |
|
20.4 |
23.8 |
27.2 |
32.3 |
35.7 |
39.1 |
42.5 |
44.2 |
45.9 |
47.6 |
|
0.85 |
|
21.7 |
25.3 |
28.9 |
34.3 |
37.9 |
41.5 |
45.2 |
47.0 |
48.8 |
50.6 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Table 2. Average Power
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Energy
(J) |
|
|
|
|
|
Efficiency |
|
|
|
|
|
|
|
|
Silicon |
|
|
GaAs |
|
|
|
|
|
|
|
Packing
Factor |
|
0.12 |
0.14 |
0.16 |
0.19 |
0.21 |
0.23 |
0.25 |
0.26 |
0.27 |
0.28 |
|
0.20 |
|
29,000 |
34,000 |
39,000 |
46,000 |
51,000 |
55,000 |
60,000 |
63,000 |
65,000 |
67,000 |
|
0.25 |
|
36,000 |
42,000 |
48,000 |
57,000 |
63,000 |
69,000 |
75,000 |
78,000 |
81,000 |
84,000 |
|
0.30 |
|
43,000 |
51,000 |
58,000 |
69,000 |
76,000 |
83,000 |
90,000 |
94,000 |
98,000 |
101,000 |
|
0.35 |
|
51,000 |
59,000 |
67,000 |
80,000 |
89,000 |
97,000 |
105,000 |
110,000 |
114,000 |
118,000 |
|
0.40 |
|
58,000 |
67,000 |
77,000 |
92,000 |
101,000 |
111,000 |
120,000 |
125,000 |
130,000 |
135,000 |
|
0.45 |
|
65,000 |
76,000 |
87,000 |
103,000 |
114,000 |
125,000 |
135,000 |
141,000 |
146,000 |
152,000 |
|
0.50 |
|
72,000 |
84,000 |
96,000 |
114,000 |
126,000 |
138,000 |
151,000 |
157,000 |
163,000 |
169,000 |
|
0.55 |
|
79,000 |
93,000 |
106,000 |
126,000 |
139,000 |
152,000 |
166,000 |
172,000 |
179,000 |
185,000 |
|
0.60 |
|
87,000 |
101,000 |
116,000 |
137,000 |
152,000 |
166,000 |
181,000 |
188,000 |
195,000 |
202,000 |
|
0.65 |
|
94,000 |
110,000 |
125,000 |
149,000 |
164,000 |
180,000 |
196,000 |
204,000 |
211,000 |
219,000 |
|
0.70 |
|
101,000 |
118,000 |
135,000 |
160,000 |
177,000 |
194,000 |
211,000 |
219,000 |
228,000 |
236,000 |
|
0.75 |
|
108,000 |
126,000 |
145,000 |
172,000 |
190,000 |
208,000 |
226,000 |
235,000 |
244,000 |
253,000 |
|
0.80 |
|
116,000 |
135,000 |
154,000 |
183,000 |
202,000 |
222,000 |
241,000 |
250,000 |
260,000 |
270,000 |
|
0.85 |
|
123,000 |
143,000 |
164,000 |
194,000 |
215,000 |
235,000 |
256,000 |
266,000 |
276,000 |
287,000 |
|
|
|
|
|
|
|
|
|
|
|
|
|
Table 3. Energy
Table 2 and Table 3 were calculated based on rotation about the major axis, a Case 2 effective area, and AM0 industry standard solar conditions (solar constant equal to 0.873 W/in2 or 1560 W/m2).