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Contract No. DE-AC36-08GO28308

Rotation Angle for the Optimum 

Tracking of One-Axis Trackers

William F. Marion and Aron P. Dobos

Technical Report

NREL/TP-6A20-58891

July 2013
NREL is a national laboratory of the U.S. Department of Energy

Office of Energy Efficiency & Renewable Energy

Operated by the Alliance for Sustainable Energy, LLC

This report is available at no cost from the National Renewable Energy

Laboratory (NREL) at www.nrel.gov/publications.

Contract No. DE-AC36-08GO28308

National Renewable Energy Laboratory

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Rotation Angle for the Optimum 

Tracking of One-Axis Trackers

William F. Marion and Aron P. Dobos

Prepared under Task No. SS13.5030

Technical Report

NREL/TP-6A20-58891

July 2013
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Cover Photos: (left to right) photo by Pat Corkery, NREL 16416, photo from SunEdison, NREL 17423, photo by Pat Corkery, NREL 

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iii

Nomenclature

β Surface tilt, angle from horizontal, 0° to +180°. 

βa Axis tilt, angle from horizontal of the inclination of 

tracker axis, 0° to +90°. 

γ Surface azimuth, angle clockwise from north of the 

horizontal projection of the surface normal, 0° to 

+360°. 

γa Axis azimuth, angle clockwise from north of the 

horizontal projection of the tracker axis, 0° to 

+360°. If the axis tilt is greater than zero, the vertex 

of the angle is at the inclined end of the axis.

γs Solar azimuth, angle clockwise from north of the 

horizontal projection of a ray from the sun, 0° to 

+360°. 

θ Incidence angle, angle between a ray from the sun 

and the surface normal, 0° to +180°. 

θz Zenith angle, angle between a ray from the sun and 

the vertical, 0° to +90°. 

R Rotation angle, angle of rotation of collector about 

axis when observed from the inclined end of axis, -

180° to +180°. Equals zero when the normal to the 

surface is in the vertical plane, clockwise is 

positive. 

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iv

Abstract

An equation for the rotation angle for optimum tracking of one-axis solar trackers is derived 

along with equations giving the relationships between the rotation angle and the surface tilt and 

azimuth angles. These equations are useful for improved modeling of the solar radiation 

available to a collector with tracking constraints and for determining the appropriate motor 

revolutions for optimum tracking. 

This report is available at no cost from the 

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at www.nrel.gov/publications.
1 

1 Introduction

The beam radiation on a tracking surface is maximized by orienting the surface, within the 

constraints of the tracking apparatus, so that the solar radiation incidence angle is minimized. 

The incidence angle, θ, is the angle between a ray from the sun and the normal to the surface. 

Braun and Mitchell (1983) provide expressions for the incidence angle in terms of the surface tilt 

and azimuth angles for fixed-tilt and for optimally tracking one- and two-axis surfaces.

This paper provides an alternative solution for one-axis trackers with the collector surface 

parallel to its axis. The minimum incidence angle is solved for by first determining the required 

rotation of the surface about its axis. Next, the surface tilt and azimuth are determined from the 

rotation angle and the tilt of the tracker axis. Finally, the value of the incidence angle is 

calculated from the surface tilt and azimuth angles and the zenith and solar azimuth angles. 

Although the rotation angle is an intermediate value for determining the incidence angle, it has 

applications of its own for the control of tracker movement and for modeling the solar radiation 

available for a collector. For a motorized tracker with fixed gearing, the tracker rotation is 

directly proportional to the number of motor revolutions; consequently, the calculated rotation 

angle can be used to determine the number of motor revolutions to move the tracker to its 

optimum position. When modeling collector solar radiation, the rotation angle can also be used 

to account for non-optimum tracking that may occur when the optimum rotation angle exceeds 

the rotation limits of the tracker.

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2 

2 Relationship Between Rotation Angle and Surface 

Tilt and Azimuth 

The surface tilt, β, and the surface azimuth, γ, are functions of the axis tilt, βa, the axis azimuth, 

γa, and the rotation angle, R. Figure 1 is used to determine the relationship between these angles. 

For the analysis, γa is the azimuth of the tracker axis when observed from the inclined end of the 

tracker axis, and R, also observed from the inclined end of the tracker axis, is positive for 

clockwise rotation and negative for counterclockwise rotation. R equals zero when the normal to 

the surface is in a vertical plane. In Figure 1, this normal is the unit normal represented by the 

line OA. Line OB is the unit normal rotated angle R about the axis. The triangles formed by the 

unit normals and the vertical axis are used to derive equations for the surface tilt and azimuth. 

Figure 1. Geometry for one-axis tracking surface

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3 

Recognizing that triangles AOC and DOE are similar triangles whose respective sides are 

proportional, the surface tilt is expressed as:

β = cos-1[cos R cos βa]. (1)

The surface azimuth differs from the axis azimuth by the angle BED. Angle BED equals 

sin-1[sin R ÷ sin β]. Consequently, the surface azimuth is expressed as:

γ = γa + sin-1[sin R ÷ sin β] For β ≠ 0, -90° ≤ R ≤ +90°. (2)

If β equals zero (horizontal surface), γ cannot be determined from equation (2). In this case, γ is

assigned any value because the surface is horizontal and assumed to have no azimuth response. 

Equation (2) will also not give the correct solution when R is outside the range of -90° to +90°

because it does not distinguish between trigonometric quadrants when performing the arcsine 

operation. R can fall outside the range of -90° to +90° when the solar azimuth differs by more 

than 90° from the axis azimuth and the axis tilt is greater than zero. Extreme cases are midnight 

sun conditions for northernmost locations, where R can range from -180° to +180° as the tracker 

follows a sun that never sets. 

For R values outside the range of -90° to +90°, either equation (3) or equation (4) applies. 

γ = γa - sin-1[sin R ÷ sin β] - 180° for -180° ≤ R < -90°. (3)

γ = γa - sin-1[sin R ÷ sin β] + 180° for +90° < R ≤ +180°. (4)

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4 

3 Rotation Angle for Optimum Tracking

Various sources give the following trigonometric relationship for the incidence angle 

(Iqbal 1983): 

cos θ = cos β cos θz + sin β sin θz cos(γs - γ). (5)

where θz and γs are the zenith and solar azimuth angles, and can be determined from the time 

and location using various algorithms, such as presented by Michalsky (1988) with Errata 

(1988). 

To introduce angle R into equation (5) and remove β and γ, substitutions for cos β, sin β, and 

cos(γs - γ) are made using equation (1) and equation (2) and various trigonometric identities. 

This procedure is shown in the appendix. The resulting expression for the cosine of the incidence 

angle is:

cos θ = cos R [sin θz cos (γs - γa ) sin βa + cos θz cos βa ] + sin R sin θz sin (γs - γa ). (6)

For optimum tracking, the value of R should give the minimum incidence angle, thereby 

maximizing the value of cos θ. This value of R is determined by differentiating equation (6) with 

respect to R, setting it equal to zero, and solving for R. 

d(cos θ )/dR = -sin R [sin θz cos (γs - γa ) sin βa + cos θz cos βa ] + cos R sin θz sin (γs - 

γa ) = 0

sin R/cos R = [ sin θz sin (γs - γa ) ] / [sin θz cos (γs - γa ) sin βa + cos θz cos βa ] 

R = tan-1( Χ ) + ψ , (7)

where:

Χ = [ sin θz sin (γs - γa ) ] / [sin θz cos (γs - γa ) sin βa + cos θz cos βa ] 

ψ = 0° if Χ = 0, or if Χ > 0 and (γs - γa) > 0, or if Χ < 0 and (γs - γa) < 0

ψ = +180° if Χ < 0 and (γs - γa) > 0

ψ = -180°, if Χ > 0 and (γs - γa) < 0 .

The variable ψ places R in the correct trigonometric quadrant. For determining which value of ψ

to use in equation (7), the difference γs - γa is evaluated as the angular displacement between two 

vectors so that it falls within the range of -180° to +180°. For example, if γs =20° and γa = 210°, 

γs - γa is evaluated as 20° + 360° - 210° = 170° and not 20° - 210° = -190°. 

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5 

4 Procedure for Determining Incidence Angle

The use of the preceding equations to determine the beam incidence angle on a one-axis tracking 

surface might be considered a series of steps: 

• Step 1 - Calculate θz and γs using algorithms such as those presented by Michalsky 

(1988) with Errata (1988) 

• Step 2 - Calculate R using equation (7)

• Step 3 - Calculate β using equation (1)

• Step 4 - Calculate γ using either equation (2), equation (3), or equation (4), as appropriate 

• Step 5 - Calculate θ using equation (5).

For concentrating collectors, the procedure can be simplified slightly by using equation (6) to 

calculate the incidence angle after R has been determined in Step 2. The surface tilt, β, although 

necessary to model the diffuse radiation for a flat-plate collector, has no role in determining the 

beam radiation for a concentrator; consequently, steps 3–5 are replaced by the use of 

equation (6).

The use of R permits better modeling of tracker performance because it can be compared to a 

tracker’s design parameters to see if R is within the range of the tracker’s physical rotation 

limits. If not, an additional step in the procedure can set R equal to the limit of the tracker 

rotation range before completing steps 3–5. A similar step could be performed using β, but R is 

more convenient because for a given tracker the physical limits of rotation, unlike the surface tilt,

do not change if the axis tilt is changed. 

The use of tracker rotation limits were evaluated for Boulder, Colorado, latitude = 40.0°N, and 

Barrow, Alaska, latitude = 71.3°N, for a flat-plate one-axis tracker with a south facing axis 

azimuth and tilted from the horizontal at an angle equal to the site latitude. Monthly and yearly 

radiation available to the collector were modeled using typical meteorological year hourly data 

(Marion and Urban 1995), the Perez diffuse radiation model (Perez et al. 1990), and the 

incidence angles calculated when two different rotation limits are imposed: -180° to +180° for 

unrestricted rotation and -70° to +70° to represent the physical limits of the tracker.

Compared to the use of unrestricted rotation limits, the use of the -70° to +70° rotation limits for 

the Barrow site reduced the available yearly radiation for the collector by 3.3% with a maximum 

monthly reduction of 5.5% occurring in June. For Boulder, the differences for the two rotation 

limits were minimal. The yearly collector radiation was reduced 0.3% and the June collector 

radiation was reduced 0.8%. Barrow showed larger differences for the two rotation limits 

because its more northerly location results in a wider range of spring and summer solar azimuths, 

which require a wider range of rotation limits for optimum tracking.

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6 

5 One-Axis Tracker With Horizontal Axis

Large one-axis trackers often have a horizontal axis because its simpler construction offers cost 

advantages. For a horizontal axis, the equations can be simplified because the range for R is -90°

to +90°, and sin βa and cos βa can be evaluated for βa = 0. This results in the following 

equations for R, β, and γ: 

R = tan-1 [ tan θz sin (γs - γa ) ] (8)

β = | R | (9)

γ = γa + sin-1[sin R ÷ sin β ] For β ≠ 0 . (10) 

If β equals zero, γ can be assigned any value because the surface is horizontal. The tracker axis 

azimuth, γa, can be assigned from either end of the tracker.

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7 

6 Conclusions

An equation for the rotation angle for optimum tracking of one-axis solar trackers has been 

derived along with equations giving the relationships between the rotation angle and the surface 

tilt and azimuth angles. General equations were presented for one-axis trackers with any axis tilt 

and azimuth as well as simplified equations for one-axis trackers with a horizontal axis.

The use of the rotation angle permits improved modeling of tracker performance because it can 

be compared to a tracker’s design parameters to see if it is within the range of the tracker’s 

physical rotation limits. If not, the rotation angle can be set equal to the limit of the tracker 

rotation range before calculating the surface tilt, azimuth, and incidence angles. As shown by the 

example, a tracker’s rotation limits are more significant for locations farther from the equator. 

This report is available at no cost from the 

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at www.nrel.gov/publications.
8 

References

Braun, J.A.; Mitchell, J.C. (1983). “Solar Geometry for Fixed and Tracking Surfaces.” Solar 

Energy (31); pp. 439–444. 

Errata. (1988). Solar Energy (41); p. 113. 

Iqbal, M. (1983). An Introduction To Solar Radiation. New York, NY: Academic Press, p. 23. 

Marion, W.; Urban, K. (1995). User’s Manual for TMY2s−Typical Meteorological Years 

Derived from the 1961−1990 National Solar Radiation Data Base. Golden, CO: National 

Renewable Energy Laboratory.

Michalsky, J.J. (1988). “The Astronomical Almanac’s Algorithm for Approximate Solar Position 

(1950−2050).” Solar Energy (40); pp. 227–235. 

Perez, R.; Inceichen, P.; Seals, R.; Michalsky, J.; Stewart, R. (1990). “Modeling Daylight 

Availability and Irradiance Components From Daylight and Global Irradiance.” Solar Energy

(44); pp. 271–289. 

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National Renewable Energy Laboratory (NREL) 

at www.nrel.gov/publications.
9 

Appendix

To introduce angle R into equation (5), substitutions for cos β, sin β, and cos(γs - γ) are made 

using equation (1) and equation (2) and various trigonometric identities.

From equation (1),

cos β = cos R cos βa . 

From the identity sin2 a + cos2 a = 1,

sin β = [ 1 - cos2 β ]1⁄2 = [ 1 - cos2 R cos2 βa ]1⁄2 . 

From equation (2) and the identity cos(a - b) = cos a cos b + sin a sin b, where a = γs - γa and 

b = ∠ BED,

cos(γs - γ) = cos (γs - γa ) cos (∠ BED) + sin (γs - γa ) sin (∠ BED) .

From equation (2), 

sin (∠ BED) = sin R/sin β = sin R/[ 1 - cos2 β ]1⁄2 

 = sin R / [ 1 - cos2 R cos2 βa ]1⁄2 . 

From the identity sin2 a + cos2 a = 1,

cos (∠ BED) = [ 1 - sin2 (∠ BED) ]1⁄2 = [ 1 - sin2 R / (1 - cos2 R cos2 βa) ]1⁄2

 = [ 1 - sin2 R / (1 - cos2 R cos2 βa) ]1⁄2 

Using a common denominator 

 = [ (1 - cos2 R cos2 βa - sin2 R ) / (1 - cos2 R cos2 βa) ]1⁄2 . 

Replacing sin2 R with 1 - cos2 R 

 = [ (1 - cos2 R cos2 βa - 1 + cos2 R) / (1 - cos2 R cos2 βa) ]1⁄2

 = [ - cos2 R cos2 βa + cos2 R) / (1 - cos2 R cos2 βa) ]1⁄2

 = [cos2 R ( 1 - cos2 βa) / (1 - cos2 R cos2 βa) ]1⁄2

 = [cos2 R sin2 βa / (1 - cos2 R cos2 βa) ]1⁄2

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10 

 = cos R sin βa / [1 - cos2 R cos2 βa ]1⁄2 . 

Therefore,

cos(γs - γ) = [ cos (γs - γa ) cos R sin βa + sin (γs - γa ) sin R ] / [ 1 - cos2 R cos2 βa ]1⁄2 . 

Replacing these expressions for cos β, sin β, and cos(γs - γ) in equation (5) yields the following 

expression for the cosine of the incidence angle: 

cos θ = cos R cos βa cos θz + sin θz [ cos (γs - γa ) cos R sin βa + sin (γs - γa ) sin R ] . 

Rearranging terms,

cos θ = cos R [sin θz cos (γs - γa ) sin βa + cos θz cos βa ] + sin R sin θz sin (γs - γa ) .

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