Tilt Factor Calculator

The tilt factor is denoted by r_b and is a geometric ratio used by solar engineers to convert the amount of radiation measured on a horizontal sensor into the amount striking an inclined surface. Most weather stations measure radiation with flat, horizontally oriented sensors while arrays installed on roofs or in the ground are tilted toward the equator so these values do not match. This calculator supports both a simple cosine formula and a format used internally by programs like PVsyst and SAM that take latitude, declination, and hour angle into account.

What is the tilt factor?

r_b is a dimensionless number that compares the (direct) radiation flux on an inclined surface to the radiation flux on a horizontal surface at the same time. Clouds or fog do not play any role. It's a purely geometrical relationship, depending on the angles which the sun rays form with each surface.

When r_b is greater than one, the tilted surface collects more energy than a flat surface. This happens on winter mornings and afternoons when the sun's altitude is low and the tilted panels are pointed directly at the sun's rays. When r_b is less than one, the panel collects less energy than a flat surface. An example of this is summer midday when the sun is nearly overhead. Because of this ratio, a 30-degree tilt angle for a panel in temperate latitudes will outperform a flat panel most of the year.

How to use this calculator:

Select the method that matches your data. If you already know the solar altitude and azimuth angles, use the angle-based method. You can calculate any of the values. Just leave one value blank and enter the other two.

For construction, you should use a location-based method. By entering the latitude, panel tilt angle, solar declination for that day and hour angle, this calculation tool will give you directly the value of r_b. When the radiation intensity map is enabled, horizontal measurements can be converted to values for inclined surfaces. By enabling the graph, you can check the tilt-angle factor over the course of a day or at different tilt angles.

Understanding of formula

The simple form is a ratio of two cosine values.

For collectors facing south in the northern hemisphere, the detailed form directly employs the geographic and geometric relationship to the sun.

In this case, phi is the latitude, beta is the panel tilt angle, delta is the solar declination and omega is the hour angle. A simple example: if a panel has an incidence angle of 35 degrees and a zenith angle of 60 degrees then r_b = cos 35 / cos 60 = 0.819 / 0.500 = 1.638. If a weather station measures the direct radiation on flat ground as 500 W/m2, then the received radiation on the tilted surface is 500 x 1.638 = 819 W/m2. At the same solar irradiance, the energy on the panel is about 64 percent higher.

Practical applications

The solar simulation tool performs this calculation for every hour of a standard year and converts the measured horizontal irradiance into numerical values for flat-plate arrays, providing the basis for a performance model for PV systems. Designers can use this tool to determine the optimal fixed tilt angle for a given location. By varying the angle β and searching for the one that gives the highest average annual value, it is usually found within a few degrees of latitude. Single-axis tracking systems perform the same process in real time to minimize the incidence angle and keep the tilt as high as possible. Off-grid system sizing and large winter tilt angles designed to shed snow from the array are also based on this relationship.

Tips to improve accuracy:

For calculating the hour angle, solar time must be used and not clock time. Because of time zone differences and the equation of time, these two values can differ by more than an hour. The declination of the sun changes daily so either Cooper's formula should be used or values from an astronomical almanac should be used. 23.45 degrees is the extreme value at solstice and not a constant.

Since a south orientation is assumed here, additional terms for the surface orientation must be added for east or west orientations. In addition, r_b only represents the light flux component. Diffuse radiation and reflected radiation each have their own geometric factors.

This calculator is for general engineering and educational purposes. To obtain results that can be used in credit decisions, model the total irradiance of a flat array including diffuse and reflected components with a dedicated solar simulation tool.