** **

Through detailed analysis, this paper clarifies that luminaires with the same beam angle may not have the same optical performance, and even have great differences. Clarified people's fuzzy understanding of the beam angle of the lamp, which is more reasonable and scientific in the lighting design.

**1 Introduction**

The beam angle is one of the most important technical parameters of the luminaire. It refers to the angle between the two directions of the light intensity equal to 50% of the maximum light intensity on one plane of the beam center line. (The definition of the beam angle is adopted by the International Commission on Illumination CIE. Instead of the North American Lighting Engineering Association IESNA's "10% Maximum Light Intensity" definition). According to the size of its value, it is usually divided into three types: narrow beam, medium beam and wide beam.

They can be visually distinguished by a light distribution curve. However, there is a potential problem here: Is the optical performance of the luminaire with the same beam angle the same? The purpose of this paper is to solve this problem through detailed analysis, which will help lighting designers to design lighting more rationally and scientifically.

**2, the mathematical expression of the light distribution curve**

In [3], the intensity of the axisymmetric light distribution luminaire in a certain direction is expressed as the following formula:

**I(Î¸)=I0(cosÎ¸)m (1)**

For the convenience of discussion, and without loss of generality, this paper takes the 30Â° beam angle as an example. In order to simplify the problem, only the symmetrical lamps with axial symmetry are discussed here.

It can be seen from the literature [3] that for the beam angle 2Î²=30Â°, Î²=15Â°, the index m:

m=lg0.5/lgcosÎ²=lg0.5/lgcos15Â°â‰ˆ19.9937

Set the lamp efficiency Î·=72%, then the center light intensity:

I0=(m+1)Î¦0Î·/2Ï€=(19.9937+1)*1000*72%/2Ï€â‰ˆ2405.7cd

Therefore, the light intensity expression:

**I(Î¸)=2405.7(cosÎ¸)19.9937 (2)**

The corresponding Cartesian coordinates and polar coordinate distribution curves are shown in Figure 1:

Figure 1 30 Â° beam angle lamp light distribution curve

In some cases, the formula (1) can well fit the light intensity IÎ¸ at the direction angle Î¸.

However, this formula does not cover all possible light distribution shapes.

Here, we discuss the following four typical light distribution curve shapes. For comparison, their beam angles are unified to 30Â°, and the power of all four lamps is 10W, and the lamp efficiency is 72%, that is, when the initial luminous flux is 1000LM, the luminous flux output is 720LM.

With the "Microsoft Mathematics" software, we fit each of the light distribution curves into a mathematical expression in order to give a precise conclusion.

For each light intensity expression of each luminaire, check whether the beam angle is 30Â°, and accurately calculate the luminous flux output by the following formula (3). The derivation process of formula (3) and the symbolic meaning are shown in [3].

**Î¦=2Ï€0Â°90Â°IÎ¸sinÎ¸dÎ¸ (3)**

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