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I have two identical scenes, one in BI and one in Cycles. They both have point lamps in the same place (or, in the case of a linked scene, it's the same lamp with different settings in each). The BI lamp's strength is controlled by the Energy field. The Cycles lamp's strength is controlled by the Strength value of it's emission node. They are both pure white. All other settings are default.

What values of Energy and Emission strength will give me the same amount of light? Will other settings, such as Size in Cycles, change this? Does it depend on materials (assuming no other emitters/lights, or reflective surfaces, etc. Light bounces will also make a difference, but we can do this without those for now.)

For example, my Cycles version currently has an emission strength of 500. My BI version has energy of 1. The BI lamp seems to be significantly brighter than the Cycles!

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Cycles is a physically based renderer, whereas Blender internal is not (it is a rasterizer I think), the two are fundamentally different, and hence no real relationship can be established between them.

Lamps in Cycles physically accurate, emission node intensity is specified as power in Watts, according to the manual, the exception being the sun lamp, which uses a power per area notation instead, as in Watts/m2, as stated in the previous link.

Blender Internal Render lamps are specified as an energy value from 0.0 to 10.0, which has no immediate real world physical connection, as far as I know.

While it is probably possible to establish some sort of "visual correspondence" between two lamps using an equivalent scene as reference, it is probably hard to establish any sort of mathematically correct one.

From there lamp power will likely progress linearly for both cases, so changing one for X units will probably have a similar effect in both scenes, so it may be possible to establish some sort of directly proportional relationship between the two.

As Cegaton mentions, other than that film settings like exposure, and color management and post processing will also have a noticeable impact in scene lighting.

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    $\begingroup$ Note that values in cycles are not limited, meaning that you can have any value you want. Aslo, the brightness of a scene will depend greatly on the materials and the color management transforms for the image. $\endgroup$ – cegaton Jun 15 '17 at 0:51
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    $\begingroup$ True, forgot about that, film exposure and color management settings will also have an impact on final result, added to answer. $\endgroup$ – Duarte Farrajota Ramos Jun 15 '17 at 0:59

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