The third post in the series – on quantifying light output – and at last an answer to the original question of how to measure, or at least calculate, the output from a lamp. Hopefully not too far “down the rabbit hole”…

### LED light output

Let’s start with the output of a sample light source – W_{rel}(*λ*) (being a Cree MHD-E 5000 K LED):

Per the previous two posts – How to measure light output – the question and Light output – lumen, lux and the candle – one watt of radiant power at 555 nm is by definition equal to 683 lumens. So given the CIE 1931 luminous efficiency function, the spectral radiant flux Φ(*λ*) in watts per nanometer for each lumen is calculated as:

This is the numerical integration equivalent of stating that the light output – the spectral radiant flux (in watts, per lumen) – equals the ratio of the **spectral** output of the LED compared to the **luminous** output of the lamp. The relevant points are:

- The calculation result is in watts per nanometer for the particular LED light wavelength.
- The luminous output is the sum of the radiant output of the LED when considering the CIE 1931 Luminous Efficiency function. So, simply: multiply the LED spectral output by the luminous efficiency function and sum over the range 400-700 nm.
- One lumen (candela) has a radiant output of 1/683 W, referenced to the CIE 1931 Luminous Efficiency function.
- Thus ultimately the spectral radiant flux, per lumen, is the sum of the output for all wavelengths.

Before moving to calculated values for a 5000 K LED lamp, some additional definitions are useful. **Luminous efficacy** is the ratio of luminous output versus input power, typically in lumens per watt (lmW^{-1}). **Luminous efficiency** meanwhile is the ratio of spectral output versus input power, a percentage value (%).

So from all the above and using some representative values for luminous output and input power:

Colour temperature: | 5000 K |

Luminous output: | 22500 lm |

Input power: | 180 W |

Φ(λ)/lm: |
3.18 mW |

Φ(λ): |
71.52 W |

Luminous efficacy: | 125 lmW^{-1} |

Luminous efficiency: | 39.73 % |

### HPS light output

By way of comparison, exactly the same calculations can be performed for a High Pressure Sodium (HPS) lamp. This is a close head-to-head test of which lighting technology is better, as HPS grow lamps are the standard, at least for certain stages of plant growth. It’s not directly comparable given differences in colour temperature, luminous output and input power, but these comparison are never straightforward.

The output spectrum from a sample HPS bulb is as follows:

The numerical values for luminous output, colour temperature and input power were taken for a sample HPS bulb – the Philips Son-t Pia Green Power 250 W, shown below. The spectrum above is therefore only representative – it’s definitely not the exact spectrum for the bulb in question – but like all things engineering it should be close enough to understand the approximate situation.

To the calculated values:

Colour temperature: | 2100 K |

Luminous output: | 33200 lm |

Input power: | 250 W |

Φ(λ)/lm: |
2.62 mW |

Φ(λ): |
87.00 W |

Luminous efficacy: | 132.8 lmW^{-1} |

Luminous efficiency: | 34.80 % |

### In conclusion

The take away from all this… Yes: the colour temperatures differ. Yes: there is a luminous output difference. And yes: the input power differs. Nonetheless a description for how to calculate the light output from a lamp when one has the necessary information. And interesting to note how close the luminous efficacy and luminous efficiency values are between the two technologies.

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