Brief Introduction of Using LEDs – A Method for Calculating Lumens Emitted form LEDs

Introduction of using LED – how to estimate luminous flux from an optical system with LED

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Background Information

A typical application of the LED with secondary optical lens is shown in the Figure 1. The theoretical loss on the perfect surface of TIR (Total Internal Reflection) should be close to zero. However, in practice, due to manufacturing process, the optical surfaces cannot be perfectly smooth, resulting in some inevitable losses. In the Figure 1, if we consider the light energy emitted from the exit surface of the TIR lens as the active lumen output, then the overall light efficiency of the optical system can be represented as: active lumen / lumen from LED.

Figure 1. A typical illustration of using LED with secondary optical lens.

At the initial design phase, one can estimate the reasonable luminous flux emitted from LEDs by the datasheet of LED and relevant thermal effect:

1. Initial lumen:
It means that the temperature of the LED is close to the ambient temperature (usually 25 degrees Celsius). When the LED is lit up, one instantly measures the lumen, it is so-called initial lumen or cold lumen.
2. Steady state lumen:
One measures the lumen emitted from LED that is continuously driven at a fixed current for a period (typically 30 minutes or 1 hour). When the junction temperature (Tj) of the LED is stable, it is so-called steady state lumen or hot lumen.

Before estimating the reasonable lumen emitter from LED, we list the necessary information shown in Table 1 below. We take a Nichia 757, however, the same approach can be applied to LEDs from other manufacturers.

➢ LED model: The LED's product name for identification purposes.
➢ LED Sorting Current (Ifs): The reference current specified in the datasheet.
➢ LED Drive Current (Ifd): The actual current used when the LED is powered by the user. Luminous Flux Bin: The upper and lower limits of a single luminous flux bin as specified in the datasheet.
➢ Luminous Flux Multiplier for Current (Mc): The multiplier determined by comparing the actual drive current to the Lm-If curve in the datasheet.
➢ Number of LEDs: Enter "1" for calculating the luminous flux of a single LED or the total number of LEDs for the entire luminaire.
➢ Factor of Thermal Impact (Ftm): The percentage of luminous flux reduction due to the Tj of LED.
➢ Factor of Optical Impact (Fopt): The percentage of luminous flux reduction caused by the secondary optical components.

Table 1. Information for LED Luminous Flux Calculation
LED Model	Nichia 757
LED Sorting Current	0.15	A
LED Drive Current	0.18	A
Luminous Flux Bin	Min	Typ.	Max
	70	80	90
Luminous Flux Multiplier for Current	1.2
Number of LEDs	1
Factor of Thermal Impact	0.9
Factor of Optical Impact	0.8

Note: Nichia is a well-known Japanese LED brand company.

To calculate the LED's luminous flux reasonably, the first step in the calculation is to start from the "base value." It is recommended in this article to use values explicitly stated in the specification sheet as the base value, as shown in the figure below.

Figure 2. Specification values from model 757*

*The specification values of Nichia's 757 LED mentioned in this article are taken from Nichia's 757 specification datasheet.

In the above figure, the relevant specification values from Nichia's specification datasheet are explained item by item as follows:
①：Ts value shows 25°C , indicating that the listed specification values are obtained at an LED junction temperature of 25°C.
②、③：When the LED is rated at If = 100 mA and Ts is 25 degrees, the luminous flux emitted by the LED is 70 lumens.
④：The value represented here indicates the official recommended standard value (typ.) of the LED. A more rigorous approach would involve looking up the min and max values from the luminous flux bin table. However, different companies in the industry have different binning rules. If it's challenging to find the formal binning rules in the specification datasheet, it is recommended to roughly estimate the min and max luminous flux by approximately ±5% deviation, as shown in the table below.

Table 2. Example of Luminous Flux Range for a Single LED Bin
Lumen range	min	typ.	max
Lumen	66.5	70	73.5

Next, the second step is to calculate the luminous multiplier based on the LED's operating current. For example, if the LED is being driven at 150 mA. First, refer to the "Current vs. Luminous Flux" curve chart in the datasheet, as shown below:

Figure 3. Lm-If curve chart of Nichia’s 757 LED model

By the lm-If curve in figure 3, one can determine the following:
1. At a standard current of 100 mA, the corresponding luminous multiplier is 1.
2. At a driving current of 150 mA, the corresponding luminous multiplier is 1.4.

A designer can check the value of multiplier, which the LED is driven at 150 mA. After checking the value is 1.4, one fills the value in the filed of “Luminous Flux Multiplier for Current” in the Table 3.

Note: The linear relationship of different driving current is calculated as 150/100 = 1.5. After checking the table, however, the luminous multiplier is 1.4. This is a common error for LED applications.

Table 3. Luminous Values at LED Driving Currents
Luminous Flux Multiplier for Current	1.4

Based on the relationship of luminous flux and temperature from the LED's datasheet, as shown in figure 4 below. The meaning of this chart is that the luminous efficiency of the LED decreases as the chip's operating temperature increases.

Figure 4. lm-Tj curve from the datasheet of Nichia's 757 LED model

①Look up the specs in the 757 datasheet which provides the curve based on "ambient temperature". There's no need to convert it to Tj value. Because this LED is measured under overall thermal stable (external heating, not self-heating due to electrical current), Ta = Tj in this condition.
②Compare the curve at 25°C, where the value of multiplier of relative luminous flux is 1 marked as circled 2.
③The 100°C position corresponding to the vertical axis is 0.86. It is the multiplier of luminous flux, which is marked as circled 3 in the figure 4. We recorded the value in Table 4.

Table 4. Luminous flux multiplier value affected by Tj of LED
Factor of Thermal Impact	0.86

After the above three steps, we obtain 3 values that are 70(typ.), 1.4 and 0.86. By multiplying them, you can derive the actual luminous flux value, 84.28 (lumens), of the LED. The following example will be used for illustration.

Practical Example - Estimating the Luminous Flux of Using Luxeon Z ES

A customer uses Luxeon Z ES LEDs with a driving current of 800 mA. According to the result of the customer's thermal design, the measured Ts temperature is 105°C, and the LED's color temperature is 2700K. We want to know:
①What is the estimated LED output luminous flux?
②What is the luminous flux after passing through the lens (with lens efficiency of 0.86)?

First, obtain the relevant specification values from the LED's datasheet, as shown below. Since the used LED with a color temperature of 2700K. We extracted the values for 2700K shown in figure 5. Please note that Luxeon Z ES's specification values are measured at Tj=85°C, which is different from the 25°C reference of Nichia used previously.

Figure 5. Partial extraction of specifications from the datasheet from Luxeon Z ES LED Model*

*The specifications related to Luxeon Z ES LED in this article are extracted from Lumileds' LED datasheet.

Next, calculate the luminous flux multiplier value at the LED's operating current through looking up the lm-If curve graph below. When If = 800 (mA), the luminous flux multiplier at 700mA is approximately 1.15. Although this value is measured at Tj=85°C, we suggest that one can neglect the cross effect of temperature and current. Supported by many actual experiences, the calculated value is still very close to the actual system measurement.

Figure 6. The Lm-If curve of Lumileds' Luxeon Z ES LED Mode

All the values mentioned earlier are based on the condition of 700mA as directly retrieved from the datasheet. Now, compare the specification values listed in the datasheet for 700mA, as shown below. Take the datasheet's 185 lm (typical value) and 170 lm (min value) for the lower limit, and for the upper limit, add 5%, which results in 194 lm (max).

Figure 7. Partial extraction of lumen values for various color temperatures of Luxeon Z ES

The values mentioned above are measured at Tj=85°C. Since most LED specifications are measured at 25°C, it is recommended in this article to start the calculation of LED luminous flux at Tj=25°C. Therefore, based on Luxeon Z ES's lm-Tj curve, one can infer the luminous flux values from Tj=85°C to Tj=25°C. According to the curve, it's approximately a 1.09 multiplier, i.e., Min: 185 lm, Typ: 201 lm, Max: 211 lm.

Figure 8. The Lm-Tj curve of Luxeon Z ES Model LED

Similarly, according to the lm-Tj curve graph above, with Ts value is 105°C given by the customer, one can calculate the Tj value using the thermal resistance formula as follows:
Tj = Rthjs*W + Ts = 3*0.8*3 + 105
Tj = 112.2°C
Here, Rthjs is 3°C/W, found in the datasheet. The current is driving at 800mA. The voltage of the LED is 3V checked from the Vf-If curve graph in the datasheet.
Based on the curve, this article’s approach is to start from the 25°C reference line. This results in a multiplier value of about 0.93 for Tj=112.2°C. It means that the factor of thermal impact caused by the delta value of the two Tj values is:
0.93/1.09=0.853
“0.853” is the calculated factor of thermal impact (relative to 25°C) for the LED in this example.
The calculated or queried values are summarized in the table below.

Table 5. Calculations for LED luminous flux
Range	Ifs	Ifd	Base Lm	Mc	LED No.	Initial Lm	Ftm	Hot Lm	Fopt	System Lm
min	0.7A	0.8A	185.3	1.15	1	213.09	0.853	181.77	1	181.77
typ.	0.7A	0.8A	201.65	1.15	1	231.89	0.853	197.80	1	197.80
max	0.7A	0.8A	211.73	1.15	1	243.48	0.853	207.69	1	207.69

Multiplying the values of "Base Lm," "Mc," "LED No.," "Initial Lm," and "Ftm" together will yield the total lumens emitted by the LED, as represented in the "LED Lm" column. After multiplying the value of "Hot Lm" by the factor of optical impact (Fopt), typically provided by optical manufacturers, one can determine the system's output lumens, which includes LED operating conditions, thermal effects, and optical effects. We summary the key items as follows:

1. Initial Lm - For a single LED, representing the LED's cold lumen output.
2. Hot Lm - Calculated value for a single LED, representing the LED's hot lumen output.
3. System Lm - Input value for Fopt is 1, assuming no optical impact from optical components.

LED Lumens Calculation Considering Optical Impact

When considering optical impact, such as secondary optical components that may be TIR lenses or reflectors, the system lumens will be influenced by these optical components (as shown in Figure 1). Assuming the factor of optical impact is 0.86, one can input this value into the "Fopt" column in Table 5 and then calculate the results. The final effect will yield the results shown in Table 6.
Typically, the values of Fopt are determined through optical simulations. Similarly, the values of Ftm are determined through thermal simulations. However, sometimes these values can also be estimated based on design experience.

Table 6. Calculations for LED luminous flux with Fopt
Range	Ifs	Ifd	Base Lm	Mc	LED No.	Initial Lm	Ftm	Hot Lm	Fopt	System Lm
min	0.7A	0.8A	185.3	1.15	1	213.09	0.853	181.77	0.86	156.32
typ.	0.7A	0.8A	201.65	1.15	1	231.89	0.853	197.80	0.86	170.10
max	0.7A	0.8A	211.73	1.15	1	243.48	0.853	207.69	0.86	178.61

Conclusion:
Through the proposed method for calculating LED luminous flux in this article, it is possible to reasonably estimate the total luminous flux emitted of a product under known conditions of LED drive current, Tj temperature, thermal impact and optical impact. The proposed method provides a theoretical basis for evaluation during the development and design phases. It also helps designer to discuss the initial measurement comparisons in the early stages of development.

Introduction of using LED – how to estimate luminous flux from an optical system with LED

Background Information

Practical Example - Estimating the Luminous Flux of Using Luxeon Z ES

LED Lumens Calculation Considering Optical Impact

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