# Designing an LED Backlight

"The only thing worse than training employees and losing them is to not train them and keep them."

— Zig Ziglar

## Introduction

Figure 1: An Active LED Backlight.

I was reading an application note by Texas Instruments yesterday on how to design an LED backlight for an LCD display (Figure 1). The article was interesting, but it did bother me because it  presented a rather involved formula and did not provided any motivation for the formula, a derivation, or even definitions of the parameters used in the formula.

Equation 1 shows the formula that bothered me. I will provide a derivation, definitions for all the parameters, and a worked example.

 Eq. 1

The principles behind designing LED backlight systems are fairly straightforward. However, Equation 1 looks rather complicated. Using basic physics, we can derive Equation 1 using a few basic principles. In the process of working through the derivation, we will acquire some insight into critical factors behind LED backlight designs.

## Background

### Backlight Basics

At the risk of grossly oversimplifying LED backlights, there are two basic types:

• Lit from the side

LEDs are mounted on the side of the display, which the advantage of giving you a very thin display. Unfortunately, the side backlighting can be uneven. Figure 2 shows an example of how this type of display is constructed.

Figure 2: Side-Mounted LEDs.

• Lit from the back

The LEDs are mounted directly behind the display. This approach gives more even lighting, but requires a thicker display. Figure 3 shows an example of how this type of display is constructed.

Figure 3: Lit from Behind Display Construction.

### Approach

Like most engineering calculations, backlight design is a form of budget calculation − so much light in, so much light out minus any losses. The process for this calculation is as follow:

• Determine the amount of backlight you need from the display for your application.
• Estimate the amount of loss the light will experience between the display and LEDs.
• Compute the total amount of light you need from the LEDs.
• Knowing the total amount of light needed, you can estimate the number of LEDs needed based on the type of LED you are using.

### Definitions

Aspect Ratio (AR)
The aspect ratio of an image describes the relationship between its width and its height.It is commonly expressed as two numbers separated by a colon, as in 16:9. Sometimes the ratio is expressed in terms of pixel counts. For example, standard HDTV's aspect ratio could be expressed as 1920:1080 instead of 16:9 (Wikipedia).
Luminous Flux (ΦV)
In photometry, luminous flux or luminous power is the measure of the perceived power of light (Wikipedia).
Display Size (LS)
Displays are usually specified in terms of their diagonal measurement. To obtain an actual length and width, you need both the display size and the aspect ratio.
VDisp
The fraction of light lost between the LEDs and the display.
MV
The illuminance of the display. You can think of the illuminance as the luminous flux density.

## Analysis

### Intuitive Viewpoint

I derive a simple relationship for the number of LEDs required to generate a specified level of screen brightness in Figure 4.

Figure 4: Intuitive Derivation of the Screen's Brightness.

You can think of the the formula for the required number of LEDs as taking the total amount of light needed (A·MV) and dividing it by the effective amount of light per LED (KX·ΦV).

### Screen Area

Display area is usually computed in terms of the display's diagonal length and its aspect area, which is not as intuitive as using the display's height and width. Figure 5 shows the formula for computing display area in terms of aspect ratio and diagonal length.

Figure 5: Derivation of Screen Area Formula.

### LED Quantity Formula

Figure 6 shows the form of the display brightness formula in terms of the variables that I think in.

Figure 6: My Formula for the Number of LEDs Required.

Equation 2 shows this equation using more conventional notation.

 Eq. 2 $\displaystyle {{N}_{{Min}}}=\left\lceil {\frac{AR}{{1+A{{R}^{2}}}}\cdot \frac{{{{M}_{V}}\cdot L_{S}^{2}}}{{{{\Phi }_{L}}\cdot {{K}_{{X}}}}}} \right\rceil$

where

• LS is the diagonal length of the screen.
• MV is the required display luminosity.
• ΦV is the luminous flux from a single LED.
• KX is fraction of light that makes it from the LED to the display.
• AR is the display aspect ratio.
• $\left\lceil {\text{ }} \right\rceil$ is the ceiling function.

### Conversion to TI Form

My Equation 2 is the same as Equation 1 given the proper substitutions, which I show in Figure 7.

Figure 7: Converting My Version to Equation 1 Form.

Equation 1 has a constant called K that simply converts inches squared to meters squared $\left( \text{m}^2 = 39.38^2 \cdot \text{in}^2 = 1550 \cdot \text{in}^2 \right)$. I just let Mathcad handle my unit conversions.

### Simple Worked Example

Figure 8 shows a worked example using common parameter values.

Figure 8: Worked Example.

## Conclusion

When I first saw Equation 1, I could not understand it. Now that I have gone through the derivation, I understand every term. Since I find the form of Equation 1 non-intuitive, I will work with Equation 2.

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