LEDs are nonlinear devices with I/V curves that resemble the curves of rectifier diodes. Designers of lighting drivers require a good approach to modeling the LED device performance in the first quadrant of the I/V plane.

For engineering applications, there are three approaches to consider: the linear, bias-voltage plus resistor equivalent; a quadratic equation; and an exponential equation. Let’s use the three techniques with Cree’s XLamp XM-L LED as the LED to be modeled *(Fig. 1)*.

### The Linear Model

This model is the simplest, but the most inaccurate. It consists of a dc bias voltage plus a resistor *(Fig. 2a)*. The equivalent mathematical form is given by:

You select two data points (v_{F}, i_{F}) in coordinate form (i.e., 3.28 V, 2.6 A and 2.71 V, 0.2 A). Then, R is evaluated and yields 0.2375. The selection also implies a = 0.2 and b = 2.71. With all three parameters (a, b, R) properly assigned, the linear model yields Figure 2b. Evidently, the linear model offers the advantage of simplicity, but suffers in accuracy.

### The Quadratic Model

The curve of Figure 1 has a concave portion that resembles one arm of a parabolic curve. It can be expressed with a quadratic equation of the form a_{2}v_{F}^{2} + a_{1}v_{F} + a_{0}. The key is the determination of three coefficients: a_{2}, a_{1}, and a_{0}.

To determine these coefficients, you can use the well-known Least Square Polynomial Curve-fitting algorithm of linear algebra. To do this, select more than a dozen data points from Figure 1 (15 in this case) and place them in two matrixes. One is a 3x15 rectangular matrix C:

The other is a 1x15 column vector i_{F}:

where vF_{j} , for j = 0 to 14, are the corresponding LED forward voltages at the selected forward currents given in vector iF. The three coefficients are then given by a column vector of:

Using software tools such as Matlab from the Mathworks or MathCAD from the Mathsoft, you can easily compute Equation 4. For this example, we obtain:

Note that that the first element of vector a is coefficient a0, etc., which leads to the quadratic model of Figure 3. This approach yields significant improvements in the model across the whole operating range of the LED.

### The Exponential Model

Figure 3 appears to bend more near the low current, which is where the exponential model may offer further improvement. This model comes in the form of ae^{b·vF} + c with three unknown parameters a, b, and c to be determined. Again, software tools such as MathCAD help you find those parameters numerically.

Under the specialized regression section, an exponential regression statement expfit ( v_{F}, i_{F}, vg) can find all three parameters, given a data set in vectors and initial guess value vg, also a column vector. For this example, a = 9.66 · 10^{–3}, b = 1.818, and c = –1.157 are obtained. This results in the exponential model of Figure 4.

Overall, the best fit near the low-current region may lie between the quadratic and the exponential models. It is unrealistic to expect a single, perfect prediction from any given model, since almost all such analytical efforts are an attempt to represent the complexities of nature.