Using Nonlinear Regression in Minitab to Model Diffusion in a High-Intensity-Discharge Lamp

When a project team testing the quality and reliability of high-intensity-discharge lamps at Philips Turnhout needed to model the chemistry of a new generation of products, they found that Minitab’s nonlinear regression functionality makes advanced data-fitting procedures very accessible. 

By P. Watté, Philips Innovative Applications, Global Technology and Development HID & SL, Belgium

Introduction

Philips Article P Watte Philips Article company shield

P. Watté, Philips Innovative Applications 

At Philips Turnhout, Minitab is widely used for various calculations in order to verify and demonstrate the quality and reliability of our high-intensity-discharge lamps. Recently, one of our project teams found additional value in the model fitting options Minitab offers. In particular, the nonlinear regression functionality makes advanced fit procedures very accessible. As detailed in this article, we applied it for understanding the chemistry inside the burner of a new generation of products.

Philips developed a new kind of high-intensity-discharge lamp, the so-called Philips Mastercolour Evolution range. This revolutionary design uses unsaturated metal halide fillings. Previous generations used an overdose of salt filling, which also created a considerable salt pool that affected the lamp’s performance. The new lamp design eliminates this problem. Consequently, a “clean beam” and longer lamp life are within reach, because there is barely any corrosion of the alumina oxide of the burner. Figure 1 shows the lamp under study, a Mastercolour Evolution 35 W T lamp with a ceramic lamp base. The discharge tube has much smaller extended plugs than a classical HID retail lamp.

Philips Article Figure 1 - Lamp

Figure 1: The Mastercolour Evolution 35W T lamp. The burner is the spherical part inside the quartz lamp envelope.

Problem Statement

The burner (the spherical part inside the lamp in which a plasma is generated) is made of polycrystalline alumina oxide and manufactured by means of injection molding. Unfortunately, elements can diffuse out of the burner matrix as oxides and react with the salt filling. In the new unsaturated lamps, this is a critical issue.   In this article, we discuss how we addressed diffusion of one particular element (we will simply refer to it as “X”) from the wall of the discharge tube. 

Data from High-resolution Spectroscopy

We include the technical information in this section for the sake of completeness. Readers who are not concerned with these details of lamp manufacture can skip this section and proceed to the next section, which discusses fitting the model in Minitab.

In a lighting application the discharge tube can reach relatively high temperatures, in the 1500-1900 K range.  This drives the out diffusion of oxides and other elements from the burner material. The amount of these elements in the discharge cannot be assessed directly, but indirect assessment is possible by means of high-resolution spectroscopy. By zooming in on one of its spectral lines in the high-resolution spectrum, we can estimate the presence of element X in the discharge.  This is illustrated in figure 2. 

The out-diffusion of X from the burner material is governed by a temperature-dependent diffusion coefficient.  By making an appropriate choice of spectral lines (we normalized a spectral line of X to a Hg spectral line), as shown in figure 3, the value X/Hg fits to a complex error function erfc(x), which is the general solution to a diffusion equation.  Six Evolution 35W lamps were measured after 100, 500, 1000, 2000, 4000, 6000, 8000 and 10000 h.  The boxplot of the values X/Hg is depicted in figure 4. We used this data to fit a complex error function. Mathematical software packages can perform this task, but it becomes rather cumbersome if you also want confidence bounds on the fitting.

Philips Article Figure 2 - Spectrum

Figure 2: Full high resolution spectrum (HRS) of the Mastercolour Evolution 35W T lamp

  Philips Article Figure 3 Spectrum Detail

Figure 3: Zoom in on a spectral line of X in the high-resolution spectrum.  We omitted the wavelength scale on purpose.

Philips Article FIgure 4 Boxlplot

Figure 4: Boxplot of the evolution of the thin optical line of X relative to a Hg line as a function of lamp life. Each box represents 6 lamps.

Data Fitting in Minitab

The high-resolution spectroscopy data was stored in two columns of a Minitab worksheet, shown in figure 5.  The first column contains the hours of lamp life at which the measurement was performed.  The second column is the height of the recorded X/Hg values. Minitab does not have a built-in erfc(x) function, but one can use Minitab to perform the calculation with an approximate formula from Abramowitz and Stegun. 

Philips Article - Equation 1

Equation 1. Abramowitz and Stegun’s formula for approximation of the complex error function.

Now we explain how the fit was elegantly done in Minitab.

Philips Article Figure 5 Worksheet

Figure 5: Arrangement of the data in the Minitab worksheet. The column C1 is the life of the lamp at which the HRS measurement was performed. The column C2 is the ratio of the X/Hg thin optical lines.

To execute the erfc fit we selected Stat Regression >  Nonlinear Regression… from the menu. The fit function equation has to be entered in the required window.  The Minitab syntax is such that the fit parameters have to be fed as θ1and θ2, as illustrated in Figure 6.

Philips Article Figure 6 Dialog

Figure 6: The fit function (containing the complex error function) is entered directly in the Expectation Function window.

When filling in the fit function, it is important to type the correct number of brackets. 

The fit will be performed using the following function:

Philips Article - Equation 2

Equation 2.

If you click ‘OK’, Minitab prompts you to enter two estimated values for the fit parameters θ1and θ2. These starting values will be used in a Levenberg-Marquardt or Newton-Gauss algorithm to find the optimum fit values. The graphical output is depicted in figure 7.

Philips Article Figure 7 Fitted Line Plot

Figure 7: Fit of the fit function in equation 2 to the HRS data of the CDM Mastercolour Evolution 35W lamp as a function of life time.

Eventually the residuals can also be plotted, in a normal probability plot or in a histogram. The residuals should follow a normal distribution for a good quality fit. This is obviously the case here, as illustrated in Figures 8 and 9. Figure 10 displays the Session window that appears after the fit. 

Philips Article Figure 8 Normal Plot

Figure 8: Normal probability plot of the residuals.

Philips Article Figure 9 Histogram

Figure 9: Residual plots of the fitting: histogram of the residuals.

Figure 10 displays the Session window that appears after the fit. It lists the fit parameters and their SE on the estimate, as well as a check for the lack of fit.  It reveals that we used the Newton-Gauss algorithm for optimization of the fit. With the starting values of θ1 = 0.005 and θ2 = 0.18, the solution was obtained after 11 iterations.

Philips Article Figure 10 Output 

Figure 10: Output of the session window after the fit.

Conclusion

Minitab is widely used to verify and demonstrate the quality and reliability of high-intensity-discharge lamps at Philips Turnhout. As we have detailed, one of our project teams found additional value in using nonlinear regression for understanding the chemistry inside the burner of a new generation of products. The team found that Minitab’s nonlinear regression functionality makes advanced fit procedures very accessible. 

References

1 M. Abramowitz and I. Stegun, “Handbook of mathematical functions”, Nat. bureau of standards, Appl Mathematics series, 55, (1972), p 299.

Acknowledgments

P. Antonis, R. Raas, D. Vandeperre, K. Denissen are acknowledged for providing the data and the discussions, J. Suijker and M. Meeuwssen for the review of this document.

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