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,
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.
Figure 1: The Mastercolour Evolution 35W T lamp. The burner is the spherical
part inside the quartz lamp envelope.
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.
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.
Figure 2: Full high resolution spectrum (HRS) of the Mastercolour Evolution
35W T lamp
Figure 3: Zoom in on a spectral line of X in the high-resolution spectrum.
We omitted the wavelength scale on purpose.
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.
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.
Equation 1. Abramowitz and Stegun’s formula for approximation of the complex
Now we explain how the fit was elegantly done in Minitab.
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.
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:
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.
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.
Figure 8: Normal probability plot of the residuals.
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
Figure 10: Output of the session window after the fit.
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.
1 M. Abramowitz and I. Stegun, “Handbook of mathematical functions”, Nat. bureau of
standards, Appl Mathematics series, 55, (1972), p 299.
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.