Document ID: chunk:federal_register_of_legislation:F2023C00229:reg:9:p69
Version: federal_register_of_legislation:F2023C00229
Segment Type: reg
Provision Reference: reg 9 (pt 69/71)
Character Range: 404068–407064

The Bessel filter itself is a recursive, second-order low-pass filter which guarantees the fastest signal rise without overshoot.

              Assuming a real time raw exhaust plume in the exhaust tube, each opacimeter shows a delayed and differently measured opacity trace. The delay and the magnitude of the measured opacity trace is primarily dependent on the geometry of the measuring chamber of the opacimeter, including the exhaust sample lines, and on the time needed for processing the signal in the electronics of the opacimeter. The values that characterise these two effects are called the physical and the electrical response time which represent an individual filter for each type of opacimeter.

              The goal of applying a Bessel filter is to guarantee a uniform overall filter characteristic of the whole opacimeter system, consisting of:

                —    physical response time of the opacimeter (tp),

                —    electrical response time of the opacimeter (te),

                —    filter response time of the applied Bessel filter (tF).

              The resulting overall response time of the system tAver is given by:

              and must be equal for all kinds of opacimeters in order to give the same smoke value. Therefore, a Bessel filter has to be created in such a way, that the filter response time (tF) together with the physical (tp) and electrical response time (te) of the individual opacimeter must result in the required overall response time (tAver). Since tp and te are given values for each individual opacimeter, and tAver is defined to be 1,0 s in this Directive, tF can be calculated as follows:

              By definition, the filter response time tF is the rise time of a filtered output signal between 10 % and 90 % on a step input signal. Therefore the cut-off frequency of the Bessel filter has to be iterated in such a way, that the response time of the Bessel filter fits into the required rise time.

        Figure a

        Traces of a step input signal and the filtered output signal
              In Figure a, the traces of a step input signal and Bessel filtered output signal as well as the response time of the Bessel filter (tF) are shown.

              Designing the final Bessel filter algorithm is a multi step process which requires several iteration cycles.   The scheme of the iteration procedure is presented below.

2.2.                            Calculation of the Bessel algorithm

              In this example a Bessel algorithm is designed in several steps according to the above iteration procedure which is based upon Annex III, Appendix 1, Section 6.1.

              For the opacimeter and the data acquisition system, the following characteristics are assumed:

                —    physical response time tp 0,15 s

                —    electrical response time te 0,05 s

                —    overall response time tAver 1,00 s (by definition in this Directive)

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