Document ID: chunk:federal_register_of_legislation:F2023C00229:reg:9:p70
Version: federal_register_of_legislation:F2023C00229
Segment Type: reg
Provision Reference: reg 9 (pt 70/71)
Character Range: 406748–409909

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)

                —    sampling rate 150 Hz
              Step 1 Required Bessel filter response time tF:

              Step 2 Estimation of cut-off frequency and calculation of Bessel constants E, K for first iteration:

              This gives the Bessel algorithm:

              where Si represents the values of the step input signal (either '0' or '1') and Yi represents the filtered values of the output signal.

              Step 3 Application of Bessel filter on step input:

              The Bessel filter response time tF is defined as the rise time of the filtered output signal between 10 % and 90 % on a step input signal. For determining the times of 10 % (t10) and 90 % (t90) of the output signal, a Bessel filter has to be applied to a step input using the above values of fc, E and K.

              The index numbers, the time and the values of a step input signal and the resulting values of the filtered output signal for the first and the second iteration are shown in Table B. The points adjacent to t10 and t90 are marked in bold numbers.

              In Table B, first iteration, the 10 % value occurs between index number 30 and 31 and the 90 % value occurs between index number 191 and 192. For the calculation of tF,iter the exact t10 and t90 values are determined by linear interpolation between the adjacent measuring points, as follows:
              where outupper and outlower, respectively, are the adjacent points of the Bessel filtered output signal, and tlower is the time of the adjacent time point, as indicated in Table B.

              Step 4 Filter response time of first iteration cycle:

              Step 5 Deviation between required and obtained filter response time of first iteration cycle:

              Step 6 Checking the iteration criteria:

              |Δ| ≤  0,01 is required. Since  0,081641 > 0,01, the iteration criteria  is not met and  a further iteration cycle has to be started. For this iteration cycle, a new cut-off frequency is calculated from fc and Δ as  follows:

              This new cut-off frequency is used in the second iteration cycle, starting at step 2 again. The iteration has to be repeated until the iteration criteria is met. The resulting values of the first and second iteration are summarised in Table A.

        Table A

        Values of the first and second iteration

         Parameter      1. Iteration      2. Iteration
     fc                 (Hz)             0,318152         0,344126
     E                  (-)              7,07948 E-5      8,272777 E-5
     K                  (-)              0,970783         0,968410
     t10                (s)              0,200945         0,185523
     t90                (s)              1,276147         1,179562
     tF,iter            (s)              1,075202         0,994039
     Δ                  (-)