Physikalisch-analytisches PraktikumAppendixExperiment ELK
FS 2009
Appendix
Preface: All statistic calculations were done with Matlab and its output is written in italic form. For the confidence interval we use always 95%.
Task 1C: Relative Error
We can calculate the relative error like this:
(19)
xw = observed value and xr = exact value
Time [ms] / Frequency regime [kHz] / Frequency on scope [kHz] / Relative error [%]1 / 1 / 1.00512 / 0.512
0.1 / 10 / 10.00487 / 0.487
0.01 / 100 / 99.99999 / -0.001
0.001 / 1000 / 999.9999 / -0.01
Tab. 2: Relative errors calculated with (19) of frequency regimes. For higher frequencies the relative error becomes smaller.
Task 2A: Ohm’s Law
For the measurement we had a frequency υ = 1947.63 Hz and for each resistor (R1 and R2) a value of 100 kΩ. The potential difference between input and output voltage is given by (15).
> linreg1
Liste mit x-Werten: [3.8,5.8,6.0,7.5,8.0,8.5,12.5,13.0,15.0,20.5]
Liste mit y-Werten: [1.8,3.0,2.6,3.2,4.0,4.0,5.8,5.5,7.4,9.8]
Liste mit Standardabweichungen: [1,1,1,1,1,1,1,1,1,1]
y-Achsenabschnitt alpha: -0.079975
Standardabweichung alpha: 0.231962
Steigung beta: 0.476141
Standardabweichung beta: 0.020772
Confidence intervals for α and β with n = 10 measurements:
8 DF for 95% ts = 2.306
c = ts · sa = 2.306 · 0.231962 = 0.5349 ± 0.5
α = - 0.08 ± 0.5
c = ts· sb = 2.306 * 0.020772 = 0.047900232 ± 0.05
β = 0.48 ± 0.05
Fig. 4 : Data points (blue dots with corresponding error bars), fitting line (red) and ideal line (black) plotted for (15). The output-voltage is proportional to the input voltage. The fitting line shows a tendency: the higher your input-voltage is the lower becomes the output voltage.
Task 2B: Ohm’s Law
For the measurement we had a frequency υ = 1947.63 Hz. For resistor R2 we had a value of 100 kΩ and for resistor R1 we changed the value within a range of 0 to 10 kΩ. The input voltage U1 was 15 V.
> linreg1
Liste mit x-Werten: [0.0891,0.595,1.606,2.933,3.602,4.770,6.470,6.830,9.700]
Liste mit y-Werten: [1/14.8,1/14.4,1/13.2,1/12.0,1/12.0,1/10.4,1/9.6,1/10.0,1/8.4]
Liste mit Standardabweichungen: [1,1,1,1,1,1,1,1,1]
y-Achsenabschnitt alpha: 0.066892
Standardabweichung alpha: 0.037318
Steigung beta: 0.005377
Standardabweichung beta: 0.007384
Confidence intervals for alpha and beta with n = 9 measurements:
7 DF for 95% ts = 2.365
c = ts · sa = 2.365 · 0.037318 = 0.08825707 ± 0.09
α = 0.07 ± 0.09
c = ts · sb = 2.365 · 0.007384 = 0.01746316 ± 0.017
β = 0.0054 ± 0.017
Fig.5: Data points (blue dots with corresponding errorbars), fitting line (red) and ideal line (black) plotted for formula (15). With growing value for R1 you will have a lower value for the output voltage Uout.
Task 3: RC circuit
For the RC circuit we took for R = 1 kΩ and for C = 10 nF. So the expected value for fg calculated with (13) is 15.915 kHz.
Fig. 6: Output-voltage plotted versus logarithmic scale of f/fg. Basically you can see for frequencies > fg the output-voltage goes down.
linreg1
Liste mit x-Werten: [14.002,14.999,16.004,17.01,18.008,20.004,22.213,24.71,30.273]
Liste mit y-Werten: [12.00,11.2,8.75,8.5,8.15,7.8,7.5,6.5,4.8]
Liste mit Standardabweichungen: [1,1,1,1,1,1,1,1,1]
y-Achsenabschnitt alpha: 15.894140
Standardabweichung alpha: 4.490915
Steigung beta: -0.382836
Standardabweichung beta: 0.221150
Confidence intervals for alpha and beta with n = 9 measurements:
7 DF for 95% ts = 2.365
c = ts · sa = 2.365 · 4.490915 = ± 10.6
α = -15.9 ± 10.6
c = ts · sb = 2.365 · 0.221150 = 0.52302 ± 0.5
β = - 0.4 ± 0.5
Fig. 7: Bode plot; G(x) verus f:fg. The blue line indicates the ideal zeroline and the green line our zero line made upon the plot. The black line indicates the tangent of the corresponding line. We've calculated a slope of
– 15.9 ± 10.6 dB in the stopbandregion.
Fig. 8: Phase plotted versus log(f:fg). With increasing frequency the phase-difference decreases.
Task 5B: Lissajous figures
Ratio f2:f1 / Sketches / Ideal figures1:1 /
2:1 /
4:1 / No figure found
5:1 / No figure found
2:3 /
Tab. 3: Input ratios, corresponding sketches and Lissajous figures
1