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Calibration with sinewaves

With a sinusoidal input, the output of a linear system is also sinusoidal, and the ratio of the two signal amplitudes is the absolute value of the transfer function. An experiment with sinewaves therefore permits an immediate check of the transfer function, without any a-priori knowledge of its mathematical form and without waveform modelling; this is often the first step in the identification of an unknown system. A computer program would however be required for deriving a parametric representation of the response from the measured values. A calibration with arbitrary signals as described later is more straightforward for this purpose.


  
Figure 24: Measuring the phase between two sinewaves with a Lissajous ellipse
\includegraphics[width=0.6\textwidth]{Fig/ellipse.eps}

When only analog equipment is available, the calibration coil or the shake table should be driven with a sinusoidal test signal and the input and output signals recorded with a chart recorder or an X-Y recorder. On the latter, the signals should be plotted as a Lissajous ellipse (Fig. 24) from which both the amplitude ratio and the phase can be read with good accuracy [Mitronovas & Wielandt 1975]. For the calibration of high-frequency geophones, an oscilloscope may be used in place of an X-Y-recorder. The signal period should be measured with a counter or a stop watch because the frequency scale of sinewave generators is often inaccurate.

The accuracy of the graphic evaluation depends on the purity of the sinewave. A better accuracy can of course be obtained with a numerical analysis of digitally recorded data. By fitting sinewaves to the signals, amplitudes and phases can be extracted for just one precisely known frequency at a time; distortions of the input signal don't matter. For best results, the frequency should be fitted as well, the fit should be computed for an integer number of cycles, and offsets should be removed from the data. A computer program "SINFIT" is offered for this purpose (see section 9). Eigenfrequency and damping of electromagnetic (as well as most other) seismometers can be determined graphically with a set of standard resonance curves on double-logarithmic paper. The measured amplitude ratios are plotted on the same type of paper and overlain with the standard curves (Fig. 25). The desired quantities can directly be read. The method is simple but not very precise.


  
Figure 25: Normalized resonance curves
\includegraphics[width=\textwidth]{Fig/res.eps}


next up previous contents
Next: Step response and weight-lift Up: Calibration Previous: Calibration of geophones
Erhard Wielandt
2002-11-08