Lissajous plot

A process is forced by sinusoidal input m,). The output is a sine wave If these two signals arc connected to an x — y recorder, we get a Lissajous plot. Time is the parameter along the curve, which repeats itself with each cycle. The shape of the curve will change if the frequency is changed and will be different for different kinds of processes. 

Figure 7.6 The interpretation of a Lissajous plot of time-domain signals in terms of impedance.

Example 7.1 Lissajous Analysis Use a Lissajous plot to find the impedance at a frequency of 100 Hz for a linear system with capacity Qi = 31 pP/cm, charge-transfer resistance Rt = 51.34 dcm, and a potential perturbation AV = 0.01 V. 

Remember 7.1 WhUe use cf Lissajous plots for numerical evaluation of impedance can he considered obsolete, it is us l to include an oscilloscope capcible of displaying a Lissajous plot in the experimental setup. 

Lissajous analysis, as an experimental approach for impedance measurement, is obsolete and has been replaced by methods using automated instrumentation. Lissajous plots, however, have great pedagogical value as a means of learning impedance spectroscopy. In addition, as discussed in Section 8.2, use of oscilloscopes is recommended for monitoring the progress of impedance measurements, and oscilloscopes capable of displaying Lissajous plots are particularly useful. 

Use a Lissajous plot to calculate the phase angle and magnitude of the impedance for the system described in Example 7.1 at frequencies of 1 Hz and 10 kHz. This approach requires calculating dte potential perturbation, the charging current, and the Faradaic current as functions of time. 

The reason for the linear response at high frequency can be seen in the Lissajous plot of surface overpotential as a function of applied potential, given in Figure 8.8. At low frequencies, the surface overpotential is large and is scaled by Rt/(Rt + Re), whereas at high frequencies the surface overpotential tends toward zero. It is interesting to note that, at low frequencies, the surface overpotential is influenced by the nonlinearity associated with the faradaic reaction. 

The optimal perturbation amplitude may be best determined experimentally. Distortions in Lissajous plots at low frequency (see Figure 8.3) may be attributed to a nonlinear response. If the shape is distorted from an ellipse, one should reduce the amplitude. A second approach is to compare the impedcince response for several amplitudes as demonstrated in Figure 8.4. If the magnitude of the impedance at low frequencies depends on amplitude of perturbation, the perturbation amplitude is too large. 

Figure 8.10 Lissajous plot for a capacitance dependent on potential following equation (8.26) at a frequency of 10 kHz.

It is strongly advised to use an oscilloscope while making impedance measurements. It is useful to monitor the time-domain signals that are processed in the impedance instrumentation. It is particulary useful to monitor the signals in the form of a Lissajous plot as discussed in Section 7.3.1. 

A) Modulated heat profile. (B) Lissajous plot. Both of these profiles can be used to verify whether the programmed temperature modulation is correctly followed by the sample for a given set of experimental conditions (courtesy of TA Instruments Inc.)... 

The modulated heat flow profile (Figure 2.11 A) and/or the Lissajous plot of modulated heat flow versus modulated heating rate (Figure 2.1 IB) provide an indication of whether the sample is able to follow the imposed temperature regime. Note that Lissajous plots are currently only available using TA Instruments Thermal Solutions software. 

Fig. 3 Elastic Lissajous plot (stress vs. strain) for a colloidal gel (33 vol% PDMS droplets in an aqueous continuous phase containing 33 vol% of bridging polymer and 230 mM sodium dodecyl sulfate) at o = lOrad/s and Yo = 0.284, reprinted with permission from [23], copyright 2014, Society of Rheology. In the SPP framework the local waveform parameters Yc and <7c have been defined by Rogers et al. [38]...

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