Saw-tooth modulation

Kwon, Y.K. Andorsch, R. Pyda, M. Wunderlich, B. Multifrequency saw-tooth modulation of a power-compensated differential scanning calorimeter. Thermochim. Acta 2001, 367 (8), 203-215. 

Wunderlich B, Androsch R, Pyda M, Kwon YK (2000) Heat Capacities by Multifrequency Saw-tooth Modulation. Thermochim Acta 348 181-190. 

Where a multiplexed sine wave or saw-tooth modulation is used the deconvolution procedure can be used to extract the response at a series of frequencies [4,10,19,20]. However, current commercial products restrict themselves to using the first component of the Fourier series, which is then, with the assumption of linearity, equivalent to using a single sinusoidal modulation. It is true that looking at the whole Fourier series, rather than just the first component, offers scope for increasing the amount of information that can be obtained from an MTDSC experiment. This applies even to single sinusoidal modulations (because non-linearities produce harmonics) as well as multiple simultaneous sine waves or saw-tooth modulations. This will be considered in greater detail below in the section on advanced MTDSC. 

Alternative modulation functions and data analysis procedures have been applied in MTDSC. Different instrument manufacturers have applied square wave or saw-tooth modulation, coupled with Fourier transform analysis methodologies. Alternative modulation functions provide some additional advant ages - for example, a square wave function ensures that a steady state is achieved over an isothermal plateau since the signal during such a period is the nonreversing contribution. The amplitude provides a measure of the reversing signal. However, there are... 

We can use not only compoimd multiphase signals, but also equivalent compound FFM signals. As example on Figure 4 is shown the signal with saw-tooth modulation [4], which is equivalent to the compound multiphase signal fi om Figure 3 (derivative of the law of phase modulation gives us... 

Mossbauer spectra are usually recorded in transmission geometry, whereby the sample, representing the absorber, contains the stable Mossbauer isotope, i.e., it is not radioactive. A scheme of a typical spectrometer setup is depicted in Fig. 3.1. The radioactive Mossbauer source is attached to the electro-mechanical velocity transducer, or Mossbauer drive, which is moved in a controlled manner for the modulation of the emitted y-radiation by the Doppler effect. The Mossbauer drive is powered by the electronic drive control unit according to a reference voltage (Fr), provided by the digital function generator. Most Mossbauer spectrometers are operated in constant-acceleration mode, in which the drive velocity is linearly swept up and down, either in a saw-tooth or in a triangular mode. In either case. 

More complex temperature programmes are sometimes useful. These might combine periods of variable heating and cooling rates with isothermal periods. For example, stepwise heating can be used to detect the onset of melting under quasi-isothermal conditions (Laye, 2002). Modulated temperature DSC (MTDSC), in which the linear temperature scan is perturbed by a sinusoidal, square or saw-tooth wave, or other modulation of temperature, has a number of potential advantages over the conventional linear scan. These include increased sensitivity and resolution, and the ability to separate multiple thermal events (Laye, 2002). 

Conventional lattices may be perturbed by functions with different periodicity, e.g. by sinusoidal or saw-tooth-like modulations, see section 1.21 in Chapter 1. In the simplest case (one-dimensional modulation), Eq. 5.1 becomes d / = h + kb + k + wq assuming that the perturbation function is periodic and has the modulation vector q. In a case of three-dimensional modulation, a total of six indices (k, k, I, m, n, and p) are required to identify every point observed in reciprocal space = ha + kb + /c +... 

The acoustic excitation in the SEM/SFM hybrid system is performed with a 15 kV electron beam locally modulated by a saw tooth function with a frequency of 76.8 kHz. This leads to a behavior comparable to an acoustic line source with a directed emission [7]. The line source has a length of 43 pm and is located in parallel to the SFM cantilever, emitting acoustic waves vertically to the cantilever axis. 

The TMDSC with Fourier analysis of the melting pentacontane and the calculations using saw-tooth analysis methods are given in the publication Wunderlich B, Boiler A, Okazaki I, Ishikiriyama K, Chen W, Pyda W, Pak J, Moon, 1, Androsch R (1999) Temperature-modulated Differential Scanning Calorimetry of Reversible and Irreversible First-order Transitions. Thermochim Acta 330 21-38. 

It is possible to use multiple sine waves [10] and so extract as a Fourier series (or other deconvolution procedure) the response to several frequencies simultaneously, as illustrated in Chapter 4. An extension of this is the use of saw-tooth temperature modulations [20]. These can be considered to be a combination of an infinite series of sine waves (though only a limited range will be available in practice). A symmetric saw-tooth (same heating and cooling rate) only has odd harmonics, but an asymmetric saw-tooth (different heating and cooling rates) is equivalent to a broad range of frequencies. 

The Wunderlich ATHAS laboratory has investigated other types of modulation, including saw-tooth and square and has shown that such modulations are likely to be incorporated into future generations of TMDSC systems for enhanced precision of measurement of heat capacity and glass transitions. 

If the amplitude of a signal varies periodically as a function of time, and if the shape of the time-varying amplitude is symmetric in time (often associated with a duty cycle of 1 1), the term AC signal is used. Most commonly, sine wave, square wave or saw-tooth wave forms are encountered in practical experiments (the former two are included in Figure 14.1). The amplitude values may alter between zero and full, but also could constitute amplitude modulations around a finite mean value (associated with amplitude modulation). 

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