Phase quadrature

This circuit is usually referred to as the Randles circuit and its analysis has been a major feature of AC impedance studies in the last fifty years. In principle, we can measure the impedance of our cell as a function of frequency and then obtain the best values of the parameters Rct,<7,C4i and Rso by a least squares algorithm. The advent of fast micro-computers makes this the normal method nowadays but it is often extremely helpful to represent the AC data graphically since the suitability of a simple model, such as the Randles model, can usually be immediately assessed. The most common graphical representation is the impedance plot in which the real part of the measured impedance (i.e. that in phase with the impressed cell voltage) is plotted against the 90° out-of-phase quadrature or imaginary part of the impedance. 

The universal interferometric response of a balanced two-port interferometer is shown in Fig. 11.2 as a function of the fixed phase offset between the two waves. The maximum slope of the intensity curve occurs when the fixed phase offset between the waves is an odd integer of = re/2. These conditions of maximum slope are called the conditions of phase quadrature. There are two quadrature conditions per cycle, with opposite slopes and hence opposite signed responses to modulated phase. These are the positions of maximum phase-to-intensity transduction and are the operating points for interferometric detection of protein or DNA on spinning discs. 

Phase quadrature is an over-arching concept for all interferometric detection. However, achieving a common-path configuration that locks in a stable quadrature condition puts constraints on possible system designs. This section reviews the several configurations of common-path quadrature that have been demonstrated so far in spinning-disc systems. These are the micro-diffraction class, the adaptive optical class, the phase contrast and the in-line class. At the end of this section, we show that the phase-contrast and in-line classes are conjugate quadratures of each other. 

A drawback of the MD-class BioCD is the microfabrication required to pattern gold spokes on the disc to set the quadrature condition. To remove the microfabrication, an alternative means to establish a quadrature condition uses adaptive optical mixers in the far field to establish and lock phase quadrature. In this case, the disc can be a high-reflectance antinode surface with protein patterned directly on the disc surface without any need for surface structuring. As the disc spins, the immobilized protein causes phase modulation that is detected in the quadrature condition set up by the adaptive mixer. 

In Figure 4 the effect of prothrombin on the differential capacity of a PS monolayer is presented. The overall capacitance, which is proportional to the out-of-phase (quadrature) ac current, increases upon interaction with the prothrombin. Moreover, a pseudocapacitance peak (at —0.7 V relative to the N-AgCl electrode) characteristic of cystine-cysteine redox potential appears. The peak potential moves as expected toward more... 

Figure S.Typical traces of the F and the 2F signals obtained in the N.F.V. method for a) a translation of the whole sample in front of the reading beams b) a sheared sample. The oscillations in phase quadrature in the two signals indicate a motion of the fluid within the region of the evanescent wave.

If, as in the case of DMA, a sinusoidal oscillating stress is applied to a specimen, a corresponding oscillating strain will be produced. Unless the material is perfectly elastic, the measured strain will lag behind the applied stress by a phase difference (c5) shown in Figure 2. The ratio of peak stress to peak strain gives the complex modulus ( ) which consists of an in-phase component or storage modulus ) and a 90° out-of-phase (quadrature) component or loss modulus ( ") ... 

It is important to recognize that data produced by the FT-FAM procedure encompass three dimensions, response, d.c. potential (Ejc), and frequency (w). Response may be the faradaic admittance (in-phase, quadrature or total) or its phase angle cotangent (cot ). Faradaic admittance or cot "polarograms" are plots of one or more of the faradaic admittance components or cot versus E(jc at constant w. Faradaic admittance or cot spectra are plots of one or more of the faradaic admittance components or cotnf> versus ail/2 at constant Ejc ... 

Figure 1-13 shows an example of a train of lock-in amplifiers used to demodulate DIRLD signals obtained with a spectrometer described in Figure 1-12. In order to obtain the time-dependent dynamic absorbance and DIRLD responses, quadrature lock-in amplifiers are used. These devices monitor signals both in phase and 90° out of phase (quadrature) with the sinusoidal strain reference signal. The monochromator is scanned one wavelength at a time through the spectrum. Data are collected on six separate channels (e.g., in-phase and quadrature dynamic dichroism, in-phase and quadrature dynamic absorbance, static dichroism, and normal IR absorbance)...

The first therm in parentheses shows the component of current in phase quadrature with the voltage as it would be in the case of lossless capacitors. The second term is a component in phase with the applied voltage and therefore it reperesents power dissipation (P), which is given by Equation 10. 

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