Phase shift angle

The period of the interaction is 360/n. The phase angle shifts the curve to the left or right. For n=l and (j)o=0, the curve represents the situation where the energy is a minimum for the tra 5-conformation with a barrier of to the highest energy cw-conformation. A... 

Further, if within the electrical circuit the ohmic resistance R can be neglected, the ic wave leads to the potential by 90°, as is known, which means that shows a positive 7t/2 phase angle shift ( between tt/2 and zero. Our main objective in AC polarography, however, is the faradaic current, so a separating condenser is placed between the amplifier and normal resistor in order to filter out the d.c. current and to evaluate the ac current component. As we want to understand the relationship between idc(i ) and iac(i ) as a function of Edc and Eac applied, we may consider Fig. 3.41(a) and (b). 

IZI=J(Z )2+(Z ), and phase angle shift,, vs. f). The electrochemical system is then simulated with an electrical circuit that gives the same impedance response. Ideally this electrical circuit is composed of linear passive elements, e.g. resistors and capacitors, each of which represents individual physicochemical steps in the electrochemical reaction. ... 

Logarithmic plot of the impedance at zero phase-angle shift as a function of electrolyte concentration obtained from the electrochemical cell with /4=314mm2, c/=112mm and 7"=298.0K. 

Logarithmic plot of the impedance at zero phase-angle shift as a function of electrode surface area obtained from the electrochemical cell with palladium and textile structure electrodes, c/=103mm, 7"=298.0Kand an electrolyte concentration of (1) 10, (2) 10 2, (3) 10 3 and (4) 10 4moll. ... 

Logarithmic relationship between measured impedance at phase-angle shift of zero and the distance between the electrodes for constant length of the electrodes (60 mm) and diameter of the electrodes (0.2mm) and different electrolyte concentrations (1) 1 x10" (2) 1x1(T2, (3) 1 x 10 3 and (4) 1x10"4molL. ... 

The expressions shown so far for harmonics have zero phase shifts with respect to the fundamental. It is not uncommon for the harmonics to have a phase-angle shift with respect to the fundamental. Figure 4.8 depicts a fifth harmonic current waveform with and without phase shift from the fundamental. Expressions for the fifth harmonics with a phase-shift angle of 0 degrees are ... 

Figure 7 Fitted voltage Vcal, current density J, and phase angle shift

The phase angle shift can be used to obtain contrast due to local differences in energy dissipation as a consequence of different surface characteristics related to materials properties. These different properties allow one to differentiate materials with different adhesion [110] or widely different Young s moduli, if these differences are related to differences in energy dissipation [111-115]. Hence the amorphous and crystalline phases in semicrystalline polymers can be clearly differentiated, as discussed in Sect. 3.2, as well as different phases in polymer blends or filled systems (see below). As an example, we show in Fig. 3.52 an intermittent contact AFM phase image of a block copolymer thin film on silicon [116]. 

From the amplitudes of stress and strain and the phase angle shift, one can obtain the following viscoelastic parameters. 

G can be resolved into two components (i) the storage (elastic) modulus G, which is the real component of the complex modulus and (ii) the loss (viscous) modulus G", which is the imaginary component of the complex modulus. The complex modulus can be resolved into G and G" using vector analysis and the phase angle shift 5,... 

Dynamic (oscillatory) measurements A sinusoidal stress or strain with amphtudes (Tjj and is appHed at a frequency a> (rads ), and the stress and strain are measured simultaneously. For a viscoelastic system, as is the case with most formulations, the stress and strain amplitudes oscillate with the same frequency, but out of phase. The phase angle shift S is measured from the time shift of the strain and stress sine waves. From a, y and S, it is possible to obtain the complex modulus j G, the storage modulus G (the elastic component), and the loss modulus G" (the viscous component). The results are obtained as a function of strain ampHtude and frequency. 

Elastic response This occurs when the maximum of the stress amplitude is at the same position as the maximum of the strain amplitude (no energy dissipation). In this case, there is no time shift between the stress and strain sine waves. Viscous response This occurs when the maximum of the stress is at the point of maximum shear rate (i.e., the inflection point), where there is maximum energy dissipation. In this case, the strain and stress sine waves are shifted by (referred to as the phase angle shift, 5, which in this case is 90°). 

Viscoelastic response In this case the phase angle shift 5 is greater than 0, but less than 90°. 

Consider the case of a viscoelastic system, for which the sine waves of strain and stress are shown in Figure 20.11. The frequency co is in rads , and the time shift between the strain and stress sine waves is At. The phase angle shift S is given by (in dimensionless units of radians). 

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