Phase angle tangent

The ratio is the tangent of a phase angle 8 by which the strain lags behind the stress. 

Fig. 2.9.12 Phase shift maps measured as the difference in the signal phases with and without current pulses. The rf, field gradient and current pulse program are shown in Figure 2.9.2. The phase shift maps must be unwrapped as the original evaluation yields phase angles only in the principal range between —7t/2 and +7r/2 as a consequence of the 7r periodicity of the tangent function formed as the quotient of the imaginary and real signal...

Tangent of the phase angle difference (< between stress and strain during forced oscillations. 

Tangent of the loss angle and the cotangent of the phase angle... 

The tangent of the capacitance phase angle has been shown to be a measure of the completeness of the anneal (2.) Based on this technique, we have data which would predict 30 minutes at 300°C is sufficient to obtain optimum dielectric properties. However, we have found chronogravimetric analysis of polyimide films to be more useful in defining optimum anneal conditions. 

Figure 10.12 Chain scission in polyethylene oxide) matrix, (a) Carbonyl region of photoaged PEO samples irradiated at lOOmWcnr2 and 75°C for various times (0, 1,2 and 4 minutes). The main band at 1725cm-1 is attributed to formate functions, whereas the shoulder at 1 750cm"1 is assigned to esters, (b) Evolution of the storage modulus (G ), the loss modulus (G"), and the tangent of the phase angle tan(8) versus time (temperature 90°C). The increase in tan(8) is evidence of chain scission, (c) Endotherms of the fusion of PEOs samples recorded at...

When using small deformation rheology there are several useful parameters that may be obtained to describe a material the complex modulus (G ), storage modulus (G ), loss modulus (G") and the tangent of the phase shift or phase angle (tan 5). These values must be taken from within the LVR, and are obtained using a dynamic oscillatory rheometer (Rao 1999). Outside the LVR, important information may be obtained such as the yield stress and yield strain. 

The ratio of the loss and storage moduli may be expressed as the tangent of the phase angle (tan5) ... 

Dielectric loss tangent The difference between 90° and the dielectric phase angle for a material. 

The tangent of the phase angle (tan 5) offers a better indicator of structural integrity than either the G or G" measurements do individually. The phase angle (8) or its tangent (tan 8) are an indicator of the system s structural behavior or integrity because it indicates whether the system behaves predominantly as a solid, liquid, or viscoelastic material. The greater the tan 8, the more liquid-like the samples will behave, and conversely, low values indicate more solid-like properties. A tan 8 value of 1 is indicative of a viscoelastic material. 

Usually each reflection forms more than one triplet and each of the triplets may be used for phase determination (estimation). In order to employ all triplets and thus, obtain the best agreement between phase angles that result from different triplets, Karle and Hauptmatf introduced a general expression for phase determination from triplets. This relationship is known today as the tangent formula ... 

Also, as seen in Fig. 6.16, the tangent of the phase angle of Ir relative to the voltage is given by the imaginary component divided by the real component ... 

The tangent of the phase angle between the stress and strain was shown to be (Chapter 2, Section C) ... 

The ratio of the loss index to its relative permittivity, the tangent of its loss angle, 5 or the cotangent of its phase angle, 0. 

Vibration forces applying a dynamic stress load to viscoelastic materials results in a phase shift by the phase angle 8 between stress a and elongation e. The tangent of 8 is called the mechanical loss factor d or mechanical damping. Damping is thus a measure of the heat produced by application of dynamic loads as a result of internal friction (dissipatiOTi) (Fig. 24). 

Closer examination of Etjs. [18.17]-[18.20] reveals important relationships between the phase and modulation values of the F and R states. For an excited-state process, the phase angles of the F and R states are additive, and the modulations multiply. Once this is understood, the complex expressions (Eqs. [18.19] and[18.20]) become easier to understand. Let be the phase angle of the R state that would be observed if this state could be excited directly. Of course, this is related to the lifetime of the directly excited R state by tan = cmOx. Using Eq. [18.19], die law for the tangent of a sum, and tan (9x f- = tani g, one finds... 

Figure 5.4 shows also the loss tangent, tan 5, the tangent of the phase angle 8 between stress and strain in a cycle as depicted in Fig. 5.3, which represents the fractional energy dissipation in a cycle, i.e.. 

The Autovibron system is designed to measure the temperature dependence of the complex modulus (E ), dynamic storage modulus (E ), dynamic loss modulus (E") and dynamic loss tangent (tan 6) of viscoelastic materials at specific selected frequencies (0.01 to 1 Hz, 3.5, 11, 35, 110 Hz) of strain input. During measurement, a sinusoidal tensile strain is imposed on one end of the sample, and a sinusoidal tensile stress is measured at the other end. The phase angle 6 between strain and stress in the sample is measured. The instrument uses two transducers for detection of the complex dynamic modulus (ratio of maximum stress amplitude to maximum strain amplitude) and the phase angle 6 between stress and strain. From these two quantities, the real part (E ) and the imaginary part (E ) of the complex dynamic modulus (E ) can be calculated. 

The effective loss factor e" is a fl equency-dependent parameter determined by the material properties that characterizes the ability of the material to absorb electromagnetic energy. It is defined as a product of the dielectric constant e and the loss tangent tan 8 (8 = phase angle between circuit currents)... 

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