Frequency-independent phase angle

This is the phenomenon of the so-called constant phase element (CPE) as it follows from eqn. (115) that Yc is composed of an imaginary and a real component, with a frequency-independent phase angle cor/2. Though the phenomenon is most clearly discovered in impedance or admittance analyses, its effect in time-domain methods should not be ignored. 

Immittance is the dependent variable in the Cole equations. For most biological systems, it is observed that the center of the impedance circular arc locus is situated below the real axis in the Wessel diagram. This was clear from the late 1920s, and Cole and Fricke published diagrams and equations based upon a frequency-independent phase angle. But in 1940, Kenneth S. Cole proposed the following empirical equation z = Zoo + (ro — rtissue impedance. The basis was findings... 

The relationship between the frequency exponent 1—p and the frequency-independent phase angle is ... 

The frequency-independent phase or loss angle tangent, defined as the ratio of imaginary and real component, is ... 

In both equations (13.1) and (13.2), the parameters a and Q are independent of frequency. When x —, Q has units of a capacitance, i.e., F/cm, and represents the capacity of the interface. When a 7 1, Q has units of s /Ocm and the system shows behavior that has been attributed to surface heterogeneity or to continuously distributed time constants for charge-transfer reactions. " Independent of the cause of CPE behavior, the phase angle associated with a CPE is independent of frequency. 

Another method often used for plotting and evaluating EIS data involves plots of log Z and 8 versus log GO. These data presentations are known as Bode plots and are illustrated by the example in Fig. 6.20, again for the simplest equivalent circuit of Fig. 6.18. Bode plots have advantages in that the impedance and impedance phase angle are shown as explicit functions of the frequency, which is the independent experimental variable. Reference to Eq 6.68 shows that at very high go values, Z approaches Rs, and at very low frequencies, Z approaches (Rs + Rp). These limits are indicated in Fig. 6.20. In analyzing intermediate frequencies, that is, when (Rs + Rp) > Z ... 

As shown by the equations for x, y, and z, v0 is the frequency of the motion. It is seen that the particle may be described as carrying out independent harmonic oscillations along the x, y, and z axes, with different amplitudes x0, yo, and z0 and different phase angles i 8V, and 82, respectively. 

Constant phase elements (CPEs) have been used in bioimpedance models since the late 1920s. A CPE can be modeled by a resistor and capacitor, both having frequency-dependent values, in such a way that the phase angle is frequency independent. A CPE is mathematically simple, but not so simple as to realize with discrete, passive components in the real world. A particular type of CPE is the Warburg element, known from electrochemistry and solid state physics. It is diffusion controlled with a constant phase angle of 45° (Warburg 1899). 

The reciprocal of the phase angle is called the quality factor Q and is often used in the description of electric circuits. The inverse of Q, i.e., Q, is often called the loss factor. The specific damping capacity of the SLS can, therefore, be expressed as AUIU=2tt8=2ttQ. Although these factors depend on frequency, they should be independent of the stress amplitude. In order to characterize the loss peak in Fig. 5.16, it is common practice to measure the peak width at 1/V2 of the maximum loss, which is approximately equal ioQ. ... 

The so-called ladder equivalent circuit shown in Figure 27.13 is characteristic of many ACs. It represents a set of several R-C parallel circuits and also Warburg diffusion impedance. Herewith, apart from the proper distributed line related to a porous structure of the studied object, one or several circuits in the ladder characterize parallel faradaic redox reactions of surface groups on the electrode. It was shown theoretically that phase angle (p = 45° independent of frequency co is observed... 

Let us calculate the characteristic properties of a general CPE as a conductor and a susceptor in parallel, both frequency dependent. The frequency dependence of the admittance Y = G + jB is sought so that die phase angle ((pq,e = arctan B/G) becomes frequency independent ... 

According to Fricke s law, there is a correlation between the frequency exponent m and the phase angle (p in many electrolytic systems when experiments show that C = CiP" , then (p = m7r/2. As pointed out by Fricke, m is often also found to be frequency dependent. However, it can be shown that Fricke s law is not in agreement with the Kramers-Kronig transforms ifm is frequency dependent (Daniel, 1967). If we therefore also presuppose m and

complex notation (remembering that j = cos aizH -H jsin aTc/2, cf. Eq. 12.6) ... 

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