Wessel diagram

Figure 3.6 Relative permittivity plotted in the wessel diagram, same data as Figure 3.5.

Figure 3.8 Complex conductivity (siemens) in the wessel diagram. A diagram with admittance variables G and B would be similar. Same data as for Figure 3.5.

In the Wessel diagram, the locus of the permittivity is a complete half circle (Figure 3.6). Also here it is easy to see that the data are purely capacitive at high and low frequencies. 

Figure 3.8 shows the conductivity characteristic frequency is >100 Hz, and in contrast to the permittivity plot with a complete semicircle locus, there is a strong deviation with the a" diverging proportional to... 

From the apex of the circular locus of e or tr in the Wessel diagram (Figure 3.6)... 

Dispersions may be described with permittivity or immittivity parameters. Complex permittivity or complex immittivity may be presented in Wessel diagrams as shown in Chapter 9. Wessel diagrams focus data in a narrow frequency range for broadband spectroscopic data, logarithmic frequency scales are better suited (Figure 3.14). 

The influence of the electron transfer resistor Ret and the slow process admittance Ysp on the total faradaic impedance is determined by which factor is reaction rate—limiting. It is possible to study this by plotting the faradaic impedance as a function of frequency in a log—log plot, or in the Wessel diagram and look for circular arcs (see Section 9.2). 

KKTs are tools brought to network theory by the work of Kramers (1926) and Kronig (1929) on X-ray optics. Just as the reciprocity theorem, they are purely mathematical rules of general validity in any passive, linear, reciprocal network of a minimum phase shift type. By minimum-phase networks, we mean ladder networks that do not have poles in the right half plane of the Wessel diagram. A ladder network is of minimum phase type a bridge where signal can come from more than one ladder is not necessarily of the minimum-phase type. The transforms are only possible when the functions are finite-valued at all frequencies. With impedance Z = R- -jX the transforms are ... 

The time derivative as well as the time integral of a sine wave is also a sine wave of the same frequency, but phase-shifted 90°. The relationship between a sine wave and the circle is seen more directly in the complex notation of a radius r rotating around the origin in the Wessel diagram ... 

Figure 9.4 Wessel diagrams for the parallel model circuit. G...

Figure 9.5 Wessel diagrams for the series model circuit. R= 100 kQ, Cs= 10 F...

The permittivity locus of a Debye dispersion in the Wessel diagram is a complete half circle with the center on the real axis. Figure 9.8(a). An ideal resistor in parallel destroys the circle at low frequencies, upper right (see Figure 9.8(b)). The conductivity locus is equally sensible for an ideal capacitor in parallel at high frequencies. Figure 9.8(d) lower right. 

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... 

In the impedance (Z-plot) Wessel diagram, the series resistance Roo of the Cole system moves the arc of the Cole element to the right along the real axis a distance equal to the value of Roo, and with no influence on the characteristic frequency (Figure 9.12). 

Wessel diagram, but move the arc along the real axis, Figure 9.14. CcpeP, a, AR, the characteristic frequency, and the frequency scale on the arc will not change, but Goo will... 

In conclusion, a circular arc locus in the immittance Wessel diagram can ... 

A Cole—Cole plot is permittivity plotted in a Wessel diagram. If the permittivity is according to the Cole—Cole equation, the locus will be a circular arc. The permittivity used in the Cole—Cole equations implies that the model is changed from regarding tissue as a conductor (2R-1C model) to regarding tissue as a dielectric (1R-2C model) with only bound charges and dielectric losses. A 2R-1C model cannot have the same spectrum as a 1R-2C model with a fixed set of component values, so an arc locus in one model will not result in an arc locus in the other model. In living tissue, there is a substantial DC conductance. Such... 

Other distributions than the Cole—Cole of Eq. 9.40 are possible for instance, a constant function extending over a limited range of time constants. Such different distributions result in Wessel diagram loci surprisingly similar to circular arcs,... 

He introduced a constant phase element (CPE), defined in the paper by the phase angle ( )3 = arccotan (m), and m = accordingly using m completely differently from Fricke ideal resistor has m = oo and < )3 = 0°, and found the impedance locus for such a system was a circular arc with the center below the real axis in the Wessel diagram. A plot of complex immittance or immittivity in the Wessel diagram with the purpose of searching for circular arcs, may according to this book, be called a Cole-plot. 

As a permittivity equation, they used 1 — a, not a, as the exponent. The equation was derived from the Debye equation simply by analogy, based upon the overwhelming amount of experimental data for all sorts of dielectrics giving impedance loci of arcs of depressed circles (and not complete Debye half circles) in the Wessel diagram. 

From this paper, a Cole—Cole plot should be defined as a plot of the complex permittivity in the Wessel diagram to search for one or more circular arcs. 

A real number G can be regarded as a position on a number line. A complex number Y can be regarded as a point in the plane of a special Cartesian coordinate system the complex plane, also called the Argand or Wessel diagram. G is an ordinary real number situated on the real x-axis. j (actually j) indicates that B is to be situated on the imaginary y-axis. B is a real number, jB is an imaginary number, Y is a complex number. 

Complex numbers sueh as Y are written in bold in this book. Y is represented by a point in the Wessel diagram determined by G and B, the locus of Y (see Figure 12.1). 

Y = G — JB is ealled the eomplex conjugate to Y. Often the complex conjugate is used to obtain positive values for the imaginary component in the Wessel diagram. Impedance loci for instance are usually plotted with the circular arcs up, so instead of Z = R + jX,... 

Caspar Wessel (1745—1818), Norwegian surveyor. Proposed the complex plane presentation in 1797, 9 years before Argand. Presentation in the imaginary plane is accordingly called a Wessel diagram in this book. 

The convention in the electrical sciences is that in the complex plane (Wessel diagram) the counterclockwise direction corresponds to a positive phase shift between two vectors. 

A Wessel diagram is the same as an Argand diagram a diagram in the complex plane. Symbol for power line ground (safety and noise) is I... 

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