The Argand Diagram

The location of z in the Argand diagram can be specified by using either Cartesian coordinates (x,y), where x = Rq z, y = lm z, or polar coordinates 

Q (a) Given that tan0 = use the addition formulae for sine 

Hint you will need to exercise a little care in determining the argument for the second of the two complex numbers. 

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Figure 7. (A, top) Simple battery circuit diagram, where Cdl represents the capacitance of the electrical double layer at the electrode—solution interface (cf. discussion of supercapacitors below), W depicts the Warburg impedance for diffusion processes, Rj is the internal resistance, and Zanode and Zcathode are the impedances of the electrode reactions. These are sometimes represented as a series resistance capacitance network with values derived from the Argand diagram. This reaction capacitance can be 10 times the size of the double-layer capacitance. The reaction resistance component of Z is related to the exchange current for the kinetics of the reaction. (B, bottom) Corresponding Argand diagram of the behavior of impedance with frequency, f, for an idealized battery system, where the characteristic behaviors of ohmic, activation, and diffusion or concentration polarizations are depicted.

Figure 4.100 shows the Argand diagram of water (curve 1) and the permittivity for 0.8 M KCl (curve 2) in water. The stractural part of the spectrum is represented by curve 3. The difference of curves 2 and 3 is the result of electrolytic conductance. 

The dispersion and absorption curves of the pure solvents undergo drastic changes when an electrolyte is added, the most important being the superposition of conductivity shown in the absorption curve q"(i ) of fig. 5 a and in the Argand diagram tj" = f(e ) of fig. 5 b. Reduction of i7"( ) to t" v) is executed with the help of measured static conductivities cr. Two relaxation processes tire corroborated by two inflexion points of e v), two maxima... 

The Argand diagram of the 2.2 M solution of Et NCl shows three relaxation processes typical for aqueous electrolyte solutions (1) ion-pair relaxation (r l = tip), (2) low frequency relaxation (rj, as 8 ps) of water, (3) high frequency relaxation a 1 ps) of water, in contrast to that of the 2 M solution of Bu NBr where the relaxation process (2) splits up into two processes. Figure 7 shows the concentration dependence of the... 

The information about the phase at is then contained in the exponent. Consequently, we can represent each term with the Argand diagram (Figure 14), where the imaginary component specifies the phase. Each term is then characterized by two parameters the amplitude Ft and the phase . 

The graphical representation of the complex number through the Argand diagram. 

Fig. 3. Construction of the Argand diagram from the dispersion of f and f". f is taken from the dispersion of apparent radius of gyration, R, of ferritin, f" is proportional to the absorption of the ferriton solution at the K absorption edge of iron at = 1.743 A...

Ftg. 110 The Argand diagram used to represent complex numbers... 

The complex conjugate, x, equals a - bi and is obtained by reflecting x in the real axis in the Argand diagram. 

In the Argand diagram, the current vector rotates 90° ahead of the voltage vector. For real dielectrics, the current leads the voltage by an angle 90° - 5, and is given by... 

Fig. 3.7 The complex-plane impedance plot representation (also called the Argand diagram or Nyquist diagram) of the ideal impedance spectra in the case of reflective boundary conditions. Effect of the ratio of the film thickness (L) and the diffusion coefiicient (D). L/D (7) 0.005 (2) 0.1 (2) 0.2 4) 0.5 and (5) 1 s /. i o = 2 Q, 7 ct = 5 Q, (7 = 50 cm fis / Cdi = 20 pFcm. The smaller numbers refer to frequency values in Hz...

IS measurements have been performed in the 253-333 K temperature range and the results from the measurement at 293 K are shown in Figure 3-43. The Argand diagram consists of two semidrcles which can be understood and represented by a best fit of the data to an ideal Debye resistance (/ 2)/capadtance (C2) parallel link (lower frequency process 2) in series with an additional Cole-Cole resistance (Ri)/CPE (constant phase element) parallel link (higher frequency process 1). From the fit data, a Cole-Cole parameter a = 2/3 results which is very typical for percolation mechanisms. The Ohmic part is the same for both processes Ri = and so = 2/ ,. The spedfic conductivity processes have Oi = 02 = 3.23 X 10 Q" m at 293 K. Since both processes are thermally... 

The current will no longer be in phase or 90° out of phase with the voltage, but can be obtained simply from the Argand diagram, Fig. 8.5. If the phase angle is denoted by 0 ... 

At higher frequencies there is a further shift when / ply to Zsc> described by Eqn. 50. However because for Case 1 p > y, the semicircle is joined when / > 1 and only a small arc is observed. Similar to Case 1, in Cases A and B the Argand diagram is dominated by the transmission line however the high-frequency behavior of Cases A and B differs from that of Cases C and D. In Cases A and B the transmission line always dominates in Cases C and D it can be shunted by the double-layer capacitance, and hence Zpc can be observed. 

Figure 12.5 Complex charging current as vectors in the Argand diagram.

The Argand diagram (Figure 7.14) shows that this parallels the complex modulus treatment. The tan 8 values are not equal for the same mechanical and dielectric process. 

FIGURE 3.4 Representation of the complex number z = x + iy ia the argand diagram. 

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