Log z plane

Instead of making root locus plots in the z plane, it is sometimes convenient to make them in the log-z plane. In the z plane, the ordinate is the imaginary part of z and the abscissa is the real part of z. In the log-z plane, the ordinate is the... 

The limit of stability becomes the imaginary axis in the log-z plane and the region of stability is the left half of the log-z plane. This is analogous to the situation in the continuous s plane. 

The first effect is obvious from the definition of the logarithm of z as given in Eq. (19.44). Inside the unit circle, the magnitude of z is less than 1. Therefore the In I z I is negative. On the unit circle, In z = 0. So the unit circle in the z plane maps into the left half of the log-z plane. 

Consider a point in the log-z plane that has these real and imaginary parts. If we draw a straight line from the origin through this point, this radial line will make an angle 0 with the horizontal axis whose tangent is the imaginary part divided by the real part (see Fig. 19.7a). 

For a constant value of damping coefficient C, tan 0 is constant. This means that a line of constant damping coefficient in the log-z plane is a radial straight line. The distance from the origin out to the point (the hypotenuse of the triangle) is... 

This is exactly the same relationship we found in the s plane. These radial lines in the log-z plane are much easier to draw than the ellipsoidal lines given by Eq. (19.17) for the z plane. 

Figure 19.7h,c compare z-plane and log-z-plane root locus plots for first-order and second-order systems. For the first-order system, the single root moves to minus infinity in the log-z plane as z goes to zero. Then the root comes back... 

The second-order system has two loci. As is increased the roots move into the right half of the log-z plane at. ... 

It should be noted that the stability limits in the s plane, the ID plane, and the log-z plane are all the same the imaginary axis. However, lines of constant damping coefficients in the ID plane are not radial straight lines as they are in the s and log-z planes. 

Sometimes other planes beside the z plane are used. The log z and ID planes offer some advantages to the z plane for some systems. We, will discuss these later in this chapter. 

Different kinds of plots based on impedance Z, admittance Z 1, modulus icoZ, or complex capacitance (z coZ) 1 can be used to display impedance data. In solid state ionics, particularly plots in the complex impedance plane (real versus imaginary part of Z) and impedance Bode-plots (log(Z) log(co)) are common. A RC element (resistor in parallel with a capacitor) has, for example, an impedance according to... 

Fig. 6. Impedance spectra in the Z plane and in the Bode plot (log(Z) vs. log(cu)) for (a) one RC element (b) two RC elements (c) three RC elements representing the situation of a polycrystal with non-ohmic electrodes and highly resistive grain boundaries and (d) three RC elements with two similar relaxation frequencies (cor2 — 3ror3) leading to overlapping semicircles.

If independent random variables x and y are uniformly distributed on a line, then their linear combination z = ax+Py is also uniformly distributed on a line. (Indeed, vector (x, y) is uniformly distributed on a plane (by definition), a set z>y is a half-plane, the correspondent probability is Vi.) This is a simple, but useful stability property. We shall use this result in the following form. If independent random variables are log-uniformly distributed on a line, then the... 

Fig. 7 Plane wave focusing by a NA = 1.35 objective lens, calculated using vectorial Debye theory, a The normalized 3D intensity distribution with the cutoff threshold at 1% intensity. The lateral cross-sections are plotted on a log-scale at the axial positions z = 0 (b) and z = 7-/2 (c), respectively. Contour lines are plotted at 0.5 (inner) and 1/e (outer) levels, respectively. Polarization of the plane wave was horizontal (along x)...

A dramatic change in the UV spectra was observed in going from the cis to the trans series of doubly-bridged ethylenes, and this can most clearly be shown between the pair of (Z) and ( )[8.8] isomers ( )[8.8], A.max 201.5 nm (log e 4.02) ( )[8.8], A.max222.5 nm (log 3.73). This bathochromic shift, as large as 21 nm, accompanied by a marked decrease in the extinction coefficient obviously reflects the unusual nature of double bond in this ( )[8.8] isomer 61c, which is caused by out-of-plane bending and rehybridization of the strained 7t bond 56). 

When some other reaction parameter, Z, such as the log of a rate constant, is plotted on to this steric and electronic map on an axis normal to the plane of the paper the comparative contributions of 6 and v should become apparent. A purely steric effect will slope north or south (the reader is encouraged to view Figures 26-28 of ref. 187 to appreciate this fully). Weimann and co-workers211 used Tolman s methodology to show the % steric effect in the oligomerization of butadiene catalyzed by nickel phosphine complexes. 

The immittance analysis can be performed using different kinds of plots, including complex plane plots of X vs. R for impedance and B vs. G for admittance. These plots can also be denoted as Z" vs. Z and Y" vs. Y, or Im(Z) vs. Rc(Z), and Im( Y) vs. Re( Y). Another type of general analysis of immittance is based on network analysis utilizing logarithmic Bode plots of impedance or admittance modulus vs. frequency (e.g., log Y vs. logo)) and phase shift vs. frequency ( vs. log co). Other dependencies taking into account specific equivalent circuit behavior, for instance, due to diffusion of reactants in solution, film formation, or electrode porosity are considered in - electrochemical impedance spectroscopy. Refs. [i] Macdonald JR (1987) Impedance spectroscopy. Wiley, New York [ii] Jurczakowski R, Hitz C, Lasia A (2004) J Electroanal Chem 572 355... 

The impedance behavior of electrode reactions is often complex but can be conveniently simulated by computer calculations, especially in the case of the method based on kinetic equations (108, 113). The forms of the frequency response represented in terms of the Z versus Z" complex-plane plots and by relations of Z or phase angle to frequency ai or log (o (Bode plots) are often characteristic of the reaction mechanism and involvement of one or more adsorbed intermediates, and they thus provide diagnostic bases for mechanism determination complementary to those based on dc, steady-state rate versus potential responses. The variations of Z versus Z" plots with dc -level potential, in controlled-potential experiments, also give rise to useful diagnostic information related to the dc Tafel behavior. 

As is well known, sources of the secondary field are induced currents, the distribution of which is defined by the frequency and the conductivity of the medium. Current lines are circles located in planes perpendicular to z-axis. In conventional induction logging, when the direction of the dipole moment coincides with the borehole axis, the main attention... 

Thus, the Koch curve has a dimension between a line (Z) = 1) and the plane (D = 2). Since D is non-integer, such stmctures are called fractals. An approximate value of D can be obtained for such structures by drawing concentric circles of different radii (R) and then counting the number of branches in each circle. A plot of N(R) against log (R) yields a straight line, the slope of which yields the fractal dimension D. In the same way if we look at the branched structure within a circle of different radii, they all look alike. 

Figure 2 X-band (9.107 GHz) CW spectrum of a powdered sample of Ni(dtc)2 doped 1 500 with Cu(dtc)2 obtained at 150K with 1,0 mW microwave power and LOG modulation amplitude and displayed as the traditional first derivative (a). Computer integration of the spectrum in A gives the absorption spectrum (b). The turning points in the powder pattern that correspond to the four copper hyperfine lines for molecules aligned with the magnetic field along the magnetic z-axis or in the perpendicular plane are marked...

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