Log Plane

As we proceed toward our ultimate objective of control loop analysis and compensation network design, we will be multiplying transfer functions of cascaded blocks to get the overall transfer function. That is because the output of one block forms the input for the next block, and so on. 

It turns out that the mathematics of gain and phase is actually easier to perform in the log plane, rather than in a linear plane. Some simple rules that will help us later are as follows  

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We are interested in plotting the gain and phase of the open-loop transfer function T(s). This is the product of the transfer functions of the G (plant) and H (feedback) blocks in cascade. Review the rules presented earlier in the section mathematics in the log plane. ... 

It actually requires a great deal of mathematical manipulation to solve the simultaneous equations and to come up with component values for a desired crossover frequency. Therefore, the derivation is not presented here, but the steps are in accordance with the basic math-in-the-log-plane tools presented in Figure 7-6. The final equations are presented below, through a numerical example, similar to what we did for Type 3 compensation. 

Next, we align all logs at the datum plane which now becomes a straight horizontal line. Note that by doing so we ignore all structural movements to which the sequence has been exposed. 

We can now correlate all events below or above the datum plane by comparing the log response. In many instances correlations are ambiguous. Where two or more correlation... 

In the excited states for the same potential, the log modulus contains higher order terms mx(x, x, etc.) with coefficients that depend on time. Each term can again be decomposed (arbitrarily) into parts analytic in the t half-planes, but from elementary inspection of the solutions in [261,262] it turns out that every term except the lowest [shown in Eq. (59)] splits up equally (i.e., the/ s are just 1 /2) and there is no contribution to the phases from these temis. Potentials other than the harmonic can be treated in essentially identical ways. 

The dynamic viscosity, or coefficient of viscosity, 77 of a Newtonian fluid is defined as the force per unit area necessary to maintain a unit velocity gradient at right angles to the direction of flow between two parallel planes a unit distance apart. The SI unit is pascal-second or newton-second per meter squared [N s m ]. The c.g.s. unit of viscosity is the poise [P] 1 cP = 1 mN s m . The dynamic viscosity decreases with the temperature approximately according to the equation log rj = A + BIT. Values of A and B for a large number of liquids are given by Barrer, Trans. Faraday Soc. 39 48 (1943). 

One can further compute the slopes b2.i and l/bi.2 of the real regression lines, drawn in the log k2 versus log ki plane and transformed into the E versus log A plane ... 

These lines are generally different from the regression lines drawn in the E versus log A plane, which have slopes h4,3 and l/b3.4... 

Both pairs of lines are identical only when r = 1 in this limiting case, all four expressions in eqs. (30) and (31) are equal, and statistics has been replaced by simple geometry. If there is no correlation at all between the original values log k2 and log ki, i.e., ri2 = 0, apparent regression lines are obtained in the E versus log A plane (Figure 10) with the slopes (when Si = S2)... 

The method outlined is quick and useful for testing isokinetic relationships described in the literature and for finding approximate values of j3 (149). It should replace the incorrect plotting of E versus log A, which gives fallacious results for the value of (3 and which usually simulates better correlations than in fact apply. Particularly, the values of correlation coefficients (1) in the E versus log A plane are meaningless. As shown objectively in Figures 9-12, the failure of this plotting is not caused by experimental errors only (3, 143, 153), nor is it confined to values of j5 near the error slope or within the interval of experimental temperatures (151). 

Fig. 2. (Left panel) evolutionary tracks using FST in the logTefj vs. log g plane (solid line non gray models with rph = 10 by Montalban et al.,2004) and 2D calibrated MLT (dashed line).(Right panel) Lithium evolution for the solar mass with different assumptions about convection and model atmospheres. The dotted line at bottom represents today s solar lithium abundance. MLT models with AH97 model atmospheres down to Tph = 10 and 100 are shown dotted for cum = 1 and dash-dotted for cpr, = 1.9. The Montalban et al. (2004) MLT models with Heiter et al. (2002) atmospheres down to Tph = 10 (lower) and 100 (upper) are dashed The continuous lines show the non gray FST models for rph = 10 and 100, and, in between, the long dashed model employing the 2D calibrated MLT.

Figure 6. Dependence of the perimeter P on the area A for the selfsimilar lakes generated by the intersection of the three-dimensional AFM image of the hot-rolled carbon steel with a plane at a height corresponding to 40% of the maximum height. Here, s means d log P/d log A...

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