Stability, marginal

The copper-rich amalgams have performed well in clinical trials in which they were compared with alloys having lower copper content. An improved marginal stability was observed, which may be associated with a longer clinical lifetime (136). These amalgams have also been called non-y2 amalgams, where refers to the compound Sn Hg comprised of Sn Hg [11092-12-9] and Sn Hg [11092-11-8]. 

Imaginary axis crossover The loeation on the imaginary axis of the loei (marginal stability) ean be ealeulated using either ... 

From Routh s array, marginal stability oeeurs at A" = 70. 

Note that method (b) provides both the erossover value (i.e. the frequeney of oseillation at marginal stability) and the open-loop gain eonstant. 

Unit circle crossover This can be obtained by determining the value of K for marginal stability using the Jury test, and substituting it in the characteristic equation (7.76). 

Sketch the root locus diagram for Example 7.4, shown in Figure 7.14. Determine the breakaway points, the value of K for marginal stability and the unit circle crossover. 

K for marginal stability. Using the Jury test, the values of K as the loeus erosses the unit eirele are given in equations (7.75) and (7.73)... 

In Example 6.4, when there was no model uneertainty, K for marginal stability was 8, and for a gain margin of 6dB, K was 4. In this example with model uneertainty, from equation (9.154) marginal stability oeeurs with K = 3.5, so this is the maximum value for robust stability. For robust performanee, equation (9.150) applies. For a speeifie step input let lV(s) = 1 /s now... 

When run, the program invites the user to seleet a point in the graphies window, whieh may be used to find the value of A" when ( = 0.25. If the last line of examp59.m is typed at the MATLAB prompt, the eursor re-appears, and a further seleetion ean be made, in this ease to seleet the value of K for marginal stability. This is demonstrated below... 

The seript file examp75.m simulates the Jury stability test undertaken in Example 7.5. With the eontroller gain K in Example 7.5 (Figure 7.14) set to 9.58 for marginal stability see equation (7.75), the roots of the denominator of the elosed-loop pulse transfer funetion are ealeulated, and found to lie on the unit eirele in the z-plane. 

Octol shows a small increase in stability over that of Cyclotol at higher temps which may be sufficient to overcome the marginal stability of Cyclotol in specific cases... 

Pu0F2(c) has not been stabilized so far and only the thorium and uranium analogs have been reported. These compounds are of marginal stability towards decomposition according to reaction (7)... 

For a more complex problem, the characteristic polynomial will not be as simple, and we need tools to help us. The two techniques that we will learn are the Routh-Hurwitz criterion and root locus. Root locus is, by far, the more important and useful method, especially when we can use a computer. Where circumstances allow (/.< ., the algebra is not too ferocious), we can also find the roots on the imaginary axis—the case of marginal stability. In the simple example above, this is where Kc = a/K. Of course, we have to be smart enough to pick Kc > a/K, and not Kc < a/K. 

Fig. 5. The first two marginal stability envelopes for VJV = 0 for the bihyperboloidal electrodynamic balance. Also shown as open circles are the experimental stability data of Taflin et a/. (1989). Reprinted, in part, with permission from Taflin, D. C, Ward, T. L., and Davis, E. J., Langmuir 5, 376-384, Copyright 1989 American Chemical Society.

The marginal stability envelopes are shifted when a bias voltage is applied, and recently Iwamoto et al. (1991) prepared a number of stability maps showing the effects of bias voltage. They solved the governing equations numerically. 

Now assuming a numerical value for Cq and computing / , the point S, should fall on the lowest marginal stability curve of Fig. 5. If not, Cq must be corrected appropriately. Tallin et a/. (1989) used this technique to determine the balance constant for a device similar to Fulton s, and their results are presented in Fig. 5 as the open circles. They determined the balance constant to be 0.79 with a standard deviation of 0.020, which is in good agreement with the numerical estimates of Philip et al and Sloane and Elmoursi. The data are seen to follow the marginal stability curve very well. 

Fig. 5. Steady-state solution of deterministic rate equations to which a stochastic term has been added. Low noise level (mean absolute magnitude of fluctuations). Note increase in noise level near lower marginal stability point.

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