Second-order real roots

The roots of the characteristic equation can be real or complex. But if they are complex they must appear in complex conjugate pairs. The reason for this is illustrated for a second-order system with the characteristic equation... 

A gain of 17 gives a closedloop damping coefficient of 0.316 and a dominant second-order closedloop time constant of 0.85 minutes. The third root is real and lies far out on the negative real axis at —2.3. Thus the largest first-order time constant is 0.43 minutes. 

We took the 4- sign on the square root term for second-order kinetics because the other root would give a negative concentration, which is physically unreasonable. This is true for any reaction with nth-order kinetics in an isothermal reactor There is only one real root of the isothermal CSTR mass-balance polynomial in the physically reasonable range of compositions. We will later find solutions of similar equations where multiple roots are found in physically possible compositions. These are true multiple steady states that have important consequences, especially for stirred reactors. However, for the nth-order reaction in an isothermal CSTR there is only one physically significant root (0 < Ca < Cao) to the CSTR equation for a given T. ... 

REM --------- SECOND ORDER MODEL - TWO DISTINCT REAL ROOTS... 

The polynomial P(s) is of second order and has two distinct roots which are not real (as in the previous case) but complex conjugates ... 

This filter has two poles and two zeroes. Depending on the values of the a and b coefficients, the poles and zeroes can be placed in fairly arbitrary positions around the z-plane, but not completely arbitrary if the a and b coefficients are real numbers (not complex). Remember from quadratic equations in algebra that the roots of a second order polynomial can be foimd by the formula (-a +/- / 2 for the zeroes, and similarly for the... 

J5 Solve the second-order secular equation (8.57) for the special case where //n = H22 and Sii = 822- Reminder fI and /2 are real functions.) Then solve for C1/C2 for each of the two roots Wj and W2-... 

Solving this cubic equation gives three values for T, i.e., the three principal values. It can be shown that if the second-order tensor is symmetric, the roots of the cubic equation must be real and the axes must be orthogonal. An example to show how Eq. (2.35) is derived will be given in the next section. 

Equation 3.19, Equation 3.29, and Equation 3.33 give the general solutions of the second order, constant coefficient, homogeneous, and linear differential equation for the respective cases of real unequal, repeated, and complex characteristic roots. However, the actual steps that are used in deriving a solution to the homogeneous problem are as follows ... 

In the above case, both eigenvalues were real and distinct, When the eigenvalues are real and repeated, the solution for a second-order equation with both roots identical is formed as follows ... 

The results are exact—we do not need to make approximations as we had to with root locus or the Routh array. The magnitude plot is the same as the first order function, but the phase lag increases without bound due to the dead time contribution in the second term. We will see that this is a major contribution to instability. On the Nyquist plot, the G(jco) locus starts at Kp on the real axis and then "spirals" into the origin of the s-plane. 

Figure 3 compares the exact solution (49) and its approximations obtained by truncating series (61) at first, second, etc. terms. It shows that the even first term (—Bq/Bj) provides reasonable approximation in the finite neighborhood of equilibrium. Addition of higher order terms increases the domain of close approximation. This is not surprising because the condition of convergence of this series is 2[Pg.74]

In principle, to study the local stability of a stationary point from a linear approximation is not difficult. Some difficulties are met only in those cases where the real parts of characteristic roots are equal to zero. More complicated is the study of its global stability (in the large) either in a particular preset region or throughout the whole phase space. In most cases the global stability can be proved by using the properly selected Lyapunov function (a so-called second Lyapunov method). Let us consider the function V(c) having first-order partial derivatives dY/dCf. The expression... 

For a sufficiently weak electric field, the second of these equations has two distinct positive roots u = Uj and u = U2, ordered such that i < Uz- v is accordingly determined by the first equation. Only the small root Ui corresponds to a local minimum of V(u,v) so that the semiclassical expansion is performed around u = Ui and v = Vi = — 4Fuf. The values i and uz approach one another with increasing strength of the electric field and eventually become equal at some F = Fq. It is then found [25] that the (real) local minimum disappears for F > Fq. The above treatment of Bender et al. has its counterpart in more conventional terms in the article by Bethe and Salpeter [27]. 

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