A second-order response

Figure 4.12 A solid-state electrode showing a second-order response. The electrode shown in Figure 4.11 can be modified by the incorporation of silver chloride into the membrane to enable the activity of chloride ions in a sample to be measured. A surface reaction between the test chloride ions and the membrane silver ions alters the activity of the latter, resulting in a change in the potential difference across the membrane.

When this happens, there are four possibilities (1) A. satisfactory result has been obtained, and further investigations are unnecessary. (2) A new direction of the steepest ascent is determined and the investigation is continued in this direction. Fig. 10.4. (3) The turning point is probably close to the optimum conditions and to locate the optimum more precisely, a second order response surface model is established to map the optimum domain. (4) The improvements obtained are not significant enough, and the investigation is abandoned. 

Tliis means that no second-harmonic generation can come from a centrosymmetric medium. Only noncentrosymmetric media will give a second-order response. 

Some general criteria for setting factor levels in a second-order response surface design are the following ... 

FIG. 5.1. All zeroth- and first-order diagrams Tor a second-order response property. 

Cascading. In most cases, the distinction between second- and third-order nonlinearities is evident from the different phenomena each produce. That distinction blurs, however, when one considers the cascading of second-order effects to produce third-order nonlinear phenomena (51). In a cascaded process, the nonlinear optical field generated as a second-order response at one place combines anew with the incident field in a subsequent second-order process. Figure 2 shows a schematic of this effect at the molecular level where second-order effects in noncentrosymmetric molecules combine to yield a third-order response that may be difficult to separate from a pure third-order process. This form of cascading is complicated by the near-field relationships that appear in the interaction between molecules, but analysis of cascaded phenomena is of interest, because it provides a way to explore local fields and the correlations between orientations of dipoles in a centros5nnmetric material (52). 

Equation (14.8) shows that the response of ST to changes in the steam temperature is a second-order response, the response to changes in and are pseudo-first-order responses. 

A del is not present in all quality measurements. Consider, for example, a pH measurement. The response of a pH electrode to a change in pH can usually be characterized by a second-order response with a small time constant of the order of a few seconds and a larger time constant of the order of 5-30 seconds, depending on the type of electrode. However, the most important characteristic of the electrode is that it gives a non-linear response the time constants for a positive change in pH and negative change in pH can be considerably different. 

There are two main differences between first- and second-order responses. The first difference is obviously that a second-order response can oscillate, whereas a first-order response cannot. The second difference is the steepness of the slope for the two responses. For a first-order response, the steepest part of the slope is at the beginning, whereas for the second-order response the steepest part of the slope occurs later in the response. 

Cascading. In most cases, the distinction between second- and third-order nonlinearities is evident from the different phenomena each produce. That distinction blurs, however, when one considers the cascading of second-order effects to produce third-order nonlinear phenomena (51). In a cascaded process, the nonlinear optical field generated as a second-order response at one place combines anew with the incident field in a subsequent second-order process. Figure 2 shows... 

The second-order nonlinear optical processes of SHG and SFG are described correspondingly by second-order perturbation theory. In this case, two photons at the drivmg frequency or frequencies are destroyed and a photon at the SH or SF is created. This is accomplished tlnough a succession of tlnee real or virtual transitions, as shown in figure Bl.5.4. These transitions start from an occupied initial energy eigenstate g), pass tlnough intennediate states n ) and n) and return to the initial state g). A fiill calculation of the second-order response for the case of SFG yields [37]... 

In order to describe the second-order nonlinear response from the interface of two centrosynnnetric media, the material system may be divided into tlnee regions the interface and the two bulk media. The interface is defined to be the transitional zone where the material properties—such as the electronic structure or molecular orientation of adsorbates—or the electromagnetic fields differ appreciably from the two bulk media. For most systems, this region occurs over a length scale of only a few Angstroms. With respect to the optical radiation, we can thus treat the nonlinearity of the interface as localized to a sheet of polarization. Fonnally, we can describe this sheet by a nonlinear dipole moment per unit area, -P ", which is related to a second-order bulk polarization by hy P - lx, y,r) = y. Flere z is the surface nonnal direction, and the... 

Equations (8-23) and (8-24) can be multiphed together to give the final transfer function relating changes in ho to changes in as shown in Fig. 8-13. This is an example of a second-order transfer function. This transfer function has a gain R Ro and two time constants, R A and RoAo. For two equal tanks, a step change in fi produces the S-shaped response in level in the second tank shown in Fig. 8-14. 

Figure E-5 Frequency response of a second-order common-mode filter L = 1 mhl).

The transient response of a second-order system is given by the general solution... 

These roots determine the transient response of the system and for a second-order system can be written as... 

Fig. 6.6 Frequency response diagrams for a second-order system.

Figure 6.1 Nonlinear optical responses, (a) Second-order SF generation, the transition probability is enhanced when the IR light is resonant to the transition from the ground state g to a vibrational excited state V. CO is the angular frequency of the vibration, (b) Third-order coherent Raman scheme, the vibrational coherence is generated via impulsive stimulated...

Hirose et al. [26] proposed a homodyne scheme to achieve the background-free detection of the fourth-order field. With pump irradiation in a transient grating configuration, the fourth-order field propagates in a direction different from that of the second-order field because of different phase match conditions. The fourth-order field is homodyned to make ffourth(td. 2 D) and spatially filtered from the second-order response hecond td, 2 D). 

Response of a Second-Order Temperature Measuring Element... 

A semi-empirical, second-order response lag is used. This consists of a first-order lag equation representing the diffusion of oxygen through the liquid film on the surface of the electrode membrane... 

Table 4 Comparison of the observed signal intensity with calculated response based on the best fit of a linear or a second-order calibration line...

We do not need to carry the algebra further. The points that we want to make are clear. First, even the first vessel has a second order transfer function it arises from the interaction with the second tank. Second, if we expand Eq. (3-46), we should see that the interaction introduces an extra term in the characteristic polynomial, but the poles should remain real and negative.1 That is, the tank responses remain overdamped. Finally, we may be afraid( ) that the algebra might become hopelessly tangled with more complex models. Indeed, we d prefer to use state space representation based on Eqs. (3-41) and (3-42). After Chapters 4 and 9, you can try this problem in Homework Problem 11.39. 

With respect to the overdamped solution of a second order equation in (3-21), derive the step response y(t) in terms of the more familiar exp(-t/xi) and exp(-t/X2). This is much easier than... 

What is the expected time response when the real part of the pole is zero in a second order function The pole can be just zero or have purely imaginary parts. 

If we assume that an oscillatory system response can be fitted to a second order underdamped function. With Eq. (3-29), we can calculate that with a decay ratio of 0.25, the damping ratio f is 0.215, and the maximum percent overshoot is 50%, which is not insignificant. (These values came from Revew Problem 4 back in Chapter 5.)... 

Example 6.2 Derive the controller function for a system with a second order overdamped process and system response as dictated by Eq. (6-22). 

In contrast to Eq. (6-22), we can dictate a second order underdamped system response ... 

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