Stepped pole

The preceding upper limit to particle size can be exceeded if more than one bubble is attached to the particle, t A matter relating to this and to the barrier that exists for a bubble to attach itself to a particle is discussed by Leja and Poling [63] see also Refs. 64 and 65. The attachment of a bubble to a surface may be divided into steps, as illustrated in Figs. XIII-8a-c, in which the bubble is first distorted, then allowed to adhere to the surface. Step 1, the distortion step, is not actually unrealistic, as a bubble impacting a surface does distort, and only after the liquid film between it and the surface has sufficiently thinned does... 

If the contact angle is zero, as in Fig. XIII-8e, there should be no tendency to adhere to a flat surface. Leja and Poling [63] point out, however, that, as shown in Fig. XIII-8/, if the surface is formed in a hemispherical cup of the same radius as the bubble, then for step la, the free energy change of attachment is... 

A hexapole assembly of rods (poles) is built similarly to the quadrupole, but now there are three sets of opposed rods evenly spaced around a central axis. The hexapole cannot act as a mass filter by applying a DC field and is used only in its all-RF mode. It is therefore a wide band-pass filter and is used to collimate an ion beam. (Like-charged particles repel each other, and an electrically charged beam will tend to spread apart because of mutual repulsion of ions unless steps are taken to reduce the effect.)... 

Synchronous speeds are calculated by Eq. (29-10). Speeds above the limits given are obtained through step-up gears large high-speed centrifugal compressors are examples. Two-pole (3600 r/min at 60 Hz) synchronous motors can be built but are uneconomical in comparison with geared drives. 

Speed control in slip-ring motors has been discussed in the previous chapter. Squirrel cage motors have limitations ill their speed control in view of their fixed rotor parameters. Speed variation, in fixed steps, however, is possible in such motors if the stator is wound for multipolcs and such motors arc known as pole changing motors. Up to four different speeds can be achieved in such motors economically, in combinations of 2/4, 4/6. 4/8, 6/8, 6/12, 2/4/6. 4/6/8. 2/4/6/12 and 4/6/8/12 poles etc, or any other similar combination. For limitation in the motor size and tlux distribution, winding sets of more than two are not recommended. I he two windings can be arranged for two. three oi (maximum) four different speeds. 

The more complicated methods of compensation, such as this, allow the designer much more control over the final closed-loop bode response of the system. The poles and zeros can be located independently of one another. Once their frequencies are chosen, the corresponding component values can be easily determined by the step-by-step procedure below. The zero and pole pairs can be kept together in pairs, or can be separated. The high-frequency pole pair appear to yield better results if they are separated and placed as below. The zero pair are usually kept together, but can be separated and placed either side of the output filter pole s corner frequency to help minimize the gain effects of the Q of the T-C filter (refer to Figure B-23). 

Fig. 7.25 Identical continuous and discrete step responses as a result of pole placement.

This produces the (pole-zero cancellation) root locus plot shown in Figure 5.18. When run, exampSlO.m allows the user to select the value of K that corresponds to ( = 0.7, and then uses this selected value to plot the step response. The text that appears in the command window is... 

Here pole placement is used to design a digital compensator that produces exactly the step response of the continuous system. 

In terms of evolutionary biology, the complex mitotic process of higher animals and plants has evolved through a progression of steps from simple prokaryotic fission sequences. In prokaryotic cells, the two copies of replicated chromosomes become attached to specialized regions of the cell membrane and are separated by the slow intrusion of the membrane between them. In many primitive eukaryotes, the nuclear membrane participates in a similar process and remains intact the spindle microtubules are extranuclear but may indent the nuclear membrane to form parallel channels. In yeasts and diatoms, the nuclear membrane also remains intact, an intranuclear polar spindle forms and attaches at each pole to the nuclear envelope, and a single kinetochore microtubule moves each chromosome to a pole. In the cells of higher animals and plants, the mitotic spindle starts to form outside of the nucleus, the nuclear envelope breaks down, and the spindle microtubules are captured by chromosomes (Kubai, 1975 Heath, 1980 Alberts et al., 1989). 

The field- and time-dependent cluster operator is defined as T t, ) = nd HF) is the SCF wavefunction of the unperturbed molecule. By keeping the Hartree-Fock reference fixed in the presence of the external perturbation, a two step approach, which would introduce into the coupled cluster wavefunction an artificial pole structure form the response of the Hartree Fock orbitals, is circumvented. The quasienergy W and the time-dependent coupled cluster equations are determined by projecting the time-dependent Schrodinger equation onto the Hartree-Fock reference and onto the bra states (HF f[[exp(—T) ... 

Draw a line from e through / and extend it to cross the line drawn in step 6, at the pole point, P. 

If the real part of a complex pole is zero, then p = coj. We have a purely sinusoidal behavior with frequency co. If the pole is zero, it is at the origin and corresponds to the integrator 1/s. In time domain, we d have a constant, or a step function. 

The transfer function has the distinct feature that a pole is at the origin. Since a step input in either q in or q would lead to a ramp response in h, there is no steady state gain at all. 

We also see another common definition—bounded input bounded output (BIBO) stability A system is BIBO stable if the output response is bounded for any bounded input. One illustration of this definition is to consider a hypothetical situation with a closed-loop pole at the origin. In such a case, we know that if we apply an impulse input or a rectangular pulse input, the response remains bounded. However, if we apply a step input, which is bounded, the response is a ramp, which has no upper bound. For this reason, we cannot accept any control system that has closed-loop poles lying on the imaginary axis. They must be in the LHP. 1... 

Furthermore, conjugate poles on the imaginary axis are BIBO stable—a step input leads to a sustained oscillation that is bounded in time. But we do not consider this oscillatory steady state as stable, and hence we exclude the entire imaginary axis. In an advanced class, you should find more mathematical definitions of stability. 

There are two important steps that we must follow. First, make sure you go through the MATLAB tutorial (Session 6) carefully to acquire a feel on the probable shapes of root locus plots. Secondly, test guidelines 3 and 4 listed above for every plot that you make in the tutorial. These guidelines can become your most handy tool to deduce, without doing any algebra, whether a system will exhibit underdamped behavior. Or in other words, whether a system will have complex closed-loop poles. 

Hence, our first step is to use root locus to find the closed-loop poles of a PI control system with a damping ratio of 0.8. The MATLAB statements to continue with Example 4.7B are ... 

Instead of spacing out in the Laplace-domain, we can (as we are taught) guess how the process behaves from the pole positions of the transfer function. But wouldn t it be nice if we could actually trace the time profile without having to do the reverse Laplace transform ourselves Especially the response with respect to step and impulse inputs Plots of time domain dynamic calculations are extremely instructive and a useful learning tool.1... 

Step 4 We can now check (if we want to) the closed-loop poles and do the dynamic simulation for a unit step change in R. 

The user has no control over the size of the x grid that is sampled in the Standard Pole Figure data collection task. The default angular ranges and step sizes have been found to be convenient for many applications. The data in Figure 4 were obtained using the Standard Pole Figure data collection function. 

Figure 4. A set of pole figures for three reflections of a highly oriented linear polyethylene specimen. The concentric circles are parallels of latitude, which represent constant values of x from 0 deg (outermost circle) to 90 deg (north pole) in 15 deg steps. The straight lines are meridians of longitude, which represent constant values of < > from 180 degrees to +180 degrees.

This paper extends previous studies on the control of a polystyrene reactor by including (1) a dynamic lag on the manipulated flow rate to improve dynamic decoupling, and (2) pole placement via state variable feedback to improve overall response time. Included from the previous work are optimal allocation of resources and steady state decoupling. Simulations on the non-linear reactor model show that response times can be reduced by a factor of 6 and that for step changes in desired values the dynamic decoupling is very satisfactory. 

An ESRI system can be built with small modifications of commercial spectrometers by, for example, gradient coils fixed on the poles of the spectrometer magnet, regulated direct current (DC) power supplies, and required computer connections [40,53,55]. Gradients can be applied in the three spatial dimensions, and a spectral dimension can be added by the method of stepped gradients. The spectral dimension is important when the spatial variation of ESR line shapes (as a function of sample depth) is of interest this situation will be described below, in the ESRI studies of heterophasic polymers. In most systems, the software for image reconstruction in ESRI experiments must be developed in-house. 

The north polar region of Mars consists of variously layered sediments, the upper layer consisting of water ice and thus having a high albedo. More detailed pictures ( 30 cm per pixel) were obtained from the High Resolution Imaging Science Experiment (HiRISE) on board the Mars Reconnaissance Orbiter (MRO). The sediment layers in this region are only about 10 cm thick but seem to be covered with a layer of dust. A detailed analysis of the HiRISE pictures of the north pole deposits indicate that complex, multi-step processes must have occurred at the polar icecaps (Herkenhoff et al., 2007). 

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