Damping time-constant

Therefore, tire dissipative force tenn cools tire collection of atoms as well as combining witli tire displacement tenn to confine tliem. The damping time constant z = is typically tens of microseconds. It is important to bear in... 

In a single coil NMR experiment, the damping has to attenuate the ringing by as much as nine orders of magnitude which will take about 20 time-constants at critical damping. Since the critical damping time-constant is 1/u)q=1/271Vq where... 

Most commercially available atomic absorption spectrophotometers have damping built in. Single beam instruments can usefully employ damping time constants up to about 20 seconds. Double beam instruments, with their greater freedom from drift, can benefit from damping time constants as high as two minutes. 

Damping (time constant), amplification, spectral bandwidth, and scan rate have a considerable effect on the signal obtained and are interdependent. For these reasons, depending on the spectral characteristics of the sample and the purpose of the experiment (exploratory, routine, or high-resolution spectrum), parameters must be optimized relative to each other. Some criteria are given below. Other information on standards and calibration methods can be found in the literature [98], [99]. 

One important parameter is the minimum force detectable in a MRFM setup. For NMR QIP applications, this will establish the effective number of qubits detectable in an experiment. So, let rrif be the effective oscillating mass of the cantilever, t its damping time-constant and B the detection bandwidth. The minimum force detectable at temperature T is [10] ... 

Fig. 2.11 Wavefunctions for particles with different lifetimes. The dotted curve is the real part of an undamped wavefunction ip expl—tEat/h) with V o = 1 this represents a particle with an infinite lifetime. The energy (E is defined precisely. The solid curve is the real part of the wavefunction -tf> exp —iEat/K)exp —t/2T) with a damping time constant 2T that here is set equal to 2hlEa, this represents a particle with an energy of but a finite Ufetime of T = h/Ea...

The damping time constants corresponding to the above ( na( ) are given by... 

Filters have a time constant r = R x C which increases the damping of the measuring instrument. The time constant depends on the required attenuation and the interfering frequency, but not on the internal resistance of the measuring instrument. The time constants of the shielding filter are in the same range as those of the electrochemical polarization, so that errors in the off potential are increased. Since the time constants of attenuation filters connected in tandem are added, but the attenuation factors are multiplied, it is better to have several small filters connected in series rather than one large filter. 

In addition, water motion has been investigated in reverse micelles formed with the nonionic surfactants Triton X-100 and Brij-30 by Pant and Levinger [41]. As in the AOT reverse micelles, the water motion is substantially reduced in the nonionic reverse micelles as compared to bulk water dynamics with three solvation components observed. These three relaxation times are attributed to bulklike water, bound water, and strongly bound water motion. Interestingly, the overall solvation dynamics of water inside Triton X-100 reverse micelles is slower than the dynamics inside the Brij-30 or AOT reverse micelles, while the water motion inside the Brij-30 reverse micelles is relatively faster than AOT reverse micelles. This work also investigated the solvation dynamics of liquid tri(ethylene glycol) monoethyl ether (TGE) with different concentrations of water. Three relaxation time scales were also observed with subpicosecond, picosecond, and subnanosecond time constants. These time components were attributed to the damped solvent motion, seg-... 

Example 5.2 Derive the closed-loop transfer function of a system with proportional control and a second order overdamped process. If the second order process has time constants 2 and 4 min and process gain 1.0 [units], what proportional gain would provide us with a system with damping ratio of 0.7 ... 

While we have the analytical results, it is not obvious how choices of integral time constant and proportional gain may affect the closed-loop poles or the system damping ratio. (We may get a partial picture if we consider circumstances under which KcKp 1.) Again, we ll defer the analysis... 

The integral time constant is x = xb and the term multiplying the terms in the parentheses is the proportional gain Kc. In this problem, the system damping ratio Q is the only tuning parameter. 

In terms of controller design, the closed-loop poles (or now the root loci) also tell us about the system dynamics. We can extract much more information from a root locus plot than from a Routh criterion analysis or a s = jco substitution. In fact, it is common to impose, say, a time constant or a damping ratio specification on the system when we use root locus plots as a design tool. 

If saturation is not a problem, the proportional gain Kc = 7.17 (point B) is preferred. The corresponding closed-loop pole has a faster time constant. (The calculation of the time period or frequency and confirmation of the damping ratio is left as homework.)... 

A forcing function, whose transform is a constant K is applied to an under-damped second-order system having a time constant of 0.5 min and a damping coefficient of 0.5. Show that the decay ratio for the resulting response is the same as that due to the application of a unit step function to the same system. 

A control loop consists of a proportional controller, a first-order control valve of time constant rv and gain Kv and a first-order process of time constant T and gain Kx. Show that, when the system is critically damped, the controller gain is given by ... 

All three are completely equivalent. The time constant and the damping coeflicient for the system are... 

It is frequently useful to be able to calculate damping coeflicients and time constants for second-order systems from experimental step response data. 

Problem 6.11 gives some very useful relationships between these parameters (damping coefTicient and time constant) and the shape of the response curve. There is a simple relationship between the peak overshoot ratio and the damping coefficient. Then the time constant can be calculated from the rise time and the damping coefficient. Refer to Prob. 6.11 for the definitions of these terms. 

Shovv that the linearized system describing the gravity-flow lank of Example 6.4 is a second-order system. Solve for the damping coellicient and the time constant in terms of the paramelers of the system. 

The dynamic performance of a system can be deduced by merely observing the location of the roots of the system characteristic equation in the s plane. The time-domain specifications of time constants and damping coefficients for a closedloop system can be used directly in the Laplace domain. 

The r and C nre the time constant and damping coefficient of the system. If the system is an openloop one, these are openloop time constant and openloop damping coefficient. If the system is a closedloop one, these are closedloop time constant and closedloop damping coefficient. 

Thus the location of a complex root can be converted directly to a damping coefficient and a time constant. The damping coefficient is equal to the cosine of the angle between the negative real axis and a radial line from the origin to the root. The time constant is equal to the reciprocal of the radial distance from the origin to the root. 

Notice that lines of constant damping coefficient are radial lines in the s plane. Lines of constant time constant are cirdes. 

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