System time constants

Experimental determination of system time constant using step response... 

Here values Ta and Tb are determined from the measured response curve and are related to the system time constants Ti and X2< by the formulae... 

Change K over a range of values and observe how this changes the time to reach equilibrium. Is there a relationship between K and the system time constant T ... 

Vary L and G and observe the influence on the dynamics. Explain the results in terms of the system time constants. 

In the event that we are modeling a process, we would use a subscript p (x = xp, K = Kp). Similarly, the parameters would be the system time constant and system steady state gain when we analyze a control system. To avoid confusion, we may use a different subscript for a system. 

The system steady state gain is the same as that with proportional control in Example 5.1. We, of course, expect the same offset with PD control too. The system time constant depends on various parameters. Again, we defer this analysis to when we discuss root locus. 

This is a question that invariably arises what is a reasonable choice of the system time constant xc Various sources in the literature have different recommendations. For example, one... 

With the IMC tuning setting in Example 5.7B, the resulting time response plot is (very nicely) slightly underdamped even though the derivation in Example 6.4 predicates on a system response without oscillations. Part of the reason lies in the approximation of the dead time function, and part of the reason is due to how the system time constant was chosen. Generally, it is important to double check our IMC settings with simulations. 

Model-based) Direct synthesis For a given system, synthesize the controller function according to a specified closed-loop response. The system time constant, xc, is the only tuning parameter. 

No a priori knowledge of the system time constants is needed. The method automatically results in a sustained oscillation at the critical frequency of the process. The only parameter that has to be specified is the height of the relay step. This would typically be set at 2 to 10 percent of the manipulated variable range. 

G( ) is the transfer function relating 0O and 0X. It can be seen from equation 7.18 that the use of deviation variables is not only physically relevant but also eliminates the necessity of considering initial conditions. Equation 7.19 is typical of transfer functions of first order systems in that the numerator consists of a constant and the denominator a first order polynomial in the Laplace transform parameter s. The numerator represents the steady-state relationship between the input 0O and the output 0 of the system and is termed the system steady-state gain. In this case the steady-state gain is unity as, in the steady state, the input and output are the same both physically and dimensionally (equation 7.16h). Note that the constant term in the denominator of G( ) must be written as unity in order to identify the coefficient of s as the system time constant and the numerator as the system... 

The denominator of this linear first-order differential equation gives the process system time constant of 20 min in the expression 1 + 20q. Likewise, the numerator gives the zero-frequency process gain of 15°F/(lb)(in2). 

This part demonstrates how deterministic models of impedance response can be developed from physical and kinetic descriptions. When possible, correspondence is drawn between hypothesized models and electrical circuit analogues. The treatment includes electrode kinetics, mass transfer, solid-state systems, time-constant dispersion, models accounting for two- and three-dimensional interfaces, generalized transfer functions, and a more specific example of a transfer-function tech-nique.in which the rotation speed of a disk electrode is modulated. 

Further light on the carbon contamination has been obtained by Hori and Schmidt during a study of CO oxidation between 430 and 1230 °C at 0.1 Torr over Pt wires. They found slow transients (>10s time constants where the system time constant was 1 s) which AES showed were due to the formation of carbonaceous films. These films were laid down between 430 and 930 °C even when the reaction mixture was oxygen rich. Simultaneously the surface became microfacetted on a 0.5 [xm size scale. 

Mahalanobis angle between a and b with vertex at origin Target for the mean, first-order system time constant MFC cost function... 

The system time constants are taken as the negative reciprocals of the real parts of the non-zero eigenvalues of the matrix A. These determine the time responses of various parts of the system in an analogous way to ri in the example above. 

The methodology and apparatus discussed above has been tested extensively on RC networks, and has performed very satisfactorily. Some results of this type are shown in Fig.3C and 3D. We have been able to measure RC time constants near 500 ns with 30 ns accuracy. These and other tests suggest that the instrument response function has a characteristic time that can be as short as about 30 ns, depending on the level of the current. Capacitive effects in the LED s junction region cause the response time of the diode to lengthen at reduced current levels, but a quite usable measurement system time constant was recorded even for a 20 nA current pulse, as shown in Fig.4. We have not been able to record current transients of this magnitude by any other means with a time resolution even close to that established in Fig.4. 

It is convenient to rearrange the equations to a standard form, which makes the system time constant explicit. Thus, we wish to rearrange so that the appearance is like... 

Next, divide by kV + q and define the system time constant as... 

Thus, suppose the CSTR flow rate is, 9 = 0.1 liter/sec and = 1 mole/liter. Now suppose we inject 100 cc of solute A of composition 1 mole/liter (= into the flowing inlet stream. We shaU model this injection as an impulse, since it was done quickly, relative to the system time constant r. The net molar... 

It is seen that (< ) - 0, and the system returns to the originid steady state. This is a clear advantage for impulse eiq)eriments, since the reactor is disturbed for only brief periods of time. Note, since Cg and e are known, the intercept at t = 0 allows the estimate of system time constant t. Again, it is also clear that the impulse response is simply the time derivative of the step response, since... 

An analogous situation can be found with the time constant t for a relaxation process. It is also a variable pertaining to space-time, defined as an eigen-value of the time derivation (without imaginary number as here), which matches a system time constant Xq defined as a... 

T Target for the mean, first-order system time constant... 

The current starts at i = Eq/R at t = 0 then decreases exponentially with time to zero as the capacitor is charged from 0 to Eq the constant current cannot flow through the capacitance. The potential step and the response of the system are displayed in Fig. 2.20. The rate at which the current decreases with time depends oti RC, which is called the system time constant t = RC, if the time constant is smaller (smaller resistance or capacitance), then the current decay is faster. 

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