First Order Lag

First-Order Lag (Time Constant Element) Next consider the system to be the tank itself. A dynamic mass balance on the tank gives ... 

Higher-Order Lags If a process is described by a series of n first-order lags, the overall system response becomes proportionally slower with each lag added. The special case of a series of n first-order lags with equal time constants has a transfer function given by ... 

The Bode diagram, given in Figure 6.10, is the mirror image, about the frequeney axis, of the first-order lag system. Note that the transfer funetion given in equation (6.35) is also that of a PD eontroller. 

Often an instrument response measurement can be fitted empirically to a first-order lag model, especially if the pure instrument response to a step change disturbance has the general shape of a first-order exponential. 

Complex models are often slow in execution owing to the large number of equations involved and the large range of time constants. Under these circumstances it is often useful to approximate the transient behaviour of the full model by a simpler model representation which is faster to compute. Such simplifications are commonly achieved by a combination of first-order lags and time delays and are often represented in Laplace transform form, especially when the sub-model is to be used as part of a control engineering application. 

Well-mixed tank systems (Fig. 2.18) are characterised by a first-order lag response. 

Thus as shown previously in Sec. 2.1.1.1, if the step response curve has the general shape of an exponential, the response can be fitted to the above first-order lag model by determining x at the 63% point. The response can now be used as part of a dynamical model, either in the time domain or in Laplace transfer form. 

A first-order lag will not fit the response. However, a combination of first-order lags in series can be used, as shown in Fig. 2.20. 

Figure 2.20. Combination of first-order lags in series.

Experience has shown, that most chemical processes can often be modelled by a combination of several first-order lags in series and a time delay (Fig. 2.22). 

L(f(c)) = e-tDS Hence for a single first-order lag with time delay... 

Figure 2.23. Fitting of measured response to a first-order lag plus time delay.

For a better fit of the system response, the method of Oldenbourg and Sartorius, as described in Douglas (1972), using a combination of two first-order lags plus a time delay, can be used. The method is illustrated in Fig. 2.24. and applies for the case... 

The temperature of both thermocouple and thermowell are each described by first-order lag equations, so that ... 

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... 

K and KH in (2-49a) are referred to as gains, but not the steady state gains. The process time constant is also called a first-order lag or linear lag. 

The so-called lead-lag element is a semi-proper function with a first order lead divided by a first order lag ... 

With the phase lag, we may see why a first order function is also called a first order lag. On the magnitude log-log plot, the high frequency asymptote has a slope of -1. This asymptote also intersects the horizontal Kp line at co = l/xp. On the phase angle plot, the high frequency asymptote is the -90° line. On the polar plot, the infinity frequency limit is represented by the origin as the Gp(jco) locus approaches it from the -90° angle. 

The magnitude and phase angle plots are sort of "upside down" versions of first order lag, with the phase angle increasing from 0° to 90° in the high frequency asymptote. The polar plot, on the other hand, is entirely different. The real part of G(jco) is always 1 and not dependent on frequency. 

Example 8.6. What are the Bode and Nyquist plots of a first order lag with dead time ... 

However, the result is immediately obvious if we consider the function as the product of a first order lag and an integrator. Combining the results from Examples 8.2 and 8.7, the magnitude and phase angle are... 

To help understand MATLAB results, a sketch of the low and high frequency asymptotes is provided in Fig. E8.9. A key step is to identify the comer frequencies. In this case, the comer frequency of the first order lead is at 1/5 or 0.2 rad/s, while the two first order lag terms have their comer frequencies at 1/10, and 1/2 rad/s. The final curve is a superimposition of the contributions from each term in the overall transfer function. 

On the Bode plot, the comer frequencies are, in increasing order, l/xp, Zq, and p0. The frequency asymptotes meeting at co = l/xp and p0 are those of a first-order lag. The frequency asymptotes meeting at co = z0 are those of a first-order lead. The largest phase lag of the system is -90° at very high frequencies. The system is always stable as displayed by the root locus plot. 

Since it was desired not to lose the advantage already gained from using a first-order lag on Q, the scheme shown in Figure 6 was actually used for the pole placement tests. Figure 6 differs from Figure 5 only in that the Q lag of Figure 2 is included. 

A proportional controller is used to control a process which may be represented as two non-interacting first-order lags each having a time constant of 600 s (10 min). The only other lag in the closed loop is the measuring unit which can be approximated by a distance/velocity lag equal to 60 s (1 min). Show that, when the gain of a proportional controller is set such that the loop is on the limit of stability, the frequency of the oscillation is given by ... 

Phase shift of each first-order lag = tan 1(-10[Pg.321]

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