Page 104 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

A New Do11ain and More Process Models 79

barely discemable and the lag is almost 90°. Notice that the scales of
the time axes on these last three plots are different. The first has a
span of 350, the second 120, and the third has a span of 3.0.
What is going on? The inertia associated with the mass of liquid
in the tank (characterized by the tank's time constant) causes the out-
put flow rate's response to be attenuated as the frequency of the input
flow rate increases. At low input frequencies, in spite of the inertia,
the output flow rate is nearly in phase with the input flow rate and
there is almost no lag. The slowly varying input flow rate gives the
mass of liquid time to respond. As the frequency increases, the mass
of the liquid cannot keep up with the input flow rate and the lag
increases and the ratio of output amplitude to input amplitude
decreases. Note, however, that the frequency of the output flow rate
is still identical to that of the input flow rate. As you might expect,
and as we will soon show, the phase lag is directly related to the pro-
cess time constant. Likewise, the attenuation in the amplitude ratio
depends on the process time constant.
From the point of view of the flow rates, the tank behaves as a low
pass filter, that is, it passes low frequency variations almost without atten-
uation with almost zero phase lag. For high frequency variations it atten-
uates the amplitude and adds phase lag. Filters as processes or processes
as filters will be dealt with later on in this chapter and in Chap. 9.

4-1-2 A Uttle Mathematical Support In the Time Domain
Let's see if some simple math can "prove" our contentions. Another
way of writing Eq. (4-2}, ignoring the constant offset value, is

U(t) =Au sin(2tr ft) =Au Re(ei2nftt

This makes use of Euler's equation that is presented in App. B. It
simply says that a sine function is the real part of a complex exponen-
tial function. If this bothers you and you do not want to delve into
App. B, then you had best skim the rest of this subsection. If not, then
temporarily forget about the "real part" and use

(4-3)

This is a common method of control engineers. It says, "make the
input flow rate a complex sinusoid (knowing full well that you are
only interested in the real part) and use it to solve a problem; then
when the solution has been obtained, if it is complex, take the real
part of the solution and you're home!" The simple algebra of complex
exponentials is often preferable to the sometimes sophisticated com-
plexity of the trigonometric relationships.
With this leap of faith in hand, feed the expression for U(t) given
in Eq. (4-3) into the differential equation describing our simple tank