Page 105 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

80 Chapter Four

of liquid, that is, Eq. (4-1), and assume that the process outlet flow
rate, Y(t}, will also be a sinusoid with the same frequency but with a
phase relative to U(t), namely,

Y(t) = Cei<2nft+8) (4-4)

Note that the amplitude of the process outlet C is as yet
unknown as is the phase 8. As mentioned in our leap of faith state-
ment, assume that the actual process outlet flow rate is the real
part of the expression in Eq. (4-4). Using Eq. (4-4) also assumes
that after making the process input U a sinusoid, all of the tran-
sients associated with that change have died out leaving the outlet
flow rate to be a complex sinusoid with a modified amplitude and
phase. If the choice of Eq. (4-4) is incorrect it will show up quickly
as we turn the crank.
With all these nontrivial preliminaries out of the way, put
Eqs. (4-2) and (4-4) into Eq. (4-1) and see if something insightful hap-
pens. A lot of the details will be left to App. B. After plugging in the
expression for U andY, Eq. (4-1) becomes

-r(j2tr f)Cei<2nft+B) + Cei<2~rft+B) = Auei<2~rft)

This messy looking expression is really a simple (but complex)
equation that has real and imaginary parts. In App. B these real and
imaginary parts are collected algebraically and the real parts on the
left-hand side of the above equation are equated to the real parts on
the right-hand side. The same thing is done with the imaginary parts.
This gives two equations that, with the help of some inverse trigono-
metric identities, can be solved for the unknown amplitude C and the
phase 8. The result is

A (4-5)
C= u
~1 + {2tr f-r) 2

This supports the earlier contention that as the frequency increases
the phase lag increases (or the phase becomes more negative) as does
the amplitude attenuation. It also tells us that as the tank size and its
associated time constant increase so does the phase lag and the ampli-
tude attenuation. Furthermore, it supports our contention that the
frequency of the process output is the same as that of the input [if this
were not the case then the idea of plugging in assumed functions for
U(t) and Y(t) would not have worked]. Finally, it suggests that the
maximum phase lag for this first-order process is 90°.