Page 106 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

A New Do11ain and More Process Models 81

Question 4-1 Why is the maximum phase lag of the first-order model 90°?

1
AniWM' As m, in 8 = -tan- (0r1 increases without bound to oo, the arctangent
1
function yields tan- (oo) = 1C /2 or 90°.
4-1-3 A Uttle Mathematical Support In the Laplace
Transform Domain
From Chap. 3 and App. F, the transfer function for the process
described by Eq. (4-1) can be obtained directly from Eq. (3-33) by set-
ting g = 1.0, resulting in

- 1 - -
Y=--U=GU
-rs+1 P
(4-6)
1
G =--
P t'S + 1
Now, another trick! Let s = j2.1Cf and find the magnitude and the
phase of the result
1
G ('2nf)-
pi --rj2.nf+1
= 1 -r j2.n f + 1
1' j2.1r f + 1 . -1' j2.1r f + 1
- -1' j2.1r f + 1
2
- (r2n f) + 1
1 . 1'2.1r f
2
2
- (-r2n /) + 1 I (-r2.n /) + 1
Here, the numerator and denominator of the complex transfer
function were multiplied by the conjugate of the denominator
-1' j2.n f + 1. This got rid of imaginary components in the denomina-
tor and allowed us to separate GP into its real and imaginary parts.
The transfer function is now a complex quantity with a magni-
tude and a phase, as in

As shown in App. B, the magnitude of a complex quantity is the
square root of the real and imaginary parts

(4-7)