Page 118 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

A New De11ain a•• More Process Models 93

and a phase. The simplest way to get the overall magnitude and
phase is to look at each factor separately and convert it into a mag-
nitude and a phase. After this conversion, the magnitudes can be
multiplied and divided as necessary and the phases can be added or
subtracted as necessary:

. :\-G (" )G(" :\- g kjro+l
G( Ja>,- P Ja> 'Jro,--.--1-.-
-r Ja>+ Ja>
_ geio .J<kro~2 + 12 eilz
- ~( -rro)2 + 1 ei~ 1ei~r/2

= g~(kro)2 + J2 ei<'z-6t-lf/2)
2
J<-rro) + 1
= 1Giei9

8=82-81-tc/2
1
8 = tan- (-rro) (4-11)
1
9 2 =tan-•(¥)

IGI = g.J(kro)2 + J2 8= tan- 1 - - -tan- (-rro)- tc
kroJ
1
2
~(-rro) + 1 ( 1 2
The development of these equations used the simple algebra of com-
plex exponentials where the magnitudes multiply and the angles add.

Question 4-4 Could you derive the appropriate equations for the magnitude and
angles for the case where I= 0?
Answer With I= 0 and k = 1, Eq. (4-9) simplifies to the equation in Sec. 4-2 for
the first-order process without control. See Eqs. (4-7) and (4-8).
Question 4-5 Could you derive the appropriate equations for the magnitude and
angles for the case where k = 0?
Answer Take the limit ask~ 0 in Eq. (4-11) and remember that the angle whose
tangent is zero is zero. The result will be