Question

Question 4 a) A simplified form of the open loop transfer function of a chemical process as shown in Figure 4 is G(s) s(s+3)(s2 + 25 + 2) where (s) = 1. i Construct the root locus. (5 marks) . From the root locus in (a), show the range of gain at the root loci so that the system will stable. Discuss your answer. (2 marks) b) State two important parameters that you got from Bode plot method and explain how you got it theoretically. Next, state the differences between Bode plot and Root Locus method (3 marks) [10 MARKS) Y(s) KG(s) Figure 4 SOALAN TAMAT / END OF QUESTIONS 10

Answer 1

I)

Im 27 N x 1 _* Re -2 -1 X 2 1

Ii) .& Iii)

8 I+G(S) HIS).-0 lit K = S(S+3) (52+25+2)+ K=0 S (5+3)( 52+28 +2) (52+35) (52+25+2) +K = 0 S4 +593 +8.52 +6S+ REO for stability -S+ 4008 Sk >O 315 6.8 5K < 40.8 S² 16.8 K K < 8.16 s 40.8-5% ko K Range of OU K < 8016 6.8 Los eo 2 1 GIS) HIS) 1 S[S+3)(S2+25+2) s +55 3+85²+65 Put s= jw G(W) HIW).- w" =j5w3 - 842+ jb w 698w+j (6w-503) 6W - 5W3 = 0 = 6 We tog 5 phase cuessovere filiquemcy We lu09 HG1st41537= [1.0961/1992 +9) 466-9492 41872) w2 geog3 Gain Margin= 1 8.093 19(3) HIS)]

Gain Margin and phase Margin we got from bodeplot .

Bode plots are used to gain insight on the frequency response of a transfer function. The magnitude plot (generally displayed in dB) is used to understand the gain of the system and the phase plot is used for phase change from input to output

A Bode Plot is a useful tool that shows the gain and phase response of a given LTI system for different frequencies. Bode Plots are generally used with the Fourier Transform of a given system. An example of a Bodemagnitude and phase plot set.

Root locus plot is used to understand how the poles of a system vary based on system parameters (e.g., controller gain, model parameters, etc.)

Root locus starts from open loop pole(K=0) to open loop zero(K=INFINITY).

The main and the foremost advantage of root locus is to check the system behaviour by adjusting the value of gain K . The adjustment in the value of K will trace the roots accordingly on the R-H plane and will give the conditions for the system to be overdamped, underdamped, crictically damped, overdamped or undampedaccording the the respective value of k. If:-

ROOTS ARE SAME AND REAL - CRITICALLY DAMPED SYSTEM

ROOTS ARE REAL AND DISTINCT- OVERDAMPED

ROOTS ARE PURELY IMAGINARY- MARGINAL STABLE SYSTEM

ROOTS ARE COMPLEX CONJUGATE PAIRS- UNDERDAMPED

Also it gives the limit of K for which it can be varied until the system becomes unstable