Question

Question 12 100(5 +2) s(s + 1D(s +5 For the system shown below, if G(S) , find the phase crossover frequency in rad/sec). (Write the answer to the nearest 2 digits after the decimal point, do not write units) G(s)

Answer 1

NOTE: i solved this problem using MATLAB.

Question. 12

MATLAB code

clc;

z=[2]

p=[0,1,5,5]

k=100

g=zpk(z,p,k)

margin(g)

[Gm,Pm,Wcg,Wcp] = margin(g)

RESULTS and BODE plot

LIVE EDITOR E Aa Tite Run Section Run and Advance Go To Control Run | Find Code es Refactor Step = 춈: Break Run to End NAVIGATE TEXT CODE SECTION RUN 8 ystemExample mbx 18e (S-2) s (5-1) (s-5)2 clc; 2 Continuous-time zero/pole/gain model 3 -[2] Bode Diagram Pm 170 deg (at 3.2 rad/s) 4 Gm Inf . 5 p-[e,1,5,5] 6 100 50 7 k-100 10 100 margin(g) [Gm, Pm, Wcg, licp]=margin(g) 12. 315 10 10 Frequency (rads) Warning: The closed-loop system is unstable GmInf Pm169.8448 wcp 3.1966

LIVE EDITOR Tite Run Section GoTo ▼ Control, 96 B I U M Run and Advance Find ▼ Text Refactor ▼ Run Step Break Run to NAVIGATE TEXT CODE SECTION RUN BodePlotOfDynamicSystemExample.mlx 50 -100 -150 270 O-315 Φ-360 405 -450 100 Frequency (rad/s) 10 10 10 Warning: The closed-loop system is unstable. Pm 169 . 8448 cp 3.1966

ANSWER to the Question 12.:

phase crossover frequency : the frequency at which the phase of the system is -180 degrees

in this example phase plot never crosses the -180 degrees .

so phase crossover frequency is not exists

ANSWER to the Question 13.:

transfer function is same as question 12.

Gain crossover frequency : the frequency at which the gain of the system is zero DB or unity.

so gain crossover frequency = 3.1966 rad/sec.

......

Question. 14

MATLAB code

clc;

z=[]

p=[-1,-10,-20]

k=200

g=zpk(z,p,k)

margin(g)

[Gm,Pm,Wcg,Wcp] = margin(g)

RESULTS and BODE plot

LIVE EDITOR Normal Run Section Go To しー:Control, % Run and Advance Find ▼ Refactor Run Step Run to End NAVIGATE TEXT CODE SECTION RUN mlx 280 1ccl 2 Continuous-time zero/pole/gain model Bode Diagram 5 P-1,-10,-20] Gm 30.8 dB (at 15.2 radis) -180 deg (at 0 rad/s) k 200 g=zpk(z, p, k) margin(g) 9 10 100 12 [Gm, Pm, Wcg,Wcp]margin(g) 135 180 10 10 Frequency (rad s) Gm 34, 6520 Pm -180 Wcg15.1662

LIVE EDITOR Run Section a Run and Advance Ag Title Control ▼ Find" Text Run Step NAVIGATE TEXT CODE SECTION RUN SystemExample.mbo 0ae ul agram Gm = 30.8 dB (at 15.2 rad/s), Pm =-180 deg (at 0 rad/s) 9 -50 100 150 90 o -135 180 225 270 10 10 10 Frequency (rad/s) Gm34.6520 wcg 15.1662

Phase margin : -180 degrees.