Question

Exercise 1: Plot the Nyquist diagram of 10 G(s) = $? + 2s + 2 and deduce the stability of the unity feedback system with the open-loop TF equal to G. Find the stability margins. Compute analytically gain and phase margins. Plot the Bode asymptotic diagram. How can we estimate the values for gain and phase margins graphi- cally?

Answer 1

Given, G(S) = 10 52 + 2S + G Ców) 1o (5wy2 + 2 jw +2 1o (2-w2) +j zw G(jw) 1o (2-w2)+j zw (2-W2)- j zw (2-w2)-jaw = 10(2-2) (2-w2)2 + 4w? j Row (2-62)²+ 402 For wcz Real → positive part Imaginary part als → Negative i.e Graph lies in 4th Quad gant. Sterling) For wy Real past → Negative te Graph lies in 3rd Quadrant gmg. part → Negative (Engling), at w= az Real part of G(ſw) gmaginary part -j 20% zon =;3.54 O + 4X2 22 ie gmaginary point of Intersection = -13.54

Nyquist plot of G(jw) :- Img Loo 180 (o)ی | - AUJO -1+jo → Re Glot) j3.54 The plot will cut imaginary axis where real part of equation is equal to Zeno. 3.54 gad/s. Given, G(s) = 1o 10 52 + 25 + 2 [s -(-1+i)][S-(-1-1)] pole (-1+1) and (al-i) p = No. of open Loop poles in the RHS of S-plane Both poles lies at 2nd and 3rd Quadrant Hence P S The point (-1 +jo) is not encircled by Nyquist plot , therefore N= 0 For Stability Creteria N = P a 0 z

are no As (2=0), thegie roots of closed Loop characteristic equation having positive real part, Hence the closed Loup system is stable. b 1 IN phase Margin calculated Margin calculated at Gian crossover Frequency Apply Magnitude condition, | G (jw) 1 = 10 1 W 2-w2)2 + 462 100 (2-w2) ²4 462 4+ wh- 462 +40 w? = 96 100 9.79 E E 3.13 rad /sec = (هلی 4 tant 2w 2*3.13 - tant 2-02 P - 9.79 38-7850 phase Margin = 1 800 + <G (jw) 180° + 38.785° 218.7855 Gain margin Calculated at phase Crossover Frequency, Apply Angle Condition <G(5w) -1869 tani (221 180° a २-02

→ 2 w 1-w2 > -1fw2+ aw 62 +13) Hence NE- = 0.414 Gad/sec is valid only, can dB) Grain Margin = /2o Login Cust =120 1 1o TW 20 Log W(2-w2) 2+ 462 W = 0.414 W(1082)2 + 4X0017! 20 1o 20 Log [ 2 l To 5 13.97 dB. Hence phase Margin 218-785° and Gam Margin 13.97 dB.