Question

Question (20 marks) Consider a control system shown in Fig.2 has an open loop TF-G(s) H(s) 49 S(S+7) A-Prove that the gain margin=infinite db at infinite rad/sec. and the phase margin= 51.8 degrees at 5.5 rad/sec.? (4)

Answer 1

Transfer function: - a(s) H(s) = 49/s(s + 7) Gain margin = -20 log (m)/w = w_pc = 20 log (1/(m)/w = m_pc w_pc is the frequency at which phi = -180 degree (phase) phase = angle G(s) H(s) = -90 degree - Jan^-1(w/T) -90 degree - Jan^-1(w/7) = -180 degree doublerightarrow Jan^-1 (w/T) = + 90 degree doublerightarrow w/7 = infinity doublerightarrow w = infinity Vertical-bar m Vertical-bar = 49/w squareroot w^2 + 49 \r\n Gain margin = infinity dB w_pc = infinity rad/sec phase margin = 180 degree + phi phi = angle G(s) H(s) /wgc w_gc is given cross area frequency at Vertical-bar m Vertical-bar = 1 Vertical-bar m Vertical-bar = 49/w squareroot w^2 + 49 = 1 doublerightarrow (49)^2 = w^2 (w^2 + 49) doublerightarrow w^4 + 49 w^2 - (49)^2 = 0 let w^2 = t doublerightarrow t^2 + 49t - (49)^2 = 0 t = 49 plusminus squareroot (49)^2 + 4(49)^2/2 = -49 plusminus 49 squareroot 5/2 t = 30.28