Question

4.8.2 For an LTIC system described by the transfer function H(s) = + 2) find the steady-state system response to a. 10u(t) b. cos (2+ + 60°) (1) c. sin (3 - 45")u(t) d. e3 u(t)

Answer 1

Sol: The Data given in the question is as - Data given in the follows! Transfer fanction is given by- HS) - 2 - 0 kihen Input is c) Mei = louch Tronstorm on both sides Weget - Now taking Loplace XC) - to Now equation ② Con also be written as - Yes (S3 S > 408) = +3) x to © Using final Value theorem the Steady State Response i given by formula I lim Yeti = lim sy lesing above Principal finding the Steady brate Response from equation ② We get a you lim s (8+3) ] o 6 405) = 10x3 - 75 [Y- 7.5 ml Hence, Steady State Response will be 7.5. When the Input is, e ne = Cos (at +600) We know that hihen sin on Cosine Input is Applied to an LT I System, the outlet is also a Sinusoidad, or Cosine kare form only Magnitude and

and Phase Changes. Steaty Stare • In that Cose we can only Genete olp but not the Response (3+2) niti - Cos (24465) Here, frequency We 2 Rad = 60 Transfor function Hes) = 5+3) Subsituting S= jw we get, s=52 H00) 25+3 C2j+3) (2 + 2)2 (25-272 Magnitude Response (4 (25) darg HCl - di smirly Phose Response (426) = tan (3) - 2x tan (²) (4(25) = tan (² – 2 x 45° Hasl = tan (37-90 Hence, lising above Results the output yet is given by - YCH - 3 Cos (2+ +60 + tan (3) -90) lyttadio Cos (21+tan (3) - 30) a @ When the Inpul is Nit Sin (3t-uso) ulty a olen Here, frequency wzgRad = ~ 450

- Transfer function - H(s) = (S13) (3+212 Now subsituting S=j3 we get- 4(31) - 33+3 Ô3+2) Now Mognitude Response- 14 Caul - data - dio gryte Simikrly those Response - LH (33= fan (3) - 24 (3) (H(ei) - 45 - 2 tan (2) Hence, the output YIt is given by- Ytt dis Sin (36-45 +45 – 2tan (2)] yet = due sin [34 - 2tar (37] ( When the Input is nit = eust lift, Note! Kinen the exponential Innut eu sot is applied to An LT I system then the outlet is givenby y ctia eu sot H (so) | Whenke HCso) is eigen Value How he have. Nette e t S023 H65) - (St3) - 0 H(3) - (3+3) - seze Hence, the outrust is given by YE) = H3 e ist Tytti est