Microsoft PowerPoint ch6 ppt 1 Signal & Systems FEEE, HCMUT – Semester 02/10 11 P6 1 Consider the signal f(t)=e 5tu(t 1) and denote its Laplace transform by F(s) a) Using analysis function, evaluate F[.]
Ch-6: Continuous-Time System Analysis Using the Laplace Transform P6.1 Consider the signal f(t)=e-5tu(t-1) and denote its Laplace transform by F(s) a) Using analysis function, evaluate F(s) and specify its ROC b) Determine the values of the finite numbers A and t0 such that the Laplace transform G(s) of g(t)=Ae-5tu(-t-t0) has the same algebraic form as F(s) What is the ROC corresponding to G(s) P6.2 Consider the signal f(t)=e-5tu(t)+e-βtu(t) and denote its Laplace transform by F(s) What are the constraints placed on the real and imaginary parts of β if the ROC of F(s) is Re{s}>-3? P6.3 For the Laplace transform of e t s in2t; t ≤ f(t)= t>0 0; Indicate the location of its poles and its ROC Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-6: Continuous-Time System Analysis Using the Laplace Transform P6.4 How many signals have a Laplace transform that may be expressed as s −1 (s+2)(s+3)(s +s+1) in its ROC? P6.5 Given that ; ROC: Re{s}>Re{-a} s+a determine the inverse Laplace transform of e − at u (t ) ↔ F(s) = 2(s + 2) ; ROC:Re{s}>-3 s +7s+12 Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-6: Continuous-Time System Analysis Using the Laplace Transform P6.6 Let g(t)=f(t)+af(-t) where f(t)=be-tu(t) and Laplace transform of g(t) is s G ( s ) = ; ROC : −1 < Re{s} < s -1 determine the values of the constant a and b P6.7 Determine the Laplace transform and the associated ROC a) f(t)=e −2t u(t) + e −3t u(t) b) f(t)=e −4t u(t) + e−5t sin(5t)u(t) c) f(t)=e2t u( − t) + e3t u( − t) d) f(t)=te −2|t| e) f(t)=|t|e −2|t| f) f(t)=|t|e 2t u(− t) g) f(t)=rect(t − 1/2) h) f(t)=t.rect(t-1/2)+(2-t)rect(t-3/2) i) f(t)=δ (t) + u (t ) j) f(t)=δ (3t) + u (3t ) Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-6: Continuous-Time System Analysis Using the Laplace Transform P6.8 Determine the Laplace transform of the following signals: a) f(t)=u(t)-u(t-1) e) f(t)=te-t u(t-t ) b) f(t)=e-(t-t0 ) u(t-t ) f) f(t)=sin[ω0 (t-t )]u(t-t ) c) f(t)=e-(t-t0 ) u(t) g) f(t)=sin[ω0 (t-t )]u(t) d) f(t)=e-t u(t-t ) h) f(t)=sin(ω0 t)u(t-t ) P6.9 Determine the function of time, f(t), for each of the following Laplace transforms and their associated ROC: s ; Re{s} > b) F(s)= ; Re{s} < s +9 s +9 s+2 s +1 ; -4< Re{s} < −3 c) F(s)= ; Re{s} < −1 d) F(s)= 2 s + s + 12 ( s + 1) + s2 − s + ( s + 1)2 f) F(s)= ; Re{s} > −1 e) F(s)= ; Re{s} > ( s + 1)2 s − s +1 a) F(s)= Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-6: Continuous-Time System Analysis Using the Laplace Transform P6.10 Determine the function of time, f(t), for each of the following one-side Laplace transforms 2s+5 g) d) a) (s+1)(s+2) s (s+2) s +5s+6 3s+5 s +4s+13 s+1) ( c) s -s-6 b) 2s+1 (s+1)(s +2s+2) s+2 f) s(s+1) e) h) s+1 s(s+2) (s +4s+5) i) s3 (s+1) (s +2s+5) P6.11 Determine the transfer function and step response of the system depicted in FigP6.11 1Ω vi (t) 1H 1F 1Ω v0 (t) FigP6.11 Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-6: Continuous-Time System Analysis Using the Laplace Transform P6.12 Determine transfer function of the system shown in FigP6.12(a), and (b) R C vi (t) C R1 R2 v (t) vi (t) (a) FigP6.12 R R1 R2 v (t) (b) P6.13 The input f(t) and output y(t) of a causal LTI system are related through the block diagram representation shown in FigP6.13 a) Determine a differential equation relating y(t) and f(t) b) Is this system stable? Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-6: Continuous-Time System Analysis Using the Laplace Transform P6.14 Draw a direct-form representation for the causal LTI system with the following system functions: a) H1 (s)= s +1 s + 5s + b) H (s)= s − 5s + s + s + 10 c) H3 (s)= ( s + 2) Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-6: Continuous-Time System Analysis Using the Laplace Transform P6.15 Realize following transfer functions: s(s+2) 3s(s+2) a) H(s)= b) H(s)= (s+1)(s+3)(s+4) (s+1)(s +2s+2) 2s-4 2s+3 c) H(s)= d) H(s)= (s+2)(s +4) 5s(s+2) (s+3) s(s+1)(s+2) s3 e) H(s)= f) H(s)= (s+5)(s+6)(s+8) (s+1) (s+2)(s+3) by canonical, series, and parallel forms P6.16 In this problem we show how a pair of complex conjugate poles may be using a cascade of two first-order transfer functions Show that the of transfer function of the block diagrams in Fig P6.16a, b, and c are s+a As+B a) H(s)= b) H(s)= c) H(s)= 2 2 (s+a) +b (s+a) +b (s+a) +b Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-6: Continuous-Time System Analysis Using the Laplace Transform Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-6: Continuous-Time System Analysis Using the Laplace Transform P6.17 Show op-amp realization of the following transfer functions: -10 10 s+2 a) H(s)= b) H(s)= c) H(s)= s+5 s+5 s+5 P6.18 Show two different op-amp realization of the transfer function: s+2 H(s)= =1s+5 s+5 P6.19 Show op-amp canonical realization of the following transfer functions: s +5s+2 3s+7 b) H(s)= a) H(s)= s +4s+13 s +4s+10 Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-6: Continuous-Time System Analysis Using the Laplace Transform P6.20 Determine the rise time tr, the settling time ts, the PO and the steady-state errors es, er and ep for each of the following systems, whose transfer functions are: 95 a) H(s)= b) H(s)= c) H(s)= s +3s+9 s +3s+4 s +10s+100 P6.21 For a position control system depicted in Fig P6.21, the unit step response shows the peak time tp=π/4, the PO=9%, and the steady-state value of the output for the unit step input is yss=2 Determine K1, K2 and a Signal & Systems - FEEE, HCMUT – Semester: 02/10-11 Ch-6: Continuous-Time System Analysis Using the Laplace Transform P6.22 For a position control system depicted in Fig P6.22, the following specifications are impose: tr≤0.3, ts≤0.1, PO≤30%, and es=0 Which of these specifications cannot be met by the system for any value of K? Which specifications can be met by simple adjustment of K? P6.23 Open loop transfer functions of four closed-loop system are given below In each case, give a rough sketch of the root locus a) H(s)= K(s+1) s(s+3)(s+5) b) H(s)= c) H(s)= K(s+1) s(s+3)(s+5)(s+7) d) H(s)= K(s+5) s(s+3) K(s+1) s(s+4)(s +2s+2) Signal & Systems - FEEE, HCMUT – Semester: 02/10-11