Download Chapter 15-Electrical Circuit Analysis-Problem Solutions and more Exercises Electrical Circuit Analysis in PDF only on Docsity! Chapter 15, Problem 1. Find the Laplace transform of: (a) cosh at (b) sinh at [Hint: cosh x = ( )xx ee −+ 2 1 , sinh x = ( )xx ee −− 2 1 .] Chapter 15, Solution 1. (a) 2 ee )atcosh( at-at + = [ ] =⎥⎦ ⎤ ⎢⎣ ⎡ + + − = as 1 as 1 2 1 )atcosh(L 22 as s − (b) 2 ee )atsinh( at-at − = [ ] =⎥⎦ ⎤ ⎢⎣ ⎡ + − − = as 1 as 1 2 1 )atsinh(L 22 as a − Chapter 15, Problem 2. Determine the Laplace transform of: (a) cos( θω +t ) (b) sin( θω +t ) Chapter 15, Solution 2. (a) )sin()tsin()cos()tcos()t(f θω−θω= [ ] [ ])tsin()sin()tcos()cos()s(F ωθ−ωθ= LL =)s(F 22s )sin()cos(s ω+ θω−θ (b) )sin()tcos()cos()tsin()t(f θω+θω= [ ] [ ])tsin()cos()tcos()sin()s(F ωθ+ωθ= LL =)s(F 22s )cos()sin(s ω+ θω−θ docsity.com ( )ttue t 4sin2− (c) ( )ttue t 2cosh3− (d) ( )ttue t sinh4− (e) ( )ttute t 2sin− Chapter 15, Solution 3. (a) [ ] =)t(u)t3cos(e-2tL 9)2s( 2s 2 ++ + (b) [ ] =)t(u)t4sin(e-2tL 16)2s( 4 2 ++ (c) Since [ ] 22 as s )atcosh( − =L [ ] =)t(u)t2cosh(e-3tL 4)3s( 3s 2 −+ + (d) Since [ ] 22 as a )atsinh( − =L [ ] =)t(u)tsinh(e-4tL 1)4s( 1 2 −+ (e) [ ] 4)1s( 2 )t2sin(e 2 t- ++ =L If )s(F)t(f ⎯→← )s(F ds d- )t(ft ⎯→← Thus, [ ] ( )[ ]1-2t- 4)1s(2 ds d- )t2sin(et ++=L )1s(2 )4)1s(( 2 22 +⋅++ = [ ] =)t2sin(et -tL 22 )4)1s(( )1s(4 ++ + docsity.com (b) [ ] = + ⋅= 5 2t-4 )2s( !43et3L 5)2s( 72 + (c) =−⋅−=⎥⎦ ⎤ ⎢⎣ ⎡ δ− )01s(4 s 2 )t( dt d 4)t(ut2 2L s4s 2 2 − (d) )t(ue2)t(ue2 -t1)--(t = [ ] =)t(ue2 1)--(tL 1s e2 + (e) Using the scaling property, [ ] =⋅⋅=⋅⋅= s2 1 25 )21(s 1 21 1 5)2t(u5L s 5 (f) [ ] = + = 31s 6 )t(ue6 3t-L 1s3 18 + (g) Let )t()t(f δ= . Then, 1)s(F = . L−′−−=⎥⎦ ⎤ ⎢⎣ ⎡ =⎥⎦ ⎤ ⎢⎣ ⎡ δ −− )0(fs)0(fs)s(Fs)t(f dt d )t( dt d 2n1nn n n n n LL L−⋅−⋅−⋅=⎥⎦ ⎤ ⎢⎣ ⎡ =⎥⎦ ⎤ ⎢⎣ ⎡ δ −− 0s0s1s)t(f dt d )t( dt d 2n1nn n n n n LL =⎥⎦ ⎤ ⎢⎣ ⎡ δ )t( dt d n n L ns docsity.com Chapter 15, Problem 6. Find F(s) given that ( ) ⎪ ⎩ ⎪ ⎨ ⎧ << << = otherwise tt tt tf ,0 21, 10,2 Chapter 15, Solution 6. 1 2 0 0 1 ( ) ( ) 2 2st st stF s f t e dt te dt e dt ∞ − − −= = +∫ ∫ ∫ 22 2 1 2 22 ( 1) 2 (1 ) 0 1 st st s se est e se s s s − − − −− − + = − − − Chapter 15, Problem 7. Find the Laplace transform of the following signals: (a) ( ) ( ) ( )tuttf 42 += (b) ( ) ( ) ( )tuetg t234 −+= (c) ( ) ( ) ( )( ) ( )tuttth 3cos83sin6 += (d) ( ) ( )( ) ( )tutetx t 4cosh2−= Chapter 15, Solution 7. (a) 2 2 4( )F s s s = + (b) 4 3( ) 2 G s s s = + + (c ) 9s 18s8 9s s8 9s 36)s(H 222 + + = + + + = (d) From Problem 15.1, 2 2{cosh } sL at s a = − 2 2 2 2 2( ) ( 2) 4 4 12 s sX s s s s + + = = + − + − 12s4s 2s)d(, 9s 18s8)c(, 2s 3 s 4)b(, s 4 s 2)a( 222 −+ + + + + ++ docsity.com Chapter 15, Problem 8. Find the Laplace transform F(s), given that f(t) is: (a) ( )42 −ttu (b) ( ) ( )2cos5 −tt δ (c) ( )ttue t −− (d) ( ) ( )τ−tut2sin Chapter 15, Solution 8. (a) 2t=2(t-4) + 8 f(t) = 2tu(t-4) = 2(t-4)u(t-4) + 8u(t-4) 4 4 42 2 2 8 2 8( ) s s sF s e e e s s s s − − −⎛ ⎞= + = +⎜ ⎟ ⎝ ⎠ (b) 2 0 0 ( ) ( ) 5cos ( 2) 5cos 5cos 2 2 st st st sF s f t e dt t t e dt te e t δ ∞ ∞ − − − −= = − = = =∫ ∫ (c) ( )t te e eτ τ− − − −= ( )( ) ( )tf t e e u tτ τ τ− − −= − ( 1)1( ) 1 1 s s eF s e e s s τ τ τ − + − −= = + + (d) sin 2 sin[2( ) 2 ] sin 2( )cos 2 cos 2( )sin 2t t t tτ τ τ τ τ τ= − + = − + − ( ) cos 2 sin 2( ) ( ) sin 2 cos 2( ) ( )f t t u t t u tτ τ τ τ τ τ= − − + − − 2 2 2( ) cos 2 sin 2 4 4 s s sF s e e s s τ ττ τ− −= + + + 5cos(2)e–2s docsity.com
