082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

Review exercises 21 679
2 Find the Laplace transformsofthe following:
(a) IfV o (s) = L{v o (t)}andV i (s) = L{v i (t)},show
seriesRC network are relatedbythe differential (d) Findi(t)if
equation
{
2e
CR dv −2t t 0
o
v(t) =
+v
dt o =v i 0 otherwise
(a) 2tsin3t (b) 1 2 (−3tcos2t)
that the transfer function, V o (s) , isgiven by
V i (s)
1
(c) e −t u(t) (d) e −t u(t −1)
sCR+1 .
(e) 3t 2 u(t −1) (f) e −t δ(t −2)
(b) IfC = 0.1 µF,R = 100 k find,and sketch a
3 Find the inverse Laplacetransformsof
graph ofthe response, v o (t),when
2s+3
{
(a)
5volts t0
(s +1)(s +2)
v i (t) =
0 t<0
4s
(b)
s 2 −9
(c) Using the component values given in (b)findthe
(c) 2(s3 +4s 2 +4s +64)
response when the input is aunitimpulse, δ(t).
(s 2 +4)(s 2 +16)
(d) Using the component values given in (b)use the
(d)e −2s 6 s 4 (e) e−ss +1
convolution theoremto determine the response
s 2
when v i (t) = 5e −100t .
4 Solvethe followingdifferentialequations usingthe 7 Express the squarewave
Laplacetransformmethod:
{
1 0<t<a
(a) x ′′ +2x ′ −3x =2t, x(0) =1 x ′ (0) =2 f(t)=
−1 a <t < 2a period 2a
(b) x ′′ −2x ′ +5x =cost, x(0) =0 x ′ (0) =1 in terms ofunitstepfunctions. Hencededuce its
(c) ẋ+ẏ+x+y= 3 2 (1+t)
Laplace transform.
ẋ−2ẏ+x+2y=2t
8 Consider the circuitshown in Figure 21.24.
x(0) =0 y(0) =0
(d) ẋ−ẏ+x=−1,x(0)=0 y(0)=2
1–
3 F 1 H 4 V
2ẋ−ẏ + y 2 −x=1
5 Thecurrent,i(t),in a seriesLC circuitis governed by
i(t)
L di
dt + 1 ∫ t
idt = v(t)
y(t)
C 0
Figure21.24
where v(t)is the applied voltage.
(a) Assumingzero initialconditions showthat
(a) Show that
LsI + 1 Cs I =V(s)
∫
di t
dt +4i+3 idt = v(t)
0
(b) If v(t) = δ(t)show that
(b) Assumingzero initialconditions, showthat
i(t) = 1 L cos t
√ s
LC
I(s) =
(s+1)(s+3) V(s)
6 Theinput andoutputvoltages, v i (t)and v o (t),ofa (c) Findi(t)if v(t) = δ(t).