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Unformatted text preview: nsform For an ideal differentiator, the input f (t) and the output yet) are related by
df = dt T he Laplace transform of this equation yields For a resistor of R o hms, t he v oltage-current relationship is vet) = R i(t), a nd i ts
Laplace t ransform is
V (s) = R I(8) [ j (0 - ) = 0 for a causal signal] y es) = s F(s) a nd
H (s) y es) = -(s) =s
F (6.55) ( c) I deal I ntegrator For an ideal integrator with zero initial state, t hat is y(O-) = 0,
yet) = l' and f er) d r
1 Y es) = - ;F(s) Therefore
'" 1 For an LTIC system with transfer function
s2 s +5
+ 4s+3 (a) Describe the differential equation relating the input f (t) and output yet).
( b) Find the system response yet) to the input f (t) = e - 2t u(t) if the system is initially in
Answers: ( a) d y + 4~ + 3 y(t) = '!!.. + 5 f(t) ( b) yet) = ( 2e- t - 3e- + e - ) u (t) "V
dt 2 dt E xample 6 .10 shows how electrical networks m ay b e a~alyzed b y writ~ng t he
i ntegro-differential equation(s) of t he s ystem a nd t hen solvmg t hese equatiOns. by
t he L aplace t ransform. We now show t hat i t is also possible t o a nalyu: electnc~l
n etworks d irectly w ithout h aving t o w rite t he i ntegra-differential equ~tiOns. T his
p rocedure is considerably simpler because i t p ermits u s t o t reat a n e lectncal network
as if i t were a resistive network. For this purpose, we need t o r epresent a network
in t he " frequency domain" w here all t he voltages a nd c urrents a re represented by
t heir L aplace t ransforms.
For t he s ake of simplicity, let us first discuss t he case w ith zero mltlal conditIOns.
I f vet) a nd i (t) a re t he v oltage across a nd t he c urrent t hrough a n i nductor of L
di vet) = L dt T he L aplace t ransform of t his e quation (assuming zero initial c urrent) is V (s) = L sI(s) m L Vj(t) = 0 a nd
j=l L ij(t) = 0
j=l (6.57) Now if Vj(t) <==> Vj(s) a nd i j(t) <==> I j(s) t hen
k L Vj(s) = 0 a nd
j=l dt 6 .4 Analysis of Electrical Networks: T he Transformed Network h enries, t hen k • E xercise E 6.7 H (s) = T hus, in t he "frequency domain," t he v oltage-current relationships of a n i nductor
a nd a c apacitor are algebraic; t hese e lements behave like resistors o f "resistance"
L s a nd l /Cs, respectively. T he generalized " resistance" o f a n e lement is called its
i mpedance a nd is given by t he r atio V (s) I I (s) for t he e lement ( under z ero initial
conditions). T he i mpedances o f a r esistor of R o hms, a n i nductor o f L h enries, a nd
a c apacitance of C f arads a re R , L s, a nd 1 /Cs, respectively.
Also, t he i nterconnection c onstraints ( Kirchhoff's laws) remain valid for voltages a nd c urrents i n t he f requency domain. To d emonstrate t his p oint, l et Vj(t)
( j=l, 2, . .. , k) b e t he v oltages across k e lements in a loop a nd l et i j(t) ( j = 1, 2,
. . . , m ) b e t he j c urrents e ntering a node. T hen (6.56) =-:; 399 Similarly, for a capacitor o f C f arads, t he v oltage-current relationship is...
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