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

21.10 Solving linear constant coefficient differential equations 651
Engineeringapplication21.3
RLcircuitwithrampinput
Recall from Engineering application 19.3 the differential equation for an RL circuit
with a unit ramp input isgiven by
iR+L di
dt
=t t0 i(0)=0
Usethe Laplace transform tosolve thisequation.
Solution
Let
L(i(t)) =I(s)
Then
( ) di
L =sI(s)−i(0) =sI(s)−0 =sI(s)
dt
From Table 21.1 wesee that
L(t) = 1 s 2
We note thatRandLare constants. We can now take the Laplace transform of the
given equation. This gives
I(s)R +LsI(s) = 1 s 2
This equation issolved forI(s):
I(s)(R +Ls) = 1 s 2
1
I(s) =
s 2 (R +Ls)
InordertotaketheinverseLaplacetransformandhencefindi(t)weexpressI(s)as
the sum of itspartial fractions. The expression
1
s 2 (R +Ls)
has a repeated linear factor,s 2 inthe denominator, giving risetopartialfractions
A
s + B s 2
The linear factor,R+Ls, gives risetoapartialfraction of the form
Hence
C
R+Ls
I(s) =
1
s 2 (R +Ls) = A s + B s + C
2 R+Ls
➔