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

544 Chapter 19 Ordinary differential equations I
Engineeringapplication19.2
RLcircuitwithstepinput
Writedownthedifferentialequationgoverningthecurrent,i,flowingintheRLcircuit
shown in Figure 19.2 when a stepvoltage of magnitudeE isapplied tothe circuitat
t = 0.Solvethisdifferentialequationtoobtaini(t).Assumethatwhent = 0,i = 0.
Solution
Applying Kirchhoff’s voltage law and Ohm’s law tothe circuitwefind
iR+L di =E
dt
for t0
thatis,
L di
dt =E−iR
so that
∫ ∫
L
E−iR di = dt
R
L
i
E/R
i
Closed
at t = 0
+ –
E
Figure19.2
Astepvoltage isapplied to the circuitat
t=0.
0.63E/R
t
Figure19.3
Responseofthe circuitofFigure 19.2 to
astepinput.
NoteinparticularthatinthisequationL,E andRareconstantsandsothevariablesi
andt have been separated. If the applied voltage,E, varied with time this would not
havebeenthecasesincethel.h.s.wouldcontaintermsdependentupont.Integrating,
wefind ∫
∫
L −L
di =
E−iR R
−R
E−iR di=−L R ln(E−iR)=t+C
Tofindtheconstantofintegration,C,aconditionisrequired.Thephysicalcondition
i = 0 att = 0 provides this. Applyingi = 0 whent = 0 we find
C = − L R lnE
Substituting thisvalue gives
− L R ln(E−iR)=t−L R lnE
Thus,
L
R [lnE−ln(E−iR)]=t
t