A 200ohm resistor, a 5000miuF capacitor a switch and a 10V battery are in series in a single circuit loop. Determine the initial and steady state current. How long will the circuit take to reach steady state?
Answers
Answer:
Solution of First-Order Linear Differential Equation
The solution to a first-order linear differential equation with constant coefficients,
a1
dX
dt + a0X = f(t) ,
is X = Xn + Xf , where Xn and Xf are, respectively, natural and forced responses of the
system.
The natural response, Xn, is the solution to the homogeneous equation (RHS=0):
a1
dX
dt + a0X = 0
The functional form of Xn is Xn = Kest (K and s are constants). Value of s can be found
by substitutingthe functional form in the homogeneous differential equation:
a1
dKest
dt + a0Kest = 0
a1Ksest + a0Kest = 0 → a1s + a0 = 0 → s = − a0
a1
Constant K is found from initial conditions. As the initial condition applies to X not Xn,
K should be found after Xf is calculated.
Some functional forms of the forced solution, Xf , are given in the table below. To find
Xf , the functional form is substituted in the original differential equation and the constant
coefficients of the functional form are found.
Trial Functions for Forced Response
f(t) Trial Function†
a A
at + b At + B
atn + btn−1 + ... Atn + Btn−1 + ...
aeσt Aeσt
a cos(ωt) + b sin(ωt) A cos(ωt) + B sin