6.1 The RL Circuit cont.
The solution (derived elsewhere) to the generic DE is: x(t) = [x(t0) - xSS(t0)]{e-(t-t0)/T} + xSS(t)
Where xSS(t) is the solution variable’s steady state response and T is a circuit parameter called the time constant (L/R for RL circuits, RC for RC circuits)
Note that we already have (via Chapters 1-5) techniques for finding the DC or AC steady-state solution xSS(t) for any single-inductor RL or single-capacitor RC circuits
If the initial condition occurs at t0 = 0, then:
x(t) = [x(0) - xSS(0)]{e-t/T} + xSS(t) or
x(t) = [x(0) - xSS(0)]e-t/T + xSS(t)
(which is text Equation 6.26 on p. 216)