 
Summary: ELEC 302 HW #2: Solutions
Problem [1]. Consider the circuit shown in the figure with R = 1 and Cap = 1F. The switch is supposed to have
been at position a for a long time (say, since t = ). At t = 0, it goes to position b. Find the current i(t), t 0,
for the following values of the two batteries:
R
Cap
V V1 2
i(t)b
a
(i) V1 = 0V , V2 = 1V ; (ii) V1 = 1V , V2 = 0V ; (iii) V1 = 1V , V2 = 1V .
Using your answers for (i), (ii), (iii), argue that the current i(t) can be considered as a sum of the circuit's zerostate
response and zeroinput response.
Solution
Choosing the state as x(t) = v(t) (i.e., the voltage across the capacitor) leads to the differential equation: x(t) +
x(t) = V (t). We choose the output as the current i(t), so y(t) = 1
R (u(t)  x(t)). In this case, the state space
matrices turn out to be A = 1, B = 1, C = 1, D = 1. The state solution is
x(t) = eAt
x(0) +
t
