Which of the following correctly represents, in an s-domain circuit including its initial condition, a capacitor C charged to a voltage V0 at time t = 0?

Answer: C. An impedance 1/(sC) with a voltage source V0/s in series

Method 1 (from the transform of a derivative): taking the Laplace transform of i = C dv/dt with the derivative rule L[dv/dt] = sV(s) − v(0) gives I(s) = sCV(s) − CV0. Solving for V(s) gives V(s) = I(s)/(sC) + V0/s: 1/(sC) in series with a voltage source V0/s (the transform of the constant voltage V0). Method 2 (equivalent parallel form): the same equation read as I(s) = sCV(s) − CV0 can also be represented as 1/(sC) in parallel with a current source CV0 (the transform of an impulse current CV0δ(t)). The initial current I0 of a coil is likewise represented by sL in series with a voltage source LI0.

• A. The value of the voltage source must also be transformed to the s-domain. A constant V0 in the time domain becomes V0/s in the s-domain.
• B. In the parallel form, the value of the current source is CV0. With V0/s even the units do not match.
• C. Correct. 1/(sC) in series with a voltage source V0/s.