1. Time Domain
2. s-Domain Equivalent
3. Fundamental Equation
i(t) = C dvCdt
↓ with initial condition
Î(s) = sC V̂(s) - C vC(0-)
↓ with initial condition
Î(s) = sC V̂(s) - C vC(0-)
4. Circuit Output Equation
Series Voltage Model:
V̂(s) = Î(s) 1sC + vC(0-)s
V̂(s) = Î(s) 1sC + vC(0-)s
Common Conceptual Pitfalls
- Initial conditions are sources, not part of the transfer function. A transfer function H(s) strictly links the input signal to the output signal. Initial energy is treated as a separate independent source in the s-domain.
- Zero-state means setting initial energy to zero. When calculating the "zero-state response", we assume all capacitors are completely discharged and inductors have zero current at t=0. The output is purely driven by the input.
- Zero-input means suppressing the external input but keeping initial energy. When calculating the "zero-input response", we turn off the external driving force. The output is entirely due to the internal energy decaying over time (the initial conditions).