Importance of initial conditions and equivalent circuits

Importance of initial conditions

A circuit equation is a differential equation. Its general solution contains arbitrary constants that can only be found from the state of the circuit at t = 0⁺, just after switching. Initial conditions are needed because:

• They fix the constants of the transient (complementary) solution, so the complete response i(t), v(t) can be written.
• Energy stored in L and C before switching decides the response: with zero stored energy the circuit is "relaxed", otherwise it has a natural response of its own.
• The currents and voltages in resistors, and the derivatives of i and v, can change suddenly at t = 0, so they must be recomputed at 0⁺.
• They are needed in the Laplace transform method (the terms L·i(0⁻) and v(0⁻)/s).
• They check the final answer: the solution must satisfy i(0⁺) and v(0⁺).

Two rules: the inductor current cannot change instantly (i_L(0⁺) = i_L(0⁻)), and the capacitor voltage cannot change instantly (v_C(0⁺) = v_C(0⁻)).

Equivalent circuits

Inductor at 0+ (I0 ≠ 0)      Capacitor at 0+ (V0 ≠ 0)

  ---->(I0)---->                 ---| + V0 -|---
  (current source)               (voltage source)

Inductor at 0+ (I0 = 0)      Capacitor at 0+ (V0 = 0)
  ---- open ----                 ---- short ----

Final dc:  L = short   C = open