Determination of initial conditions

DETERMINATION OF INITIAL CONDITIONS

The procedure generally followed to determine the initial conditions consists of two steps in sequence.

Step 1: Solving for initial values of variables at t = (0+).

For this, the following rules are to be followed:

(i) Replace every inductor by an open circuit or by a current generator having source current equal to that flowing at t = (0+).

(ii) Replace every capacitor by short circuit or by a voltage generator having source voltage

V_co = q₀ / C

 (iii) Leave every resistor in the network unchanged.

Step 2: Solving for the initial values of derivatives at t = 0+.

The details and the order of mathematical manipulation involved for the derivatives will be different for each different case.

Consider a series RLC network shown in fig. 3.2(a). It is connected to a DC voltage source V₀ at t = 0. Assume that no voltage has been applied to the network before t = 0.

It is required to find it i (0+), di / dt (0+) and d'i / dt2 (0+).

F

Step 1: The equivalent network of fig.3.2 (a) is as shown in fig. (b), at t=0+. Here inductor is shown by an open circuit and capacitor is shown by short circuit. It is by assuming that initial current through inductor and voltage across capacitor are equal to zero.

From fig. (b). We see that i (0+) = 0. Applying Kirchoff's voltage law to the given network we get,

L di / dt + Ri + 1/C ∫ idt = V₀ ... (11)

Equation (11) holds at t = 0+. Also at t = 0+, i = 0 and 1/C ∫ idt = 0. This last term is the voltage across the capacitor at t = 0+. Putting these values in equation (11) at t = 0+, we get L di / dt (0+) = V₀

di / dt (0+) dt = V₀/L ... (12)