Ex:           Evaluate the state vector for the circuit shown below at t = 0+.

     

         The voltage vg(t) changes instantly from −vo to vo at t = 0.

Ans:                         [i1(0+), v1(0+), v2(0+)] = [0, −vo, −vo]

Sol'n:      The state variables are always the inductor currents and capacitor voltages (which are also the variables we use to calculate stored energy). Thus, our state variables are iL1, vC1, and vC2. We denote these as i1, v1, and v2.

                  Because their values cannot change instantly, the state variables have the same values at time t = 0− as they do at time t = 0+.

                  Because the circuit has had the same DC current source for an infinitely long time at t = 0−, (from time −∞ to 0−), the circuit will have reached equilibrium and time derivatives of state variables will be zero. In other words, currents and voltages are no longer changing.

                  Thus, we have that vL1, iC1, and iC2 are all zero based on the basic component equations:

                       

                       

                  This means the inductors look like wires and the capacitors look like open circuits. We get the equivalent circuit shown below at time t = 0−.

                 

                  Because the inductor is in series with an open circuit, the inductor current will be zero. Similarly, i2 = 0, and it follows that i3 = 0. Thus, there is no current in any of the resistors, and no voltage drop across any of the resistors. Consequently, vg(t) appears across C1 and across C2. At time t = 0, vg(t) = −vo.

                       

                       

                       

                  Because they can't change instantly, these are the same values as the initial conditions at t = 0+.