Ex:           Evaluate the state vector x at t = 0+ for the circuit below in terms of symbolic component names.

                 

              Third-order circuit. ig(t) switches from −i0 to i0 at t = 0.

Ans:        

                 

                 

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, iL2, and vC1. We denote these as i1, i2, and v1.

                  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, vL2, and iC1 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−.

                 

                  The circuit is now a simple current divider, and the capacitor voltage is given by Ohm's law (using ig and R1||R2).

                       

                       

                       

                  These are the same values as the initial conditions at t = 0+.