Ex:           Write the state-variable equations for the circuit shown below.

     

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

Ans:                         di1/dt = (v2 − v1)/L − i1 R1/L

                dv1/dt = i1/C1

                dv2/dt = −i1/C2 + (vo − v2)/(R2 C2)

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.

                  We use the basic component equations to translate derivatives of state variables into non-derivatives:

                       

                       

                  Application of these equations reduces the problem to that of writing equations for vL1, iC1, and iC2. Each of these equations must have only the state variables, iL1, vC1, and vC2, on the other side so the final equations (in terms of the derivatives of state variables) involve only state variables.

                  The circuit diagram below shows vL1, vL2, and iC1. We now apply Kirchhoff's laws—voltage loops and current sums at nodes—to find our state-space equations.

                 

                  The equation for vL1 must come from a voltage loop, and the voltage loop around the outside will suffice in this case. We use Ohm's law to express the voltages across R1.

                       

                  (Note that the inner voltage loop that includes L1 would pose difficulties with expressing the voltage for R2. We could express the voltage across R2 as v2 − vg(t), however, and obtain the same equation as above.)

                  The equation for iC1 is simple:

                       

                  The equation for iC2 must come from a current summation. In this circuit, there are only two nodes: the top and the bottom rails. Because they effectively yield the same current summation equation, we only use one of these nodes. For the top node, summing currents flowing out of the node poses the problem of writing the current in the middle branch in terms of state variables. The solution is to the inner voltage loop on the right side to solve for the current through R2:

                       

                  Using this equation for iR2 and summing currents yields an equation for iC2:

                       

                  To complete the derivation, we use the basic component equations to change vL1, vL2, and iC1 back into derivatives of state variables.

                       

                       

                       

                  Note that in the third equation we have substituted the value of vo for vg for t > 0.