Ex:           Write the first-order coupled differential equations for the circuit below in the form dx/dt = f(x, t) where x is the state vector and t is time.

                 

              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.

                  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, vL2, and iC1. Each of these equations must have only the state variables, iL1, iL2, and vC1, 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 on the left will suffice in this case. We use Ohm's law to express the voltage across R1:

                       

                  (Note that other voltage loops that include L1 would pose difficulties with requiring a voltage for a current source or requiring a second inductor voltage. If the C were missing, however, we would be forced to write an equation including both vL1 and vL2. In that case, we would also write a current summation equation for the top center node, allowing us to express i1 in terms of i2 and eliminate one state variable. Because we already eliminated one state variable by eliminating C, this would leave us with only one state equation plus an equation relating the two state variables.)

                  The equation for vL2 must also come from a voltage loop, and a voltage loop thru the C will once again suffice. We use Ohm's law to express the voltage across R2:

                       

                  The equation for iC1 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 gives the following equation:

                       

                  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 ig for t > 0.