1. Higher Order Circuits – How To Obtain State Equations? Consider a circuit with capacitor, inductors, n-terminal resistors and independent sources. Aim is to obtain to obtain state equations in the following form: state variables - some capacitor voltages and some inductor currents (mostly all) output variables - currents and voltages under consideration input variables - values for independent sources Method: Step 1: Draw the circuit graph and choose an appropriate tree: a-) Choose voltage sources as twigs, if tree is not completed continue with (b). b-) Choose capacitors as twigs (you may have to exclude some capacitors), if tree is not completed continue with (c). c-) Choose current-controlled edges of resistors as twigs, if tree is not completed continue with (d).

2. d-) choose inductors as twigs. Step 2: Determine the state variables: capacitor voltages in twigs and inductor currents in chords. Write element equations for state variables: Step 3: Calculate state equations: Write linearly independent current equations: KCL’s for fundamental cut-sets Write linearly independent voltage equations: KVL’s for fundamental loops Using these equations obtain currents of capacitors in twigs and voltages of inductors in chords in terms of state variables and inputs. Which elements are these?

3. + R2R2 L e(t) j(t)C R1R1 An Example Obtain the state equations for the following circuit.

4. An Example Write state equations for the following circuit. L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York 2i L

5. An Example Write state equations for the following circuit. L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York

6. Solutions of 2. Order Differential Equations Solution can be obtained as the sum of the solution of the homogeneous equation and the particular solution: Homogeneous equation: exponential solution as a guess What should we find now ? S=0 is of course a solution, but can we find a nonzero solution too? Characteristic Equation

7. Roots of the characteristic polynomial: eigenvalues Also need to find : eigenvectors An element of Solve this to find the eigenvector for Particular Solution: (should be found by quess method) Total solution: Solve this to find the eigenvector for

8. State Transition Matrix zero-input solution zero-state solution

9. An Example Solve the state equations for the following circuit. L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York

10. An Example Write state equations for the following circuit! L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York

11. An Example Draw the voltage of the capacitor for different initial values! L.O. Chua, C.A. Desoer, S.E. Kuh. “Linear and Nonlinear Circuits” Mc.Graw Hill, 1987, New York

12. A constant solution of a dynamical system: Equilibrium How many equilibria are there? What happens near the equlibrium? Definition: Lyapunov stability Let be an equilibrium of the system given by. The equilibrium is Lyapunov stable if for every there exists a such that A Lyapunov stable equilibrium is asymptotically stable if there exists a such that.

13. Zero-input solution: eigenvectors eigenvalues How do eigenvalues and eigenvectors affect the solution?.............................................................................................................

14. Solutions of Linear Systems What is common in all these systems? S. Haykin, “Neural Networks- A Comprehensive Foundation”2 nd Edition, Prentice Hall, 1999,New Jersey.