endof-

Chapter 5 State Variables and State Equations2 2 + 1 2 + 3 2 = 141 2 + 1 2 + 2 2 = 6The unit eigenvectors are(–1) 2 + 1 2 + 1 2 = 3---------14------6Unit X λ 1 =1--------- Unit X λ 2 =1------146Unit X λ 3 =32---------------146=2=1– 1------31= ------31------3(5.83)We observe that for the first unit eigenvector the sum of the squares is unity, that is,⎛---------2 ⎞ 2 1⎝ 14 ⎠+ ⎛---------⎞ 2 ---------3⎝ 14 ⎠+ ⎝⎛ 14 ⎠⎞2= -----4+ -----1+ -----9= 114 14 14and the same is true for the other two unit eigenvectors in (5.83).(5.84)5.6 Circuit Analysis with State VariablesIn this section we will present two examples to illustrate how the state variable method is used incircuit analysis.Example 5.10For the circuit of Figure 5.7, the initial conditions are i L ( 0 −) = 0 , and v C ( 0 −) = 0.5 V . Use thestate variable method to compute i L () t and v C () t .RL1 Ω 1⁄ 4 HC+ − it ()v S () t = u 0 () t4 ⁄ 3 F+v− C () tFigure 5.7. Circuit for Example 5.105−22Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fourth EditionCopyright © Orchard Publications