tailieunhanh - DISCRETE-SIGNAL ANALYSIS AND DESIGN- P35

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P35:Electronic circuit analysis and design projects often involve time-domain and frequency-domain characteristics that are difÞcult to work with using the traditional and laborious mathematical pencil-and-paper methods of former eras. This is especially true of certain nonlinear circuits and sys- tems that engineering students and experimenters may not yet be com- fortable with. | 156 DISCRETE-SIGNAL ANALYSIS AND DESIGN N 8 n 0 1. N T Vc IL R 3 L 1 C VC x n IL if n 0 T . -1 0 C 10 1 R 01 L L x n - 1 if n 0 Solution to matrix differential equation for initial conditions of Vc Il Figure A-2 LCR Circuit differential equation solution for initial values of Vc and Il lgen 0. dvc ic C u Il diL vL L vc RiL A-3 VOUT RiL ADDITIONAL DISCRETE-SIGNAL ANALYSIS AND DESIGN INFORMATION 157 Rewrite Eq. A-3 in state-variable format vc 0vc 1 1L Lc iL --u C L C R ir 0u L A-4 Vo RiL A nodal circuit analysis confirms these facts for this example. R L and C are constant values but they can easily be time-varying and or nonlinear functions of voltage and current. The discrete analysis method deals with all of this very nicely. We now add in the initial conditions at time zero VCo and ILo vc 0 vc VC0 iL Ilo u 1L L vc VC0 L iL ILO 0u A-5 Vo RiL The two derivatives appear on the left side. Note that if vC VCo is multiplied by zero the rate of change of vC does not depend on that term and the rate of change of iL does not depend on u if the u is multiplied by zero. The options of Eqs. A-4 and A-5 can easily be imagined. Description of flow-graph methods in Dorf and Bishop 2004 Chaps. 2 and 3 and in numerous other references are excellent tools that are commonly used for these problems. We will not be able to get deeply into that subject in this book but Fig. A-4 is an example. The next step is to rewrite Eq. A-5 in matrix format. Also vC is now called X1 and IL is now called X2. X A _ 0 À V L 1 c R L j_ 0 u A-6 Vo RX2 158 DISCRETE-SIGNAL ANALYSIS AND DESIGN Now write the A-6 equations as follows Xi 0Xi - -X2 -u 1 1 C C X2 1 Xi - RX2 0u L L Vo RX2 A-7 Next we will solve Eq. A-6 same as Eq. A-7 for Xi vc X2 iL and V o . Component u is the input signal generator. This general idea applies to a wide variety of practical problems see Dorf and Bishop 2004 Chap. 3 and many other references . The methods of matrix algebra and matrix calculus operations