DISCRETE-SIGNAL ANALYSIS AND DESIGN- P35

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P35:Electronic circuit analysis and design projects often involve time-domainand frequency-domain characteristics that are difÞcult to work with usingthe traditional and laborious mathematical pencil-and-paper methods offormer eras. This is especially true of certain nonlinear circuits and sys-tems that engineering students and experimenters may not yet be com-fortable with.
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DISCRETE-SIGNAL ANALYSIS AND DESIGN- P35156 DISCRETE-SIGNAL ANALYSIS AND DESIGN N := 8 n := 0,1.. N T := 0.1 VC := 1.5 IL := 1.0 R := 3 L := 1 C := 0.5 VC x(n) := if n = 0 IL −1 0 C 1 0 T. + x(n − 1) if n > 0 1 R 0 1 L L 1.5 V(C) 1.2 I(L) x(n)0 0.9 x(n)1 0.6 0.3 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 n.T Solution to matrix differential equation for initial conditions of VC = 1.5, IL = 1.0 IL L + VL − + u(t) C + VOUT R VC − −Figure A-2 LCR Circuit differential equation solution for initial valuesof VC and IL , Igen = 0. dvC iC = C = u − iL dt diL vL = L = vC − RiL (A-3) dt vOUT = RiL ADDITIONAL DISCRETE-SIGNAL ANALYSIS AND DESIGN INFORMATION 157 Rewrite Eq. (A-3) in state-variable format: · 1 1 vC = 0vC − iL + u C C 1 R õ· = vC − iL + 0u L (A-4) L L vO = RiL A nodal circuit analysis conÞrms these facts for this example. R, L,and C are constant values, but they can easily be time-varying and/ornonlinear functions of voltage and current. The discrete analysis methoddeals with all of this very nicely. We now add in the initial conditions at time zero, VC0 and IL0 : · 1 1 vC = 0(vC + VC0 ) − (iL + IL0 ) + u C C 1 R õ· = (vC + VC0 ) − (iL + IL0 ) + 0u L (A-5) L L vO = RiL The two derivatives appear on the left side. Note that if (vC + VC0 ) ismultiplied 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 multipliedby zero. The options of Eqs. (A-4) and (A-5) can easily be imagined.Description of ßow-graph methods in [Dorf and Bishop, 2004, Chaps.2 and 3] and in numerous other references are excellent tools that arecommonly used for these problems. We will not be able to get deeplyinto 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, v C is nowcalled X 1 , and I L is now called X 2 . ˙ X1 0 −1 X1 1 ˙ = 1 C −R + C (u) (A-6) X2 L L X2 0 VO = RX2158 DISCRETE-SIGNAL ANALYSIS AND DESIGNNow write the (A-6) equations as follows: ˙ 1 1 X1 = 0X1 − X2 + u C C ˙ 1 R X2 = X1 − X2 + 0u (A-7) L L VO = RX2 Next, we will solve Eq. (A-6) [same as Eq. (A-7) for X 1 ( = v c ), X 2( = i L ), 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 meth-ods of matrix algebra and matrix calculus operations are found in manyhandbooks (e.g., [Zwillinger, 1996]). The general format for state-variable equations, similar to Eqs. (A-4)and (A-5), is ˙ x = Ax + Bu (A-8) y = Cx + Duin which A, B, C, and D are coefÞcient matrices whose numbers maybe complex and varying in some manner with tim ...