6 003 Signals and Systems Second Order Systems October 8 2009 Last Time We analyzed a mass and spring system x t F K x t y t M y t y t x t K M y t A 1 K A2 Y M K A2 X 1 M y t A y t Last Time We also analyzed a leaky tanks system r0 t h1 t 1 r 1 t r0 t r1 t r1 t 2 r 2 t r1 t r2 t h2 t r2 t r0 t 1 1 r 1 t A r1 t 1 2 R2 A 1 A 2 R0 1 A 1 1 A 2 r 2 t A r2 t Second Order Systems Today Look more carefully at growth and decay of oscillatory responses by studying an analogous electrical circuit vi R L C vo But First The canonical forms for CT and DT differ X A Y s0 H X Y z0 Y A X 1 s0 A H Delay Y 1 X 1 z0 R h t e s0 t u t h n z0n u n A 1s R z1 s plane s plane z plane z plane Check Yourself What if we had used the DT canonical form for CT X Y s0 A What is the impulse response of this system 1 s0 es0 t u t 2 st0 u t 3 1 s0 es0 t u t 4 t s0 es0 t u t 5 none of the above Check Yourself What is the impulse response of this system X Y s0 Y 1 s0 A 1 X 1 s0 A 1 s0 A Therefore the impulse response is h t t s0 es0 t u t A Check Yourself Check by tracing signal flow paths X Y s0 X Y 1 s0 A k X 1 s0 A k 0 A Check Yourself Check by tracing signal flow paths X Y s0 A X Y 1 s0 A k X 1 s0 A k 0 If x t t then y t t y t s0 0 t Check Yourself Check by tracing signal flow paths X Y s0 A X Y 1 s0 A k X 1 s0 A k 0 If x t t then y t t s0 u t y t s0 0 t Check Yourself Check by tracing signal flow paths X Y s0 A X Y 1 s0 A k X 1 s0 A k 0 If x t t then y t t s0 u t s20 tu t y t s0 0 t Check Yourself Check by tracing signal flow paths X Y s0 A X Y 1 s0 A k X 1 s0 A k 0 If x t t then y t t s0 u t s20 tu t 21 s30 t2 u t y t s0 0 t Check Yourself Check by tracing signal flow paths X Y s0 A X Y 1 s0 A k X 1 s0 A k 0 If x t t then y t t s0 es0 t u t y t s0 0 t Check Yourself What if we had used the DT canonical form for CT X Y s0 A What is the impulse response of this system 1 s0 es0 t u t 2 st0 u t 3 1 s0 es0 t u t 4 t s0 es0 t u t 5 none of the above 4 Second Order Systems Today Look more carefully at growth and decay of oscillatory responses by studying an analogous electrical circuit vi R L C vo Second Order Systems Solve with state variable approach State variables represent the minimum knowledge of the past t t0 needed to propagate the output into the future t t0 Check Yourself iL iC vC iC C dvC dt vL di vL L L dt Which of the following can be state variables 1 vC and vL 3 iC and iL 5 iC and vC and iL and vL 2 iC and vL 4 vC and iL 6 none of above Check Yourself iC vC iC C dvC dt Integrate from a starting point t0 to a stopping point t1 Z t1 iC t dt C vC t1 vC t0 t0 and solve for vC t1 Z 1 t1 vC t1 i t dt vC t0 C t0 C Calculating vC t1 requires vC t0 Knowing iC t for t0 t t1 is not enough Therefore vC can be a state variable iC cannot Analogously iL can be a state variable vL cannot Check Yourself iL iC vC iC C dvC dt vL di vL L L dt Which of the following can be state variables 1 vC and vL 3 iC and iL 5 iC and vC and iL and vL 4 2 iC and vL 4 vC and iL 6 none of above Second Order Systems State variable approach determine expressions for derivatives of state variables in terms of undifferentiated state variables vL vi 1 1 dvC iC iL dt C C R iL iC L C KCL diL 1 1 vL vi RiL vC dt L L KVL v C vo Second Order Systems Determine the system functional vL R vi iL iC L C v C vo 1 dvC iL dt C diL 1 vi RiL vC dt L Use first equation to eliminate iL from the second equation C 1 d2 vC dv vi RC C vC 2 L dt dt Integrate twice ignoring initial conditions why 1 CvC A2 vi RCAvC A2 vC L 2 A Vo V LC C 1 2 Vi Vi 1 R L A LC A Check Yourself Alternatively determine system functional from block diagram dvC 1 iL dt C vi w1 1 diL vi RiL vC dt L w2 1 L w3 A w4 1 C w5 y A R Which node corresponds to iL 1 w1 2 w2 3 w3 4 w4 7 none of the above 5 w5 6 y Check Yourself Alternatively determine system functional from block diagram dvC 1 iL dt C vi diL 1 vi RiL vC dt L iL 1 L A iL 1 C vC vC A R Which node corresponds to iL 1 w1 2 w2 3 w3 4 4 w4 7 none of the above 5 w5 6 y Second Order Systems Finding the system functional from the block diagram vi w1 iL w2 1 L A R 1A IL LR W1 1 LA Vo Vi 1 LA 1 A C 1 R LA 1 A2 1 LCR 1 L A 1 1 2 LC A R A 1 A2 L LC iL 1 C vC A vo vC Second Order Systems Analogous systems RLC circuit Vo Vi 1 1 2 LC A R A 1 A2 L LC 02 A2 1 Q0 A 02 A2 02 A2 0 1 Q A 02 A2 02 A2 0 1 Q A 02 A2 r 1 LC r K M r 1 1 2 0 Mass and spring Y X K 2 MA K A2 1 M 0 Leaky tanks 1 2 R2 1 2 …
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