Signal Processing and Linear Systems-B.P.Lathi copy

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Unformatted text preview: Fig. 13.2. In t he network in Fig. 13.2, we replace t he i nductor b y a c urrent source o f c urrent X l a nd t he c apacitor by a voltage source of voltage X 2, as shown i n Fig. 13.3. T he r esulting network consists o f five resistors, two voltage sources, a nd o ne current source. We can determine t he voltage v L across t he i nductor and t he c urrent i c t hrough t he c apacitor by u sing t he principle o f s uperposition. This s tep c an be accomplished by inspection. For example, V L h as t hree c omponents arising from three sources. To compute t he c omponent d ue t o f , we a ssume t hat X l = 0 ( open circuit) a nd X 2 = 0 ( short circuit). Under these conditions, all of t he network t o t he r ight of t he 2 0 r esistor is opened, a nd t he c omponent o f V L due t o f is t he voltage across t he 2 0 resistor. T his v oltage is clearly ~f· Similarly, t o find t he c omponent of V L due t o X l, we s hort f a nd X 2. T he source X l sees a n equivalent resistor of 1 0 across it, a nd hence V L = - Xl. C ontinuing t he process, we find t hat t he c omponent of V L d ue t o X 2 is - X2. Hence VL . I = X l ="2 f - Xl - X2 (13.12a) Using the same procedure, we find (13.12b) I n S ec. 6.6 w e s aw t hat a g iven t ransfer f unction c an b e r ealized i n s everal w ays. C onsequently, w e s hould b e a ble t o o btain d ifferent s tate-space d escriptions o f t he s ame s ystem b y u sing d ifferent r ealizations. T his a ssertion will b e c larified b y t he f ollowing e xample. • E xample 1 3.4 D etermine t he s tate-space description of a s ystem specified by t he t ransfer function Hs _ ( ) - s3 2 s + 10 + 8s 2 + 19s + 12 (13.15a) (13.15b) ! 2 ~ = _3_ ___ + _3_ s +1 8 +3 s +4 (13.15c) tThis procedure requires modification if the system contains all-capacitor voltage source tie sets or all-inductor current source cut sets. In the case of all-capacitor voltage source tie sets, all capacitor voltages cannot be independent. One capacitor voltage can be expressed in terms of the remaining capacitor voltages and the voltage source(s) in that tie set. Consequently, one of the capacitor voltages should not be nsed as a state variable, and that capacitor should not be replaced by a voltage source. Similarly, in all-inductor current source tie sets, one inductor should not be replaced by a current source. If there are aII-capacitor tie sets or all-inductor cut sets only, no further complications occur. In all-capacitor-voltage source tie sets and/or a11-inductor-current source cut sets, we have additional difficulties in t hat the terms involving derivatives of the input may occur. Thls problem can be solved by redefining the state variables. The final s tate variables will not be capacitor voltages and inductor currents. a 792 13 S tate-Space A nalysis 13.2 A S ystematic P rocedure for D etermining S tate E quations 7 93 Using t he p rocedure developed in Sec. 6.6, we s hall realize H{s) i n Eq. (13.15) w ith four different realizations: (i) t he c ontroller canonical form [Eq. (13.15a)], (ii) t he o bs...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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