110_1_Desoer&Kuh_Chap12 - When t he differential...

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When the differential equations of a lumped network are written in the form x = f(x,w,t) (where x is a vector, say, with n components; w represents the set of inputs, and t represents the time), we say that the equations are in the state form and that x represents the state of the network. There are three basic reasons for writing the equations in this form: (1) this form lends itself most easily to analog and/or digital computer programming, (2) the extension of the analysis to nonlinear and/or time-varying networks is quite easy (whereas this extension is not easy in the case of loop, mesh, cut-set, or node analysis), and (3) in this form a number of system- theoretic concepts are readily applicable to networks. In Sec. 1 we use examples to illustrate how state equations are written for simple linear time-invariant networks. In Sec. 2 we review and extend some per- tinent concepts of state. In Sec. 3 simple time-varying and nonlinear networks are treated. Finally, in Sec. 4 we give a general method of writing state equations for a broad class of linear time-invariant networks. Linear Time-invariant Networks - Consider the linear time-invariant network shown in Fig. 1.1. It has three energy-storing elements: one capacitor C and two inductors Ll and L.!. . Any response ofthis network is thus closely related to the behavior of the capacitor voltage D and the inductor currents it i2. If we wish to use these variables for our analysis and also wish to write equations in the state form, (1.1) x = f(x,w,t) we may choose D, iI, i2 as the state variables; that is, we choose (1.2) x = m as the state vector. Note that in the vector differential equation (1.1), x, 501
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Chap. 12 State Equations 502 il i2 Ll L2 jc + + VR2 + R2 v C Fig.
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This note was uploaded on 03/28/2011 for the course EE EE 110 taught by Professor Gupta during the Winter '09 term at UCLA.

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110_1_Desoer&Kuh_Chap12 - When t he differential...

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