8.4 The Natural Response of a Series/Parallel RLC Circuit

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8.5 The Step Response of a Series/Parallel RLCCircuitProblem : (a) Time consuming to eliminate the othervariable to get a higher order differentialequation.(b) It is also necessary to obtain the desiredinitial conditions.(c) As the order gets higher when thenetwork contains more energy storageelements, the process gets morecomplicated.The difficulty can be overcome by using state equationformulation.C.T. Pan518.6 State EquationWhen the differential equations of a circuit is written inthe following form:d x = f ( xut , ,)dtTx = [ x x .... x ] state vectorT1 2u = [ u u .... u ] input vector1 2Tf = [ f f .. f ] vector function1 2nnmC.T. Pan52
8.6 State EquationIt is said that the circuit equations are in the stateequation form.(a) This form lends itself most easily to analog or digitalcomputer programming.(b) The extension to nonlinear and/or time varyingnetworks is quite easy.(c) In this form, a number of theoretic concepts ofsystems are readily applicable to networks.C.T. Pan 53For a linear time-invarying circuit , a simpler formd x = A x + B u , state equationdty= Cx+D u , output equationA : n×n matrix.B : n×m matrix.C :8.6 State Equationl×n matrix.D: l × m matrix.Note that on the right hand side of the state or outputequation, only x and u are allowed.C.T. Pan54
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8.4 The Natural Response of a Series/Parallel RLC Circuit

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