V3R C3 R 2 L4 R2 RLA L4 This is the state equation in normal form for the active network N of Fig. 7.1 Suppose that resistor voltage v, and capacitor current i are the output variables. Then from Eqs. (7.5b)and (7.9) +R1 R2 In matrix form, the output equation of the network becomes 2R R1 R1 R R2 quations(7.11)and(7. 13)together are the state equations of the active network of Fig. 7.1 7.5 State Equations for Networks described by scalar Differential equations In many situations we are faced with networks that are described by scalar differential equations of order higher than one. Our purpose here is to show that these networks can also be represented by the state equations in normal Consider a network that can be described by the nth-order linear differential equation dy (7.14) Then its state equation can be obtained by defining 15 mn= in-l e 2000 by CRC Press LLC© 2000 by CRC Press LLC (7.11) This is the state equation in normal form for the active network N of Fig. 7.1. Suppose that resistor voltage v1 and capacitor current i3 are the output variables. Then from Eqs. (7.5b) and (7.9) we obtain (7.12a) (7.12b) In matrix form, the output equation of the network becomes (7.13) Equations (7.11) and (7.13) together are the state equations of the active network of Fig. 7.1. 7.5 State Equations for Networks Described by Scalar Differential Equations In many situations we are faced with networks that are described by scalar differential equations of order higher than one. Our purpose here is to show that these networks can also be represented by the state equations in normal. Consider a network that can be described by the nth-order linear differential equation (7.14) Then its state equation can be obtained by defining (7.15) ˙ ˙ v i R C C L R R L R L v i R C R R L R L v i g g 3 4 2 3 3 4 1 2 4 1 4 3 4 2 3 1 2 4 1 4 1 1 1 2 1 0 È Î Í Í ˘ ˚ ˙ ˙ = - - - - È Î Í Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ ˙ È Î Í Í ˘ ˚ ˙ ˙ + - È Î Í Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ ˙ È Î Í Í ˘ ˚ ˙ ˙ v R R v R i R i v R g g 1 1 2 3 1 4 1 2 = + 2 + - Ê Ë Á ˆ ¯ ˜ i v R i v R g 3 3 2 4 2 = - - + v i R R R R v i R R R R v i g g 1 3 1 2 1 2 3 4 1 2 1 2 2 1 1 1 0 È Î Í Í ˘ ˚ ˙ ˙ = - - È Î Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ È Î Í Í ˘ ˚ ˙ ˙ + - È Î Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ È Î Í Í ˘ ˚ ˙ ˙ d y dt a d y dt a d y dt a dy dt a y bu n n n n n + + n + + n n + = - - - 1 - - 1 1 2 2 2 1 . . . x y x x x x n n 1 2 1 1 = = = - ˙ . . . ˙