《电路》（英文版）17-5 State variables and normal-form equations

817-5 State variables and normal-form equations The set of variables is a hybrid set that may include both currents and voltages. They are the inductor currents and the capacitor voltages. Each of these quantities may be used directly to express the energy stored in the inductor or capacitor at any instant of time. They collectively describe the energy state of the tem. They are called the state variables State variables L9C19C2· L--KVL C--KCL 十D diL xuc L C2 R R duci+n(Dc2 -of +Ucu+i=0 dt R du (Uc2-U,+Uc1)=0 dt

§17-5 State variables and normal-form equations The set of variables is a hybrid set that may include both currents and voltages. They are the inductor currents and the capacitor voltages. Each of these quantities may be used directly to express the energy stored in the inductor or capacitor at any instant of time. They collectively describe the energy state of the system. They are called the state variables. State variables-- , , . L C1 C2 i   L--KVL C--KCL + C2 − s = 0 L dt di L   ( ) 0 1 2 1 1 1 + C − s + C + s = C i dt R d C     ( ) 0 1 2 1 2 2 − L + C − s + C = C R i dt d C      s L i − +  C 2 s i R i C2 C1 + C1 − L − + R

The state equations are said to be in normal form when the derivative of each state variables is expressed as a linear combination of all the state variables and forcing functions. The ordering of the equations defining the derivatives and the order in which the state variables appear in every equations must be the same dii tact 0 dt L due+n(2-U+1)+i=0 d CI CIRC C2RC1 DS-cIs dt R dt RCI dtLL+(Uc2 -D, +Uc1)=0 2 R dt RCa RC 1 vc2 RCL L du C1 dt RC1BC∥an/+/7 RC ducr D 0 C RC RC RC

The state equations are said to be in normal form when the derivative of each state variables is expressed as a linear combination of all the state variables and forcing functions. The ordering of the equations defining the derivatives and the order in which the state variables appear in every equations must be the same. + C2 − s = 0 L dt di L   ( ) 0 1 2 1 1 1 + C − s + C + s = C i dt R d C     ( ) 0 1 2 1 2 2 − L + C − s + C = C R i dt d C     C s L dt L L di   1 1 = − 2 + C C s s C i dt RC RC RC C d 1 1 2 1 1 1 1 1 1 1 1 = −  −  +  −  L C C s C RC RC RC i dt C d     2 2 2 1 2 2 2 1 1 1 1 = − − +                         + −                             − − − − − =                 s s C C L C C L i RC RC C i L C RC RC RC RC L dt d dt d dt di      0 1 1 1 0 1 1 1 1 1 1 0 1 0 0 2 1 1 2 1 2 2 2 1 1 2 1

The normal form equations: di 0 0 dt 0 dt RCI RCI RC du dt C2 RC2 RC RC2 state vector or q=aq+b q'--derivative of the state vector f'-- forcing function vector a,b--system matrix du d t RCI RCI RC duci U,| RC2 RC2 C2 RC i d t L 0

The normal form equations:                         + −                             − − − − − =                 s s C C L C C L i RC RC C i L C RC RC RC RC L dt d dt d dt di      0 1 1 1 0 1 1 1 1 1 1 0 1 0 0 2 1 1 2 1 2 2 2 1 1 2 1 • • • • • or q'= a q+ b f q − − state vector • q − −derivative of the state vector • ' f − − forcing function vector • a b− −system matrix • • ,                       − +                           − − − − − s s L C C i L RC RC C i L RC RC C RC RC    0 1 0 1 1 1 0 1 0 1 1 1 0 1 1 2 1 1 2 1 2 2 2 1 1 =                 dt di dt d dt d L C C 2 1  