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Unformatted text preview: Lecture 1  EE743 Lecture 1  EE743 Dynamic Systems Professor: Ali Keyhani Professor: Ali Keyhani 2 Dynamic Systems ■ Consider a system that can be modeled by the following equivalent electric circuit Figure 1 3 Dynamic Systems ■ Using KVL, Loop 1 : dt di L Ri R R i v dt di L i R i i R v 1 1 2 1 1 1 1 1 1 2 1 ) ( ) 1 ( ) ( + + + = + + + = 4 Dynamic Systems ■ Using KVL, Loop 2 : dt di L Ri R R i v dt di L i R i i R v 2 2 1 2 2 2 2 2 2 2 1 ) ( ) 2 ( ) ( + + + = + + + = 5 Dynamic Systems ■ Solving Equation (1) for i 2 , ) 4 ( ) 3 ( 2 1 2 1 1 1 2 1 1 1 1 2 + = + = dt i d R L dt di R R R dt di dt di R L i R R R R v i 6 Dynamic Systems ■ Substituting (3) in (4), we will obtain equation (5) ( 29 ( 29 2 2 1 1 2 2 1 2 1 2 1 2 1 1 1 2 2 1 ) 5 ( vR u R R R R RR c L R RL L R RL b L L a u ci dt di b dt di a = + + = + + + = = = + + 7 Dynamic Systems ■ For the dynamic system described by equation (5), we have two energy storing elements: L 1 and L 2 . Therefore, the system is represented by a second order differential equation. ■ Let , then eq.(5) can be written as, dt d p = ) 6 ( 2 = + + c bp ap 8 Dynamic Systems ■ Equation 6 represents the characteristic equation of the system described by Figure 1. ■ The roots of eq.(6) are called eigenvalues eigenvalues of the system, and are determined by, β α ± = ± = 2 1 2 2 1 , 2 4 , p p a ac b b p p 9 Dynamic Systems ■ The value of β strongly depends of the kind of elements of the system. The following combination of storage elements give an idea of the numerical value of β : – For RLL RLL systems β is real – For RCC RCC systems β is real – For RLC RLC systems : ●...
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 Spring '08
 Keyhani
 Lyapunov stability, phase space, DYNAMIC SYSTEMS, Discrete Solution Method

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