LECTURE 6-7 (full) - LECTURE 6-7. FREQUENCY RESPONSE AND...

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Unformatted text preview: LECTURE 6-7. FREQUENCY RESPONSE AND SYSTEM CONCEPTS MAE307 Applied Electronics © Inkyu Park, 2010 Instructor: Prof. Inkyu Park Department of Mechanical Engineering Korea Advanced Institute of Science and Technology (KAIST) Review of Lecture #4-5 MAE307 Applied Electronics © Inkyu Park, 2010 Transient Analysis- When you turn on/off the switch, the circuit does not instantly follow your command. Instead, it needs some time period to approach to the command. MAE307 Applied Electronics © Inkyu Park, 2010 First-order System Equation- General equation form for the linear circuit containing single energy storage element can be written as: ) ( ) ( ) ( 1 t f b t x a dt t dx a = + First-order linear ordinary differential equation other form, this equation can be MAE307 Applied Electronics © Inkyu Park, 2010- In other form, this equation can be rewritten as: ) ( ) ( ) ( t f K t x dt t dx S = + τ Time constant DC gain Differential Equation for Circuit w/ Two Energy Storage Elements (One Capacitor & One Inductor)- General equation form for the linear circuit containing two energy storage elements (capacitor & inductor) can be written as: ) ( ) ( ) ( ) ( 1 2 2 2 t f b t x a dt t dx a dt t x d a = + + Second-order linear ordinary differential equation MAE307 Applied Electronics © Inkyu Park, 2010- In other form, this equation can be rewritten as: ) ( ) ( ) ( 2 ) ( 1 2 2 2 t f K t x dt t dx dt t x d S n n = + + ω ζ ω Natural frequency DC gain Damping ratio More energy storage elements → higher order differential equation Transient Response of First-Order Circuits- Solution of differential equation: ) , ( ,...) , , ( t x g x x x f = ′ ′ ′ ) ( ) ( ) ( t X t x t x + = Homogeneous solution Non-homogeneous (i.e. particular) solution- Procedures: A. Find the homogeneous solution . ind the particular solution MAE307 Applied Electronics © Inkyu Park, 2010 B. Find the particular solution C. General solution = homogeneous solution + particular solution Transient Response of First-Order Circuits- Given the equation for the first-order circuit: F K t x dt t dx S = + ) ( ) ( τ ≥ t- First, we find the natural response (homogeneous solution): ) ( ) ( = + t x dt t dx N N τ- The solution for this equation is τ α / ) ( t e t x- = MAE307 Applied Electronics © Inkyu Park, 2010 N- This natural response shows an exponential decay: n X(t)/X0=exp(-n τ/τ) 1 1 0.3679 2 0.1353 3 0.0498 4 0.0183 Transient Response of First-Order Circuits, cont’d- Second, find the forced response (particular solution): F K t x dt t dx S F F = + ) ( ) ( τ ≥ t- Assume that F is a constant for ≥ t Then, the particular solution is F K t x S F = ) ( This is equal to the DC steady-state solution ) ( ) ( ∞ → = = t x F K t x S F MAE307 Applied Electronics © Inkyu Park, 2010- The complete response is the sum of natural and forced responses....
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This note was uploaded on 11/21/2010 for the course MECHANICAL mae302 taught by Professor Jang during the Spring '10 term at Seoul National.

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LECTURE 6-7 (full) - LECTURE 6-7. FREQUENCY RESPONSE AND...

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