enee204Lectures_21_22Gomez

enee204Lectures_21_22Gomez - Welcome to ENEE 204 Lecture 21...

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1 Lecture 21&22 1 Welcome to ENEE 204 Lecture 21 Chapter 7: TRANSIENT ANALYSIS -understand time varying signals - differential equations (transient and steady state solutions) -1 st and 2 nd order circuits (Differential Equations) Lecture 21&22 2 Transient Analysis – Simple Picture (Series Connection) 1 st order: 1 resistor and 1 cap (or 1 inductor) in series – Solution without source – Solution with source, dc or ac 2 nd order - Undamped: 1 cap and 1 inductor in series (LC) - Damped: 1 cap, 1 inductor, 1 resistor in series - underdamped -overdamped - critically dampled …with and without sources t = 0 v s (t) t = 0 Lecture 21&22 3 Transient Analysis - Big Picture II (Parallel) 1 st order: 1 resistor and 1 cap (or 1 inductor) in parallel – Solution without source – Solution with source, dc or ac 2 nd order: - Undamped: 1 cap and 1 inductor in parallel (LC) - Damped: 1 cap, 1 inductor, 1 resistor in parallel - underdamped - critically dampled - with and without sources Lecture 21&22 4 Main steps in transient analysis: 1. Given a circuit, use KVL or KCL equations. 2. Use terminal relations for the elements. 3. Use 1 and 2 to derive a differential equation. 4. Solve the differential equation. 5. Use initial conditions* to determine the constants. Main Goal: To derive and solve the differential equations that describe the time dependence of the voltage, v(t) and current, i(t) from the initial state (t=0) to steady state t *initial conditions are defined before hand, do not worry on how the circuit got to that state.
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2 Lecture 21&22 5 1 st Order Circuit 0 = dt (t) dv RC + (t) v C C RC t c Ae t v / ) ( = 0 ) ( ) ( = + t v t v C R t = 0 - + v R (t) v C (t) i (t) R C Terminal relations: + _ KVL: dt (t) dv C = (t) C i R t i t v R ) ( ) ( = Differential equation of the circuit: Solution of the circuit: Initial condition: o V t v = = ) 0 ( 0 / 0 ) 0 ( V Ae v RC c = = 1 RC t o c e V t v / ) ( = Solution of the form: Lecture 21&22 6 Graph of 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 RC RC t o C e V = (t) v t 1.0 V 0 (t) v C e -1 2RC 3RC 4RC e -2 e -3 RC t o c e V t v / ) ( = This gives the familiar result that the voltage on a capacitor, initially charged to voltage, Vo, will discharge exponentially with time. The rate of decay is given by the RC time constant. Lecture 21&22 7 Another example: an inductor charged with current is connected in series with a resistor. Initial condition (switch closed) L (0 ) = I o What is i(t), t>0? _ + v R (t) _ + v l (t) i (t) L R 0 i(t)= L R dt di(t) + 0 = dt di(t) i(t)R + L 0 ) ( ) ( = + t v t v L R KVL: dt di(t) (t) = L v L R t i t v R ) ( ) ( = T.R’s: ( ) L R t Ae t i / ) ( = Solution of the form: Initial condition: o L R o I A Ae )=I i( = = 0 0 ( ) L R t o e I t i / ) ( = Solution: Lecture 21&22 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 L/R 2L/R 3L/R 4L/R t 1.0 I 0 (t) L e -1 e -2 e -3 Graph of the answer L Rt o e I = (t) i
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3 Lecture 21&22 9 1 st Order circuits with no driving sources and excited only by initial conditions produce exponential decay in time τ = RC for capacitor τ = L/R for inductor L Rt o L e I = (t) i RC t o C e V = (t) v The rate of decay is determined by the time constant τ : Lecture 21&22 10 1 st Order excited by a voltage source.
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This note was uploaded on 04/07/2008 for the course ENEE 204 taught by Professor Gomez during the Fall '04 term at Maryland.

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enee204Lectures_21_22Gomez - Welcome to ENEE 204 Lecture 21...

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