enee204Lectures_21_22Gomez

# enee204Lectures_21_22Gomez - Welcome to ENEE 204 Lecture 21...

This preview shows pages 1–4. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 04/07/2008 for the course ENEE 204 taught by Professor Gomez during the Fall '04 term at Maryland.

### Page1 / 14

enee204Lectures_21_22Gomez - Welcome to ENEE 204 Lecture 21...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online