SecondOrderCircuits

SecondOrderCircuits - SECOND-ORDER CIRCUITS vC THE BASIC...

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SECOND-ORDER CIRCUITS THE BASIC CIRCUIT EQUATION Single Node-pair: Use KCL R i L i C i 0 = + + + C L R S i i i i ) ( ); ( ) ( 1 ; ) ( 0 0 t dt dv C i t i dx x v L i R t v i C L t t L R = + = = S L t t i t dt dv C t i dx x v L R v = + + + ) ( ) ( ) ( 1 0 0 Differentiating dt di L v dt dv R dt v d C S = + + 1 2 2 Single Loop: Use KVL + R v + C v 0 = + + + L C R S v v v v ) ( ); ( ) ( 1 ; 0 0 t dt di L v t v dx x i C v Ri v L C t t C R = + = = dt dv C i dt di R dt i d L S = + + 2 2 S C t t v t dt di L t v dx x i C Ri = + + + ) ( ) ( ) ( 1 0 0 LEARNING BY DOING LY RESPECTIVE FOR EQUATION AL DIFFERENTI THE WRITE ), ( ), ( t i t v > < = 0 0 0 ) ( t I t t i S S S i MODEL FOR RLC PARALLEL dt di L v dt dv R dt v d C S = + + 1 2 2 0 ; 0 ) ( > = t t dt di S 0 1 2 2 = + + L v dt dv R dt v d C + S v > < = 0 0 0 ) ( t t V t v S S MODEL FOR RLC SERIES dt dv C i dt di R dt i d L S = + + 2 2 0 ; 0 ) ( > = t t dt dv S 0 2 2 = + + C i dt di R dt i d L
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THE RESPONSE EQUATION ) ( ) ( ) ( ) ( 2 1 2 2 t f t x a t dt dx a t dt x d = + + EQUATION THE FOR SOLUTIONS THE STUDY WE solution ary complement solution particular : KNOWN c p c p x x t x t x t x ) ( ) ( ) ( + = 0 ) ( ) ( ) ( 2 1 2 2 = + + t x a t dt dx a t dt x d c c c SATISIFES SOLUTION ARY COMPLEMENT THE IF THE FORCING FUNCTION IS A CONSTANT solution particular a is 2 ) ( a A x A t f p = = A x a dt x d dt dx a A x p p p p = = = = 2 2 2 2 0 : PROOF ) ( ) ( ) ( 2 t x a A t x A t f c + = = FUNCTION FORCING ANY FOR 0 ) ( 4 ) ( 2 ) ( 2 2 = + + t x t dt dx t dt x d LEARNING BY DOING 0 ) ( 16 ) ( 8 ) ( 4 2 2 = + + t x t dt dx t dt x d FREQUENCY NATURAL AND RATIO DAMPING EQUATION, STIC CHARACTERI THE DETERMINE COEFFICIENT OF SECOND DERIVATIVE MUST BE ONE 0 ) ( 4 ) ( 2 ) ( 2 2 = + + t x t dt dx t dt x d 2 n ω n ςω 2 0 4 2 2 = + + s s EQUATION STIC CHARACTERI DAMPING RATIO, NATURAL FREQUENCY 2 = n 5 . 0 = ς THE HOMOGENEOUS EQUATION 0 ) ( ) ( ) ( 2 1 2 2 = + + t x a t dt dx a t dt x d 0 ) ( ) ( 2 ) ( 2 2 2 = + + t x t dt dx t dt x d n n FORM NORMALIZED 2 1 1 2 2 2 2 2 a a a a a n n n = = = = ratio damping frequency natural (undamped) n 0 2 2 2 = + + n n s s EQUATION STIC CHARACTERI
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ANALYSIS OF THE HOMOGENEOUS EQUATION 0 ) ( ) ( 2 ) ( 2 2 2 = + + t x t dt dx t dt x d n n ω ςω FORM NORMALIZED 0 2 ) ( 2 2 = + + = n n st s s Ke t x iff solution a is If and only if s is solution of the characteristic equation st st Ke s dt x d sKe t dt dx 2 2 2 ; ) ( = = : PROOF st n n n n Ke s s t x t dt dx t dt x d ) 2 ( ) ( ) ( 2 ) ( 2 2 2 2 2 + + = + + 0 2 2 2 = + + n n s s EQUATION STIC CHARACTERI roots) distinct and (real : 1 CASE 1 > ς t s t s e K e K t x 2 1 2 1 ) ( + = roots) conjugate (complex : 2 CASE 1 < d n n j s j s σ ± = ± = 2 1 ( ) t A t A e t x d d t sin cos ) ( 2 1 + = t j t t j st d n d n e e e e ± = = ) ( : HINT t j t e d d t j d sin cos = roots) equal and (real 1 : 3 CASE = n s = ( ) t n e t B B t x + = 2 1 ) ( ) 0 2 2 ( ) 0 2 ( 2 2 = + = + + n n n st s AND s s te iff solution is : HINT t s t s e K e K t x 2 1 2 1 ) ( + =
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This note was uploaded on 12/01/2011 for the course EE 2120 taught by Professor Aravena during the Fall '08 term at LSU.

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SecondOrderCircuits - SECOND-ORDER CIRCUITS vC THE BASIC...

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