SecondOrderCircuitSp03(1pp)

SecondOrderCircuitSp03(1pp) - SECOND-ORDER CIRCUITS THE...

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Unformatted text preview: SECOND-ORDER CIRCUITS THE BASIC CIRCUIT EQUATION Single Node-pair: Use KCL R i L i C i = + + + C L R S i i i i ) ( ); ( ) ( 1 ; ) ( t dt dv C i t i dx x v L i R t v C L t t L R = + = = S L t t i t dt dv C t i dx x v L R v = + + + ) ( ) ( ) ( 1 Differentiating dt di L v dt dv R dt v d C S = + + 1 2 2 Single Loop: Use KVL + R v + C v + L v = + + + L C R S v v v v ) ( ); ( ) ( 1 ; 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 LEARNING BY DOING LY RESPECTIVE FOR EQUATION AL DIFFERENTI THE WRITE ), ( ), ( t i t v > < = ) ( 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 ; ) ( > = t t dt di S 1 2 2 = + + L v dt dv R dt v d C + S v > < = ) ( 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 ; ) ( > = t t dt dv S 2 2 = + + C i dt di R dt i d L 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 ) ( ) ( ) ( + = ) ( ) ( ) ( 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 : PROOF ) ( ) ( ) ( 2 t x a A t x A t f c + = = FUNCTION FORCING ANY FOR ) ( 4 ) ( 2 ) ( 2 2 = + + t x t dt dx t dt x d LEARNING BY DOING ) ( 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 ) ( 4 ) ( 2 ) ( 2 2 = + + t x t dt dx t dt x d 2 n n 2 4 2 2 = + + s s EQUATION STIC CHARACTERI DAMPING RATIO, NATURAL FREQUENCY 2 = n 5 . = THE HOMOGENEOUS EQUATION ) ( ) ( ) ( 2 1 2 2 = + + t x a t dt dx a t dt x d ) ( ) ( 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 2 2 2 = + + n n s s EQUATION STIC CHARACTERI ANALYSIS OF THE HOMOGENEOUS EQUATION ) ( ) ( 2 ) ( 2 2 2 = + + t x t dt dx t dt x d n n FORM NORMALIZED 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 + + = + + 2 2 2 = + + n n s s EQUATION STIC CHARACTERI roots) distinct...
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SecondOrderCircuitSp03(1pp) - SECOND-ORDER CIRCUITS THE...

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