Hwk_Soln_3-1

Hwk_Soln_3-1 - 332: 305 Numerical Modeling in Biomedical...

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332: 305 Numerical Modeling in Biomedical Systems Fall 2010 Homework Solutions #3 1) Prob. 2.4.3 2 2 C C C S d d R d RC v v v e dt dt L dt + + = Dividing the differential equation of the circuit by RC and comparing it to the standard form 2 2 1 0 0 1 2 2 2 c c c s s s d d d d v a v a v b e b e b e dt dt dt dt + + = + + 1 0 2 1 0 1 1 1 , 0, 0 a a b b b RC LC RC = = = = = The simulation diagram in the text reduces to Hwk Soln #3 Fall, 2010 Dr. Shoane 1 S e C v R C L + _ + _ 1 RC - 1 LC - 1 RC - c v s e
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2) Prob. 2.4.5 c) This simulation diagram represents a 2 nd order system with 2 inputs. 1 0 1 1 2 1 2 2 1 2 x a x u x a x u x x x + = + = = + & & Transforming to the s-variable, or Laplace, domain gives X X X U X a sX U X a sX = + = + = + 2 1 2 2 1 2 1 1 0 1 Method 1 - Direct Algebraic Solution. Solving for X 1 and X 2 gives [ ] 2 0 2 1 1 1 1 0 1 0 2 2 2 0 1 1 1 0 1 0 2 1 0 1 0 2 2 0 1 1 1 2 0 1 2 1 1 2 2 0 1 1 ) ( d gives ng Transformi Inverse or u a dt du u a dt du x a a dt dx a a dt x ) U a (s ) U a (s X a a )s a (a s a a )s a (a s ) U a (s ) U a (s a s U a s U X X X a s U X a s U X + + + = + + + + + + = + + + + + + + + + = + + + = + = + = + = Hwk Soln #3 Fall, 2010 Dr. Shoane 2 x 1 u 0 a - 1 a - 2 u 1 x
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Method 2 - Cramer’s Rule for 2x2 matrix for X 1 and X 2 [ ] before. as answer same the is Which ) ( d gives ng Transformi Inverse or ) )( ( ) ( ) ( ) )( ( ) ( , ) )( ( ) ( 0 0 0 , 0 0 0 obtain we Rule, s Cramer' Applying U U X X 0 0 2 0 2 1 1 1 1 0 1 0 2 2 2 0 1 1 1 0 1 0 2 1 0 2 0 1 1 2 1 1 0 2 0 2 1 0 1 1 1 1 0 2 1 0 2 1 0 1 2 1 1 2 1 2 1 0 u a dt du u a dt du x a a dt dx a a dt x ) U a (s ) U a (s X a a )s a (a
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Hwk_Soln_3-1 - 332: 305 Numerical Modeling in Biomedical...

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