ECE 202 Lesson 18

# ECE 202 Lesson 18 - ECE 202 Circuit Theory 2 Lesson 18...

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ECE 202 Circuit Theory 2 Lesson 18 Partial Fraction Expansion for Multiple-Order Real Poles

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Multiple-Order Real Pole ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 1 3 1 2 2 3 1 1 ! 0,1, , 1 m s a m m m s a F s F s s a F s K K K K F s s a s a s a s a d F s K ds m s - - - = + = + + + + + + + + + = = - l l l K K l l K Pole of order m at s = -a 2
The Time Function f(t) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 3 1 2 2 3 1 2 1 1 1 1 2 3 2! 1 ! 1 1 ! m m n at n m at m m K K K K F s s a s a s a s a t e u t n s a f t K K t K t K t e u t - + - - - = + + + + + + + + + + = + + + + + K K K K 3

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Class Problem ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 3 3 1 2 4 2 3 1 2 3 1 2 3 4 4 3 4 4 Solve for , , s s F s s s K K K K F s s s s s K K K + + = + + = + + + + + + + 4
Pole of Order 3 at s = -4 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 3 1 1 3 1 2 3 4 1 2 3 1 ! 0,1,2 s a s s F s s s s s F s s d F s K ds - - + + = + + + + = + = = l l l l l 5

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Pole of Order 3 at s = -4 ( 29 ( 29 ( 29 ( 29 1 3 4 3 1 4 1 2 4 2 1 1 1 2 2 4 1 ! 0,1,2 s s s s d F s K ds K F s dF s K ds d F s K ds - - - - - = = = = = l l l l l 6
Determination of K 3 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 3 1 4 1 3 3 1 2 3 4 1 4 2 3 2 4 3 1 6 s K F s s s F s s K K s - = + + = + - + - + - - = = - + - = - 7

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Determination of K 2 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 2 4 2 1 2 1 2 2 1 2 1 2 3 2 3 3 3 2 3 3 2 3 6 7 3 s dF s K ds s s s s F s s s s s s s dF s ds s dF s s s ds s s - = + + + + = = + + + + - + + = + + + = + 8
Determination of K 2 ( 29 ( 29 ( 29 ( 29 1 2 4 2 1 2 2 2 2 6 7 3 16 24 7 1 1 s dF s K ds dF s s s ds s K K s - = + + = + - + = - = - 9

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Determination of K 1 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 1 1 1 2 2 4 2 1 2 2 2 2 1 4 2 2 2 2 1 3 2 2 1 3 2 6 7 3 3 2 6 6 7 2 3 3 3 6 7 2 3 4 3 s d F s K ds dF s s s ds s s s s s s d F s ds s s s s d F s ds s d F s ds s s - = + + = + + + - + + + = + + - + + = + = + 10
Determination of K 1 ( 29 ( 29 ( 29

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