Partial Fraction Expansion

Question

Answer 1

The expression

1/2{s / [ (s^2 + 1)(s + 1/2) ]} + 1/10 [s / (s + 1/2) ]

Consider 1st term

s / [ (s^2 + 1)(s + 1/2) ]   =   [ A / (s + 1/2) ] + [ (Bs + C) / (s^2 + 1) ]  ---------->(T)

Take the LCM

s / [ (s^2 + 1)(s + 1/2) ]   =   [ A (s^2 + 1) +  (Bs + C)(s + 1/2) ] / [ (s^2 + 1)(s + 1/2) ]

s   =   A s^2 + A +  Bs^2 +Bs/2 + Cs + C/2

s   =   s^2 (A + B) + s (B/2 + C) + (A + C/2)

Multiply above equation with "2"

2s   =   s^2 (2A + 2B) + s (B + 2C) + (2A + C)

Compare s^2, s and constant terms

2A + 2B   =   0 --------------------> (1)

2B  =  - 2A

B   =   - A ---------------> (2)

B + 2C   =   2  ---------------> (3)

2A + C   =   0

C   =   - 2A ---------------> (4)

Substitute B   =   - A and C   =   - 2A in equation (3)

(-A) + 2(-2A)   =   2

-5A   =   2

A   =   - 2/5

From the equation (2)

B   =   2/5

From the equation (4)

C = -2(-2/5)

C   =   4/5

Substitute A  =  - 2/5, B  =  2/5 and C = 4/5 in equation (T)

s / [ (s^2 + 1)(s + 1/2) ]   =   [ (-2/5) / (s + 1/2) ] + [ (2s/5) + (4/5)] / (s^2 + 1)

s / [ (s^2 + 1)(s + 1/2) ]   =   [ -2/ 5(s + 1/2) ] + [ (2s/5)/(s^2 + 1) ]+ [ (4/5) / (s^2 + 1) ]

s / [ (s^2 + 1)(s + 1/2) ]   =   [ -2/ 5(s + 1/2) ] + [ 2s/5(s^2 + 1) ]+ [ 4/5(s^2 + 1) ]

Hence

1/2{s / [ (s^2 + 1)(s + 1/2) ]} + 1/10 [s / (s + 1/2) ]   =   1/2{[ -2/ 5(s + 1/2) ] + [ 2s/5(s^2 + 1) ]+ [ 4/5(s^2 + 1) ]} + 1/10 [s / (s + 1/2) ]

1/2{s / [ (s^2 + 1)(s + 1/2) ]} + 1/10 [s / (s + 1/2) ]   =   1/5[s/(s^2 + 1)]+ 2/5[1/(s^2 + 1)] + -1/ 5[1/(s + 1/2) ] + 1/10 [s / (s + 1/2) ]