Inverse Laplace transform using partial-fraction expansion

Find the inverse Laplace transform of $G(s) = \frac{10s²}{(s+1)(s+3)}, σ > 0$ .

Question

Inverse Laplace transform using partial-fraction expansion

Find the inverse Laplace transform of [latex]G(s) = \frac{10s²}{(s+1)(s+3)}, σ > 0[/latex].

Accepted Answer

This rational function is an improper fraction in s. Synthetically dividing the numerator by the denominator we get

[latex]\begin{gathered}s^2+4 s+3 )\overset{\underline{\quad \quad \quad 10}}{10s^2 \quad \quad} \\quad \quad \quad \quad \quad \quad \quad \quad\frac{10 s^2+40 s+30}{-40 s-30}\end{gathered} \Rightarrow \frac{10 s^2}{(s+1)(s+3)}=10-\frac{40 s+30}{s^2+4 s+3}[/latex]

Therefore

[latex]G(s) = 10-\frac{40s+30}{(s+1)(s+3)}, σ > 0[/latex].

Expanding the (proper) fraction in s in partial fractions,

[latex]G(s)=10-5\left(\frac{9}{s+3}-\frac{1}{s+1} ight), \sigma>0[/latex].

Then, using

[latex]e^{-a t} \mathrm{u}(t) \stackrel{\mathcal{L}}{\longleftrightarrow} \frac{1}{s+a} \quad ext { and } \quad \delta(t) \stackrel{\mathcal{L}}{\longleftrightarrow} 1[/latex]

we get

[latex]g(t) = 10δ(t) − 5(9e^{−3t} − e^{−t} )u(t)[/latex]

(Figure 8.14).

Question 8.5: Inverse Laplace transform using partial-fraction expansion F......

Related Answered Questions

Solution of a differential equation with initial conditions using the unilateral Laplace transform Solve the differential equation x(t)+7x(t)+12 x(t) = 0 for times t > 0 subject to the initial conditions ...

Verified Answer:

Response of a bridged-T networknIn Figure 8.18 the excitation voltage is vi(t)= 10u(t) volts. Find the zero-state response vRL(t). ...

Verified Answer:

Frequency response of a system from its pole-zero diagram Find the magnitude and phase frequency response of a system whose transfer function is ...

Verified Answer:

Using the time integration property to derive a transform pair In Example 8.11 we used complex-frequency differentiation to derive the Laplace transform pair ...

Verified Answer:

Using s-domain differentiation to derive a transform pair Using s-domain differentiation and the basic Laplace transform u(t)↔L 1/s, σ > 0, find the inverse Laplace transform of 1/s² , σ > 0. u(t) ↔L 1/s, σ > 0 ...

Verified Answer:

Laplace transform of a noncausal exponential signal Find the Laplace transform of x(t) = e^−t u(t) + e^2t u(−t). ...

Verified Answer:

Laplace transforms of two time-scaled rectangular pulses Find the Laplace transforms of x(t) = u(t) − u(t − a) and x(2t) = u(2t) − u(2t − a). ...

Verified Answer:

Use of the s-domain shifting property If X1(s) = 1/s+5, σ > -5 and X2 (s)=X1(s-j4)+X1(s+j4), σ > -5 find x2 (t). e^-5tu(t)↔ L 1/s+5, σ > -5 ...

Verified Answer:

Verified Answer:

Partial fraction expansion using MATLAB’s residue function Expand the expression H(s) = s²+3s+1/s^4+5s³+2s²+7s+3 in partial fractions. ...

Verified Answer: