Question

An electrical network is shown in the figure below :

Which one of the following statements is true regarding the system considering a causal system ?

• A. The system is BIBO stable.
• B. The system is BIBO unstable.
• C. None of these
• D. The system stability depends on values of R and L.

Answer 1

The correct option is A The system is BIBO stable.

System transfer function using laplace transform would be,

$H\left(s\right)=\frac{Y\left(s\right)}{X\left(s\right)}=\frac{Ls}{R+Ls}=1-\frac{R/L}{\left(R/L\right)+s}$

taking inverse laplace transform

$h\left(t\right)=\delta \left(t\right)-\frac{R}{L}{e}^{-\left(R/L\right)t}$

For stability of any system,

${\int }_{-\infty }^{\infty }|h\left(t\right)|dt<\infty$

${\int }_{-\infty }^{\infty }|h\left(t\right)|dt={\int }_0^{\infty }|\delta \left(t\right)-\frac{R}{L}{e}^{-\left(R/L\right)t}|dt=1-{\left[\frac{R{e}^{-\left(R/L\right)t}}{L(-R/L)}\right]}_0^{\infty }$

= 1 - (0+1) = 0

So ${\int }_{-\infty }^{\infty }|h\left(t\right)dt|<\infty$

So system is BIBO stable.