Question

• A. \(\dfrac{s^2 + 1.5s + 1}{s(s+1)}\)
• B. \(\dfrac{s^2 + 3s + 1}{s(s + 1)}\)
• C. \(\dfrac{2s^2 + 3s + 2}{s(s + 1)}\)
• D. \(\dfrac{2s^2 + 3s + 1}{2s(s + 1)}\)

Hint:

Always reduce parallel combinations first, and then proceed with series combinations. Convert each element to its Laplace domain representation before solving.

Answer 1

The Correct Option is A

The circuit has the following elements:
Capacitor: \(C = 1\ \text{F} \Rightarrow \text{Impedance } Z_C = \dfrac{1}{s}\)
Inductor: \(L = 2\ \text{H} \Rightarrow \text{Impedance } Z_L = 2s\)
Two resistors: each \(R = 1\ \Omega\)
Step 1: Identify series-parallel combination
- The resistor and inductor (1 Ω + 2s) are in series.
- This series is in parallel with the other 1 Ω resistor.
- The result is in series with the capacitor.
Step 2: Compute parallel branch impedance
\[ Z_{parallel} = \left[\left(1 + 2s\right) \parallel 1\right] = \dfrac{(1 + 2s) \cdot 1}{(1 + 2s) + 1} = \dfrac{1 + 2s}{2 + 2s} \] Step 3: Add capacitor impedance in series
\[ Z(s) = \dfrac{1}{s} + \dfrac{1 + 2s}{2 + 2s} \] Step 4: Simplify to single fraction
Take LCM: \[ Z(s) = \dfrac{(2 + 2s) + s(1 + 2s)}{s(2 + 2s)} = \dfrac{2 + 2s + s + 2s^2}{s(2 + 2s)} = \dfrac{2s^2 + 3s + 2}{s(2 + 2s)} \] Factor denominator: \[ Z(s) = \dfrac{2s^2 + 3s + 2}{2s(s + 1)} \] Divide numerator and denominator by 2: \[ Z(s) = \dfrac{s^2 + 1.5s + 1}{s(s + 1)} \]