A series loop is driven by an independent voltage source v_s whose negative terminal is…

A series loop is driven by an independent voltage source \(v_s\) whose negative terminal is grounded. From the positive source terminal one meets, in series, an inductor \(L=1.00\,\mathrm{H}\), a resistor \(R=4.00\,\Omega\), and a capacitor \(C=1/5\,\mathrm{F}\) returning to ground. The output voltage \(v_o\) is the drop across the resistor in the direction of the mesh current that leaves the positive source terminal. All initial conditions are zero when the transfer function is formed. (a) Derive \(H(s)=V_o(s)/V_s(s)\) as a ratio of coprime real polynomials. Identify every finite pole and every finite zero, including multiplicity. (b) Evaluate \(H(0)\) and \(\lim_{s\to\infty}H(s)\). Classify the network as low-pass, high-pass, band-pass, band-reject, or all-pass at the level of these two limits. Compute the undamped natural frequency \(\omega_0\), the damping ratio \(\zeta\), and the quality factor \(Q=1/(2\zeta)\). (c) Invert \(H(s)\) to obtain the impulse response \(h(t)\) for \(t>0\). For the step input \(v_s(t)=u(t)\,\mathrm{V}\), determine \(v_o(t)\) for \(t>0\) and confirm \(v_o(0^+)\) and \(v_o(\infty)\) by the initial-value and final-value theorems. (d) Decide BIBO stability from the pole locations.

Output across the series resistor: \(H=R/Z_{\mathrm{series}}\), which produces a factor of \(s\) in the numerator after clearing \(1/(sC)\).