Circuits II · Resonance and passive filter design
An independent current source i_s injects into node a from the reference node 0
Problem
An independent current source \(i_s\) injects into node \(a\) from the reference node \(0\). Connected in parallel from \(a\) to \(0\) are \(R=4.00\,\mathrm{k}\Omega\), \(L=16.0\,\mathrm{mH}\), and \(C=250\,\mathrm{nF}\). The output is \(v_o=v_{a0}\). Initial conditions are zero. Define the transimpedance \(H(s)=V_o(s)/I_s(s)\). (a) Derive \(H(s)\) as a ratio of coprime real polynomials, with units of ohms stated for the overall gain constant. Identify \(\omega_0\), \(Q\), and \(\Delta\omega\). (b) Show that \(H(s)\) is band-pass in form. Evaluate \(H(0)\), \(H(\mathrm{j}\omega_0)\), and \(\lim_{s\to\infty}H(s)\). (c) If \(i_s(t)=2.00\cos(\omega_0 t)\,\mathrm{mA}\), determine the steady-state \(v_o(t)\). Report the half-power frequencies in hertz.
Hint
The transimpedance of a parallel \(R\)-\(L\)-\(C\) tank vanishes at DC and at infinite frequency.
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