Two coils of L_1=16.0 mH and L_2=9.00 mH, with M=6.00 mH, are connected in parallel…

Two coils of \(L_1=16.0\,\mathrm{mH}\) and \(L_2=9.00\,\mathrm{mH}\), with \(M=6.00\,\mathrm{mH}\), are connected in parallel between nodes \(p\) and \(q\). Coil 1 joins \(p\) to \(q\) with its dotted terminal at \(p\). Coil 2 also joins \(p\) to \(q\). An independent current source \(i_s(t)=3.00\cos(5000 t)\,\mathrm{A}\), with \(t\) in seconds, is injected into node \(p\) from node \(q\), so that the source voltage \(v\) is the voltage of \(p\) with respect to \(q\). No other elements are present. Determine (a) the coupling coefficient \(k\). For connection P the dotted terminal of coil 2 is also at \(p\) (parallel-aiding). Determine (b) the equivalent inductance \(L_{\mathrm{eq},P}\) seen by the current source, the peak phasor \(\mathbf{V}\) of \(p\) with respect to \(q\), and \(v(t)\). For connection Q the dotted terminal of coil 2 is moved to \(q\) (parallel-opposing). Determine (c) \(L_{\mathrm{eq},Q}\), the corresponding peak phasor \(\mathbf{V}\), and \(v(t)\). Then (d) compute the stored magnetic energy at an instant when \(i_s=3.00\,\mathrm{A}\) in each connection, using coil currents consistent with the common voltage \(v\), and confirm that both energies equal \(\tfrac12 L_{\mathrm{eq}} i_s^2\).

Parallel coils share one voltage; aiding versus opposing changes how \(M\) appears in the denominator of \(L_{\mathrm{eq}}\).