(Solved): Show that the loop gain for an oscillator of the form shown in the figure is: ...

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Show that the loop gain for an oscillator of the form shown in the figure is: \[ L=\frac{-A X_{2} X_{3}}{-X_{3} X_{1}-X_{2} X_{3}+j R_{o}\left(X_{1}+X_{2}+X_{3}\right)} \] where \( \mathrm{A} \) is the real and finite open-loop gain and \( \mathrm{R}_{\mathrm{o}} \) is the finite output resistance of the otherwise ideal amplifier. Prove that the phase and magnitude conditions for oscillation are satisfied if: \[ \left\{\begin{aligned} X_{1}+X_{2}+X_{3} & =0 \\ \frac{X_{3}}{X_{2}} & =-A \end{aligned}\right. \] What determines the value of the ratio \( \mathrm{R}_{2} / \mathrm{R}_{1} \) in the Hartley oscillator for oscillation to occur? Assume that \( \mathrm{R}_{1} \) is much larger than the reactance of \( \mathrm{L}_{1} \) at the frequency of oscillation. \[ f_{n}=\frac{1}{2 \pi \sqrt{\left(L_{1}+L_{2}\right) C_{1}}} \]