Question 3. [10 points in the cross-coupled oscillator circuit shown below, Ro represents the loss of each inductor. Each transistor is operating at a transconductance gm and has an output resistance ro. Find the oscillation frequency cop and explain why the circuit oscillates at this frequency. Also, derive an expression for the minimum value of ge needed to obtain sustained oscillations. - w mm 11 C فففف LC w +0 0 2 5

NCH2IV The Asker · Electrical Engineering

Transcribed Image Text: 3. [10 points in the cross-coupled oscillator circuit shown below, Ro represents the loss of each inductor. Each transistor is operating at a transconductance gm and has an output resistance ro. Find the oscillation frequency cop and explain why the circuit oscillates at this frequency. Also, derive an expression for the minimum value of ge needed to obtain sustained oscillations. - w mm 11 C فففف LC w +0 0 2 5

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The small signal equivalent circuit of the cross-coupled oscillator is drawn asThe resonance frequency of each of the two tank circuits is \omega=\omega_{0}=\frac{1}{\sqrt{L C}}The load of each of Q_{1} and Q_{2} reduces to a resistance R_{P}=\omega_{0} L Q, where Q is the quality factor of the inductance.considering the output resistance r_{0} of each of the Q_{1} and Q_{2}, we can write the gain of each of the two stages at \omega=\omega_{0} as A_{1}=A_{2}=-q_{m}\left(R_{p} \| \gamma_{0}\right)Thus each stage exhibits a 180^{\circ} phase shift, for a total phase shift around the loop of 360^{\circ}.Thus the circuit will provide sustained oscillations at w_{0}=\frac{1}{\sqrt{L C}}provided \left|A_{1} A_{2}\right|=\left|q_{m}\left(R_{p} \| \psi_{2}\right)\right|^{2}=1 \Rightarrow q_{m}\left(R_{p} \| b_{0}\right)=1Thus the minimum sequired value of q_{m}=\frac{1}{\left(R_{p} \| l_{0}\right)} at which each of Q_{1} and Q_{2} is operated for oscillations to be sustained ...