Question 7.14: Prove the above lemma using Thevenin equivalents.

We construct a Thevenin model for the circuits in Figs. 7.39 (a) and (b) but exclude Z_L, as depicted in Figs. 7.40 (a) and (b). With I_n=0 and V_n=0, the two topologies are identical, and hence Z_{\text {Thev1 }}=Z_{\text {Thev2 }}. We thus need only find the condition under which V_{T h e v 1}=V_{T h e v 2}.

To obtain the Thevenin voltages, we must replace Z_L with an open circuit [Fig. 7.40(c)]. ^{10} Since the current flowing through Z_S is zero in both circuits, we have V_{T h e v 1}=I_n r_O and V_{T h e v 2}=g_m V_n r_O. It follows that V_n=I_n / g_m.