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Mathematical Analysis  

Basically, the hybrid-pi model represents the dynamic incremental variation of the stored charge in the base upon the incremental change of the applied emitter voltage. A change of DV_E gives rise to an increase of DQ_B and thus DI_C. The transconductance g_m is defined by:

g_m = dI_c / dV_E

We have,

n_p(0) = n_p0e^(VE ^{/ V}T⁾, and

|I_C| = {qAD_nn_p(0)} / x_B        for x^B << Lⁿ

Thus, we derive,

g_m = |I_C| / V_T = qI_C / kT

The input resistance r_p relates v_be to i_b, i.e.,

rp = dV_BE / dI_B = kT / qI_B

The output resistance r₀ relates i_c to v_ce, i.e.,

r0 = dV_CE / dI_C

The capacitance are designated by

Cm = C_dBC

and

C_p = C_dBE + C_DE