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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 DVE gives rise to an increase of DQB and thus DIC. The transconductance gm is defined by:

gm = dIc / dVE

We have,

np(0) = np0e(VE / VT), and

|IC| = {qADnnp(0)} / xB        for xB << Ln

Thus, we derive,

gm = |IC| / VT = qIC / kT

The input resistance rp relates vbe to ib, i.e.,

rp = dVBE / dIB = kT / qIB

The output resistance r0 relates ic to vce, i.e.,

r0 = dVCE / dIC

The capacitance are designated by

Cm = CdBC

and

Cp = CdBE + CDE