Working with basic emitter follower

This is a schematic of the basic emitter follower circuit. Just a transistor and a single resistor from the emitter to ground.

Writing the voltage equation around the input loop,

Vi - Vbe - Vo = 0 ---------- 1

Making Vbe very small or the gain large, we have, Vi =(approx.) = Vo

The output follows the input. thats why this circuit is called an emitter follower.
The voltage gain Vo/Vi is less than but near unity in magnitude.

From the load in the emitter circuit,

Vo = Re (Ib + Ic) = (1+hfe)*Re * Ib -----Since Ic = hfe Ib

At the input we have Vbe = hie * Ib

Substituting these back to our equation 1,

Vi = [hie + (1 + hfe)*Re ]* Ib ------------ 2

The input resistance of the amplifier follows as,

Rie = Vi/Ib = hie + (1 + hfe)*Re = approx. = hie + hfe*Re ---- 3

To find the output resistance,

(refer to the attached figure) The output resistance at the 2,2 port is found by making the independent source Vi = 0.

With hie parallel with Re,

Output resistance, Roc = hie * Re/[hie + Re] --------4

Voltage gain, Avf = -Ai Re/Ri

= hfe *Re/hie[1 + gm Re] where gm = hfe/hie

= hfe*Re/hie*(1+hfe*Re/hie)

= gm*Re/[1+gm*Re] ---------5

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