Problems with deriving output resistance without using a model

Question

I was reading different texts about input and output resistance of the above circuit and trying to understand the Rin and Rout without using any transistor models such as pi-model or T-model. I think I understand how can we comprehend what input resistance is. Here is the summary from what I understand for Rin without using any model:

Lets assume the voltage at the point Vb is increased by ΔV.

This means the voltage at the point Ve also will increase by ΔV.

If we want to write an equation for the input resistance Rin(the resistance seen from the point Vb); we can use the β relation between the base current Ib and the emitter current and the fact that the same ΔV will appear both at Vb and Ve. So Rin can be derived by using the following steps:

ΔIb = ΔV / Rin

ΔIe = ΔV / (Re//Rload)

since Ie = (1+β)*Ib

Rin = (1+β)*(Re//Rload)

So Rin input resistance can be written in terms of Re, Rload and β.

Question:

I cannot find a similar step by step comprehension derivation of the Rout without using a model. I mean I would like to clearly write down how Rout is obtained as I have written for Rin above.

The texts say that the Rout(looking back at the emitter) is:

Re // (Rsource/β)

But I'm very confused at this point how this is derived How they obtain Rsource/β conceptually.

How can we step by step explain Rout here as in Rin case?

Answer 1

How they obtain Rsource/β conceptually.

let Rsource=Rb

• if Rb is biased from some voltage Vcc

• such that the value of Rb controls Ie using KVL;

  • Ib=(Vcc-Vbe-Ve)/Rb
  • thus Ie/β =(Vcc-Vbe-Ve)/Rb
  • or (Vcc-Vbe-Ve)/Ie = Rb/β
  • let Vcc and Vbe be constants and Zout=Ve/Ie

• the change in emitter voltage from an external current can be nulled by an opposing base current amplified by Rb/β thus Zout = Rb/β // Re

q.e.d.

in conclusion the Emitter Follower Zin = β Re and ( for any Ze ) Zout = Rb/ β ( or any complex Zb parts)

For future reference

• a transistor amp with negative feedback using Rf from collector to base with Rseries to a source voltage input,
• we get voltage gain Av= -Rf/Rb and it turns out that the input and output impedance becomes a function of negative feedback gain such that Zin is reduced by feedback ratio and Zout at collector is reduced by feedback R ratio //Rc

The same is true for Op Amps but the gain is so high we consider the differential input impedance to be zero and likewise for output impedance until current limiting saturates the driver.